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Computer graphical and semiclassical approaches to molecular rotations and vibrations
COMPUTER GRAPHICAL AND SEMICLASSICAL APPROACHES TO MOLECULAR ROTATIONS AND VIBRATIONS
William G. HARTER Theoretical Division T-12, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA
NORTH-HOLLAND
- AMSTERDAM
W.G. Harter / GraphicaI approaches to molecular rotations and vibrations
320
Contents 322 1. Introduction. .............................................. 324 1.1. The rotational energy (RE) surface. ........................... 324 1.2. The constant energy (CE) surface. ............................ 325 ........................................ 1.3. Other techniques 325 ................... 2. RE surfaces for tensor Hamiltonians of rigid rotors 326 2.1. Spherical, symmetric, and asymmetric RE surfaces ................ 2.2. Properties of classical and quantum RE surface levels and eigensolutions . 327 331 2.3. Symmetry properties of eigensolutions ......................... 334 .................................. 2.4. Semklassical quantization 342 3. RE surfaces for tensor Hamiltonians of semi-rigid rotors. ............... 342 3.1. Fourth rank octahedral tensor Hamiltonians ..................... 347 3.2. Symmetry properties of eigensolutions ......................... 348 3.3. Semiclassical quantization .................................. 350 3.4. Mixed-rank octahedral tensor levels ........................... 353 4. Quasi-spin RE surfaces for two dimensional vibrations ................. 353 4.1. Spinors, rotors, and coupled oscillators ......................... 360 4.2. Quasi-spin RE surfaces .................................... 362 4.3. Higher dimensional oscillators ............................... 363 5. RE surfaces for generic Hamiltonians of non-rigid rotors ............... 363 5.1. The generic Hamiltonian in the body frame ..................... 365 5.2. X, centrifugal distortion energy .............................. 370 5.3. RE surfaces for vibronically excited states. ...................... 370 6. Multiple RE surfaces and Coriolis coupling effects .................... 371 6.1. Semiclassical mechanics on multiple surfaces. .................... 373 6.2. Asymmetrically coupled spin and rotor systems. .................. 378 ................... 6.3. Octahedral scalar and tensor Coriolis coupling 386 7. Conclusion. ............................................... 387 Appendix A. Computation of rotation matrices and products .............. 389 Appendix B. Body frame transformations and momenta. ................. 390 Appendix C. Classical equations of rotational motion ................... 391 ........................... Appendix D. Tensor operator construction 393 References ..................................................
321
Computer Physics Reports 8 (1988) 319-394 North-Holland, Amsterdam
COMPUTER GRAPHICAL AND SEMICLASSICAL ROTATIONS AND VIBRATIONS
APPROACHES
TO MOLECULAR
William G. HARTER TheoreticaI Division T-12, Los Alamos National Loboratory, Las Alamos, NM 87.545, USA and Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA Received
15 January 1988
Rotation-vibration dynamics and eigensolutions for several models are visualized using computer generated rotational energy surface and semiclassically quantized trajectories. Several numerical approximations to rotation-vibration eigensolutions are developed which use information gained from graphical presentations. Techniques are given for finding essential physical phenomena in complex numerical rovibronic eigenvalue problems. Possible alternatives to large scale numerical diagonalizations are introduced.
0167-7977/88/$26.60 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
322
W.G. Harter / Graphical approaches to molecular rotations and vibrations
1. Introduction
The development of computers has strongly influenced the field of atomic and molecular physics. Most problems involving the electronic structure, vibrations, or rotations of polyatomic molecules have been too complicated to have exact analytic solutions, and so numerical solutions have been developed using various methods. The most common method involves numerical diagonalization of the large matrix representations of Hamiltonian operators, and this has led to considerable progress in the study of molecular structure and dynamics. As the computers increased in size it has been possible to choose bigger and better basis sets in which to carry out the numerical diagonalizations using many different techniques. Also, as computers became more accessible researchers began developing completely different numerical treatments of molecular quantum mechanics. As a result there has been a renaissance of semiclassical treatments [1,2]. The semiclassical approaches can in many cases avoid the numerical difficulties associated with setting up a large quantum mechanical basis set. Classical equations of motion are generally easier to solve numerically and several techniques for quantizing these solutions have been developed. The resulting solutions also suggest better ways to visualize the physics of the molecular dynamics. It is the last point which will be the main subject of this report. Until recently, the development of raw computing power has been more rapid than the development of the interface between the machine and its user. Now the introduction of sophisticated graphical input and output makes it easier to interpret and visualize complicated computer solutions. This report shows some new ways to use modern computer graphics to help understand molecular rotational and vibrational dynamics. Some of the methods to be described here are based upon quantum and semiclassical techniques which have been developed and applied elsewhere. A number of the most recent techniques are described in the articles by De Leon and Mehta [l] and Skodje and Cary [2] in companion articles. However, the main topic of present work will be use of the geometry of rotational energy (RE) surfaces for efficiently visualizing and computing classical and quantum effects associated with molecular rotations and vibration. The rotational energy (RE) surface was developed by Harter and Patterson [3] as a method for directly visualizing and calculating certain types of rotational spectral fine structure and for clearly understanding the mechanics of energy level clustering or superfine structure. Previously, level clustering had been noted in computer diagonalizations of tensor Hamiltonians by Lea, Leask, and Wolf [4], Domey and Watson [5], and Fox, Galbraith, Krohn, and Louck [6]. An explanation of the clustering on quasi-degeneracy based upon classical correspondence was given by Domey and Watson [5]. A detailed quantum theory for the structure of level clusters was given by Harter, Patterson [7-lo] and de Paixao [ll]. The semiclassical calculation of fine or superfine structure based upon RE surface trajectories and tunneling paths has been done more recently [3] and this will be reviewed in the sections 2 and 3 of this article. The discussion in section 2 is based upon the well known quantum mechanics of the rigid rotor. This provides one of the simplest examples of an RE surface and semiclassical quantization. (Probability the simplest .example is the quasi-spin RE surface for a vibrator, but this is a more abstract use of the concept which will be discussed in section 4.) Previous work on semiclassical quantization of the asymmetric top is given by Ring, Hainer, and Cross [12,13] and
W. G. Harter / Graphical approaches to molecular rotations and vibrations
323
more recently by Colwell, Handy, and Miller [14]. Approximation methods and discussion by Brown [15] should also be noted. The rigid asymmetric top provides a relatively simple pedagogical aid and testing ground for a number of concepts and techniques, and hence this is the longest section in this report. The discussion in section 3 uses a model which has had success in explaining certain phenomena in experimental spectra and in predicting new spectroscopic effects. It is based upon an effective Hamiltonian introduced by Hecht [16] for semirigid tetrahedral molecules such as methane (CH,), and it has been applied extensively to high resolution spectroscopy of a number at tetrahedral, octahedral, and cubic molecules. The SF, molecular spectrum has been extensively studied since it is the prototype for an isotope separation scheme [17,18]. It has also been the target of the highest resolution 10 pm spectrometer built by Borde and coworkers [19,20]. SF, provided an example in which a complicated spectrum and related dynamics can be understood easily by appealing to graphical constructions such as RE surface geometry. Such understanding was used to predict further properties of SF, nuclear-spin-rotor eigensolutions [21,22] which have been confirmed by the analysis the Borde experiments [20,23]. The discussion in section 4 is a more abstract application of an RE surface to describe normal mode and local mode phase trajectories of harmonic and anharmonic vibrators with two degrees of freedom. Algebraically, this is the simplest possible example of an RE surface. The model uses the mapping between the unitary U(2) group of a two-dimensional oscillator and the rotational group R, of a rotor. The RE surface picture helps to clarify some aspects of the SU(2) models introduced by Lehmann [24], and Kellman [25,26] and the elementary vibron model of von Roosmalen, Iachello, Levine, Benjamin, and Dieperink [27,28]. One of the possible future directions for the RE surface technology is introduced in section 5. It is proposed that a useful description of certain non-rigid rotors might be obtained numerically directly from an ab initio Hamiltonian. This may provide an alternative to the usual procedure in which an effective molecular tensor Hamiltonian such as the Hecht operator is obtained analytically by an algebraic perturbation expansion using the Van Vleck or Birkhoff-Gustaveson transformations [29]. Some numerically generated sets of RE surfaces are produced for some simple harmonic and anharmonic rotor-vibrators of the simplest polyatomic X, structures. The behavior of some systems with eigenvalue-generated RE surfaces is discussed in section 6. These are analogous to the multiple sheets of potential energy (PE) surfaces in vibronic models such as Jahn-Teller or Renner Hamiltonians. Two systems are discussed which are extensions of the systems treated in sections 2 and 3. The first involves a spin asymmetrically coupled to asymmetric rotor. The second is an octahedral XV, model with variable scalar and tensor Coriolis coupling between the rotor and a triply degenerate vibrational mode. The first of these systems is studied in regions where the classical model exhibits non-regular motion. Both models are used to compare classical RE surface geometry and dynamics to quantum eigensolutions. The discussions of the last three sections 4, 5, and 6 are very preliminary developments. One objective is to extend some of the advantage of the basic RE surface methods given in sections 2 and 3 to the more difficult problem of non-rigid rotors or highly excited rotor-vibrators. A more ambitious objective is to provide a numerical shortcut for treating the generic or ab initio Harniltonians of non-rigid molecules and calculating their spectroscopic properties in a way that provides as much understanding of their rotation-vibration dynamics as possible. It is unlikely that any single technique will achieve these objectives. Rather, a satisfactory
324
W.G. Harter / Graphical approaches to molecular rotations and vibrations
treatment of the rotation-vibration problem will involve a combination of analytical, numerical, and graphical techniques. For a long time it will be necessary to compare new approaches with the more standard approaches which diagonalize various representations of analytic Hamiltonians. These methods have been successful for many problems involving semi-rigid and non-rigid molecules [29] since before computational physics became a named subject [30]. However, potential algebraic and numerical bottlenecks appear ready to frustrate the theorists attempts to treat highly excited [31] or very floppy [32] species. Also, there are questions about what to do with a molecular spectral analysis after it is completed. One needs more ways to visualize and think about a given molecule’s structural and dynamic properties in order to plan more revealing experiments and invent new processes. A better picture is useful if it is suggestive or if it provides an image and a language with which to discuss complicated numerical and experimental results effectively. Before beginning the discussion of visualisation devices a number of the most useful ones will be listed and described briefly. The principal device is the first one listed below which is the RE surface. 1.1. The rotational energy (RE) surface The RE surface is a plot of a molecule’s energy as a function of the direction of an angular momentum vector in a molecule fixed frame for fixed magnitude of the momentum. The techniques for obtaining this plot and the detailed definition of the quantities involved will vary for different examples in section 2 thru 6. Usually the energy will be the total molecular energy of some state and the angular momentum will be the total angular momentum which is laboratory fixed in a classical model. The energy will generally be plotted radially outward and form a closed surface of radius E where E=H(J),
forJ=
]J( =constant.
(1.1)
The RE surface plays for rotational dynamics some of the roles which a potential energy (PE) surface plays for vibrational dynamics. However, the RE surface is derived from the PE surface, and it is a total energy. It includes information about properties of the potential energy surface as well as the kinetic energy of motions on it. This connection is elaborated in sections 3 through 6. The RE surfaces compresses information about the vibrational dynamics and PE surface. An RE surface is a two-dimensional surface embedded in a 3-space, while a PE surface is a (3N - 6)-dimensional object. A general RE surface is composed of multiple sheets with one sheet for each relevant vibronic state. 1.2. The constant energy (CE) surface The CE surface is a concept that is perhaps more familiar in mechanics. It is simply the locus of the angular momentum vector in molecule fixed (J,J,J,)-space for constant energy. E = H(J)
= constant.
(1.2)
One may require this type of surface when the quantity J is an action variable whose definition
W. G. Harter / Graphical approaches to molecular rotations and vibrations
325
changes with the value of energy. However, for rotors discussed in this article the J is well defined for all energies so we shall not use this construction. For spectroscopic applications it is better to have a display of a range of energies for a fixed magnitude of J than a range of J-values for a fixed energy. 1.3. Other techniques
Several types of auxiliary techniques may be employed to compare and discuss the information provided by the RE surface. These include plots of standard matrix diagonalizations, several semiclassical quantizations techniques, and trajectories derived by integration of classical equations of motion. Computer graphic visualization of molecular physics is something of an art. One should not expect a single technique to exist at the exclusion of all others. The computer graphics shown in this report is mostly done by a program called GRAFIC written by Melvin Prueitt of Los Alamos National Laboratory and modified for these specialized applications by the author. The program is capable of plotting realistic three dimensional views of multiple objects with each defined by an arbitrary set of surface coordinates. The objects may have level contours drawn on them with respect to an arbitrarily given point of origin. The program written in FORTRAN is efficient enough that most of the figures produced for this report can be done in a second or two on a mainframe computer. The program is being developed for use on an IBM PC and a MacJntosh computers by Steven Parker and the author, and in its present form takes several minutes to generate surface mesh plots on personal microcomputers. Assembly language implementations of some of the more repetitive calculations should improve the speed greatly. Some of figures in this report are exerpted from interactive animation programs. One of these in an animated 3D (stereoptic) view of of motion in a four-dimensional phase space. These advanced techniques are particularily valuable for use on an interactive-graphics oriented computer such as the Macintosh machines, where one can escape the usual “batch” processing that has become the rule on mainframe machines. One should look for a large number of innovative methods as personal workstations come available.
2. RE surfaces for tensor Hamiltonians of rigid rotors The simplest Hamiltonian H=AJ,2+
BJ;+
for a massive free rotor is that of the rigid top. (2.la)
Cl;.
The rotational constants A, B, and C are each one-half of the inverse of the principal inertia I,, I,,, and 1,. A = ,uL,/2 = l/21,,
B = p/2
= l/21,,,
C= pL,/2 = l/21,.
(2.lb)
The coordinate axes are fixed to the directions of principal inertia in the body. The tensor form of the rigid top Hamiltonian uses second rank multipole operators Tq2 to replace Jx2 terms in
326
W.G. Harter / Graphical approaches to molecular rotations and vibrations
(2.la). The necessary rank-2 multipole operators are given below. A brief review of general rotational transformations and tensors is given in appendices A-D.
J2=J,2+J;+J;,
G2= -(J,z+J;-2JL’)/2,
T;+T!,=3(J,2-J;)/&.
(2.2)
The Hamiltonian expressed in terms of these operators is as follows.
H= A+B+CJ2+ 3
2C-A-BT2+ 3
0
A-B -(T; fi
(2.3)
+ T!,).
To construct a rotational energy surface each tensor operator Tl is replaced by a corresponding multipole function Ci or Wigner D-function harmonic which is described in appendix A
T,‘-,D,‘,(.,P,Y)IJI’=C~~(--P,
(2.4)
-v)lJI’.
These are functions of standard Euler spherical polar coordinates which include the polar angle 8 = -B and azimuthal angle 9 = - y of the J-vector in the body frame. Substitution (2.4) is equivalent to replacing the original J,, JY, and J, in (2.1) by the following polar coordinate expressions.
J,= IJlsin(-P)cos(-y),
JY= lJlsin(-B)sin(-y),
J,= (Jlcos(-p).
(2.5)
The result is a spherical polar rotational energy (RE) surface function.
A+B+C
+ 2C-A-B 6
(3 cos2p - 1) + q
(sin2 B cos 2y)].
(2.6)
This is to be plotted radially as a function of B and y for fixed ) J I. Some examples of this RE surface will now be discussed for several different values of A, B, and C. 2.1. Spherical, symmetric, and asymmetric RE surfaces A rigid spherica top has three equal constants A = B = C, and only the first term in (2.3) or (2.6) contributes. This is a scalar (Too) operator. The resulting RE surface is a sphere Jx2+ JY2+ J: = 21H. A symmetric top has two equal constants. The case A = B > C corresponds to a prolate symmetric top with a z-symmetry axis, and only the first two terms in (2.3) or (2.6) survive. The resulting RE surface is shown in fig. 2.la for values of A = B = 0.2 and of C = 0.6. In this case only the cylindrically symmetric parts Tz and Tz contribute. The general rigid rotor is the asymmetric top in which all three constants are nondegenerate and all three terms in (2.3) contribute. An asymmetric top RE surface is shown in fig. 2.lb for constant values of A = 0.2, B = 0.4, and C = 0.6. Finally, fig. lc shows and example of an oblate symmetric top RE surface in which B is set equal to C and both are greater than A. Only the value of the B-constant is different for figs. 2.la, b, and c. Each of the RE surfaces in fig. 2.1 contains equal energy contour lines or level curves. These curves are intersections of the RE surface with equal energy spheres of progressively greater
W. G. Harter / Graphical approaches to molecular rotations and vibrations
327
Fig. 2.1. Rigid rotor RE surfaces. Bands are semiclassical J-vector paths for J=lO. Thicker paths are classically slower. (a) Prolate top: A = B = 0.2, C = 0.6. (b) Asymmetric top: A = 0.2, B = 0.4, C = 0.6. (c) Oblate top: A = 0.2, B = C = 0.6.
energies, and they are computed by the graphics program. Since the RE surface corresponds to a constant magnitude 1J ) of angular momentum, the constant-E lines on the surface are possible classical trajectories of the J-vector in the body frame. It will be shown that thicker curves represent classically slower paths and regions with greater wavefunctions. The curves drawn in fig. 2.1 are the ones that correspond to the quantum values for J = 10. It is important to see how RE surface trajectories correspond to the exact quantum eigensolutions of the asymmetric top since it is a good prototype for more complicated rotor Hamiltonians. 2.2. Properties of classical and quantum RE surface levels and eigensolutions The exact eigensolutions are obtained for comparison by diagonal&zing an matrix representation of the tensor Hamiltonian for a rotor. It is conventional to use a basis set of (2 J + 1) eigenvectors 1:) of J,, where 41;)
=KI$),
(k=J,
J-l
,...,
-J)-
These span a block matrix representation theorem.
of each tensor term according to the Wigner-Eckart
(2.7a) Here C$$ is a Clebsch-Gordan coupling coefficient which determines the form of each (2 J + 1) dimensional matrix. This is multiplied by an overall reduced matrix element factor.
328
W.G. Harter / Graphical approaches to molecular rotations and vibrations
The asymptotic value of this factor is 1J 1r for high J since Ti is an th degree polynomial in J. For a second rank tensor (2.3) and (2.7) yield (2.8a) and (2.8b) The matrix representation of the Hamiltonian (2.3) is then given
(A +B)
(JK,(H(;) =
I
(it
J(J+l)+
(2C-A-B)K2 2
=0
(2.9a)
@&(JU)(J+IGl)(J~K+l)(JkK+2)
[
(K’=Kf2).
(2.9b)
The matrix is diagonal if one sets A = B (this is the prolate limit using the z-axis of quantization), or if one switches A and C and sets C = B (this is the oblate limit using the x-axis of quantization). Then the asymmetric terms Ti2 do not contribute. The correspondence between the exact J = 10 eigenlevels and asymmetric RE surface trajectories is given in fig. 2.2.. Pairs of eigenlevels are shown in the lower portion of the figure in magnifying circles. Each pair of levels corresponds to a pair of equivalent level curves on the RE surface. The level splittings of each pair are called superfine structure splittings since they are usually very small. Only the levels near the separatrix region in the center of the fine structure pattern have splittings which are visible on the scale of the drawing. The separatrix region divides the trajectories and the corresponding levels into two distinct types. One type of level pairs lie at energies above the separatrix and correspond to pairs of trajectories surrounding the z-axis which is centered on the hills of the RE surface. The corresponding quantum levels fall into pairs of A, and B, levels or A, and B, levels depending upon whether the quantum number K,(z) is even or odd, respectively. These levels are shown on the righthand side of fig. 2.2. Another type of level pairs which he below the separatrix correspond to pairs of trajectories surrounding the x-axis which is centered in the valleys of the RE surface. The low energy levels fall into pairs of A, and B, levels or A, and B, levels depending upon whether the quantum number K,(z) is even or odd, respectively. Odd and even symmetry properties will be discussed in section 2.3. The z-type trajectories go around the high points of the asymmetric RE surface. They are related to the prolate symmetric top trajectories shown in fig. la. J-vectors on trajectories around high points precess in a clockwise sense at a rate that is proportional to the RE surface according to Hamilton’s equations of motion (see appendix C) o = aH/iYIJ.
(2.10)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
VISUALIZING THE J = IO LEVELS OF AN ASYMMETRIC TOP
329
?OTATION AXES UEAR f AXIS
h
IO=KJz)
62 At
‘1 ’ \\!/I
C,(z)-Type Clusters 8 6x1&d =26 kHz
a i
77x1&
2,6xf&‘5
IX&,
=230kHz
=64MHz
= I.5GHz
C&x)-Type Clusters II,,
II,,
,,I,
20
LO
I,,,
i II,,
LO
I,,,
,,I,
LO
(I,,
I,,,
60crri’
Fig. 2.2. J = 10 asymmetric top energy levels and related RE surface paths (A = 0.2, B = 0.4, C = 0.6) (after ref. [3]). Clustered pairs of levels are indicated in magnifying circles which show superfine splittings.
RE surface levels are thicker in regions of less slope where precession is slow. Precessional rates are also related to energy differences in the prolate symmetric top spectrum. These energy levels follow from (2.9a) with A = B.
E;=
BJ(J+
1) + (C-
B)K'.
(2.11)
330
W.G. Harter / Graphical approaches to molecular rotations and vibrations
The average difference between nearest neighboring K-levels is
AE =
[(&+I -E;)+(E;-E;_1)]/2=2(C-B)K
(2.12)
and this is exactly the classical precessional frequency. (The classical rotational frequency 2BJ is proportional to the difference between neighboring J-levels with the same K.) If (C - B)K is positive as it is for the trajectories in the upper half of fig. la then wZ= 9 is positive. Since the body azimuthal angle of J is -y, all the upper paths carry J-motion from right to left in fig. 2.la; that is, clockwise. The lower paths have negative K and therefore carry motion in the opposite direction of left to right. The negative-K motion is clockwise, too, when viewed from the high point at the center of the lower paths. In between the two kinds of prolate top paths in fig. la is a single line at the lowest energy corresponding to K = 0. The (K = 0)-line contains fixed points. The J vector cannot move if K = 0 since according to (2.12) the precessional frequency is zero for these points. The fixed points correspond to a pure end-over-end rotation. (Incidentally, this is the only possible motion for a diatomic rotor.) The (K = 0)-line of fixed points gives rise to two new valleys of precessing trajectories as B becomes larger than A and approaches C. Then the asymmetric Ti2 tensor terms in (2.3) contribute a separatrix consisting of two intersecting circles. It opens like a pair of scissors to make room for new precessional paths around the x-axis and to cut into the phase space occupied by the paths around the z-axis. The x-paths are related to the oblate paths which they become when B = C as in fig. lc. Around each valley the sense of precession will be counterclockwise when viewed from above the valley center. Paths along the separatrix are headed in the same direction on either side as indicated by arrows in fig. 2.2. The angle 8 between the separatrix circles and the z-axis is a measure of the phase space available for the z-type paths 8=arctan[(C-
B)/(B-A)].
(2.13)
The separatrix angle is 9 = 90 O, 45 O, and 0 o for figs. a, b, and c, respectively. The number of quantizing trajectories must remain constant as the separatrix angle increases. Paths are forced to change from z-type paths to x-type paths as the separatrix closes in on the z-axis. Paths nearest the separatrix are distorted, or “ squeezed” considerably from the circular shape which they had for the symmetric top case. The squeezing spoils the symmetric top path’s K-conservation since the projection of the J-vector along the z-axis varies as it moves along a warped path. As 8 decreases each z-path must meet its mate on the separatrix and then cross over and break-up into a new pair of x-paths. Each crossover and break-up corresponds to a crossover and break-up of quantum energy levels clusters. The exact quantum energy levels for J = 10 are plotted in fig. 2.3 for the same range of B values (0.2 < B < 0.6) used in going from fig. la to fig. lc. The levels shown in fig. 2.2 are plotted vertically above the point B = 0.4 in fig. 2.3. At this point the K = 5 quantizing paths are occupying the separatrix and together account for just one A, level in the center of figs. 2.2 and 2.3. Most of the levels in fig. 2.3 are paired or clustered. Only the levels near the separatrix region split up, and this region is clearly visible running form the lower lefthand to upper righthand corner of fig. 2.3. In the separatrix region one level always exhibits much more sensitivity to the change of B then the other member of the pair. The difference in sensitivity is due to the
W.G. Harter / Graphical approaches to molecular rotations and u&rations
331
Fig. 2.3. J = 10 energy levels as a functionof 33(0.2 zzB I; 0.6) betweenprolatetop and oblate top limits.(A = 0.2 and C = 0.6 are constant)LevelsaboveB = 0.4 correspondto thosein fig. 2.2. difference in symmetry properties properties will be described shortly. Energy level correlations such as eigensolutions for a whole class of particular type of level cluster and corresponding evolving RE surface.
of the corresponding
quantum
wavefunctions.
Symmetry
fig. 2.3 provide useful information about the nature of the Hamiltonians. One em often spot regions belonging to a this can be related to particular regions of stability iu the
One learns which eigensolutions are relatively localized and insensitive to small changes of the Hamiltonian, i.e., levels for which perturbation theory could easily accommodate the changes. On the other hand the eigenfunctions which are caught in the neighborhood of a separatrix are relatively delocalized and extremely sensitive to changes. Their level correlation plots are characterized by destruction, and rebuilding of level clusters with relatively rapid changes in between. One expects perturbation theory to be less useful in describing this case when the wavefunctions and their corresponding classical trajectories are being radically changed. 2.3. Symmetry properties of eigensolutions The rotational symmetry of a generic (A + B + C) rigid rotor Hamiltonian is described by the four-group D, = (1, R,, R,, R,) which containes the identity (1), and three 180° rotations R,, R, and R, around the body axes X, y, and z, respectively. The irreducible characters of D, are labeled AI, A *, B,, and B, in the character table AI AZ Br %
1 1 1 1 1
R,
R, 1
-1 1 --I
R, 1 1
-1 -1
1 -1 -1
(2.14) 1
332
W.G. Harter / Graphical approaches to molecular rotations and vibrations J t
J
&
x
y
?-
m K,(x)
I
\
c0 2y
D
K&Y
)
e3 K,(z)
Fig. 2.4. Different choices of rotation axes for asymmetric rotor corresponding to local symmetry (a) C,(x), and (c) C,(z). Tables correlate global D, symmetry species with the local ones.
(b) C,(y),
The subscripts 1 and 2 label even and odd symmetry, respectively, with respect to x-rotations R,. The A and B label even and odd R, symmetry, respectively. We shah see that the R, symmetry determines which levels are more sensitive to perturbations in the separatrix region. Note that R, symmetry is a product of R, and R, symmetry for each of the four D, species. Symmetry correlations of D, and the three separate axial subgroups { 1, R,} = C,(x), { 1, R,,} = C,(y), and (1, R,} = C,(z) are summarized by tables in fig. 2.4. Even and odd axial C,-symmetry is denoted by 0, (0 mod 2) and 1, (1 mod 2), respectively. The figures above each table show an XY, asymmetric top in a state of rotation which corresponds to the J vector being near the x, y, or z body axis, respectively. The C,(x) rotations correspond to the lowest energy asymmetric top trajectories in fig. 2.2 which are nearest to the x-axis, while the C,(z) rotation corresponds to the highest energy trajectories which are nearest to the z-axis. The C,(y) motion corresponds to being on the separatrix and is therefore not a simple rotation. The quantum levels belonging the x and z paths are paired according to the columns of the C,(x) and C,(z) tables, respectively. The even or O2 column of the C,(x) table contains the pair (A,B,) which are the species contained in the even-K,(x) levels on the lefthand side of fig. 2.2. They alternate with the pairs (A,B,) which occupy the odd or 1, column of the C,(x) table. Both these are pairs of levels that belong to trajectories that have C,(x) symmetry on the RE surface and arise from the oblate top spectrum on the lower righthand side of fig. 2.3. The same explanation holds equivalently for the alternating (A,B,) and (A,B,) pairs arising from the prolate part of the spectrum. Each classical z-trajectory resides on one of the prolate ends of the RE surface and each closed path by itself has a local C,(z) symmetry. The relation between pairs of classical paths and pairs of eigensolutions such as (A,B,) needs to be defined more precisely. Equal combinations of the two eigenstates in a pair correspond to
W. G. Harter / Graphical approaches to molecular rotations and vibrations
333
eigenfunctions which are more nearly localized onto a single RE surface trajectory. For example, the following combinations of the highest (A1B2) pair of J = 10 levels in fig. 2.2 or 2.3 yields (2.15a) (2.15b) where the coefficients a10 = (Y_~~are nearly equal to one and all the other components cr* a, (Y* 6, (Y* 4; and so forth are much smaller. The odd coefficients (YT 9, (Ye 7, etc. vanish because of symmetry; only the states with local symmetry of 0, can contribute to (2.15). The positive combination (2.15a) corresponds to a wavefunction centered on the upper trajectory in fig. 2.2, and the negative combination (2.15b) is on the lower one. Similarily, the next highest pair of energy levels labeled (A,B,) in fig. 2.2 yields the following combinations of states with odd (12) local C,( 2) symmetry (2.16a)
( I A,)
-
I B,))/O
=
a.9
1’;)
+
ci_,I’$
+
* * * .
(2.16b)
ak5, etc., are small. The next pair (A,B,) has the same form as (2.15) except that (Y+ 8 are the dominate coefficients. These combinations correspond to wavefunctions centered on the K,(z) = 8 or the K2( z) = 8 trajectory, respectively in fig. 2.2. An approximate calculation of these states will be discussed in section 2.4~. In order to obtain wavefunctions that are localized on a single classical trajectory it is necessary to mix states of different D, symmetries as in eqs. (2.15, 16). In this way a wavepacket is made which has some given local symmetry, say, 0, or 1, from C,(z) by combining two species such as A, and B, or A, and B,, respectively, of global symmetry D,. Since the different species have different energies the resulting combinations are non-stationary states, i.e. the wavepackets move from one location to another. However, if the energy differences are small this motion can be very slow. This is the case for the states localized far from the separatrix, i.e., high-K states. In fig. 2.2, the K = 10 states on the x or z axes have energy differences of only 26 kHz which is very small compared to their 150 GHz precessional frequency. Here it is easy to distinguish faster precession from the much slower tunneling motion. For states in the neighborhood of the separatrix the wavefunctions are not so localized onto the classical paths. Instead, the wavefunctions tend to pile up around the saddle region where the classical trajectories slow down and turn around. A J-vector on a trajectory that is closer to the separatrix will take longer to turn around and come out of the saddle region. (If it is exactly on the separatrix it will take forever!) The sensitivity of a given eigenwave to the change in the term BJ; in (2.la) depends on the wave amplitude near the y axes. Waves B, and B, which have odd C,(y) symmetry must vanish at the saddle turning point and be generally smaller in that neighborhood than the A, and A, waves. Therefore it is always A-levels which rise more quickly as a function of B in the separatrix region of fig. 2.3.
Here ak7,
334
W.G. Harter / Graphical approaches to molecular rotations and vibrations
2.4. Semiclassical quantization A number of semiclassical methods are being developed for approximating quantum rotor eigensolutions directly from the classical rotational energy surface. Such methods might be applied without having to calculate an effective tensor Hamiltonian for a non-rigid rotor. So far, these techniques have only been used to study rotor eigensolutions where well defined effective Hamiltonians exist and can be compared to the approximations. Here three examples of semiclassical methods based upon rigid top RE surfaces are described briefly. The first method is just a very simple visualization aid based upon angular momentum cones which provides quick rough estimates of certain spectroscopic features. The second and third methods are based upon simple semiclassical path integration and wavepacket propagation, respectively.
(a) Angular momentum come quuntization The angular momentum cone method is quite accurate for nearly symmetric tops and for non-rigid rotor surfaces with local regions of C,, C,, or higher symmetry. As we will see it is not so accurate for the asymmetric top, but it is based upon the properties of a symmetric top for which it is exact. Hence it will be introduced here. The angular momentum cones are defined as the loci of the angular momentum vector J obtained from a “literal interpretation” of the symmetric top base state 1;) and its quantum eigenequations
JzIi) = K(IK) 7
The interpretation magnitude of
J’i;) = J(J+
1) k).
(2_17a,b)
of these equations is that J is constrained to have a z-projection of K and a
(2.18)
(JI =/m.
This constrains the J-vector to lie on a cone of altitude K and slant height 1J 1 = /m. The polar angle 0; of a quantizing cone is defined by the arccosine of the altitude to slant height ratio (2.19) The base of the cone provides a measure of the uncertainty (A J,)( A J,) of the two unquantized components of angular momentum. This uncertainty is minimum for the highest magnitude K values which are K = f J. In this sense a state 1;) (or I-“,)) is a minimum uncertainty state. The symmetric top Hamiltonian can be written as a function of the polar angle -p = 0 using (2.6). H(O)=
lJ12[(2B+C)+(C-B)(3~~~20-l)]/3.
Substituting quantum values (2.18) and (2.19) for magnitude and angle of J yields the correct eigenvalues
H(@;)=BJ(J+l)+(C-B)K’.
(2.20)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
335
This implies that the quantizing trajectories on the symmetric top RE surfaces in figs. 2.la and c are precisely the intersections of the angular momentum cones with the RE surface for J = 10 and K= 10, 9, 8 ,..., - 10. This provides a graphic relation between the angular momentum uncertainty and the semiclasscial precessional paths of the RE surface. The cone quantization will be a useful approximation in the neighborhood of maxima or minima of any RE surface which is locally nearly symmetric. The symmetric neighborhood is the region where the relative contributions of off-diagonal tensor terms TJ with 4 not equal to zero are small compared to those of the diagonal terms TL. An asymptotic expression for the Wigner-Eckart coupling coefficient in the (4 = 0)-tensor metrix element (2.7a) yields a Wigner D-function or a Legendre function of the cone angle
(;IT$) =D&(.@;.)) 1Jl’=P,(cos
@;) 1JI’
(Jx=- 1)).
(2.21)
Hence, the diagonal or isotropic part of the matrix is asymptotically equal to the RE surface altitude at the cone intersection. This is true since the RE surface is defined by replacing each multipole operator T; by its corresponding multiple function, that is, Ti + II,& ( - ,p, y) 1J 1r, according to (2.4), and only the (4 = O)-functions are nonzero in the neighborhood of p = 0. This approximation is better for higher rank (r > 2) tensors as we will see in the next section. For the asymmetric top in figs. 2.lb and 2.2 the majority of paths are “squeezed” by the anisotropy arising from the off-diagonal tensors Ti2 which take effect quickly. An angular momentum cone with 0: = 45 O which just reaches the separatrix corresponds to the following K-value K= \I[lool
cos 45” = 7.4.
From this one might conclude that only states with K = 10, 9, 8, and (with difficulty) 7, could fit in the available phase space between the separatrix circles. However, we have neglected the squeezing which makes it possible for paths labeled by K = 6 and 7 to fit easily in the quadrant of the RE surface shown in fig. 2.2. Of course, squeezing ruins the strict conservation of K and off-diagonal tensors T: z mix in states with K-values that differ by multiples of two.
(b) Semiclassical path integration For highly squeezed RE surface trajectories it is necessary to use more general forms of quantization. The Bohr-Sommerfeld action integral quantization formula is useful
#
p*dq=nh
(n=0,1,2
,... ).
For free rotors the three Euler angles and their conjugate momenta are the 4’s and p’s of the action integral (2.22)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
336
However, for a free rotor, the Euler angle cx is an ignorable coordinate and J, = ALAB is a constant integer which can be left out. Furthermore, for J-paths with bilateral xy symmetry the /3-integral will vanish identically. This leaves only a one-dimensional quantization condition involving the body z-component J, = JzBoDY 1 2TJYdy=K %e J
(unitsof
ti).
(2.23)
For the asymmetric top J, can be expressed in terms of H = E by solving (2.la) using (2.5) J, = Jy (z-axis)
(2.24a)
=
For paths that encircle the x-axis a change of the quantization coordinate is equivalent to simply exchanging A and C J, = J., (x-axis)
=
//*
(2.24b)
The integrals (2.24) are evaluated numerically for a sequence of energies E which converge onto a solution to the quantization condition (2.23) for each desired integer K. The resulting semiclassical E-values are accurate to between three and four significant figures for J = 10 and between four and five figures for J = 20 (see table 2.1). A slight modification of the path integral can be used to estimate tunneling rates which account for the superfine splittings in fig. 2.2. The necessary integral for the (K(z) = lO)-levels, for example, is along the dashed circle that connects the (K= lO)-path to the (K= - lO)-path and crosses the saddle point at the y-axis. The integral is evaluated between the closest approach points y_ and y + of the two z-paths at the correct energy value for the (K = lo)-level. Since the integral is around the x-axis the x-expression (2.24b) is used in the integral and limits are the points where J, = 0, i.e., the points where the z-path crosses the yz-plane. Between these limits J, is imaginary and the e-integral below is real.
e=i
‘+dyJ,=i
JY-
IJ12(Ccos2y+B
sin2y)-I&,,,,,
C cos2y + B sin’y - A
dv.
(2.25)
The tunneling amplitude is given by a standard semiclassical approximation S = (l/T)
e-*
(2.26)
where l/T is the classical precessional or “attempt” frequency on the z-path. The tunneling amplitude must be doubled since the two dashed tunneling paths connecting the z-paths in fig. 2.2. The tunneling matrix
(W=(& ‘E”)
(2.27)
W. G. Harter / Graphical approaches to molecular rotations and vibrations Table 2.1. Semi-classical Qua&ration
for Asymmetric Top (in cm-‘)
EK (QM)
J%(SC)
9 8 7 6
63.181 57.851 53.147 49.113 45.833
19 18 17 16 15 14 13 12 11
246.352 235.360 224.977 215.214 206.083 197.603 189.801 182.721 176.448 171.132
337
A = 0.2, B = 0.4, C = 0.6
SK(QM)
Si4C)
63.176 57.842 53.133 49.084 45.781
2.16x 1O-7 1.93 x 10-s 6.98 x 1O-4 1.27x10-* 1.11x10-’
1.96~10-~ 1.87 x 1O-5 6.91 x 1O-4 1.29x lo-* 1.20x10-’
246.347 235.354 224.970 215.205 206.071 197.586 189.777 182.684 176.382 171.029
2.84x lo-‘* 2.55 x10-” 1.42x10-* 5.33 x lo-’ 1.41 x1o-5 2.71 x 1O-4 3.76x1O-3 3.64x10-* 2.18 x10-’
1.25 x 1014 2.66x10-l2 2.49x10-” 1.40x 1o-8 5.29x lo-’ 1.41 x 1o-5 2.72~10-~ 3.80~10-~ 3.75 x lo-* 2.38x10-l
J=lO K=lO
J=20 K=20
QM - Quantum mechanical diagonalization. SC - Semi-classical treatment.
gives A and B eigensolutions EA=E+2S, EB=E-2S
(2.28)
which have a total superfine splitting of 4s. The semiclassical superfine splittings calculated using (2.25) thru (2.28) vary over six to twelve orders of magnitude yet are correct to one or two figures of accuracy for J = 10 or J = 20 (see table 2.1). (c) Wavepacket propagation and quantization Another method of semiclassical approximation yields approximate eigenfunctions as well as eigenvalues. This method is an adaptation to the rotational problem [33] of a vibrational quantization method developed by De Leon and Heller [34] which is described in this volume [l]. The wavepacket quantization methods often use a minimum uncertainty state ]MU(q, p)) or wavefunction ( ]MU( q, p)) which is localized around a certain point (q, p) in classical phase space. In the original formulation of this problem for vibrational mechanics, this state was chosen to be a (q, p)-translation of an oscillator ground state. In this case the corresponding wavefunction ( 1MU( q, p)) is a translated complex gaussian packet with minimum value (h) for the uncertainty (Aq)( Ap) for all degrees of freedom. The method uses classical equations of motion to derive the time evolution of the phase variables (p(t), q(t)). It starts with a judicious choice of initial values (p(O), q(0)) which have a
338
W.G. Harter / Graphical approaches to molecular rotations and vibrations
classical energy reasonably close to the eigenvalues that are desired. Then a path propagation state ) q(t))
= )MU(q(t),
p(t)))
e(i/h)sp(t)
(2.29)
is obtained by numerically integrating the classical equations to obtain the phase variables (q(t), and the classical action or Hamilton’s Principle Function S,.
p(t))
(2.30) Generally, one defines a classical translation operator CT which has the effect of changing the (q, p) parameters of the wavepacket to the correct classical values for each time t IMU(
p(t))>
= CT(q(t),
p(t):
q(O),
P(O))
IMU(
P(O)))-
(2.31)
The same CT operator can be used to set the initial wavepacket, as well. The initial wavepacket expectation value of this operator, including the action phase, is called the time autocorrelation function.
AC(t) =
PfU(dO)7P(O))
ICTkdt),
p(t): q(O),P(O)) IMU(q&%P(O))) e(“h)Sp(r) (2.32)
= ( W) I W)) *
This function oscillates or “beats” each time the evolving wavepacket returns on a classical path that is close enough to overlap with the initial packet. The energy Fourier transform of the autocorrelation FTAC( E) provides information which can be used to generate eigenfunctions. The values E = E, which are peaks of the following transform function FTAC( E) = &
can be used to obtain an approximate propagation state. 1EFT)
= &
/:dt
(2.33)
/TTdt e(‘lhjE’AC( t) semiclassical eigenfunction
e(i/h)ERr I ‘k( t ))
by a Fourier integral of the
.
Finally, the desired semiclassical energy is computed by an expectation Hamiltonian.
Esc=(E,lHIE,)/(E,lE,). This last step is necessary since, as we will see, the FTAC peaks {E,, slightly from the desired quantum energy values { Es,-, Ei,, . . . } .
(2.34) value of the quantum (2.35) EfT,. . . } may differ
W. G. Harter / Graphical approaches to molecular rotations and vibrations
339
One way to understand the wavepacket method is to study the conditions which would cause (2.33) to have a peak or to make (2.34) keep accumulating a stationary state over longer and longer integration times. Such a peak or accumulation implies that E has been chosen so as to obtain a phase coherence or constructive interference each time the evolving wavepacket returns to the neighborhood of the initial packet. This occurs when the action for a closed path satisfies (EBK) certain quantization conditions. In general, these are the Einstein-Brillouin-Keller quantization conditions applied to Hamilton’s Characteristic Function S, or “reduced action” S,=S,+Et=
/
(2.36)
p*dq.
For a closed path the EBK quantization
conditions are
p dq = (n + a)2nA LfJ l
(2.37)
where ar is called the Maslov constant. In terms of the classical action this gives Esc7 + S, = (n + a)2llft,
(2.38)
where 7 is a classical period for the closed path and Es, is the desired semiclassical approximation to the energy. This may be written EFr7 + SP = (n)2nA, where E, E,
(2.39)
is a peak value in the FTAC (2.33) given by = Es, + 2aaA/7.
(2.40)
If the Maslov constant is not zero, then the true value Es, is shifted. From (2.32) one sees that condition (2.39) makes the phase in (2.33) or (2.34) come out to a multiple of 2ai after classical period 7. Hence the wavepacket method amounts to a numerical resonance or spectroscopy experiment in which one finds stationary states by “tuning” E to get a peak in (2.33) and then “painting” the eigenfunction using (2.34). The resulting eigenfunction consists of a sum of a series of MU packets set out on a classical trajectory. As explained in refs. [33] and [34] the last step can be accomplished with relatively small number of time steps. It is also helpful to “tune” the initial conditions (q(O), p(0)) in (2.31) to get the strongest peaks. The rotational adaptation of wavepacket propagation involves a different type of minimum uncertainty state, phase variables, and classical translation operator. The latter is a rotation operator which affects the Euler angles { (upy }, and these angles are the rotational equivalent of phase variables. Recall that the angles -p, and -y are the RE surface coordinates. The wavepacket propagation can be viewed as taking place on the RE surface shown in fig. 2.lb.
340
W.G. Harter / Graphical approaches to molecular rotations and vibrations
A form for the rotational wavepacket is suggested by the discussion of angular momentum cones in section 2.4a. A choice for a rotational minimum uncertainty wavepacket is the wavefunction associated with the narrowest angular momentum cone, that is,
IMU(0,090))= 1;).
(2.41)
A good choice for an initial wavepacket is one centered on some part of a quantizing RE surface path. Assuming that the point (/3 = 0, y = 0) is the center of an RE surface hill or valley, then the Kth quantizing path would be near to the point { & = - Oi, y0 = 0). A suitable initial wavepacket is therefore obtained by the following y-rotation of initial state (2.41) ]MU(O, -O;,
0)) =R(O,
-O;,
(2.42)
O)];).
(Negative Euler angles are used since the rotation is defined with respect to the body fixed frame.) The choice of initial angles is usually not so critical, but a good initial guess can reduce computational time somewhat. The classically propagating wavepacket state (2.29) has the following form for the rotational problem. (2.43)
The Euler angles and the action are obtained Hamiltonian’s equations. . aH a=a~,,
.
aH
Jzi= --as,
a=a, P,
by numerical
Runge-Kutta
[35] solutions
of
(2.44a)
Y,
and (244b) Detailed examples of these equations are given in appendix C. The initial conditions (a(O), /3(O), y(O)) given by eq. (2.42) may be used along with S,(O) = 0 for the action. The autocorrelation function is given in terms of the initial and final Euler angles AC(t) = V(O) I W)) = (.$W#),
P(O), y(WWt)9
IW),
Y(NIJ,)
exp[(Vh)S,(t)].
(2.45a)
The rotation product is reduced and represented by a Wigner D-function AC(t) = (J”I+,,
&
Y,$)
ew[(i/h&(t)]
= S(~,,
BP9
Y,> exp[(i/h)~,(t)].
(2.45b)
341
W. G. Harter / Graphical approaches to molecular rotations and vibrations 60 000
A- 020 a*0.00
8- 0.40 &--30.00
c-
WO E,,,-
0 80 To’ 0.00
50.000
A-
0.20 0.00
B= 040 /3,--30.00
CP 0 60 ,p 0.00
Time
Fig. 2.5, Wave packet autocorrelation function and Fourier transform for rigid asymmetric top (a) Function shows beats (b) Function FAC( w) has peaks which correspond to eigensolutions.
AC(t)
The group product rule and the D-function formula are given in appendix A. An example of a rotor autocorrelation function and its Fourier transform (2.33) are plotted in figs. 2.5a and b, respectively. The beats due to the returning wavepacket are shown in fig. 2.5a. A peak in the transform at energy E,(K= 8) has been singled out in fig. 2Sb. Each of the larger peaks E,(K) indicates a possible eigensolution. According to eqs. (2.34) and (2.43) the following state is an approximate eigenstate associated with that peak
IJ%rW) = >ir
$/_rrdf
ex~[(i/h)(%r(K)
+ s,(t))]R(+),
B(t),
A(t)) 1;).
(2.46)
The representation of this state in the angular momentum basis reduces to a Fourier transform of D-functions
(2.47) It can be shown that the coefficients (2.47) are always real [33]. The coefficients chosen in fig. 4.5b are the following =
=
10, 0.118,
8, 0.929,
6, - 0.339,
4, 0.075,
2, -0.012,
for the example
0 0.001,
1. . ...
(2.48)
342
W. G. Harter / Graphical approaches to molecular rotations and vibrations
These are to be compared with the exact values for a localized state of the form (2.15) which are the following (ll& --0.129,
0.922,
-0.347,
Finally, the semiclassical according to eq. (2.35). &c(K=
8) = G%(8)
0.106,
-0.062,
0.018 ,....
energy value for the wavepacket
IH I ~%r(8))/(Gr(8)
I E&8))
generated
= 53.134.
(2.49) state (2.48) is given
(2.50)
This to be compared with the quantum mechanical values of 53.149 and 53.146 for the (K= 8)cluster. The accuracy is comparable to that of the elementary path integration result in table 2.1. This completes an example of the most elementary “bare bones” wavepacket propagation technique. More sophisticated treatments [33] give more accurate wave functions and energy values by taking tunneling into account.
3. RE surfaces for tensor Hamiltonians
of semi-rigid rotors
Some examples of generalizations of the RE surface techniques to semirigid rotors of higher symmetry will be considered in this section. Semirigid rotors are deformable bodies whose distortion energy is readily described by simple analytic tensor Hamiltonians. A brief discussion of the derivation of the tensors, their limitations, and some possible alternatives will be given in sections 5 and 6. Much of the development of the RE surfaces for semirigid rotors follows the development given for the rigid rotors in section 2. Analogies between the rigid and semirigid rotors will be emphasized as will be the points where they differ. 3.1. Fourth-rank octahedral tensor Hamiltonians The simplest non-trivial rotor Hamiltonians having tetrahedral (Td), octahedral (0,) or cubic symmetry is the fourth rank Hecht Hamiltonian [16]. This may be expressed in polynomial form
H = Hscalar+ lot,,
[ J,4 +
J; + J; - (3/5) J"] .
(3.la)
or in tensorial form (3.lb) where the scalar part contains the undistorted spherical top rotor kinetic energy B 1J ) 2 and a fourth order correction D 1J 14 due to isotropic distortion.
H scalar = BJ2 + DJ4.
(3.lc)
The detailed dynamics and spectroscopy of this type of rotor depends upon the anisotropic distortion effects which are described by tensor parts of Hamiltonian such as the t,,&erm in the
W. G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 3.1. Octahedrally
symmetric RE surfaces for Hecht Hamiltonian (eq. (3.2)) with H_,, (b) tw 1J4 1= - 1.0.
343
= 1.0, (a) tw 1J4 1= 1.0;
Hamiltonian (3_la, b). To help visualize these effects the tensor part of (3.1) is converted to an RE surface function using the substitution (2.4) which was quoted in the discussion of rigid tops. The RE surface function follows from (2.4) and (3.lb). (3.2a)
=
+tC14M4[(35 Hscalar
c0s4p
-
30 cos3p + 3) + (5 sin”p cos 4y)].
(3.2b)
This RE surface is drawn in fig. 3.1 for arbitrarily chosen values of HSCdar= 1.0 and 4t,, 1J I 4 = 1.0 in fig. 3.la. The form of the RE surface levels and the corresponding eigenfunctions are independent of the values chosen for either HSCdaror I,,.+,.The eigenvalue fine structure patterns belonging to each J value will be shifted by adjusting HScalar and expanded uniformly by increasing t,, . A positive value 4t 044) J 14 = 1.3 is used in fig. 3.la, while negative value 4t,, = - 1.0 is used to give fig. 3.lb. The positive value corresponds to an octahedral XY, molecule which is more susceptable to bending its radial bonds than stretching them. One could imagine the x, y, and z body axes to lie along the X-Y equilibrium radial bonds of an XY, molecule. The axes are centered on the six peaks of the RE surface in fig. 3.la (or in the corresponding six valleys in fig. 3.lb). The second term in (3.2b) gives an RE surface height of 4.0 units on the z-axis (/I = 0). These are units of t,, I J I 4 above Hdar. The rotational energy is the following. RE(0, y) = HSdar + t,,
I J ( 4(4.0).
(3.3)
The third term in (3.2b) does not contribute in the neighborhood of the z-axis (/3 = 0 mod IT), but it is mostly responsible for the four peaks along the x and y axes (@ = + n/2) in fig. 3.la.
344
W.G. Harter / Graphical approaches to molecular rotations and oibratrons
04
A,
I
A,
.
I
E T,
I I
I 2
T2j2
14
24
34
.
’
Fig. 3.2. Different choices of rotation axes for octahedral rotor corresponding to local symmetry (a) C,, (b) C,, and (c) C,. Tables correlate global octahedral symmetry species with the local ones.
The locations of topography lines indicate that all six peaks are equivalent so the surface has a global Oh symmetry. The local symmetry of each peak is indicated by the shapes of the topography curves. Near the tip the curves have a nearly circular (R,) symmetry while further down they change into a square or C, symmetry near the separatrix. Beyond the separatrix the curves assume a triangular or C, symmetry which becomes more nearly circular again at the valley bottoms. Trajectories near the C, valley bottoms of fig. 3.la correspond to an XY, molecular rotation that is similar to that which is shown in fig. 3.2a. On the other hand trajectories near the C, peaks correspond to an XY, rotation pictured in fig. 3.2~. In the latter (C,) case centrifugal force would have a tendency to stretch the spinning bonds while in the former (C,) case it would bend the bonds. If bending is easier than stretching then the motion which causes bending will have lower energy for a given magnitude ( J 1 of the angular momentum. Hence there will be energy valleys for the XY, RE surface on the C, symmetry axes, and the C, symmetry axes will be on hills as shown in fig. 3.la. This would correspond to positive to&. For tetrahedral (XY,) on cubic (XY, or X,Y,) structure this constant may be negative. Then the RE surface has a cubical shape as shown in fig. 3.lb. As for the rigid tops it often happens that the RE surface is shaped something like the molecules that it represents. The topography lines in fig. 3.1 represent possible classical trajectories of the lab-fixed-J-vector or lab z-axis. This is like the rigid top surfaces in fig. 2.1 with one very important difference: The XY, molecule is constantly changing shape as the J vector precesses around any of the topography lines in fig. 3.1. Each point on this RE surface corresponds to definite amounts of the vibrational coordinate distortions induced by the angular momentum J at that point. The effective Hamiltonian (3.1) is a useful description only if the rotational distortion follows the precessional motion of J udiabatically. That is, only “virtual” mode polarizations are allowed; a significant amount of “real” or resonant mode excitation by rotation signals a departure from the simple RE surface.
W.G. Harter / Graphical approaches to molecular rotations and vibrations
345
Note that these adiabatic requirements apply to the effective tensor Hamiltonian. The RE surfaces in fig. 3.1 are merely ways to visualize effective Hamiltonians and to clarify the adiabatic requirements. An analogous set of adiabatic requirements exist in order that a potential energy (PE) surface be a useful description of vibrational motion in the Born-Oppenheimer approximation. In that case the electronic orbital distortion must adiabatically follow the vibrational motions without undergoing real or resonant transitions. Ultimately, it is hoped that the RE surface can be used to carry the Born-Oppenheimer approximation to the next level of detail and treat rotational motion which is adiabatically driven by vibrational and electronic (or vibronic) distortions. This may involve RE surfaces obtained directly from generic Hamiltonians as discussed in section 5 and multiple RE surfaces which are introduced in section 6. We consider now how the RE surface mechanics is related to the eigensolutions of the fourth rank tensor Hamiltonian of the SF, molecule. The RE surface levels in fig. 3.la are the appropriate quantizing curves for angular momentum quantum number J = 30. (fig. 3.lb shows levels for J = 36.) The lowest levels in the C, valleys and the two or three highest levels on the C, hills are very close to the intersections of the RE surface with angular momentum cones. The angles of these cones are found using (2.19). They are given for J = 30 and a range of K-values as follows. K
=
9;
=
30,
29,
28,
10.3”,
18.0”
23.3”,
27, 27.7”,
26, 31.5”,
25, 34.9”,
...
...
(3.4)
Simple geometrical analysis shows that the separatrix in fig. 3.1 approaches to within 35.3” for the C, axes and to within 19.5O of the C, axes. Using this along with (3.4) one can estimate which K-states will be bound by each region. The C, valley regions can only hold K = 30 and 29, while the C, hills can hold K = 30 thru 25. This prediction is verified by the exact level spectrum for J = 30 which is displayed in fig. 3.3. This is drawn using the same format as the asymmetric top spectrum in fig. 2.2. The six magnifying circles on the righthand side show clusters of levels associated with K-values of 30 through 25 while the two circles on the lower lefthand show the clusters associated with K-values of 30 and 29. Furthermore, the cubic symmetry species contained with in each of these circles can be predicted using the selected columns of the correlation tables in fig. 3.2. The use of these tables is the same as for the rigid top tables in fig. 2.4. For example, the lowest cluster of levels in the C, valley is labeled by K, = 30 and has local symmetry 0 mod 3 or 0,. The O3column of the C, table predicts that one each of the A,, A,, T,, and T2 species would appear in the cluster. The K = 29 = 2 mod 3 cluster contains one each of T,, E, and T2 species which are exactly the ones listed in the 2, column. Note that A, E, and T levels are one, two, and three degenerate levels, respectively. Thus the (AITrTzAZ) and (TrET,) clusters each contain eight levels and this corresponds to having eight equivalent classical trajectories for each K-value. The clusters above the separatrix are each associated with six trajectories having C, local symmetry. The trajectories belonging to K4 = 28 are indicated by arrows which show the direction of precession in fig. 3.3 Since this K-value is 0 mod 4 it is correlated with the 0, column of the C, correlation table in fig. 3.2, and the species E, Tr and A,, account for the six quantum levels in this cluster. The other C, clusters also contain six levels each but with different species. Odd-K, clusters contain Tr and T,, while A,, T,, and E show up for K = 2 mod 4.
346
W G. Harter / Graphical approaches to molecular rotations and vibrations
VISUALIZING THE J = 30 LEVELS
OF A
SPHERICAL
c, Clusters -4)
,-31
,
-21
TOP
C4 Clusters
i :
_,I
,
7 I
,
II
, Relative
12
,
Units
13
of
,
(4
,
(5
10‘4cmS’=3.0MHz
84.77319
Fig. 3.3. J = 30 Octahedral rotor levels and related RE surfaces paths (after ref. [3]).
The six-hold and eight-fold near-degeneracies of the C, and C, clusters are analogous to the C,(x) and C,(y) clusters of the rigid asymmetric top as can be seen by comparing figs. 3.2 and 2.4. We noted that asymmetric top clusters could be traced back in fig. 2.3 to the K-doublets associated with the irreducible representatives of the symmetric top symmetry R,. However, this connection can be misleading. First, there is no obvious symmetry like R, which has six or eight dimensional irreducible representations. Second, the cluster degeneracies are not perfect but have exponentially small superfine splittings that are a function of a tunneling amplitude between semiclassical trajectories. This tunneling amplitude depends upon the RE surface geometry in the
W. G. Harter / Graphical approaches to molecular rotations and vibrations
saddle point regions of the separatrix and not necessarily K-conservation of a particular trajectory.
347
on the degree of symmetry
or
3.2. Symmetry properties of eigensolutions A more detailed picture of the eigensolutions on the RE surfaces in fig. 3.1 emerges if one takes account of the geometry of the tunneling mechanisms. An example involving the superfine level structure of the 0, type clusters should demonstrate this point. For an O&pe cluster such as the (K = 28)-levels on the right hand side of fig, 3.3 one can imagine six equivalent angular momentum cones touching the RE surface at angle of approximately 82: = 23.3” near the classical trajectories. Centered on each of these one could imagine a wavepacket consisting mostly of a symmetric top wavefunction with M = J = 30
(a, P Y(;t) =~%**(% P Y) defined with responect to the local axis of quantization. These six states may serve as a basis { II>, 12)Y.V 16)) of a tunneling matrix of the following form.
m =
H
0
0
H
-S
-H
-S -S -S .-S
-S -S -S -S
-S -S H
-S -S 0 0 H
H 0 -S
-S
-S -S
-S -S -S -S -S H
0
-S -S -S -S -S 0 H
11) 12) 13) 14)
(3.5)
I? 16)
Here the base states 11) and 12) are associated with the wavepackets on the +z and -z axes, respectively, while 13) and 14) are similarly related to +X axes, and 15) and 16) go with the +y axes. A tunneling amplitude - S is assumed to exist between nearest neighboring trajectories, i.e., from +z to +X or to + y but not from + z to -z. Eigenvectors of this are easily obtained analytically. The totally symmetric eigenvector is
(A,) = (1, 1,1,1,1,1>/6
(3.6a)
with eigenvalue ,?(A,) = H - 4s. Three antisymmetric
(3.6b)
eigenvectors
ITIn)= ((4% 1, -1, 0,0)/d% lT,y) = (0, CO, Cl, IT,z) =
-1)/d%
(1, -1, O,O, 0,
0)/d%
(3.7a)
348
W.G. Harter / Graphical approaches to molecular rotations and vibrations
each share the triply degenerate eigenvalue E(T,)
= H.
(3.7b)
Finally, two eigenvectors ]El) = (l,l,
-l/2,
]E2) = (0, 0, l/2,
-l/2, -l/2,
-l/2, l/2,
-l/2)/&
-l/2),
(3.8a)
each share the doubly degenerate eigenvalue E(E)=H+2S.
(3.8b)
These eigenvalues explain the ordering and spacing that is observed repeatedly for each ETA cluster in the magnifying circles of fig. 3.3. A similar analysis can be done for all superfine structures in that figure [8]. A wavefunction of type-O, localized onto a single trajectory of, say, the +z-state 11) would involve a non-stationary combination of symmetry-defined states analogous to the rigid top states (2.15) or (2.16). Inverting (3.6a) thru (3.7a) gives the following combination at time t = 0 1q(o))
= 11) =
k
IA,) + +
IT,4 + + I’%.
(3.9)
At a later time the state would evolve into a delocalized combination e-i(H-4S)t
IW))
=
fi
e-i(H)t
IAd + - fi
of the following form
e-i(H+2S)r
IW
+
0
El).
(3.10)
However, this combination would relocalize perfectly again and again. Perfect recurrance occurs with an angular frequency of w = 2s since the eigenvalue (3.6b) through (3.7b) are integral multiples of this values plus a constant H. The values of tunneling amplitudes S decrease quasi-exponentially as clusters move away from the separatrix. In the case of the SF, rotor levels with J = 30 shown in fig. 3.3, the total superfine splitting varies between 4.8 Hz on the highest ATE cluster to about 2.0 MHz near the separatrix. For higher J the superfine structure splittings cover an even larger spectroscopic range. We consider now numerical approximations for fine and superfine levels. 3.3. Semiclassical quantization The semiclassical techniques which were used to approximate the asymmetric top spectrum can also be applied to higher rank tensor eigensolutions. Here we discuss the angular momentum cone and action path integral approximations to the fourth rank tensor solutions. The maximum points on the RE surface in fig. 3.la are the [lOO] directions of the coordinate axes. Substituting J= (0, 0, I J I) into (3.la) or Engler angles (p = 0, y = 0) into (3.2b) gives E ,,_=E(O,
0) =4( J14
(3.11)
349
W. G. Harter / Graphical approaches to molecular rotations and vibrations
Table 3.1. Comparison of exact cluster centroid energies and tunneling amplitudes with classical RE surface and semiclassical approximations. (Units of [ J( f + l)j2 for tw = 1) (a) J= 30
E(exact diag.)
E(RES-cone)
E(Semi-class)
S(exact)
S(SC)
K=30 k = 29 kc=28 K=27 K=26 K=25
3.3829967 2.256160 1.302880 0.51132 -0.1272 - 0.616
3.373 2.246 1.292 0.495 -0.157 - 0.677
3.371 2.249 1.293 0.503 - 0.140 - 0.637
1.68E-07 6.083X- 06 9.758- 05 9.05E-04 .5.30E-03 1.49E - 02
1.59E-07 6.10E-06 9.94E - 05 9.378-04 5.51E-03 204E - 02
(b) J= 88
E(exact diag.)
E(RES-cone)
E(Serni-class)
S(exact)
S(SC)
1.55E-21 1.95E- 19 l.l8E- 17 4.60E - 16 1.29E - 14 2.80E- 13 4.86E- 12 6.93E- 11 8.26E - 10 8.36E - 09 7.22E - 08 5.38E-07 3.47E - 06 1.93E-05 9.348-05 3.80E - 04 1.29E-03
1.46E-21 1.91E- 19 1.17E - 17 4.59E- 16 1.29E- 14 2.81E- 13 4.89E - 12 6.98E - 11 8.348- 10 8.45E-09 7.32E-08 5.47E - 07 3.53E-06 1.98E-05 9.52E - 05 3.97E-04 1.40E-03
K=88
K=87 K=86 K=85 K=84 K=83 K=82 K=81 K=80 K=79 K=78 K=77 K=76 K=75 K=74 K=73 K=72
-
3.778691053 3.349362373 2.942069530 2.556379795 2.191699743 1.847493649 1.523240328 1.218437672 0.932~5088 0.665288448 0.416068852 0.18456690 0.0295167 0.226355 0.~6~7 0.56832 0.7104
-
3.777 3.348 2.941 2.555 2.190 1.846 1.521 1.215 0.928 0.659 0.408 0.174 0.044 0.246 0.433 0.604 0.762
-
3.778 3.348 2.940 2.555 2.190 1.846 1.522 1.217 0.932 0.664 0.415 0.183 0.031 0.228 0.408 0.571 0.717
where we set t,, = 1 and H,,, = 0. The ~~rnurn points are the [ill] directions. Setting J= (l/0)( 1J 1, 1J f, 1J 1) or (/3 = 54.7*, y = 4S”) gives the following energy for the lowest points on the fourth rank RE surface. =E(54.7”, enin
45’)
= -(8/3)
1J14=
-(2/3)E_.
(3.12)
The saddle points occur at the [llO] directions where J = (I/a)( I J I, 1J 1, 0) or (/3 = 90 O, y = 45 * ) and the energy is
E sadd,e=E(900,450)=
-lJ14=
-(l/4)&_.
(3.13)
This is the separatrix energy with divides the fine structure interval E,, - E_ into two parts. The upper three-quarters of the region is devoted to C, levels while the lower quarter of the fine structure pattern is devoted to C, levels. By substituting the J = 30 cone angles p = 8;’ from (3.4) into the RE expression (3.2b) with y = 0 one obtains the approximate values shown in the second column of table 3.la. These agree within a few percent with the exact values for the C, level clusters in units of t,, 1J I 4 with Hscalar= 0. Note that the highest cluster value 3.38 is well
350
W.G. Harter / Graphical approaches to molecular rotations and vibrations
below the classical maximum (E,, = 4.0). The difference is the rotational zero point energy associated with the angular momentum cone uncertainty angle. This zero point energy decreases as J increases. The highest value of the J = 88 level clusters is 3.78 as seen in table 3.lb. This is closer to the classical maximum. Also, the RE surface-cone approximation is more accurate for higher J. Semiclassical path integration formula (2.23) applies to the fourth rank RE surface paths, as well, and the numerical results are quoted for J = 30 and J = 88 in the third column of table 3.la, b. Some improvement in accuracy over the simple cone approximation is obtained for lower K-values. However, the main strength of the action path integration is the ability to estimate the tunneling amplitude or S-values in the superfine pattern formulas like (3.6b) thru (3.7b). The procedure used to give the semiclassical results in the S(SC) column of table 3.1 and analogous to the rigid top formulation in (2.25) thru (2.28). Details are given in ref. [3]. 3.4. Mixed-rank
octahedral tensor levels
So far we have considered only the lowest order rotational tensors which exhibit the symmetries D, of the rigid rotor and 0, of the semi-rigid cubic or octahedral rotor. We consider now the effect of the sixth-rank normalized octahedral tensor operator
PI=
(l/fi)[g-
(d7/Jz)(T46+P4)].
(3.14)
This will be added in varying amounts to the normalized fourth rank tensor (3.15) contained in (3.1). A sixth rank centrifugal distortion may be necessary in the presence of anharmonic and other higher order effects. The magnitude of the Tt6] contribution would vary according to a higher power of J than that of Tt4] and might be significant at higher J values. Here the magnitudes of their respective contributions are varied artificially through an angle parameter v in a combination which maintains the overall normalization T4,6( v) = Tt41 cos v + PI
(3.16)
sin v.
The exact quantum (J = 30)-eigenvalues for this mixed [4,6]-rank tensor operator are plotted as a function of the mixing angle v in fig. 3.4. The plot begins on the lefthand side (v = 0) with a scaled copy of the T [41level spectrum in fig. 3.3 and ends on the righthand side (v = V) with the same spectrum inverted. Between these limits the level clusters become completely reorganized [36,37]. Certain values of the v parameter in fig. 3.4 are marked (b), (c), (d), and (e). At these values the RE surface of the combination tensor (3.16) is drawn in fig. 3.5. The RE surface function used for Tt4] is as follows Et”(/3, y) = (7/12)“2(9/4+2(35
cos4b - 30 cos2p + 3 + 5 sin4p cos 4y)/8.
(3.17)
W.G. Harter / Graphical approaches to molecular rotations and vibrations
(4
09 00
10
(4 ..,“t,”
20
v III
40
351
(4 JO
60
of 7r/6
Fig. 3.4. J = 30 eigenvalues of varying mixtures of 4th and 6th rank tensors. (Y = 0) corresponds to levels in fig. 3.3.
For
TL6] the RE function is as follows
I?(p,
y) = (1/8)“2(13/4+2(231
co@
- 315 cos4/3
+105 cos3p - 5 - 21 sin4p(11 cos2p - 1) cos 4y)/16. The tensors
(3.18)
T[‘] and RE functions E [rl have a spherical harmonic normalization factor ([2r + 11/4a) ‘I2 that was not included in previous definitions (2.4) or (3.2). This factor is used here to slightly enhance the effect of the 6th rank tensor for this particular example. Also, the 1J 1r factors are deleted in (3.17) and (3.18) so that the higher rank tensor effects are not J dependent. The eigenlevels marked by (b) in fig. 3.4 correspond to the RE surface drawn in fig. 3.5b. The later shows that the separatrix has taken over the regions that formerly held C, symmetric trajectories, and only C, trajectories remain. (Note that these are equally spaced contours and are not quantized paths.) The result is the destruction of C, clusters in the spectrum which is composed almost entirely of C, clusters above the (b) point in fig. 3.4. Beyond this point a remarkable new type of cluster is formed. Just above the points marked (c) and (d) in fig. 3.4 lie two clusters which contain twelve levels each. These correspond to trajectories which encircle twelve equivalent valleys which lie on the C, symmetry axes in figs. 3.5~ and d. The symmetry species within each of these clusters are exactly the ones contained in the C, correlation table in the center of fig. 3.2. The lowest cluster in fig. 3.4 would correspond to K = 30 and hence to the even local symmetry or the 0, column of the table in fig. 3.2 which contains species A, E, T,, and 2T2. The next cluster has K = 29 and contains the five species A,, E, 2T2, and Tr listed in the odd column 1,. The superfine splittings between these five levels are
352
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 3.5. RE surfaces corresponding to selected v-values in fig. 3.4. (a) v = 0.0; (b) v = 0.4 (T/t); v = 4.0 (n/6); (e) v = 4.6 (q/6); (f) v = 5.0 (IT/~).
(c) v = 2.0 (~/6);
(d)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
353
actually visible in the scale of fig. 3.4. As Y changes the levels as seem to change order within this cluster. This is result of competition between tunneling mechanisms. Between the (b) and (d) points in fig. 3.4 there is another phenomenon which occurs in the upper energy levels. There are a number of crossings or Fermi-like resonances between accidentally coinciding C, and C, clusters. This is because there are two kinds of mountains on the RE surfaces in figs. 3.5~ and d: the C, mountains which are shrinking and C, mountains which are growing with Y. For certain values of v quantizing paths on one type of mountain are bound bo te in resonance with different kinds of paths on the other. The result is an extraordinary kind of tunneling in which eigenfunctions are delocalized over both kinds of paths at once and a peculiar sort of hybrid superfine structure occurs in the eigenlevels. The spectral region containing the unusual fine structure is bounded on the righthand side by the (e) point in fig. 3.4 which corresponds to the RE surface in fig. 3.5e. At this point the eight C, mountains dominate the surface geometry entirely and the eigenlevels are composed entirely of very strong C, clusters of eight levels each. The final fig. 3.5f shows the situation at v = 5.0( ITS) where the C, trajectories begin to return. Now they are occupying the valleys. By v = 6.0( 76) the RE surface resembles the one shown in fig. 3.lb.
4. Quasi-spin RE surfaces for two-dimensional
vibrations
The following is a method for using RE surfaces to help visualize anharmonic vibrations [38]. The method is based upon an analogy between rotor motion, two-dimensional oscillation, and quantum evolution of a spin-l/2 or spinor system. This analogy has a long history, some of which will be mentioned in section 4.1. 4.1. Spinors, rotors, and coupled oscillators
(a) Spin and rotors The quantum mechanics of spin-l/2 operators is related to the algebra of the special unitary group SU(2) and to the three dimensional rotation group R, which is homomorphic to SU(2). The evolution of spin-l/2 or general two-level systems is described by the Rabi-Ramsey-Schwinger [39] or Feynman-Vernon-Hellwarth [40] formulations. These formulations treat the two-state Schrijdinger equation
@/at) I+> = H I SV
(for A = 1)
(4.1)
by interpreting the solution I +W>
= cHf
I WN
(for constant H)
as a rotation. If the representation
(4.2)
of (4.1) has the following form
(4.3)
354
W. G. Harter / Graphical approaches to molecular rotations and vibrations
then the Hamiltonian operators as follows.
matrix may be expanded in terms of Pauli spin angular momentum
+%4
+w$
32
(4.4)
The first component w. = (A + D)/2 is th e angular velocity of the overall phase. The remaining three components can be viewed as Cartesian components of an angular velocity vector w = (o,, u,,, wz) which can be written in terms of a set of polar coordinates { o, 8, +} as follows. w, = w cos t#~sin 9
wr = w sin $3 sin 8
= 2B,
o, = o cog e
= 2c,
(4.5)
=A-D.
The time evolution in eq. (4.2) is viewed as a rotation combined with an overall phase change. = e-i(wo-oeJ)t I*(O)).
1*k(t))
(4.6)
A similar three-dimensional vector description is given for the state 1\k) or for the two-level system or for the density operator which represents that state. Each state can be written as an Euler angle rotation of the first base state 11)
A spin-l/2
representation of this construction has the following form (see appendix A.) e-‘(” +y)/2cos( B/2)
(I ( *I
‘k2 =
e1(a-u)/2 sin( B/2)
- e-1(a-v)‘2
sin( B/2)
e’(* +7)/2 cos( B/2)
1 i!i 0
=
eeia12 Cos( B/2) i ela12 sin( B/2) I
e_iy,2 .
(4.8) The expectation value of the spin angular momentum vector J gives an expected spin vector S.
~x=Pl4l~) =+ cosasinp,
~y=(~lJ,lW =- i sinasinp,
~z=(~l4IW =- : cos P,
(4.9)
This spin vector can be viewed as the main axis of a rotor which will be acted upon by the evolution operator in eq. (4.6). The third Euler angle ( y) is the “ twist” of this rotor, and it is proportional to the overall phase of the spin state (4.8). The rotor rotates and twists around the w-vector given by eq. (4.5) as the spin state evolves according to eq. (4.6).
355
W. G. Harter / Graphical approaches to molecular rotations and vibrations
(b) Spinors and oscillators Spinor state evolution can also be related to that of a two Dimensions oscillator. The real and imaginary parts of the spinor amplitudes { JI,, ttz ) can viewed as oscillator phase space variables h P17 x27 P2 1. These in turn can be related to the Euler angle definition in eq. (4.8). G1 = x1 + ipr
(4.10a)
+,=x,+ipz
= JFe-‘(or+y)‘z cos( /3,/2),
= Jie-‘(y-“f/2
(4.10b)
sin( b/2).
Here an intensity variable I or normalization factor fi has been included so that the Euler angle definition will have a total of four variables which is the same number as the oscillator phase space dimension. We now consider the following coupled oscillator Hamiltonian H HO = (A/2)(p:
+x:>
+ (o/2)(
p; + x2’) + B(P,P,
+ Xl%)
+ C(XlP2
- X2PJ
(4.11)
This Hamiltonian yields classical equations of motion that are identical to the quantum equations that result from substituting eq. (4.10a) into the SchrGdinger equation (4.3). Either (4.3) or (4.11) yield the following . Xl
. x
=hK3
=-=Ap1fBp2-Cxz, 8P1 %iO
2 =-=BpI+Dpz+Cx,, 3P1
a&i0
-A
=
-+Ax,+BxZ+Cp2, ax, (4.12)
-j2
=
a%i0
-=BxI+Dx2-CpI_ 3x2
In general, an n-level quantum system has the same equations of motion as an n-dimensional classical harmonic oscillator. This idea goes back to the Planck and DeBroglie hypotheses at the beginning of quantum mechanics. However, it is only recently that this idea has been used as a visualization aid for molecular vibrations 1381. The idea of spinorial ch~acte~zation of two dimensional oscillators goes back even farther. Over a century ago, Stokes developed a theory of electromagnetic polarization ellipsometry [41]. In this development the complex electric field amplitudes E, and EY correspond to spinor amplitudes #i and #2, respectively. The spin expectation value 5’ corresponds to the Stoke’s vector which defines the state of elliptical polarization. In fig. 4.1. the three-dimensional Stokes vector space is shown on the righthand side. The Stokes “quasi-spin” vector components (S,, S’, S,) are labeled (B, C, A) and they characterize the shape of a polarization ellipse shown on the lefthand side of fig. 4.1.. The ellipse is a plot of the real parts x = x1 and y = x2 of the spinor amplitudes in eq. (4.10a) for a state of constant S. One characterization of the ellipse is its major to minor axis ratio b/a = tan JI and the inclination angle cp of the major axis. These two angles determine the orientation of the Stokes vector relative to the C-axis. The latitude and longitude of S relative to the C-axis are GM& these angles, that is 2lc/ and 2+, respectively. If the C-axis is chosen as the z = axis of quantization in the rotor-spin analogy, then the polar angles in eq. (4.9) would be ((Y = ~QYJ, p = 7~/2 - 2$). The C-axis would be a convenient choice if the oscillator was to be described in terms of left and right handed circular polarization states. In the molecular vibrational model
356
W.G. Harter / Graphical approaches to molecular rotations and vibrations
a
Fig. 4.1. Ellipsometric characterization of oscillator trajectories for spin-rotor analogy. (a) Two-dimensional spinor or SU(2) oscillator space; (b) three-dimensional vector or R(3) rotor space.
this would be an appropriate choice in the presence of a strong Coriolis term, that is, a large value of C in the Hamiltonian H,, in eq. (4.11). The choice of z-axis for the present description is the A-axis. This is a convenient choice if the oscillator is to be described in terms of x- and y-polarization states. In the molecular vibration model this would be an appropriate choice when ) x) and 1y) correspond to normal modes, and the value of (A - D) is large compared to B or C in H,,. The polar angles of S relative to the A-axis in fig. 4.1 are ((Y = 29, p = 2~). This choice corresponds to another way to characterize the polarization ellipse on the lefthand side of fig. 4.1. Here the ratio X/Y = tan v of maximum x and y amplitudes and the angle 9 determine the ellipse. The geometrical interpretation of the angle 19 in the two dimensional oscillator picture is tricky, and will be discussed shortly. However, the azimuthal angle 26 is easily visualized in the three-dimensional part of fig. 4.1. Also, the algebraic interpretation of 29 is simple; it is the relative phase of 1x) and 1y) in the following state 1‘k) = (X e-”
1x) + X e-”
1y)) = (cos v e-”
I x)
+ sin v eis 1y))
e-‘ofi.
(4.13)
The third Euler angle y = 20 is proportional to the overall phase as in eq. (4.8). Also, the amplitude factor J? is necessary since a classical oscillator can have arbitrary amplitude. The geometric interpretation of fi and I as the radius of a circle and a sphere, respectively, is shown in fig. 4.1. The expectation values for Pauli spin angular momentum operators in the state (4.13) can be expressed in terms of the spin-rotor parameters 9, v, and I, or else in terms of the oscillator phase space variables xi, x2, pi, and p2
~x=cwJ~k)
=xlxz+PlP2
=
(1/2) sin(2v) cos(29),
(4.14a)
S,=(‘k
IJYIW
=xlP,-x,P,
=
(1/2) sin(2v) sin(26 )9
(4.14b)
S,=(!P
MIW
=(P:+xx:
JO=(+
luwk)
=
(P:+
- p2’ x; +pz’+
x22)/2 = (1/2) cos(2v),
(4.14c)
x,2)/2 = (I/2).
(4.14d)
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Then the oscillator Hamiltonian quasi-spin vector S. H HO =
[CA+ D)/21JO+ ‘&
(4.11) can be expressed in terms of the components
+ cs,
357
of the
(4.15a)
=
woJo
(4.15b)
=
woJo
(4.1%)
This appears to have the form _ZQ,J, of an action angle Hamiltonian, neither the classical Poisson brackets such as (S,, S,,) nor the quantum commutators such as [J,, J,] will vanish. One may use Euler angle variables to obtain a legitimate action angle form as discussed in appendix B. However, the form of eqs. (4.15) is useful for comparing the angular rates of the oscillator models. It is seen that the angular rates w, of the three-dimensional spin vector or rotor are double the correspondence rates 1(2, of the two-dimensional oscillator. This doubling was also seen when comparing angles between the two parts of fig. 4.1. Roughly speaking, this is because spin “up” and “down” are only 90 o apart in the two-dimensional space but are 180 o apart in three dimensions. The angular doubling is an important part of the correspondence between the rotor and oscillator dynamics. Some examples which compare the two ways to visualize spinor or oscillator motion are shown in fig. 4.1. Two archetypical cases are used. The first case is shown in fig. 4.2a, and it corresponds to isotropic (A = D), uncoupled (B = 0), oscillators under the influence of a Coriolis force. The resulting motion is called a Foucault Pendulum precession in mechanics, or Faraday rotation of the polarization in optics. The w-vector for this case points along the C-axis as shown in the three-dimensional side of fig. 4.2a, and this is consistent with eq. (4.5) and our choice of C as the y-axis in (4.15). S-vectors which are parallel or anti-parallel to the C-axis correspond to the left or right circularly polarized optical states, respectively. These vectors would not precess, and circular orbits would remain circular. However, the speed of the circular orbits depends upon whether the orbit is right or left handed. All other states correspond to elliptically polarized orbits which precess. Their S-vectors go around at an angular rate of w,=2Cin{A, B,C}spacewhileanellipsegoesaroundatarateofonly~~=Cin{ lx), Iv)} space. The second archetypical case is shown in fig. 4.2b, and its corresponds to an anisotropic (A # D), uncoupled (B = 0), and Coriolis-free (C = 0) two dimensional oscillator, or to two one-dimensional oscillators in which 1x) and 1y) represent symmetric and antisynunetric normal modes. The resulting motion is called beating in mechanics and electronics, or polarization birefringence in optics. The w-vector for this case points along the A-axis as shown in the three-dimensional side of fig. 4.2b, and this is consistent with eq. (4.5) and our choice of A as the z-axis in (4.15). S-vectors which are parallel or anti-parallel to the A-axis correspond to x and y linearly polarized optical states, respectively. These vectors would not precess, and linear orbits would remain linear. However, the frequency of the linear oscillations would depend upon whether they were x or y polarized; the former is A and the latter is D. All other states which are mixtures of 1x) and 1y) will exhibit beating and generate a Lissajous pattern in the
358
W.G. Harter / Graphical approaches to molecular rotations and vibrations (a ) Faraday Rotation
fb 1 Birefringence
Fig. 4.2. Two diiensional
oscillator trajectories and corresponding three dimensional rotations. (a) C-axial rotation or Faraday precession; (b) A-axial rotation or birefringence.
two-dimensional space as shown in fig. 4.2b. At the same time the S-vector will precess around the A-axis at an angular rate of w, = (A - D) which is commonly called the beat frequency. This is the frequency of the beat intensity, but the beat amplitude oscillates at half this frequency: s2, = (A - 0)/2, in agreement with eq. (4.15). The visualization of beating as a rotation is a useful concept in discussing models of molecular vibrations. Suppose the 1x) and 1y) represent symmetric and antisymmetric combinations of localized models I&) and l (f2) I4
= IA,) = (10
where 15,) and I&) molecular system IEd =(I4
+ ltz))/fi,
IY) = I&)
= (It,>
- I&%‘~~
(4.16a)
represent motions confined to atom-l or atom-2, respectively, in a
+ IYWfi~
152) ‘(IX)
- IYW.
(4.16b)
The I &) and I(,) correspond to optical polarization at 3145 * to the fast and slow axes. The S-vectors for these states lie along the f B axes. The rotation around the A-axis of such a vector causes the atoms to take turns vibrating or “beating” as the S-vector rotates from + B through
359
W.G. Harter / Graphical approaches to molecular rotations and vibrations CORIOLIS
HARMONIC
OSCILLATOR
NH=2
CORIOLIS
NL=l A
HARMONIC
OSCILLATOR
NH=2
NL=l
A
Fig. 4.3. Phase space trajectory on torus in {x, y, px} space for a mixed (ABC) rotation, (Stereoptic figure.)
+ C to -B an then back to + B after passing - C. As the S-vector passes the + C axes each time the atoms will be vibrating with equal amplitude but differing in phase by *a/2. These are the points where the resonant transfer of energy between the atoms is maximum. Anharmonic perturbations can spoil this resonance and trap the S-vector near one side of the B-axis. This corresponds to local mode trapping which is discussed below. Computer animation of the beating motion provides an alternative view to a time trace such as the one given in fig. 4.2b. A properly animated view is seen immediately as a rotation, albeit a rather curious one. It is rotation of the angle 6 with angle Y held constant around 45”. What is curious is that the viewer sees it as an optical illusion which rotates around either the x-axis or the y-axis of the two-dimensional screen! This is an interesting observation since the three dimensional {A, B, C} picture of this motion is a simple rotation about the A-axis, but that is the axis which contains the x and y polarizations. The three-dimensional picture is always simpler, but the two-dimensional one has more information. The latter information exists in a subtle form which the human visual system can quickly interpret given a little practice. The &rotation is one example in which animated views of the spinor or two-dimensional oscillator picture are effective. Another way to enhance the oscillator picture is to display more of the oscillator phase space. An example of this is shown by a stereo plot of three of the four phase variables xi, x2, and p1 in fig. 4.3. This figure may be viewed with a standard stereopticon, or by simply relaxing one’s eyes so that the left (right) image is seen by the left (right) eye. The example is a Hamiltonian for which the w-vector has components on all three axes as shown by the {A, B, C} plot of the w and S vectors in the upper righthand corner of fig. 4.3. In stereo one can clearly see an invariant torus for this Hamiltonian and a closed phase path on that surface. A careful inspection of the torus shows that it is not just a “doughnut”. More details of this type phase picture are given in ref. [38].
360
W. G. Harter / Graphical approaches to molecular rotations and vibrations
4.2. Quasi-spin RE surfaces (a) Harmonic oscillator surfaces The analogy between rotor dynamics and vibrational motion suggests that the RE surface picture used for rotors in sections 2 and 3 might also be of some value for describing vibrational dynamics and spectra. The Harmonic oscillator Hamiltonian (4.15) yields the simplest type of RE surface since it is linear in the quasi-spin components S,. By replacing these components with their polar coordinate form given in eq. 4.14a-c, one obtains the following expression for an RE surface E HO=(1/4)[(A+D)+2BcoscrsinB+2CsincysinB+(A-D)cosB].
(4.17)
Here the Euler angles (Y= 29 and B = 2v are used as the azimuthal and polar angles, respectively. The surface which results is a quasi-spherical or balloon-shaped object which is displaced and stretched in the direction of the w-vector (2B, 2C, A - 0) as shown in fig. 4.4a. The level curves on the surface are possible trajectories for the quasi-spin vector S, and these are seen to be a set of circles around the w-vector. This is consistent with the {A, B, C} pictures given in figs. 4.2 and 4.3. The S-orbit corresponding to birefringence or beating in fig. 4.2b is one of the circles in fig. 4.4a. The orbits on the harmonic oscillator surface all move in the same direction with the same angular frequency. The poles of the surface are the only fixed points. Note that orbits move counterclockwise when viewed from above a maximum point which is the positive A-axis in fig. 4.2b or 4.4a. This is opposite to the convention used for describing body fixed RE surfaces in sections 2 and 3. Here we are using lab fixed polar angles { (Y, B } instead of { - y, - B } . Note, also, that the vibrational RE surfaces do not necessarily have to have in inversion symmetry since their may be odd powers of the quasi-spin components in the oscillator Hamiltonian. (b) Anharmonic oscillator surfaces The effects of anharmonic perturbations can be visualized using the energy surface construction. For example, the following Hamiltonian will be examined as a function of the anharmonicity parameter a, (4.18) The quasi-spin RE surface function is the following E A0
=
E,,
+ a, ( 12/4)
cos*a sin*B.
(4.19)
Hamiltonians of this form have been introduced in refs. [24-281 as models for local mode behavior in quantum systems. The classical variable defined in eqs. (4.10) and (4.14) can be changed to quantum operators by replacing the phasor variables 4, = x, + ip, and J/,* = x, - ip, by boson operators a, = x, + ip, and ai = x, - ip,, respectively. The energy surfaces for a, = 1.0 and 3.0 are shown in figs. 4.4b and c, respectively. The harmonic constants are fixed to the values A = 4.0, B = 0.0 = C, and D = 2.0 for each case in
W.G. Harter / Graphical approaches to molecular rotations and vibrations
361
Fig. 4.4. Vibration quasi-spin RE surfaces and level paths drawn using eq. (4.17) with: J = 1, A = 4.0, B = 0 = C, d = 2.0. (a) Harmonic case: a, = 0.0; (b) Threshold anharmonicity: a, = 1.0; (c) Local mode due to large anharmonicity: a, = 3.0.
fig. 4.4. The RE surface for the anharmonicity constant a, = 1.0 is shown in fig. 4.4b. There is some squeezing of the circles that were near the positive A-axis in the harmonic case of fig. 4.4a, but the topology of the orbits is unchanged. However, as shown in fig. 4.4c, higher values of a, yield a new pair of fixed points surrounded by figure-eight separatrices which grow out of the positive A-axis as a, increases. The new fixed points approach the f B axes as a, becomes large. We recall that the f B axes are in the neighborhoods of the local mode state I&) or ) t2) given in eqs. (4.16). The ellipsometry embodied in fig. 4.1 allows one to precisely visualize the shape of the oscillator trajectory in {x1, x2 }-space corresponding to any given fixed point in { A, B,
362
W.G. Harter / Graphical approaches to molecular rotations and vibrations
C }-space. From eqs. (4.17-19) it follows that the extrema are located on the A&plane (a=29=Omod 71) at an angle (& = 21) to the A-axis for which the following derivative is zero W40
- ysin O= w a-o=
/3+ aXGsin
p cos p
i
$+_L!+.
(4.20a)
In general, there are the following four zeros for the derivative p. = 0,
7,
kcos-1%.
(4.20b)
x
The first two solutions p = 0 and p = IT correspond to the normal modes A, and A,, respectively, in eq. (4.16a). The second pair of solutions are valid only if a, is greater than the ratio a/I of the true normal mode beat frequency L? and the intensity of the oscillation. For the example shown in fig. 4.4 this threshold occurs at a, = 1.0, which is the value used in fig. 4.4b. Beyond this value the A, normal mode becomes unstable, and the new fixed points approach p = f 71/2 which belong to the local modes. The A, mode corresponding to p = IT remains stable, but the paths surrounding the negative A-axis become more and more squeezed as a, increases and the local modes occupy more of the RE surface. The shape of the RE surface and its level paths provide an indication of the form of the quantum level spectrum for each value of the oscillator intensity I or the principal quantum number N(afa, + a$a2) = (24).
(4.21)
In the case of fig. 4.4a the vibrational levels are nondegenerate and uniformally spaced. In the case of fig. 4.4~~the lower levels levels corresponding to the A, mode are still singly degenerate, but they become more crowded near the separatrix region in the middle of the level pattern. The upper levels corresponding to the pairs of local mode loop paths are nearly degenerate doublet cluster similar to the asymmetric top clusters seen in fig. 2.2. 4.3. Higher dimensional oscillators A number of anharmonic oscillators models have been investigated for local mode behavior. Many of them such as the well known Henon-Heiles [42] or Barbannis models [43] are two-dimensional, but do not necessarily conserve the principal quantum number given in eq. (4.21). Jaffe and Rheinhardt [42] have used Birkhoff-Gustaveson transformations to derive effective action functions of the Henon-Heiles Hamiltonian whose RE surface equivalent is like the symmetric top surface in fig. 2.la but with three local mode valleys around the equator. Patterson [44] has constructed vibrational energy (VE) surfaces for the anharmonic three dimensional model of SF, vibrations which resemble the RE surface in fig. 3.lb. This model also exhibits chaotic classical motion. It is possible that a graphical analysis of multiple N-surfaces might prove worthwhile for visualizing the irregular or chaotic motions found in these models. Elementary examples of multiple surfaces and the resulting irregular motions are discussed in section 6.
W. G. Harter / GraphicaI approaches to molecular rotations and vibrations
5. RE surfaces for generic Hamiltonians
363
of non-rigid rotors
The tensor Hamiltonians for semi-rigid rotors are effective operators that are derived from a fundamental or generic Hamiltonian of the form H=KE+PE.
61)
The effective operators such as the fourth order Hamiltonian (3.1) or the sixth order tensor (3.14) are just the first two terms in a series expansion of (5.1) involving arbitrarily high powers of the rotor angular momentum J. If higher order terms contribtite negligibly to the distortion of the rotor then its rotational dynamics and fine structure energy levels will be be well described by just the fourth order tensors. However, if many higher order terms are necessary, the algebra needed to derive them and their subsequent numerical treatment becomes increasingly laborious. As an alternative approach, one might derive the rotational distortion energy numerically as a function of the rotor momentum vector. This approach would avoid the algebra of the tensor expansion while, in effect, carrying the expansion to infinite order. Furthermore, it would work just as well for cases in which the potential energy of the Hamiltonians (5.1) was highly anharmonic; in fact the potential could be given numerically. For non-rigid or highly excited molecules, it is unlikely that any usable analytic forms will approximate their potentials. An example of this approach is shown in this section for an elementary model of the simplest polyatomic symmetric rotor X,. The numerical approach will be compared to the fourth order expansion for a harmonic model of the X, rotor, and RE surface plots will be shown for harmonic and anharmonic models. As shown in sections 2 and 3, it is possible to approximate rotational energy levels by quantizing paths on the RE surface. So far this has not been done for any of the numerically generated RE surfaces. However, since the RE quantization is already a numerical process, it will not matter if the input is given numerically instead of by a formula that has to be numerically evaluated anyway. A completely numerical quantization example should be done for a non-trivial model which can be solved by other means in order to compare the different approaches. A number of possibilities can be suggested. One example is the well known Hougen-Bunker-Johns non-rigid bender model [45] and it numerical implementation by per Jensen [30]. Another example is the stretch bender model which have been studied classically by Fredrick, McClelland, and Brummer [46] and Eua [47] and semi-classically by Frederick and McClelland [48]. It is likely that these models will involve multiple RE surfaces of the type that will be discussed in section 6. 5. I. The generic Hamiltonian
in the body frame
The present discussion is just meant as a preliminary study of the generic RE surface as an aid to visualizing the properties of numerical rotor models. The approach is based upon using a body-fixed version of the generic Hamiltonian such as the classical Wilson-Howard Hamiltonian. [49]. This is given below in a form similar to the one described by Papousek and Aliev [50] H vib--rot= ;R,&.R
+ Hcb.
(5.4
W.G. Harter / Graphical approaches to molecular rotations and vibratrons
364
Here the rotor momentum is R=
J-Ltib,
(5.3)
where J and Ltib are the total and vibrational angular momenta, respectively. The latter is defined in terms of Coriolis S coefficients of the first kind and vibrational normal coordinates and momenta which will be defined further below (Note that indices which are repeated on only one side of an equation are to be summed.) (5.4) The normal coordinates are linearly related to body fixed Cartesian displacement ~,Jv) {k=x, y, z; v=l...N} ofeachof Natoms ‘b =
Bb,k(v)dk(v)9 dj(
v,=B-l I(Y)*aSa
coordinates
(5.5)
’
The vibrational part of the Hamiltonian contains the Born-Oppenheimer approximate potential and an inverse G-matrix kinetic term that depends upon the B-coefficients in the normal coordinate definition Htib = G;;s,sb
+ PE( S) .
(5 -6)
The G-matrix is defined by
where m(v) is the mass of the v th atom. The B and G coefficients of the two kinds of Coriolis coefficients l and 6 as follows
also appear in the definition
where cink is a standard c-tensor. These in turn are used to define the canonical vibrational momenta (5 -9) where w is the instantaneous angular velocity of the rotor frame. Finally, tensor components $ involve both kinds of Coriolis coefficients
the inverse inertial
(5.10) where the nuclear inertia tensor is written as follows
365
W.G. Harter / Graphical approaches to molecular rotations and vibrations
in terms of body frame defined Cartesian coordinates x,(y)
=a&>
(5.12)
+ d,(v).
These include the positions of static equilibrium a,(v) as well as the displacements d,(v). The definition of the generic Wilson-Howard Hamiltonian is based upon having a subset of normal coordinates which are “genuine” vibrations, that is, they contain no components of translation or rotation. This leads to the first and second Eckart conditions on the molecular displacements m(v)di(v)
= 0,
m(v)+a,(v)dk(
v) = 0,
(5.13a)
(i = x, Y, 2).
The center of mass is understood to be the origin, that is ??2(v)a,(v)
(5.13b)
= 0.
The object of the following discussion will be to determine the Eckart-allowed displacements for a given angular moments .I. This involves finding a set of dynamic equi~b~um points for the genuine normal coordinates s, (a = 1 . . .3N - 6.) which, in general, differ from the static (J = 0) equilibrium points s, = 0. It should be noted here than Watson [Sl] has shown that the quantum mechanical form of the Wilson-Howard Hamiltonian is formally identical except for a small correction term E = -( tt2/8) Trace pR. Since we are here interested in semiclassical approximations it will not be necessary to invoke this fully quantum mechanical formalism. 5.2. X, centrifugal distortion energy A simple model for a C,, symmetric X, rotor will be considered here. The genuine normal coordinates s1 = sc, s2 = s:, and s3 = SF are defined by the Cartesian displa~ments in the first three columns of the B-‘-matrix.
Wl) d2(1)
d,(2) = d2 (2) dlW \d2c%
l/O 0 -J;5/6 l/2 -o/6 - l/2
l/G
0
0
l/a
0
0
-l/G
1/a
0
l/G
-O/6
-l/2
-l/2
- l/2
o/6
--G/6
-l/2
G/6
l/2
G/6
l/J?-
\ / Sl \
s2
0
s3
0
l/G
s4
l/2
l/0
0
s5
-J?-/6
0
l/0
I
\
(5.14)
%f
This relates all the in-plane coordinates shown in fig. 5.1. The out of plane motions are a rigid z-translation and two rigid rotations which are not genuine vibrations. The other non-genuine coordinates s4, sg, and $6 are a rigid rotation and two translations, respectively, in the molecular plane.
366
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 5.1. Normal coordinates of an X, molecule labeled by C,” symmetry.
The vibrational Hamiltonian will be constructed to have the C,, symmetry of the assumed equilibrium positions which lie on the vertices of an equilateral triangle (5.15) The harmonic frequency an is assumed to be the same for the two genuine modes s2 and sj that belong to the E species of C,,. The C,, symmetric anharmonic potential is assumed to be in the form of the Henon-Hieles cubic function. (a) Harmonic case The desired dynamic equilibrium values for the genuine normal coordinates si, s2, and s3 may Hamiltonian (5.2) and (5.15) for a range be found by minimizin g the generic rotation-vibration of values of the J-vector with zero vibrational angular momentum (L = 0). It is instructive to compare the minimum values obtained numerically with a fourth order approximation. This is easily done in the harmonic case with A,, = 0 in (5.15). The minimum energy distortion values are found by including (5.15) in (5.2) and setting to zero the derivative with respect to coordinate s,
+
w,zs, .
(5.16)
367
W. G. Harter / Graphical approaches to molecular rotations and vibrations
The resulting coordinate displacements
for dynamic equilibrium are then given
-1 a/-%, s, = -JJ.2a,2 ’ J as, * The change in the inverse inertia tensor can be expanded in the displacement
(5.17)
coordinates (5.18)
If second or higher order terms in this expansion are negligible then the Hamiltonian the following
becomes
(5.19a) Using eq. (5.17) gives the following H = :J,JIpiJ(0)
+
This result has the form of a rigid rotor plus a fourth degree J-polynomial Hamiltonian (3.la).)
(5.19b)
(Recall the Hecht
(5.19c)
It can be written in terms of the harmonic potential energy of the centrifugal mode distortions using eq. (5.17) H = :JQqj(0)
- &a&,2.
(5.20)
There is a gain in potential energy due to stretching according to the third term in eq. (5.19b). However, in this approximation, the rotor loses twice that much kinetic energy due to the expansion according to the second term. The result is net loss in total energy equal in magnitude to the gain in the potential energy as stated in eq. (5.20). The displacements (5.17) can be rewritten in terms of the derivatives of the inertia tensor I = p-l (5.21) In the approximation
which neglects quadratic or higher terms in the inverse inertia (5.18), it is
368
W. G. Harter / Graphical approaches to molecular rotations and vibrations
only necessary to evaluate the inertial quantities in eq. (5.21) at static equilibrium (s, = 0). This yields a simple approximate expression for the X, centrifugal distortions 5
“=s,=(J,‘+J,‘+~J,*)k/~~,
(5.22a)
sf = s* = (J,2+ JyZ)k/L&
(5.22b)
SF = sj = (-2J,J,)k/&.
(5.22~)
The constant factor depends upon molecular mass (m) and equilibrium radius (a) k = 2(3m)-3’2a-3.
(5.22d)
For a numerical example, let us choose the mass and molecular radius to be a unity. (m = 1, u = 1). Let us also pick a unit angular momentum vector, and direct it along the (x = x,)-axis. (J, = 1, Jy = 0 = J,). Then the centrifugal distortions of the normal coordinates beyond static equilibrium will be calculated for two different sets of the vibrational normal mode frequencies. A comparison will be made between a “semirigid” rotor for which wA= 50 and wE = 30, and a “ semifloppy” rotor for which wA= 5 and wr = 3. The approximate eqs. (5.22) give normal mode distortion values for the floppy rotor that are exactly a hundred times those for the rigid rotor. This follows from eq. (5.22) since their frequencies differ by a factor of ten. The approximate values given in the top two rows of table 5.1 are to be compared with the corresponding numerical values given in the bottom two rows. The coordinate SF is identically zero by symmetry for this choice of J-vector, and this provides a first check for the numerical routine. It is seen that the approximate equations (5.22) are accurate to three places for the semirigid rotor but are considerably off for the semifloppy rotor. The accuracy of eqs. (5.22) deteriorates further in the presence of higher J or with the inclusion of anharmonicity. Note that eq. (5.22) overestimates the distortion since equilibrium values of I- ’ were taken in eq. (5.21). In fact, I is generally larger, and so 1-l is generally smaller when the correct expanded distortions are used. The numerical results were calculated using standard Powell-Brent minimization routines [35] of the Hamiltonian given by eqs. (5.2) and (5.15). These routines do rapid calculations of global
Table 5.1. X, Distortion values due to angular momentum J = (LO, 0) Semirigid rotor (WA= 50, wa = 30)
Semifloppy rotor 5, WE = 30) cwA=
Approximation Distortion value (from eq. (5.22))
sA = 1.5396 x 1O-4
sA =1.5396x10-’
sl” = 4.2767 x 1O-4
~1”= 4.2767 x 1O-2
Exact numerical Distortion values
sA =1.5381 x1O-4 SF = 4.2721 x 1O-4
~~=1.4062~10-~ s1”= 3.9011 x 1o-2
W. G. Harter / Graphical approaches to molecular rotations and vibrations
369
b
a
, Fig. 5.2. Equatorial (xy) section of numerically generated X, RE surface for centrifugal distortion model. mA = 5, tiE = 3. J-vector is of constant magnitude ( 1J 1= 3/2) and variable direction in xy plane. Distortions are magnified six times. (a) Harmonic model: X Haron= 0; (b) anharmonic model: XHenon= 1.5. minima, but are compact enough to easily be used on a Macintosh or other personal computer. They are also rapid enough that the J-vector can be swept over a circle or sphere to quickly produce RE surfaces. The speed of the numerical minimization is usually not affected by the inclusion of anharmonicity, and so it is easy to show the effects of anharmonic forces on rotations. The two plots in fig. 5.2 show the scaled equatorial sections of an X, RE surface for harmonic nenon = 0) and anharmonic (hn_, = 1.0) Hamiltonians of the form given in eqs. (5.2) and (A (5.15). These include scaled drawings of the actual rotor distortions belonging to select points on the RE contour. The scale of the distortion is magnified sixfold for each figure. The harmonic RE surface is circular, but the anharmonic RE surface develops hexagonal anisotropy in the (J, J,)-plane. Note that a triangular RE surface is impossible for the rotor since the Hamiltonian must be invariant to angular momentum inversion (J + -J) or time reversal. The scale of the hexagonal anisotropy in fig. 5.2 is exaggerated compared to the scale of the anisotropy of the oblate planar symmetric top. The complete RE surface corresponding to fig. 5.2 is shown in fig. 5.3a. This resembles the surface for the oblate top in fig. 3.lc, except for the beginnings of six small resonance islands that distort the equatorial contours. By increasing the magnitude of J, the islands are seen to grow in figs. 5.3b, c until they occupy a significant fraction of the surface. Also, as J increases, the angular momentum cone angles decrease while the angular size of the islands increases. At some critical value of J three will be room for one or more @-cones (recall eq. (2.19).) inside the equatorial separatrices of the surfaces in fig. 5.3. At or shortly below this critical J-value, the rotor will cease to behave like an oblate symmetric top. It will then be possible for it to become stuck rotating like a spinning coin around one of the six hills on its edge or equator. The symmetric top rotational spectrum will become extremely perturbed, and the simple doublets at the top of each J-manifold will be replaced by clusters of six levels. The C,, species in those sextets will be in the form (A,EEA,). The splittings are likely to resemble the projections of a hexagon according to the simplest tunneling models [lO,ll].
370
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 5.3. Numerically generated anharmonic model X, RE surfaces and level paths showing the effects of increasing the total angular momentum. Harmonic constants are U* = 5, aE = 3. Anharmonic constant is XHenon= 1.0. (a) J = 2; (b) J= 5; (c) J=lO.
The semiclassical methods described in section 2 can be developed to derive approximate energy levels and eigenfunctions directly from the numerical RE surfaces. The same graphics routines which draw contour paths can be modified to satisfy quantization conditions as well. Classically propagated wavepackets can be used to approximate eigenfunctions in the neighborhood of quantizing trajectories. 5.3.
RE surfaces for vibronically excited states
The minimization methods which gave the RE surfaces in figs. 3.3 were chosen to deal with the special case of a rotor in a vibronic ground state. In the first try of a generic approach to RE surfaces one needed to have a rapid search for global minima more than a general algorithm for treating excited states. Now we speculate on alternative methods which might yield usable RE surfaces in excited states, as well. One alternative is to run classical equations of vibrational motion while changing the J-vector adiabatically. This would involve procedures related to those discussed by Skodje [2]. One could pick a quantized vibrational action and slowly turn on the J-vector as well as the anharmonicities. One would obtain a quantized vibrational state polarized by the frozen rotation. Then the J-vector direction could be changed slowly enough to trace out an adiabatic RE surface. It would also be possible to determine to what extent the true J-vector precession was adiabatic. Another alternative is to solve quantum vibronic equations to obtain effective adiabatic RE surfaces. The RE surface or surfaces would be plots of quantum vibrational eigenvalues as a function of J-vector directions. As mentioned previously, this is analogous to the derivation of Born-Oppenheimer vibrational PE surfaces from quantum electronic eigenvalues. In the following section, we consider some examples of rotational dynamics driven by vibrational or vibronic eigenvalues. 6. Multiple RE surfaces and Coriolis coupling effects This section contains a discussion of some simple models for Coriolis coupling between rotors and their internal degrees of freedom. Much of molecular physics involves an analysis of coupled
W.G. Harter / Graphical approaches to molecular rotations and oibrations
371
systems involving a rotor and some passengers such as orbiting electrons, electron spins, nuclear spins, vibrational angular momentum, or combinations of any of these. Here we will discuss ways to visualize coupling effects using RE surfaces. It is a common practice to analyze coupled angular momentum systems in terms of special cases in which certain components of rotor or passenger momenta are conserved quantities or good quantum numbers. Certain types of coupling operators may conserve certain quantum numbers; and their eigensolutions are classified as archetypical cases. For example, there are the LS and jj coupling cases for multielectron problems in which spin-orbit coupling is weak or strong, respectively. There are the Hunds coupling cases (a) through (d) for levels of diatomic or linear molecules. These depend on which type rotor, spin, or orbit coupling is dominant. [52] The connection between energy levels of two archetypical cases are often shown using eigenvalue correlation diagrams. [53] This is particularly useful if the desired eigensolutions are close to one of the archetypical cases. For polyatomic molecular problems, the classification or archetypical cases is more difficult. Polyatomic molecules do not generally have the infinite R, symmetry or the associated quantum numbers found in symmetric diatomic or linear rotors. At best they have a finite permutational symmetry, but these symmetries are labeled by a finite number of ‘“quantum letters” such as A,, A,, T,, etc., instead of quantum numbers. These letters usually cannot label states uniquely since each label must be used over and over again. Associated with an inadequate labeling system is an increased difficulty to visualize the eigensolutions and to identify what, if anything, is being conserved or nearly conserved. Also most finite symmetry Hamiltonians yield spectra which must be classified region by region or even level by level. Most of the levels diagrams in sections 2 and 3 exhibited two or more archetypes in the same spectrum, and simple quantum letters or numbers do not give enough info~ation by themselves. In sections 2 and 3 the labeling and visualization problem was solved for finite symmetry rotors by using RE surface trajectories and their associated semiclassical quantum numbers or quasi-conserved momenta. The effects of perturbations on level correlations could be visualized continuously as the perturbation strength increased, not just when the perturbation was very strong or very weak. A generalization of RE surface visualization methods is introduced in this section for describing Coriolis coupling between the angular momenta of a rotor and its passengers. Again, it is sometimes possible to visualize the level correlations continuously between archetypical cases of strong or weak couplings. However, this generalization involves the use of multiple or overlapp~g RE surfaces, and it leads to more complicated semiclassical dynamics. These ideas are introduced in the following section 6.1. Then examples are treated in sections 6.2 and 6.3. 6.1. Semiclassical mechanics on multiple surfaces In preceeding sections the rotational has been described using a single rotational energy surface. The analogy has been made between a rotational energy (RF) surface and a vibrational potential energy (PE) surface. A PE surface is a measure of electronic and electrostatic energy, and it depends upon having an electronic state which follows the vibrational motion adiabatically. In the case of pure vibrational excitation this would be the ground electronic state. Electronic adiabaticity is necessary in order to have an effective potential energy as a function of
372
W. G. Harter / Graphical approaches to molecular rotations and vibrations
the classical vibrational coordinates q = ( ql, q2, . . . ). In analogous fashion, an RE surface depends upon the adiabatic following of rotational motion by a vibrational or vibronic state. In the case of a deformable rotor, this would be the ground vibronic state. Vibrational adiabaticity is necessary in order than an RE surface can accurately described the total effective rovibronic energy as a function of the direction of the classical rotational momentum vector J = ( J1, J2, J3). This section contains a description of models which involve two or more neighboring RE surfaces. Multiple RE surfaces are needed if two or more vibrational or vibronic states are degenerate in the absence of rotation (J = 0). This is analogous to well known Jahn-Teller or Renner models [54] which involve two or more neighboring PE surfaces. Multiple PE surfaces are needed if two or more electronic states are degenerate in the absence of vibrational distortion (4 = 0). The multiple Jahn-Teller or Renner PE surfaces are each plots of different electronic eigenvalues as a function of vibrational coordinates q. By analogy, the multiple RE surfaces are each plots of different vibrational or vibronic eigenvalues as a function of the direction of total angular momentum J. Each set of eigenvalues gives one point on each PE (or RE) surface, and these are evaluated in the adiabatic approximation at constant q (or J). However, each PE (or RE) surface determines a possible classical path of motion for q (or J) that is, q (or J) are usually not constant, after all. Each PE (or RE) surface belongs to a different electronic (ro vibronic) eigenstate, and it provides a different set of marching orders for q (or J). The adiabatic approximation is valid if each electronic (or vibronic) eigenstate can remain nearly an eigenstate as q (or J) follow their matching orders. Then the electronic (or vibronic) quantum states are said to be adiabatically following the q (or J) motions and the configuration is said to reside on a single surface as far as its own dynamics are concerned. However, the presence of nearly coupled PE (or RE) surfaces belonging to different electronic (or vibronic) eigenstates allows for more general types of motion. It is possible to consider PE (or RE) surfaces belonging to mixtures of electronic (or vibronic) eigenstates as well as the base surfaces belonging to eigenstates. With two or more electronic (or vibronic) states, there will be a continuum of possible different mixtures of states and a corresponding continuum of possible different PE (or RE) surfaces. Each mixture of eigenstates corresponds to another surface which lies between the base surfaces belonging to eigenstates. If adiabatic following occurs on each of the PE (or RE) surfaces belonging to electronic (or vibronic) eigenstates, then it may be possible for adiabatic following to occur on surfaces belonging to mixed states. Eigenstates are not necessarily the only states capable of adiabatic following. At this point, it is necessary to temporarily abandon the analogy between PE and RE surfaces in order to reemphasize their differences. A single adiabatic RE surface is a two-dimensional manifold, and its intersection with the constant energy sphere defines a one-dimensional classical J-trajectory. In contrast, a PE surface is a (3N - 6)-dimensional manifold, and its intersection with the constant-energy (hyper) sphere defines only the outside boundary of classical motion. The classical rotation J-motion or precession is determined directly from RE surface topography lines, but the determination of vibrational motion on a PE surface requires a detailed (generally numerical) solution Hamiltonian’s equation of motion. Motion of on a single RE surface is simple integrable and periodic, while PE surface motion may be irregular or chaotic even for a single two-dimensional surface. However, rotational motion also becomes complicated if an eigenstate cannot follow J-motion
W. G. Harter / Graphical approaches to molecular rotations and vibrations
313
adiabatically. Then the J-vector will no longer be constrained to lie along the intersection of the energy sphere and a single constant RE surface. This will happen if an original RE surface path changes the J-vector too rapidly for the original eigenstate to follow, and a mixture of eigenstates results. As the state mixture evolves, so does the RE surface, and the precessional mechanics is no longer a simple one-dimensional problem. The energy sphere is constant but the RE surface deforms and vibrates through a continuum of possible shapes. The paths of J-motion can still be thought of as the energy sphere-RE surface intersections, but these paths can change dramatically as the surfaces change even a little. As a result, the J-vector is not confined to any particular region of J-space, and there is a possibility for classically chaotic rotational motion. Nevertheless, the RE surfaces can still provide useful information even when parts of them show a breakdown of the adiabatic approximation. Each RE surface has regions surrounding stable fixed points. These are separated by separatrix regions that connect unstable fixed points. There may be little or no tendency for the J-vector to move near the center of a fixed point region. The vibronic state corresponding to that part of the RE surface may be able to keep up with the J-vector if it is close enough to the fixed point. Hence, one might expect those portions of RE surfaces to contain regular motion and provide a useful semiclassical characterization of a subset of rovibronic eigensolutions. In separatrix regions, on the other hand, the single constant surface is most likely to fail and irregular motion may occur. Still, the simple RE surface calculation is useful for locating the irregular regions. Some examples will now be given to demonstrate these ideas. 6.2. Asymmetrically
coupled spin rotor systems
One of the simplest polyatomic models for Coriolis coupling is an asymmetric rotor coupled to a spin-l/2. This is a model which involves just two RE surfaces, and it could serve as a model for coupling of an electronic or nuclear spin S to a rotor. It could also be a model for a two-dimensional vibrational Coriolis coupling in which the vibration was represented by a quasi-spin S as described in section 4. Here the model Hamiltonian will be restricted to have just first and second order terms in the coupled spin momentum and total momentum J=R+S
(6.1)
where R is the angular momentum of the rotor by itself. The body defined components of J satisfy the usual commutation relations with the spin
1J& I =
-kzjkJk,
[J,9 S,] =O,
[&S,]
=ic,jkSk.
(6.2)
The general second order Hamiltonian has the following form (6.3) Here we will consider special cases of the Hamiltonian in which the spin is assumed to have zero rotational inertia (E,, = 0). Only the diagonal inertial and coupling terms which have D2
374
W.G. Harter / Graphical approaches to molecular rotations and vibrations
symmetry will be kept. For an example of an asymmetric spin coupled rotor, only the quadratic terms are kept in order to preserve time-reversal symmetry. This special case is the following: HR= C,,J,‘+
C,,J;
+ C,,J,‘+
D,,J,S, +DyyJ,Sy + Dz,JzSz.
However, the simplest nontrivial spin Hamiltonian following example, which will be discussed first.
(6.4
which requires two RE surfaces is the
(6.5)
H, = A, J, + B,S, + D,, J,S,
This represents two precessing zero-inertia rotors or spins asymmetrically coupled along one component. A form like this was studied by Feingold and Perez [55] as an example of classical chaos. Their objective was to study the correspondence between exact quantum eigensolutions and classical dynamics the exhibited chaos. The following describes the constriction RE surfaces which help to visualize both the classical and quantum dynamics. (a) Asymmetric coupled spins As usual, the total angular momentum vector components (J,, JY, J,) will be treated as classical functions (2.5) of Euler direction coordinates for RE surfaces. The spin components (S,, SY, S,) will be treated as quantum mechanical operators whose eigenvalues determine points on each RE surface along the direction of a given J-vector. This is accomplished by first replacing each spin operator in the Hamiltonian by its two-by-two Pauli matrix representation. For the Hamiltonian (6.5) the result is the following matrix
’ (Hs) =
A, 1J 1cos p + Bz2ISI -D
\
xx “J I2I ’
-D XX ]J]]S]sinpcos 2
I sin /3cos p
A, I J 1cos p -
ISI Bz2
(6.6a)
Here the spin magnitudes are I J I = \i[J(J+l>l and ( S ] = \i[s(sm. The resulting Hamiltonian can be viewed as a two-state Hamiltonian that is a function of the polar angles (- /3) and (- y) of the J-vector. For each J there is a precession vector Ic2 that is analogous to the w-vector derived for the general two-state Hamiltonian in eqs. (4.4) through (4.6). This derivation gives the following D vector for eq. (6.6a). 52,= -D,,I
JllSl
sin /3cos y,
Q,,,=O,
Q2,=BzIS(
(6.6b)
This determines the angular velocity vector for the expectation value of the spin vectors. The two eigenstates of (H,) will have spin vectors that are aligned “up” or “down” along the L&vector, and these are the only directions for which S is stationary relative to 0. However, the Q-vector is a function of J, and both will move according to the Hamiltonian H,. The possible J-motion can be determined approximately by plotting the “up” and “down” eigenvalues of the (H,) matrix in (6.6) as a function of polar angles ( -p) and (- y) of J. Some examples of the resulting pairs of surfaces are shown for a range of values of the coupling
W. G. Harter / Graphical approaches to molecular rotations and vibrations
ig. 6.1. RE
surfaces for asymmetricallycoupled spins J = 5/2
and S =1/2.
(a) D,, =l.O;
375
(b) D,, = 3.0; (c)
D,, = 9.0.
parameter D,, in figs. 6.la-c. If the S vector could follow the motion of 52 and J, then the latter would be confined to the equal-energy contours which are drawn on the surfaces. The contours on each of the two RE eigensurfaces start as a set of parallel circles. Then as the coupling constant DXX increases, the circles around one pole on each surface become very squeezed. This is shown happening at the north pole of the outer (high energy) surface and the south pole of the inner (low energy) surface in fig. 6.la. Then the squeezed pole of each surface becomes an unstable fixed point or saddle point at the center of a figure-eight separatrix which surrounds a pair of nearly circular loops. One loop of each pair is visible on each surface in fig. 6.lb. The contours on each surface resemble those of the local mode paths in the anharmonic oscillator of fig. 4.5~. The pole opposite to the saddle point remains a stable fixed point, but as D,, increases, the paths around the stable pole get squeezed by the figure-eight loops which are approaching the equator on each surface as shown in fig. 6.la-c. For high D,,, the figure eight centers on both surfaces approach the x-axis, and symmetric top surface like figs. 2.la,c emerge. The stable pole for the high energy RE surface is above the unstable pole for the low energy one, and visa versa. Also in fig. 6.lb, the stable figure eight loops of the high energy surface lie above the separatrix region of the low energy surface. A J-vector that is initially inside one of the top loops in fig. 6.lb will have much less tendency to move rapidly if the initial spin vector is “up” than if it is “down”. To test these ideas the exact classical equations derived from the Hamiltonian Hs were solved numerically. The resulting trajectories for the vectors J, S, and 1(2 were ploted in 3D stereograms in fig. 6.2. The stereo pair in fig. 6.2a show the small amplitude precessional motion of J for the case that the S-vector is initially “up” along 9. For the stereo pair in fig. 6.2b, the initial conditions are the same except the S-vector is “down” along Sz. The result is irregular motion with large amplitude precession of J. It appears that J takes turns going around curves that are close to one or the other of the figure-eight loops on the spin-down surface in fig. 6.lb. Since the S cannot follow the rapidly moving J and 9, it is possible for the system to classically jump from one loop to the other. Such a classical jump would be impossible on a single RE surface, but which resonance between two surfaces the J-vector will sometimes find the saddle point pass “open” while other times the pass will be “closed”. The preceeding example shows that while the J-vector does not stick precisely to RE eigensurface paths, one can still use the paths to visualize qualitatively what the resulting motion
376
W G. Harter / Graphical approaches to molecular rotations and vibrations a
J= 1.0S,= 1.00
A,= 000 A,= 0.00 A,= 1.00 B,= 0.00 By= 0.00 B,= 1.00 c = 0.00 c,= 000 c = 0.00 D",=3.00 D,= 0 00 d",=0.00 aO= 0.00 &= 45.00yO= 000
a J= 1.0S,= 1.00 Emiti.lly= 3.052315 *to EO p"te=3.052315
A,=
0.00
A,=
0 00
A,=
, 00
B,= 0.00 B,= 0.00 B,= 1.00 c,= 0.00 c,= 0.00 c = 0.00 D,= 3.00 D,= 0.00 I$,-0.0~ ao= 000 &= 4500 -yO=000
II 0
b J= 1.0S,=-100 E =-1636101 inithlly
AX=000 A,= 000 A,= 100 000 000 100 c B,= = 000 c,= By= 000 c= B,= 000 D",=300 D,= 000 dz= 000 aO= 000 &= 4500 yO= 000
b J= 10 S,=-100
A,= B,= c,= D,= aO=
000 000 100 000 A,= El,=0.00 A,= B,= 100 000 c,= 0.00 c = 000 3.00 D,= 000 o",=0.00 000 p,= 4500 To= 000
Fig. 6.2. Examples of classical motion for J near an upper surface fixed point in fig. 6.lb. (Stereoptic figures.) (a) Initial spin is “up” along &?-vector; (b) initial spin “down” results in irregular motion.
will be. It is also possible to obtain qualitative information about the quantum energy levels. One can see that the high D,, surfaces (figs. 6.lb, c) contain two pairs of equivalent figure-eight loops which become more localized at very high or very low energies. These loops should correspond to level cluster pairs similar to asymmetric top clusters. The clusters should become more nearly degenerate in the higher and lower regions of the spectrum. This clustering is seen in the quantum energy level correlations shown in fig. 6.3. The energy levels in the center correspond to irregular near-separatrix motion and are unclustered.
317
W. G. Harter / Graphical approaches to molecular rotations and vibrations
(b) Asymmetrically coupled spin-rotor Some examples of RE surfaces and level correlations will be shown for the elementary asymmetric rotor coupled to spin - l/2 through the Hamiltonian HR in eq. (6.4). The spinor representation of HR is the following h+D
‘J”s’cms~
(HR)=
-(D,,
cm y -*:D,,,,:in
u)ysin
-(D,,
/3
cm y+iDy,,
I J I I s I sin p
sin y)--~_
h - Dz,~cos
/3
(6.7a) ’
Here h is the free asymmetric top energy which is similar to eq. (2.1) h = 1J 1*(Cxx sin*/3cos*y + C,, sin*P sin*y + C,, cos’p).
(6.7b)
The eigenvalues of the matrix in eq. (6.7a) are shown by the RE surfaces in figs. 6.4a-c. The figures represent a range of coupling parameters in which the ratio D,, = 3D,, and D,,,, = 2 D,, are maintained as 0,. increases, while the inertial parameters C,, = 2.0, C,, = 4.0, and C,, = 6.0 are kept constant with similar ratios. These are also similar to the values used for the simple asymmetric top in section 2. This choice of ratios yields a peculiar set of trajectories on the inner (low energy) RE surface. The outer surface is very similar to the simple asymmetric top RE surface in fig. 2.1, but the
Fig. 6.3. J = 5/2
energy levels for asymmetrically couples spins plotted as a function of coupling constant Dxx. Clustering is most pronounced for the highest and lowest levels.
W. G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 6.4. RE surfaces for asymmetric spin coupled rotor. (C,, = 2.0, C,,y = 4.0, C,, = 6.0; Dyy = 2D,,, (a) D,, = 30.0; (b) Dxx = 40.0; (c) D,, = 50.0.
and D,, = 3D,,)
inner surface becomes more nearly spherical with a much more complex topography. For one thing there are sets of four equivalent paths in the form of two figure eight valleys. These appear on the x-axis and move toward the z-axis as OX, increases. This indicates the possibility of an extraordinary level cluster made of four D, species {A,A,B,B,}. Such a cluster is formed in the lowest level of the J = 21/2 spectrum in the middle of fig. 6.5. Above and to the right or left are levels which resemble the J = 10 asymmetric top pattern of figs. 2.2 and 2.3. The upper pattern corresponds to the outer surface in each of figs. 6.4 and is very much like a single asymmetric top. As OX, becomes very large, the lower surface must also approach a simple asymmetric top. For very large OX,, the rotor momentum quantum numbers R = 10 and R = 11 become good quantum numbers for the upper and lower RE surfaces, respectively. However, for intermediate values (30.0 < 0,. c 50.0) in fig. 6.5, there is a clustering of two ordinary cluster doublets into a fourfold near-degeneracy in the lowest levels. This is the region described by the RE surfaces in fig. 6.4. In addition to the unusual clustering, there is a considerable crowding of all the levels in the lower energy branch of fig. 6.5. This is related to the low hills and shallow valley of the inner surface of fig. 6.4. Also, the inner surface has a complicated separatrix region. Together these properties tend to make the classical motion irregular or chaotic. However, if J is large compared to S, it cannot be upset as easily as it was in figs. 6.2 or 6.3 where they were given comparable magnitudes. Hence, the levels for J = 21/2 and S = l/2 are fairly orderly. 6.3. Octahedral scalar and tensor Coriolis coupling The fundamental vector vibrational modes vj and vq of SF, are described by a triplet of base states { &, &, #- 1> which have a total vibrational angular momentum L = 1 and cubic symmetry label T1. The coupling of the momentum L with an octahedral rotor of momentum R yields states of total momentum J = R + L as in eq. (6.1). However, the three L = 1 states will give rise to a triplet of RE surfaces for each J-value. Here a set of surfaces will be derived from a semiclassical analysis of the following effective Hamiltonian due to Hecht [16] H, = v + BJ* + 2B{J*L
+ t,,[v*(rotation)
8 u*(vibration)]i,.
(6-g)
379
W. G. Hurter / Graphical approaches to molecular rotations and vibrations A,= 0.00
B,= 0.00
A,=
0 .OO
A,=
0.00
B,= 0 .OO B = 0.00 4.00 ‘cm= 6.00
R= 11 I
0
8
0.0
10.0
0
20.0
30.0
I
40.0
50.0
Dxx
60.0
70.0
80.0
90.0
10
With D,,=2Dxx and D,,=3D,,
Fig. 6.5. J= 21/2 energy levels for asymmetric spin-coupled rotor plotted as a function of Ox,. The levels in the crowded region in the lower branch correspond to the innermost RE surfaces shown in fig. 6.4. The lowest two clusters in that region contain four levels each and correspond to a quartet of equivalent paths on the RE surfaces.
The first two terms are constants for a given J which give the vibration and scalar rotational energy. We shall be concerned with the scalar and tensor vibration-rotation coupling in the third and fourth terms, respectively. The tensor operator is a fourth rank combination of vibrational and rotational quadrupole operators u2(vib) and u2(rot). A brief description of tensor construction and notation is given in appendix D [ u2(rot) @ u’(vib)]i,
= V% [ u2 @ U’]: + 5([ u2 QDZI’]: + [u2 c3 d]‘q.
(6.9)
Except for an overall factor this is the same combination of [ 1: tensor terms as found in the T,” expression in eq. (3.15). The vibrational quadrupole tensors are quadratic functions of vector boson creation and destruction operators
380
W.G. Harter / Graphical approaches to molecular rotations and vibrations
The rotational quadrupole quadrupole tensors quadratic function of J-vector form given in eqs. (2.2). ui (rot) =
operators of the
Tq2
The necessary Clebsch-Gordon
[T*@v*];=&(
products of the rotational and vibrational tensors are as follows
T,2?,+
T:,u;)+
@(
T,2&+
T~l~~)+~(T~v;),
[T*@u*]~~= T:,u;,. Assembly of these terms in eq. (6.9) yields the desired operator = &Tt( alal- 2&z,,
[ v*(rot) @ U*(vib)]i,
+ T2,( &_,
+ u~~Q)
+ 5u;u_,)
+ 2fiT!,(
&_r
- a$~,) (6.10)
+ C.C.
*+= (- l)qT,* (q # 0)are included in where the complex conjugate of the off-diagonal tensors T_, the c.c.-term to make the operator self-conjugate. The details for construction of quantum mechanical matrix elements of the Hamiltonian Hv are given in ref. [56]. Here we shall only consider the semiclassical matrix whose eigenvalues give the RE surfaces. A semiclassical matrix follows by replacing boson operators in eq. (6.10) by their fundamental 3 x 3 representations, L by its vector representations, and the tensor operators by their Euler angle functions according to eq. (2.4). The result is the three-dimensional matrix which is the representation in the basis of body-fixed vibrational states which are labeled using the standard Z and II notation IW
= I&>
IX) = I&o)
IW
= Ih>*
The matrix in this basis is the following 1
0
0
0
1
0
0
0
1
3 cos*p - 1 - J8elY sin p cm /I si&?(6 cm 2y - i4 sin 2~)
-\/8e-‘Y sin ,9 cm p -2(3 cos*/?- 1)
sin*j3(6cos 2y + i4 sin 2y) J8e-‘Y sin B cos /3
fielv sin fi cos p
3 cd/3 - 1
1.
(6.11)
381
W.G. Harter / Graphical approaches to molecular rotations and vibrations
If the I: and II states are eigenstates, it means that the projection of the vibrational angular momentum on the body axis of quantization is conserved and (L,) = A = 0, & 1 is a good quantum number. For a very strong scalar Coriolis constant (B3 B tzz4 1J I) it may be advantageous to use bases in which the rotor momentum R is conserved and the good quantum numbers are R = J + 1, J or J - 1. These bases are labeled 1P), 1Q), and 1R) respectively, after the corresponding branches of the infrared spectrum to which they belong. In this basis, the scalar Coriolis operator is diagonal
(%)
PQR =(v+BIJ/2)
(6.12a)
The off-diagonal basis
tensor components in the PQR basis are more complicated
Hro = 5 sin /I[ 7 cos2@ - 3 cos fl-
sin2/3(cos p cos 4y + i sin 4y)] /2fi
than in the XII
= H& ,
HPR = 5[ - 7 COS”~+ 8 cos2/3+ (1- cos4/3) cos 4y + 2i cos B sin”/3sin 4y - l] /4 = H&, . (6.12b) However, the diagonal tensor components in the PRQ basis have precisely the same form as the simple RE surface expressions in eq. (3.2b) HPP = HRR = (35 cos4p - 30 cos2/3-t 5 sin4p cos 4y + 3)/4 = -J&&2.
(6.12~)
This means that the P and R branch RE surfaces for high B{ and positive t, (< B{ will be two nearly perfect copies of the single surface in fig. 3.la. The Q branch surface will resemble the surface in fig. 3.lb. It will have twice the amplitude of the P and R branch surfaces and lie between them. The high-( B{) case is the one usually encountered. Nevertheless, it is instructive to consider RE surfaces in the neighborhood of B{ = 0. The eigenvalues of either representation (6.11) or (6.12) yield the RE surfaces shown in fig. 6.6a-d. Each set of three eigensurfaces are shown as stereo pairs (a), (b), (c), and fd). These correspond to B{ = 0.0, 0.5, 1.0, and 1.5, respectively, with t,,, = 1. The four sets of surfaces are to be compared with corresponding points labeled (a), (b), (c), and (d) in the exact (J = 60) eigenvalue correlations in figs. 6.7a, b. A magnified view of the low-( B{) region (BS I 1.5) is shown in fig. 6.7b. In the center of this region the Z and II labels apply since the Coriolis force is weak and does not cause the L-vector to deviate rapidly from its
382
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 6.6. RE surfaces for Coriolis coupled octahedral rotor-vibrator in the low B{ region. (t,,, 1J 12 = 1.0). (a) B{ = 0.0; (b) BP = 0.5; (c) Bl = 1.0; (d) B3 = 1.5.
precession around body-fixed axes. The lowest ten or eleven levels in fig. 6.7b at Bl = 0 each contain six levels in clusters of the 4-fold (C,) symmetry type which were shown in fig. 3.3. These clusters can be associated with the six sets of C,-symmetric valley paths on the innermost RE surface “cube” in fig. 6.6a. It can be shown that the L-vector precession is mostly normal to the instantaneous axis of quantization for the C, states, and so they are labeled 4-fold Z-clusters in fig. 6.7b. Above the C4-C levels is a separatrix region containing a few unclustered levels that create a dark band in fig. 6.7b at B{ = 0. The five or six level clusters just above the separatrix band each contain eight nearly degenerate levels and are associated with equivalent paths on the eight C,-symmetric comer hills of the innermost cube in fig. 6.6a. Note that the level curves in figs. 6.6 are equally spaced topography lines and are not actual quantizing paths for J = 60 levels. Hence, the number of paths in figs. 6.6 does not necessarily correspond to the number of levels in fig. 6.7. The cube comers are sharp points which are very different from the rounded hills on the simple fourth-rank RE surface in fig. 3.lb. Furthermore, the corner hills of the inner surface actually meet comer valleys of the middle RE surface to make a conical point. It can be shown
W.G. Harter / Graphical approaches to molecular rotations and vibrations
383
Fig. 6.6 (continued).
[56] that the paths on these corners correspond to states made mostly from combinations of ( II+K,) and 1IIR + 3) where K, is the rotor momentum component (R,) along the C,-axis. There is one state 1n-59) whose mixing partner 1l-I-62) does not exist since its total J,-component ( KJ = A + K, = 61) exceeds J. Its lone level cluster belongs to the straight line crossing the middle of the 3-fold II * cluster region in fig. 6.7b. This “lone state” is a quantum mechanical entity that does not follow the usual RE surface mechanics. All the other Q-II, levels correspond to paths on the cones in fig. 6.6a. These levels appear to “repel” each other in fig. 6.7 because their angular uncertainty angles prevent their paths from approaching the conical intersection. The middle surface for (B[ = 0) in fig. 6.6a has an octahedral shape. Its C, symmetry axes lie in valleys that form eight conical intersections with the lowest energy surface. The C, symmetry axes lie on hills that form six oscullating (i.e., “kissing”) intersections with the outermost and highest energy surface. The C, intersections are surrounded by a very peculiar arrangement of paths belonging to II-states. A careful inspection of fig. 6.6a reveals that the middle surface has three square shaped stable path around each C, axis while the outer surface has no stable C,
384
W G. Harm / Graphicalapproachesto molecularrotationsand vibrations
(a)
r-,
EXACT ,
DIAGONALIZATION ,
,
,
J = 60
lb),
EXACT
DIAGONALIZATION
J=60
Fig. 6.7. J = 60 energy levels of octahedral rotor plotted as a function of B[ corresponding to RE surfaces of fig. 6.6. (a) Higher B{ values yield normal PQR level patterns similar to those in fig. 3.4; (b) low B[ values yield extraordinary XII level patterns and two-fold symmetry clusters (see fig. 6.6).
paths at all. There are only C, symmetric paths on the hills of the outermost surface while a separatrix goes through the C, axes. Another way to plot multiple RE surfaces is shown in figs. 6.8a-d, where the energy is plotted vertically instead of radially. The upper hemisphere of each surface is plotted as a function of projected polar coordinates. The cubic symmetry is not so obvious in these plots, but the surface topography is easier to see. For example, the four C, hills are clearly visible on the top surface of fig. 6.8a. (This corresponds to the outer surface in fig. 6.6a.) The outer surface has only C, paths and only C,-type clusters show up in the highest levels of the (B[ = O)-spectrum in fig. 6.7b. These are Z-type 3-fold symmetry levels which go along with the (C,, II) pairs of levels discussed previously. Directly below the (C,, II)-levels he the (C,-II) levels. However, unlike the (C,-II) stack, the (C,-II) levels only contain II+ level from each IT-pair. The II_ state has no stable paths and no band of quantum levels. Only the band of levels corresponding to III-states belong to stable paths. This is similar to the spin-rotor effects seen in figs. 6.1, 2 for which spin “up” is stable and spin “down” is unstable. The stability is very sensitive to the value of Bc. At point (b) in fig. 6.7 where B{ = 0.5, the upper levels change dramatically. The middle and upper RE surfaces in figs. 6.6b or 6.8b show how the changes come about. The outermost surface has &-paths replaced by Cd-paths which lie above C,-paths. The latter are difficult to see in fig. 6.6b, since much of the outer surface is cut away. The C,-paths are easier to see in fig. 6.8b, and they correspond to 2-fold clusters indicated on the bottom of the upper set of levels above point (b) in fig. 6.7b. Note also, that the 3-fold Z-clusters have changed into a new set of 4-fold symmetry clusters above the 2-fold clusters. Also, the middle surface has a region of instability near the Q-axes and all stable paths surround the C,-axes. At this value of coupling (BS = 0.5) the 2-fold clusters begin to form on the middle surface, they become very strong at B[ = 1.0 as seen by examining figs. 6.6~ and 6.8~ or the (c) point in fig. 6.7b.
W. G. Harter / Graphical approaches to molecular rotations and vibrations
385
a
b
b
Fig. 6.8. Vertical projection RE surfaces for octahedral rotor-vibrator in the low B{ B{ = 0.0; (b) Bl = 0.5; (c) B{ = 1.0; (d) B{ = 1.3.
region. (tZZ41J 1’ =l.O) (a)
It is remarkable to have stable 2-fold or C, symmetry paths on the twelve axes that normally contain saddle points on a fourth rank tensor RE surface. In each case a near degeneracy of order twelve appears in the exact eigenvalue spectrum just as happened for sixth rank tensors in fig. 4.6. Here it happens when the scalar Coriolis coefficient BS is comparable to the fourth tensor coefficient t,,. Coriolis stability on a saddle point is analogous to ion cyclotron stability in a quadrupole field. For values near Bl = 1.5 (see fig. 6.6d or 6.8d), the lower surface becomes nearly spherical and develops stable C, symmetric paths as well as the usual C, and C, trajectories. This corresponds to an extremely crowded band containing all three types of clusters in the lower part of fig. 6.7b. This band crowding was also seen in fig. 6.5 for the spin rotor. As B{ increases further one eventually recovers the normal situation which can be described
386
d
W.G. Harter / Graphical approaches to molecular rotations and vibrations
d
Fig. 6.8 (continued).
accurately by single independent RE surfaces. In fig. 6.7a, one can see the recovery of three separate P, Q, and R bands of the type discussed in section 3. There it was shown that roughly 25% of a band is devoted to C3 clusters and the remaining 75% belongs to the C, clusters with a single C, separatrix in between. This recovery of standard bands is expected, given the form of the representation (6.12). It is worth noting that a revealing RE surface picture is possible even when the level structure is not so simple! 7. Conclusion The adiabatic rotational energy surface has been shown to be a useful visualization and approximation device for semiclassical treatments of diverse rotational and vibrational phenom-
387
W.G. Harter / Graphical approaches to molecular rotations and vibrations
ena. Presently, it complements the usual numerical tour de force diagonalizations on a large basis sets by providing insights into dynamics and spectral phenomena that might otherwise be unexplained or missed entirely. It does this by providing phase portraits of the eigenvector properties as well as those of the eigenvalues. By combining these techniques with wavepacket dynamics, it is now possible to calculate both eigenvalues and eigenfunctions semiclassically. An introduction has been given to some generalizations of the energy concept including vibrational quasi-spin surfaces, generic or ab initio numerically generated surfaces, and multiple Coriolis coupling surfaces. With the concepts, there is possibility for visualization and approximate numerical solution of problems involving highly fluxional molecular or ionic complexes for which the standard basis set analysis may be impossibly complicated.
Acknowledgements I would like to thank Dr. Phil Bunker for inviting me to begin this article and to Dr. Chris Patterson for telling me to get it finished. I would also like to thank Drs. Chris Patterson, Daniel Huber, Nelson De Leon, and Eric Heller for past and present collaborations and many helpful suggestions. For typing and editorial assistance I would like to thank Margot Harter and Lea Hermann. This work is supported in part by the Atomic and Molecular Theory Division of the National Science Foundation (Grant PHY-86-96052), and by Los Alamos National Laboratory Theoretical Division T-12 through a grant from the Department of the Army. Graphics equipment used in the later stages of this project were provided by a grant from the Arkansas Science and Technology Authority (ASTA).
Appendix A. Computation
of rotation matrices and products
The Euler angler parameters are defined here according to conventions found in most texts on rotation groups [57]. A pictorial view of the standard definition is contained in refs. [3] and [38]. All rotations are products of a Z-rotation by y, followed by a Y-rotation by p, and finally, another Z-rotation by cr.
R(% P, Y) = R(% O,~)NO,P?wm 0, VI.
(A-1)
Always the Y and Z are lab-fixed axes. The fundamental SU(2) representation is defined using the same parameters, however the volume of the SU(2) parameter space is twice that of the R(3) space. The fundamental or spinor representation is e-ia/2
@‘2(%8, Y) =
o
i
where
co4P/2)
0
e i42
i sin( b/2) ii
-i
e-iy’2
sin( p/2) cos(P/2)
i(
0
0 eir/2
1
(A-2)
388
W. G. Harter / Graphical approaches to molecular rotations and vibrations
This was used in section 4 to derive the general spinor state in eq. (4.8). For R(3) it is only necessary to consider P-values between 0 and 7~. A binomial expansion [57] of the spinor representation yields the general W-matrix for all integer and half integer values of j. e-_i(ma+my)
(j+m-k)!k!(j-n-k)!(n-m+k)!
k=O
(A-3)
x(COS(j3/2))2’+m-n-2k(COS(~/2))n-m+2k. In computing D ‘, the limits of the sum over k are set first. They are determined requirement that the arguments of the factorials must be non-negative. The limits are max(O, m-n)
by the
j-n).
(A-4)
The resulting sum is all that is needed for the RE surfaces in sections 2. and 3. The wave packet minimum uncertainty function is section 2.4 is
DA( c@y) = e-iJ(u+Y)(cos /3/2)2”.
(A.5)
Other D-functions in the wavepacket calculation follow easily from (A.3). An expansion of the spin-l/2 rotation matrix (A.2) in terms of Pauli spin matrices provides a convenient way to compute rotation (or SU(2)) group products. The desired expansion is the following.
W2(a, j3,
y) = exp( T)
cos( t)(
i
-isin
sin(f)(~
-i
cos( t)(
sin(T)
y) k)-ices(y)
t
sirr($
-d)
_!I)).
If one wishes to combine two rotation operators R( a,a,a,)
(A.6) and R( b,b,b,)
into a product
then the expansion (A.6) of each operator has the following forms. R(u,u,u,)=A,I-iA*a=A,I-i(A,u,+A,u,+A,u,),
(A.8a)
R(b,b,b,)
= BoI - i(BXuX + BYuY+ B,u,),
(A.8b)
R( c1c2c3) = Co1 - iC* u = Co1 - i( C,u, + CYuY+ C,u,).
(A.%)
= B,I - iB*u
389
W.G. Harter / Graphical approaches to molecular rotations and vibrations
The coefficients identity.
in the third expansion are computed from those of the first two using Pauli’s
[A*a][B*a]
=A*B+i(A
XB)=a.
(A-9)
The results are as follows. c, = cos[ c/2]
cos[(c3+c&2]
(A.lOa)
=A,B,-A,B,-A,B,-A,B,,
C, = sin[ c,/2] sin[(c,-c,)/2]
=A,Bo+AoB,+A,B,-A,By,
(A.lOb)
C, = sin[ c/2]
COS[($-cr)/2]
=A,B,+A,B,-A,B,+A,B,,
(A.lOc)
c, = cos[ c/2]
sin[(c,+c,)/2]
=A,B,+A,B,+A,B,-A,B,.
(A.lOd)
The relation between the Euler angles and the four-vector (C,,, C) follows from eq. (A.6), and applies to the A and B quantities as well. The inverse of this supplies the desired Euler angles in terms of the four-vector components. c2 = 2 cos-l[
cr=c+-c_,
C,/sin
c,] ,
c3 =
c,
+
(A.lOe)
c_
Here c, and c_ are expressed in terms of a four-quadrant FORTRAN ATAN ~+=(c,+cr)/2=tan-~[C,/C~],
arctangent
function such as the
C-=(c3-c1)/2=tan-‘[C,/C,].
Appendix B. Body frame transformations
(A.lOf)
and momenta
The angular momentum vector operators may be expressed in terms of lab-fixed components { JxJyJz} or in terms of body-fixed components { J, Jy J,}. The two systems are related by Euler rotation given by eq. (A.l) using the Cartesian vector representation. cos (Y (R(a,B,y))=
(
sina 0
-sin a cosa 0
0 0 1 Ii
msP 0 -sin/3
1 0
x
= x
cosacos~cosy-sinasiny
Y sinacos/3cosy+cosasiny Z i
-cm
y sin /I
sin /3 0 cos/3 ii
cos y
-siny
0
sin y 0
cos y
0
0
1i Z
Y -cosacos/3sin~-sinacosy
cm d: sin /3
-sinacosfisiny+cosacosy
sin a sin /3
sinysinp
cos /!3
’
(B.1)
i
leads to the following relation [3] between the lab-fixed components of angular velocity ( wx, wY, wz) and Euler angle velocities { ai, p, i, } . This
-sin
(Y
*X
0
WY =
0
cos (Y
cos (Y sin p sin (Ysin p
OZ
1
0
cos j3
III
& fi 11 9 1
.
(B-2)
390
W. G. Harter / Graphical approaches to molecular rotations and vibrations
The body-fixed components of o involve the internal polar angles ( - /3) and ( - y) only *X WV =
- cos y sin p
sin y
0
d!
sin y sin fl
cos y
0
#8 .
cos p
0
ij(@z
(B-3)
1 )i 9 i
These matrices contain Jacobian transformation components. The inverse matrices coefficients for relating Cartesian angular momenta to Euler momenta.
contain
The lab-fixed momenta depend on Euler angles a and p I-cosacotp
‘Jx
4
-sin
\A
\
a cot /I 1
- sin a
cos a/sin /3 i J,
cos a
sin a/sin
0
/3
Jp _
0
)I
(B.9
J-r1
The body-fixed momenta depend on the body-defined polar coordinates ( - y) and ( - p) 1 - cos y/sin /3
Jy JZ I*)
sin y/sin /_I
= \
0
sin p
cos ycot p
cos y
-sin
0
(J,
y cot /3 1
Js .
(B.6)
I J-7I
The commutation relations are different for each set of momenta. [ J,J,]
= iJ, etc,
[ JxJy] = -iJ,
etc.,
[J,,
Jo] = 0 etc,
(B.7)
The Euler momenta are completely mutually commuting. Euler coordinates are holonomic and canonically conjugate to the momenta { J, JaJy }. There are no holonomic coordinates conjugate to Cartesian momenta.
Appendix C. Classical equations of rotational motion Numerical solution of classical rotational equations is generally easier to accomplish using Euler coordinates ( a/3y} and momenta { J, Ja Jy } . The relations in appendix B are used to convert Cartesian angular velocities, and momenta into Euler quantities. In some cases the analytic expressions are more revealing, too. The Hamiltons equations for Euler angles S = a, p, y are as follows
W. G. Harter / Graphical approaches to molecular rotations and vibrations
For the rigid symmetric top Hamiltonian aH
2B(Ja
-Jy
(2.1) with A = B # C this become the following
~0s P)
sin*P
&=aJ,=
(C.2a) ’
. aH p=aJ,=2BJa,
aH
391
(C.2b)
-2B(Ja-J,cosP)cosP
‘=aJ,=
sin*p
+2ccos J
Y’
(C2a)
The equations simplify considerably if the J vector is assumed to lie along the lab axis. If we let J,=J,=J,
Ja=O,
J,=J,=Jcosp,
(C-3)
then the coordinate equations yield ai = 2BJ,
,d=O,
jl=2(C-B)Jcosp.
(C.4)
The y-equation describes the classical precessional motion discussed after eq. (2.10). The motion of azimuth ( - y) is negative (clockwise) if C > B and J cos j? > 0. The equations for Euler momenta and the action are used in eq. (2.44)
The righthand expression is valid only if the Hamiltonian is an explicit function of body components of Cartesian momenta only. This is the case for rotors free of external torques. External fields may introduce an explicit coordinate dependence. This requires H to be re-expressed in terms of Euler coordinates and momenta before the coordinate derivitive in eq. (C.5) is evaluated. Numerical solutions for general free rotors using eqs. (C.l) and (C-5) are easier to program if the Cartesian A-derivatives of the Hamiltonian are evaluated separately in a numerical subroutine. This makes it easy to change the Hamiltonian or to derive the classical mechanics. It also facilitates the use of general Hamiltonians to obtain classical equations of motion directly from an ab initio RE surface. However, analytic expressions such as were given in the examples by eqs. (C.2) will usually run faster.
Appendix D. Tensor operator construction The members of a set of r th rank irreducible tensor components { c, T,‘_i, . . . , Tl,} are each composed of certain polynomial combinations of r th degree monomial products V;:,V;.,. . . V;.r of vectors. Generally, a vector set { V,, VY, VY} or, equivalently, a rank-l tensor set { V:, Vi, I? 1} is
W.G. Harter / Graphical approaches to molecular rotations and vibrations
392
the starting point for the development of higher rank tensors. It is first necessary to specify the relation between Cartesian or “linearly polarized” {x, y, z}-components and the angular or “circularily polarized” { 1, 0, - 1}-components as follows
The phase convention is the standard Condon-Shortley convention for spherical harmonics YA = { Yt, Y,‘, Yyl}. Examples of vectors V include operators of spatial position r = {x, y, z}, angular momentum J = { J,, Jy, J,}, or three-dimensional vibrational boson creation operators a+={,+ .+ a! }. The fundamental symmetry properties of rank-r tensor operators are that they tral;lsfo’rm according to the following D-matrix form
Rank-2 tensor operators T2 may be made from Clebsch-Gordan vector operators V(a) and V(b) as follows T4’= [V’(a)
Q V’(b)];=
~C;,~2,V&z)V,‘2(b).
products of any pair of
(D-2)
..’ “’ are the standard Clebsch-Gordan angular momentum coupling Here the coefficients C~J,,j,~f coefficients for coupling outer products of states 1;) and I,$) into a state I( j X j’);:!) of definite total angular momentum j”.
Examples of rank-2 tensor products are the following. T2 = J7’JT1 2
The 4 = 0 also used operators n = C&z, vibrational T4’=
1
1,
T2, = V:,V’,,
and 4 = + 2 terms are proportional to the tensors in eq. (2.2). The rank-2 tensors are in eq. (6.9). For the latter application one needs the fact that boson destruction al, transform like a!,( - 1)“. This is necessary in order that the number operator is a scalar operator. This implies that the diagonal (number-preserving) rank-2 tensor operators are constructed using the following form of eq. (D.2)
[u’+@ul]Z,=cc~~~,,u~:U’,,(-1)42.
tw
W. G. Harter / Graphical approaches to molecular rotations and vibrations
393
Higher rank tensors are derived in a similar manner. For example, rank-4 tensors could be made from products of any two rank-2 factors as follows
(D-6) This type of tensor product is used explicitly in eqs. (6.8) to (6.10). This structure is implicit in the simple rank-4 tensors which are described in section 3. Other examples are described by Hecht [ 161.
References 111D. De Leon and M.A. Mehta, Comput. Phys. Rep. 8 (1988) 293. PI R. Skodje and J.R. Cary, Comput. Phys. Rep. 8 (1988) 221. 131 W.G. Harter and C.W. Patterson, J. Chem. Phys. 80 (1984) 4241. 141 K.R. Lea, M.J.M. Leask and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. 151 A.J. Domey and J.K.G. Watson, J. Mol. Spec. 42 (1972) 1. 161 K. Fox, H.W. Galbraith, B.J. Krohn and J.D. Louck, Phys. Rev. Al5 (1977) 1363. 171 W.G. Harter and C.W. Patterson, Phys. Rev. Lett. 58 (1977) 224. PI W.G. Harter and C.W. Patterson, J. Chem. Phys. 66 (1977) 4872. 191 C.W. Patterson and W.G. Harter, J. Chem. Phys. 66 (1977) 4886. WI W.G. Harter and C.W. Patterson, Int. J. Quantum. Chem. 11 (1977) 479. 1111 W.G. Harter, C.W. Patterson and J.J. da Paixao, Revs. Mod. Phys. 50 (1978) 37. WI G.W. King, J. Chem. Phys. 15 (1947) 820. 1131 R.M. Hairier, P.C. Cross and G.W. King, J. Chem. Phys. 17 (1949) 826. P41 W.M. Colwell, N.C. Handy and W.H. Miller, J. Chem. Phys. 68 (1978) 745. 1151 J. Brown, J. Mol. Spectrosc. 31 (1969) 118. WI K.T. Hecht, J. Mol. Spectrosc. 5 (1960) 355. [I71 R.S. McDowell, H.W. Galbraith, C.D. Cantrell, N.G. Nereson and F.D. HinkIey, J. Mol. Spectrosc.
68 (1977) 288. WI R.S. McDowell, M.J. Reisfeld, M.W. GaIbraith, B.J. Krohn, H. Flicker, R.C. Kennedy, J.P. Aldridge and N.C. Nereson, J. Mol. Spectrosc 83 (1980) 440. 1191 J. Borde, Ch.J. Borde, C. Salomon, A. Van Lerberghe, M. Ouhayoun and C.D. Cantrell, Phys. Rev. Lett. 45 (1980) 14. J. PO1 Borde and Ch.J. Borde, Chem. Phys. 71 (1982) 417. PI W.G. Harter and C.W. Patterson, Phys. Rev. Al9 (1979) 2277. P21 W.G. Hatter, Phys. Rev. A24 (1981) 192. 1231 W.G. Harter, in: Proceedings of the XVth Colloquium on Group Theoretical Methods in Physics, ed. R. Gihnore (World Scientific, Singapore, 1987). ~241 K.K. Lehmann, J. Chem. Phys. 79 (1983) 1098. 1251 M.E. Kellmann, J. Chem. Phys. 76 (1982) 4528. WI M.E. Kellmann, (a) J. Chem. Phys. 83 (1985) 3843; (b) Chem. Phys. Lett. 113 (1985) 489. 1271 O.S. van RoosmaIen, I. Benjamin and R.D. Levine, J. Chem. Phys. 81 (1984) 5986. PI O.S. van RoosmaIen, F. Iachello, R.D. Levine and A.E.L. Dieperink, J. Chem. Phys. 79 (1983) 2515. Spectra (Elsevier Scientific, New York, 1982) [291 D. Papousek and M.R. Ahev, Molecular Vibration-Rotational chap. III. p. 14. 1301 P. Jenson, Comput. Phys. Rep. 1 (1983) 1. [311 E. Abranson, R.W. Field, D. Imre, K.K. Innes and J.L. Kinsey, J. Chem. Phys. 83 (1985) 453. Y. Matsumoto, L.H. Spangler and D.W. Pratt, Chem. Phys. Lett. 95 (1983) 343.
394 [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]
W. G. Harter / Graphical approaches to molecular rotations and vibrations D.J. Nesbitt, Chem. Rev. (in press). D. Huber, E.J. Heller and W.G. Harter, J. Chem. Phys 87 (1987) 1116. N. De Leon and E.J. HelIer, J. Chem. Phys. 81 (1984) 5957. W.H. Press, B.P. Flannery, S.A. Tenkolsky and W.T. Vetterhng, Numerical Recipes, (Cambridge University Press, Cambridge, 1986). W.G. Harter and C.W. Patterson, J. Math Phys. 20 (1979) 1453. W.G. Harter, J. Stat. Phys. 46 (1984) 749. W.G. Harter, J. Chem. Phys. 85 (1986) 5560. 1.1. Rabi, N.F. Ramsey and J. Schwinger, Rev. Mod. Phys. 26 (1954) 167. R.P. Feynman, F.L. Vernon Jr. and R.W. Hellwarth, J. Appl. Phys. 26 (1957) 167. G. Stokes, Proc. Roy. Sot. (London) 11 (1862) 545. C. Jaffe and W.P. Reinhardt, J. Chem. Phys. 77 (1982) 5191. E.J. HelIer, in: Potential Energy Surfaces and Dynamics Calculations, ed. D.G. Truhlar (Plenum, New York, 1981) p. 127. C.W. Patterson, J. Chem. Phys. 83 (1985) 4618. J.T. Hougen, P.R. Bunker and J.W.C. Johns, J. Mol. Spectr. 34 (1970) 136. J.H. Frederick, G.M. McClelland and Paul Brumer, J. Chem. Phys. 83 (1985) 190. G.S. Ezra, Chem. Phys. Lett. 127 (1986) 492. J.H. Frederick and G.M. McClelland, J. Chem. Phys. 84 (1986) 867, 84 (1986) 4347. E.B. Wilson Jr. and J.B. Howard, J. Chem. Phys. 4 (1936) 200. D. Papousek and M.R. Aliev, ref. [29] pp. 42-54. J.K.G. Watson, Mol. Phys. 15 (1968) 479. G. Hetzberg, Molecular Spectra and Molecular Structure, vol. I. Spectra of Diatomic Molecules (Van Nostrand/ Reinhold, New York, 1950) pp. 218-237. C.H. Townes and A.L. Schawlow, Microwave Spectroscopy (Dover, 1975) p. 187. (A correlation plot is used as a cover design in this edition.) C. Herzberg, Molecular Spectra and Molecular Structure, vol. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand/Reinhold, New York, 1966) pp. 20-68. M. Feingold and A. Peres, Physica D 9 (1983) 433. W.G. Harter, C.W. Patterson and H.W. Galbraith, J. Chem. Phys. 69 (1978) 4896. L.C. Biedenham and J.D. Louck, Encyclopedia of Mathematics and its Applications, vol. 18. Angular Momentum in Quantum Physics, ed. G. Rota (Addison Wesley, New York, 1981) pp. 46-55.
William G. HARTER Theoretical Division T-12, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA
NORTH-HOLLAND
- AMSTERDAM
W.G. Harter / GraphicaI approaches to molecular rotations and vibrations
320
Contents 322 1. Introduction. .............................................. 324 1.1. The rotational energy (RE) surface. ........................... 324 1.2. The constant energy (CE) surface. ............................ 325 ........................................ 1.3. Other techniques 325 ................... 2. RE surfaces for tensor Hamiltonians of rigid rotors 326 2.1. Spherical, symmetric, and asymmetric RE surfaces ................ 2.2. Properties of classical and quantum RE surface levels and eigensolutions . 327 331 2.3. Symmetry properties of eigensolutions ......................... 334 .................................. 2.4. Semklassical quantization 342 3. RE surfaces for tensor Hamiltonians of semi-rigid rotors. ............... 342 3.1. Fourth rank octahedral tensor Hamiltonians ..................... 347 3.2. Symmetry properties of eigensolutions ......................... 348 3.3. Semiclassical quantization .................................. 350 3.4. Mixed-rank octahedral tensor levels ........................... 353 4. Quasi-spin RE surfaces for two dimensional vibrations ................. 353 4.1. Spinors, rotors, and coupled oscillators ......................... 360 4.2. Quasi-spin RE surfaces .................................... 362 4.3. Higher dimensional oscillators ............................... 363 5. RE surfaces for generic Hamiltonians of non-rigid rotors ............... 363 5.1. The generic Hamiltonian in the body frame ..................... 365 5.2. X, centrifugal distortion energy .............................. 370 5.3. RE surfaces for vibronically excited states. ...................... 370 6. Multiple RE surfaces and Coriolis coupling effects .................... 371 6.1. Semiclassical mechanics on multiple surfaces. .................... 373 6.2. Asymmetrically coupled spin and rotor systems. .................. 378 ................... 6.3. Octahedral scalar and tensor Coriolis coupling 386 7. Conclusion. ............................................... 387 Appendix A. Computation of rotation matrices and products .............. 389 Appendix B. Body frame transformations and momenta. ................. 390 Appendix C. Classical equations of rotational motion ................... 391 ........................... Appendix D. Tensor operator construction 393 References ..................................................
321
Computer Physics Reports 8 (1988) 319-394 North-Holland, Amsterdam
COMPUTER GRAPHICAL AND SEMICLASSICAL ROTATIONS AND VIBRATIONS
APPROACHES
TO MOLECULAR
William G. HARTER TheoreticaI Division T-12, Los Alamos National Loboratory, Las Alamos, NM 87.545, USA and Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA Received
15 January 1988
Rotation-vibration dynamics and eigensolutions for several models are visualized using computer generated rotational energy surface and semiclassically quantized trajectories. Several numerical approximations to rotation-vibration eigensolutions are developed which use information gained from graphical presentations. Techniques are given for finding essential physical phenomena in complex numerical rovibronic eigenvalue problems. Possible alternatives to large scale numerical diagonalizations are introduced.
0167-7977/88/$26.60 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
322
W.G. Harter / Graphical approaches to molecular rotations and vibrations
1. Introduction
The development of computers has strongly influenced the field of atomic and molecular physics. Most problems involving the electronic structure, vibrations, or rotations of polyatomic molecules have been too complicated to have exact analytic solutions, and so numerical solutions have been developed using various methods. The most common method involves numerical diagonalization of the large matrix representations of Hamiltonian operators, and this has led to considerable progress in the study of molecular structure and dynamics. As the computers increased in size it has been possible to choose bigger and better basis sets in which to carry out the numerical diagonalizations using many different techniques. Also, as computers became more accessible researchers began developing completely different numerical treatments of molecular quantum mechanics. As a result there has been a renaissance of semiclassical treatments [1,2]. The semiclassical approaches can in many cases avoid the numerical difficulties associated with setting up a large quantum mechanical basis set. Classical equations of motion are generally easier to solve numerically and several techniques for quantizing these solutions have been developed. The resulting solutions also suggest better ways to visualize the physics of the molecular dynamics. It is the last point which will be the main subject of this report. Until recently, the development of raw computing power has been more rapid than the development of the interface between the machine and its user. Now the introduction of sophisticated graphical input and output makes it easier to interpret and visualize complicated computer solutions. This report shows some new ways to use modern computer graphics to help understand molecular rotational and vibrational dynamics. Some of the methods to be described here are based upon quantum and semiclassical techniques which have been developed and applied elsewhere. A number of the most recent techniques are described in the articles by De Leon and Mehta [l] and Skodje and Cary [2] in companion articles. However, the main topic of present work will be use of the geometry of rotational energy (RE) surfaces for efficiently visualizing and computing classical and quantum effects associated with molecular rotations and vibration. The rotational energy (RE) surface was developed by Harter and Patterson [3] as a method for directly visualizing and calculating certain types of rotational spectral fine structure and for clearly understanding the mechanics of energy level clustering or superfine structure. Previously, level clustering had been noted in computer diagonalizations of tensor Hamiltonians by Lea, Leask, and Wolf [4], Domey and Watson [5], and Fox, Galbraith, Krohn, and Louck [6]. An explanation of the clustering on quasi-degeneracy based upon classical correspondence was given by Domey and Watson [5]. A detailed quantum theory for the structure of level clusters was given by Harter, Patterson [7-lo] and de Paixao [ll]. The semiclassical calculation of fine or superfine structure based upon RE surface trajectories and tunneling paths has been done more recently [3] and this will be reviewed in the sections 2 and 3 of this article. The discussion in section 2 is based upon the well known quantum mechanics of the rigid rotor. This provides one of the simplest examples of an RE surface and semiclassical quantization. (Probability the simplest .example is the quasi-spin RE surface for a vibrator, but this is a more abstract use of the concept which will be discussed in section 4.) Previous work on semiclassical quantization of the asymmetric top is given by Ring, Hainer, and Cross [12,13] and
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323
more recently by Colwell, Handy, and Miller [14]. Approximation methods and discussion by Brown [15] should also be noted. The rigid asymmetric top provides a relatively simple pedagogical aid and testing ground for a number of concepts and techniques, and hence this is the longest section in this report. The discussion in section 3 uses a model which has had success in explaining certain phenomena in experimental spectra and in predicting new spectroscopic effects. It is based upon an effective Hamiltonian introduced by Hecht [16] for semirigid tetrahedral molecules such as methane (CH,), and it has been applied extensively to high resolution spectroscopy of a number at tetrahedral, octahedral, and cubic molecules. The SF, molecular spectrum has been extensively studied since it is the prototype for an isotope separation scheme [17,18]. It has also been the target of the highest resolution 10 pm spectrometer built by Borde and coworkers [19,20]. SF, provided an example in which a complicated spectrum and related dynamics can be understood easily by appealing to graphical constructions such as RE surface geometry. Such understanding was used to predict further properties of SF, nuclear-spin-rotor eigensolutions [21,22] which have been confirmed by the analysis the Borde experiments [20,23]. The discussion in section 4 is a more abstract application of an RE surface to describe normal mode and local mode phase trajectories of harmonic and anharmonic vibrators with two degrees of freedom. Algebraically, this is the simplest possible example of an RE surface. The model uses the mapping between the unitary U(2) group of a two-dimensional oscillator and the rotational group R, of a rotor. The RE surface picture helps to clarify some aspects of the SU(2) models introduced by Lehmann [24], and Kellman [25,26] and the elementary vibron model of von Roosmalen, Iachello, Levine, Benjamin, and Dieperink [27,28]. One of the possible future directions for the RE surface technology is introduced in section 5. It is proposed that a useful description of certain non-rigid rotors might be obtained numerically directly from an ab initio Hamiltonian. This may provide an alternative to the usual procedure in which an effective molecular tensor Hamiltonian such as the Hecht operator is obtained analytically by an algebraic perturbation expansion using the Van Vleck or Birkhoff-Gustaveson transformations [29]. Some numerically generated sets of RE surfaces are produced for some simple harmonic and anharmonic rotor-vibrators of the simplest polyatomic X, structures. The behavior of some systems with eigenvalue-generated RE surfaces is discussed in section 6. These are analogous to the multiple sheets of potential energy (PE) surfaces in vibronic models such as Jahn-Teller or Renner Hamiltonians. Two systems are discussed which are extensions of the systems treated in sections 2 and 3. The first involves a spin asymmetrically coupled to asymmetric rotor. The second is an octahedral XV, model with variable scalar and tensor Coriolis coupling between the rotor and a triply degenerate vibrational mode. The first of these systems is studied in regions where the classical model exhibits non-regular motion. Both models are used to compare classical RE surface geometry and dynamics to quantum eigensolutions. The discussions of the last three sections 4, 5, and 6 are very preliminary developments. One objective is to extend some of the advantage of the basic RE surface methods given in sections 2 and 3 to the more difficult problem of non-rigid rotors or highly excited rotor-vibrators. A more ambitious objective is to provide a numerical shortcut for treating the generic or ab initio Harniltonians of non-rigid molecules and calculating their spectroscopic properties in a way that provides as much understanding of their rotation-vibration dynamics as possible. It is unlikely that any single technique will achieve these objectives. Rather, a satisfactory
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
treatment of the rotation-vibration problem will involve a combination of analytical, numerical, and graphical techniques. For a long time it will be necessary to compare new approaches with the more standard approaches which diagonalize various representations of analytic Hamiltonians. These methods have been successful for many problems involving semi-rigid and non-rigid molecules [29] since before computational physics became a named subject [30]. However, potential algebraic and numerical bottlenecks appear ready to frustrate the theorists attempts to treat highly excited [31] or very floppy [32] species. Also, there are questions about what to do with a molecular spectral analysis after it is completed. One needs more ways to visualize and think about a given molecule’s structural and dynamic properties in order to plan more revealing experiments and invent new processes. A better picture is useful if it is suggestive or if it provides an image and a language with which to discuss complicated numerical and experimental results effectively. Before beginning the discussion of visualisation devices a number of the most useful ones will be listed and described briefly. The principal device is the first one listed below which is the RE surface. 1.1. The rotational energy (RE) surface The RE surface is a plot of a molecule’s energy as a function of the direction of an angular momentum vector in a molecule fixed frame for fixed magnitude of the momentum. The techniques for obtaining this plot and the detailed definition of the quantities involved will vary for different examples in section 2 thru 6. Usually the energy will be the total molecular energy of some state and the angular momentum will be the total angular momentum which is laboratory fixed in a classical model. The energy will generally be plotted radially outward and form a closed surface of radius E where E=H(J),
forJ=
]J( =constant.
(1.1)
The RE surface plays for rotational dynamics some of the roles which a potential energy (PE) surface plays for vibrational dynamics. However, the RE surface is derived from the PE surface, and it is a total energy. It includes information about properties of the potential energy surface as well as the kinetic energy of motions on it. This connection is elaborated in sections 3 through 6. The RE surfaces compresses information about the vibrational dynamics and PE surface. An RE surface is a two-dimensional surface embedded in a 3-space, while a PE surface is a (3N - 6)-dimensional object. A general RE surface is composed of multiple sheets with one sheet for each relevant vibronic state. 1.2. The constant energy (CE) surface The CE surface is a concept that is perhaps more familiar in mechanics. It is simply the locus of the angular momentum vector in molecule fixed (J,J,J,)-space for constant energy. E = H(J)
= constant.
(1.2)
One may require this type of surface when the quantity J is an action variable whose definition
W. G. Harter / Graphical approaches to molecular rotations and vibrations
325
changes with the value of energy. However, for rotors discussed in this article the J is well defined for all energies so we shall not use this construction. For spectroscopic applications it is better to have a display of a range of energies for a fixed magnitude of J than a range of J-values for a fixed energy. 1.3. Other techniques
Several types of auxiliary techniques may be employed to compare and discuss the information provided by the RE surface. These include plots of standard matrix diagonalizations, several semiclassical quantizations techniques, and trajectories derived by integration of classical equations of motion. Computer graphic visualization of molecular physics is something of an art. One should not expect a single technique to exist at the exclusion of all others. The computer graphics shown in this report is mostly done by a program called GRAFIC written by Melvin Prueitt of Los Alamos National Laboratory and modified for these specialized applications by the author. The program is capable of plotting realistic three dimensional views of multiple objects with each defined by an arbitrary set of surface coordinates. The objects may have level contours drawn on them with respect to an arbitrarily given point of origin. The program written in FORTRAN is efficient enough that most of the figures produced for this report can be done in a second or two on a mainframe computer. The program is being developed for use on an IBM PC and a MacJntosh computers by Steven Parker and the author, and in its present form takes several minutes to generate surface mesh plots on personal microcomputers. Assembly language implementations of some of the more repetitive calculations should improve the speed greatly. Some of figures in this report are exerpted from interactive animation programs. One of these in an animated 3D (stereoptic) view of of motion in a four-dimensional phase space. These advanced techniques are particularily valuable for use on an interactive-graphics oriented computer such as the Macintosh machines, where one can escape the usual “batch” processing that has become the rule on mainframe machines. One should look for a large number of innovative methods as personal workstations come available.
2. RE surfaces for tensor Hamiltonians of rigid rotors The simplest Hamiltonian H=AJ,2+
BJ;+
for a massive free rotor is that of the rigid top. (2.la)
Cl;.
The rotational constants A, B, and C are each one-half of the inverse of the principal inertia I,, I,,, and 1,. A = ,uL,/2 = l/21,,
B = p/2
= l/21,,,
C= pL,/2 = l/21,.
(2.lb)
The coordinate axes are fixed to the directions of principal inertia in the body. The tensor form of the rigid top Hamiltonian uses second rank multipole operators Tq2 to replace Jx2 terms in
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
(2.la). The necessary rank-2 multipole operators are given below. A brief review of general rotational transformations and tensors is given in appendices A-D.
J2=J,2+J;+J;,
G2= -(J,z+J;-2JL’)/2,
T;+T!,=3(J,2-J;)/&.
(2.2)
The Hamiltonian expressed in terms of these operators is as follows.
H= A+B+CJ2+ 3
2C-A-BT2+ 3
0
A-B -(T; fi
(2.3)
+ T!,).
To construct a rotational energy surface each tensor operator Tl is replaced by a corresponding multipole function Ci or Wigner D-function harmonic which is described in appendix A
T,‘-,D,‘,(.,P,Y)IJI’=C~~(--P,
(2.4)
-v)lJI’.
These are functions of standard Euler spherical polar coordinates which include the polar angle 8 = -B and azimuthal angle 9 = - y of the J-vector in the body frame. Substitution (2.4) is equivalent to replacing the original J,, JY, and J, in (2.1) by the following polar coordinate expressions.
J,= IJlsin(-P)cos(-y),
JY= lJlsin(-B)sin(-y),
J,= (Jlcos(-p).
(2.5)
The result is a spherical polar rotational energy (RE) surface function.
A+B+C
+ 2C-A-B 6
(3 cos2p - 1) + q
(sin2 B cos 2y)].
(2.6)
This is to be plotted radially as a function of B and y for fixed ) J I. Some examples of this RE surface will now be discussed for several different values of A, B, and C. 2.1. Spherical, symmetric, and asymmetric RE surfaces A rigid spherica top has three equal constants A = B = C, and only the first term in (2.3) or (2.6) contributes. This is a scalar (Too) operator. The resulting RE surface is a sphere Jx2+ JY2+ J: = 21H. A symmetric top has two equal constants. The case A = B > C corresponds to a prolate symmetric top with a z-symmetry axis, and only the first two terms in (2.3) or (2.6) survive. The resulting RE surface is shown in fig. 2.la for values of A = B = 0.2 and of C = 0.6. In this case only the cylindrically symmetric parts Tz and Tz contribute. The general rigid rotor is the asymmetric top in which all three constants are nondegenerate and all three terms in (2.3) contribute. An asymmetric top RE surface is shown in fig. 2.lb for constant values of A = 0.2, B = 0.4, and C = 0.6. Finally, fig. lc shows and example of an oblate symmetric top RE surface in which B is set equal to C and both are greater than A. Only the value of the B-constant is different for figs. 2.la, b, and c. Each of the RE surfaces in fig. 2.1 contains equal energy contour lines or level curves. These curves are intersections of the RE surface with equal energy spheres of progressively greater
W. G. Harter / Graphical approaches to molecular rotations and vibrations
327
Fig. 2.1. Rigid rotor RE surfaces. Bands are semiclassical J-vector paths for J=lO. Thicker paths are classically slower. (a) Prolate top: A = B = 0.2, C = 0.6. (b) Asymmetric top: A = 0.2, B = 0.4, C = 0.6. (c) Oblate top: A = 0.2, B = C = 0.6.
energies, and they are computed by the graphics program. Since the RE surface corresponds to a constant magnitude 1J ) of angular momentum, the constant-E lines on the surface are possible classical trajectories of the J-vector in the body frame. It will be shown that thicker curves represent classically slower paths and regions with greater wavefunctions. The curves drawn in fig. 2.1 are the ones that correspond to the quantum values for J = 10. It is important to see how RE surface trajectories correspond to the exact quantum eigensolutions of the asymmetric top since it is a good prototype for more complicated rotor Hamiltonians. 2.2. Properties of classical and quantum RE surface levels and eigensolutions The exact eigensolutions are obtained for comparison by diagonal&zing an matrix representation of the tensor Hamiltonian for a rotor. It is conventional to use a basis set of (2 J + 1) eigenvectors 1:) of J,, where 41;)
=KI$),
(k=J,
J-l
,...,
-J)-
These span a block matrix representation theorem.
of each tensor term according to the Wigner-Eckart
(2.7a) Here C$$ is a Clebsch-Gordan coupling coefficient which determines the form of each (2 J + 1) dimensional matrix. This is multiplied by an overall reduced matrix element factor.
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
The asymptotic value of this factor is 1J 1r for high J since Ti is an th degree polynomial in J. For a second rank tensor (2.3) and (2.7) yield (2.8a) and (2.8b) The matrix representation of the Hamiltonian (2.3) is then given
(A +B)
(JK,(H(;) =
I
(it
J(J+l)+
(2C-A-B)K2 2
=0
(2.9a)
@&(JU)(J+IGl)(J~K+l)(JkK+2)
[
(K’=Kf2).
(2.9b)
The matrix is diagonal if one sets A = B (this is the prolate limit using the z-axis of quantization), or if one switches A and C and sets C = B (this is the oblate limit using the x-axis of quantization). Then the asymmetric terms Ti2 do not contribute. The correspondence between the exact J = 10 eigenlevels and asymmetric RE surface trajectories is given in fig. 2.2.. Pairs of eigenlevels are shown in the lower portion of the figure in magnifying circles. Each pair of levels corresponds to a pair of equivalent level curves on the RE surface. The level splittings of each pair are called superfine structure splittings since they are usually very small. Only the levels near the separatrix region in the center of the fine structure pattern have splittings which are visible on the scale of the drawing. The separatrix region divides the trajectories and the corresponding levels into two distinct types. One type of level pairs lie at energies above the separatrix and correspond to pairs of trajectories surrounding the z-axis which is centered on the hills of the RE surface. The corresponding quantum levels fall into pairs of A, and B, levels or A, and B, levels depending upon whether the quantum number K,(z) is even or odd, respectively. These levels are shown on the righthand side of fig. 2.2. Another type of level pairs which he below the separatrix correspond to pairs of trajectories surrounding the x-axis which is centered in the valleys of the RE surface. The low energy levels fall into pairs of A, and B, levels or A, and B, levels depending upon whether the quantum number K,(z) is even or odd, respectively. Odd and even symmetry properties will be discussed in section 2.3. The z-type trajectories go around the high points of the asymmetric RE surface. They are related to the prolate symmetric top trajectories shown in fig. la. J-vectors on trajectories around high points precess in a clockwise sense at a rate that is proportional to the RE surface according to Hamilton’s equations of motion (see appendix C) o = aH/iYIJ.
(2.10)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
VISUALIZING THE J = IO LEVELS OF AN ASYMMETRIC TOP
329
?OTATION AXES UEAR f AXIS
h
IO=KJz)
62 At
‘1 ’ \\!/I
C,(z)-Type Clusters 8 6x1&d =26 kHz
a i
77x1&
2,6xf&‘5
IX&,
=230kHz
=64MHz
= I.5GHz
C&x)-Type Clusters II,,
II,,
,,I,
20
LO
I,,,
i II,,
LO
I,,,
,,I,
LO
(I,,
I,,,
60crri’
Fig. 2.2. J = 10 asymmetric top energy levels and related RE surface paths (A = 0.2, B = 0.4, C = 0.6) (after ref. [3]). Clustered pairs of levels are indicated in magnifying circles which show superfine splittings.
RE surface levels are thicker in regions of less slope where precession is slow. Precessional rates are also related to energy differences in the prolate symmetric top spectrum. These energy levels follow from (2.9a) with A = B.
E;=
BJ(J+
1) + (C-
B)K'.
(2.11)
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
The average difference between nearest neighboring K-levels is
AE =
[(&+I -E;)+(E;-E;_1)]/2=2(C-B)K
(2.12)
and this is exactly the classical precessional frequency. (The classical rotational frequency 2BJ is proportional to the difference between neighboring J-levels with the same K.) If (C - B)K is positive as it is for the trajectories in the upper half of fig. la then wZ= 9 is positive. Since the body azimuthal angle of J is -y, all the upper paths carry J-motion from right to left in fig. 2.la; that is, clockwise. The lower paths have negative K and therefore carry motion in the opposite direction of left to right. The negative-K motion is clockwise, too, when viewed from the high point at the center of the lower paths. In between the two kinds of prolate top paths in fig. la is a single line at the lowest energy corresponding to K = 0. The (K = 0)-line contains fixed points. The J vector cannot move if K = 0 since according to (2.12) the precessional frequency is zero for these points. The fixed points correspond to a pure end-over-end rotation. (Incidentally, this is the only possible motion for a diatomic rotor.) The (K = 0)-line of fixed points gives rise to two new valleys of precessing trajectories as B becomes larger than A and approaches C. Then the asymmetric Ti2 tensor terms in (2.3) contribute a separatrix consisting of two intersecting circles. It opens like a pair of scissors to make room for new precessional paths around the x-axis and to cut into the phase space occupied by the paths around the z-axis. The x-paths are related to the oblate paths which they become when B = C as in fig. lc. Around each valley the sense of precession will be counterclockwise when viewed from above the valley center. Paths along the separatrix are headed in the same direction on either side as indicated by arrows in fig. 2.2. The angle 8 between the separatrix circles and the z-axis is a measure of the phase space available for the z-type paths 8=arctan[(C-
B)/(B-A)].
(2.13)
The separatrix angle is 9 = 90 O, 45 O, and 0 o for figs. a, b, and c, respectively. The number of quantizing trajectories must remain constant as the separatrix angle increases. Paths are forced to change from z-type paths to x-type paths as the separatrix closes in on the z-axis. Paths nearest the separatrix are distorted, or “ squeezed” considerably from the circular shape which they had for the symmetric top case. The squeezing spoils the symmetric top path’s K-conservation since the projection of the J-vector along the z-axis varies as it moves along a warped path. As 8 decreases each z-path must meet its mate on the separatrix and then cross over and break-up into a new pair of x-paths. Each crossover and break-up corresponds to a crossover and break-up of quantum energy levels clusters. The exact quantum energy levels for J = 10 are plotted in fig. 2.3 for the same range of B values (0.2 < B < 0.6) used in going from fig. la to fig. lc. The levels shown in fig. 2.2 are plotted vertically above the point B = 0.4 in fig. 2.3. At this point the K = 5 quantizing paths are occupying the separatrix and together account for just one A, level in the center of figs. 2.2 and 2.3. Most of the levels in fig. 2.3 are paired or clustered. Only the levels near the separatrix region split up, and this region is clearly visible running form the lower lefthand to upper righthand corner of fig. 2.3. In the separatrix region one level always exhibits much more sensitivity to the change of B then the other member of the pair. The difference in sensitivity is due to the
W.G. Harter / Graphical approaches to molecular rotations and u&rations
331
Fig. 2.3. J = 10 energy levels as a functionof 33(0.2 zzB I; 0.6) betweenprolatetop and oblate top limits.(A = 0.2 and C = 0.6 are constant)LevelsaboveB = 0.4 correspondto thosein fig. 2.2. difference in symmetry properties properties will be described shortly. Energy level correlations such as eigensolutions for a whole class of particular type of level cluster and corresponding evolving RE surface.
of the corresponding
quantum
wavefunctions.
Symmetry
fig. 2.3 provide useful information about the nature of the Hamiltonians. One em often spot regions belonging to a this can be related to particular regions of stability iu the
One learns which eigensolutions are relatively localized and insensitive to small changes of the Hamiltonian, i.e., levels for which perturbation theory could easily accommodate the changes. On the other hand the eigenfunctions which are caught in the neighborhood of a separatrix are relatively delocalized and extremely sensitive to changes. Their level correlation plots are characterized by destruction, and rebuilding of level clusters with relatively rapid changes in between. One expects perturbation theory to be less useful in describing this case when the wavefunctions and their corresponding classical trajectories are being radically changed. 2.3. Symmetry properties of eigensolutions The rotational symmetry of a generic (A + B + C) rigid rotor Hamiltonian is described by the four-group D, = (1, R,, R,, R,) which containes the identity (1), and three 180° rotations R,, R, and R, around the body axes X, y, and z, respectively. The irreducible characters of D, are labeled AI, A *, B,, and B, in the character table AI AZ Br %
1 1 1 1 1
R,
R, 1
-1 1 --I
R, 1 1
-1 -1
1 -1 -1
(2.14) 1
332
W.G. Harter / Graphical approaches to molecular rotations and vibrations J t
J
&
x
y
?-
m K,(x)
I
\
c0 2y
D
K&Y
)
e3 K,(z)
Fig. 2.4. Different choices of rotation axes for asymmetric rotor corresponding to local symmetry (a) C,(x), and (c) C,(z). Tables correlate global D, symmetry species with the local ones.
(b) C,(y),
The subscripts 1 and 2 label even and odd symmetry, respectively, with respect to x-rotations R,. The A and B label even and odd R, symmetry, respectively. We shah see that the R, symmetry determines which levels are more sensitive to perturbations in the separatrix region. Note that R, symmetry is a product of R, and R, symmetry for each of the four D, species. Symmetry correlations of D, and the three separate axial subgroups { 1, R,} = C,(x), { 1, R,,} = C,(y), and (1, R,} = C,(z) are summarized by tables in fig. 2.4. Even and odd axial C,-symmetry is denoted by 0, (0 mod 2) and 1, (1 mod 2), respectively. The figures above each table show an XY, asymmetric top in a state of rotation which corresponds to the J vector being near the x, y, or z body axis, respectively. The C,(x) rotations correspond to the lowest energy asymmetric top trajectories in fig. 2.2 which are nearest to the x-axis, while the C,(z) rotation corresponds to the highest energy trajectories which are nearest to the z-axis. The C,(y) motion corresponds to being on the separatrix and is therefore not a simple rotation. The quantum levels belonging the x and z paths are paired according to the columns of the C,(x) and C,(z) tables, respectively. The even or O2 column of the C,(x) table contains the pair (A,B,) which are the species contained in the even-K,(x) levels on the lefthand side of fig. 2.2. They alternate with the pairs (A,B,) which occupy the odd or 1, column of the C,(x) table. Both these are pairs of levels that belong to trajectories that have C,(x) symmetry on the RE surface and arise from the oblate top spectrum on the lower righthand side of fig. 2.3. The same explanation holds equivalently for the alternating (A,B,) and (A,B,) pairs arising from the prolate part of the spectrum. Each classical z-trajectory resides on one of the prolate ends of the RE surface and each closed path by itself has a local C,(z) symmetry. The relation between pairs of classical paths and pairs of eigensolutions such as (A,B,) needs to be defined more precisely. Equal combinations of the two eigenstates in a pair correspond to
W. G. Harter / Graphical approaches to molecular rotations and vibrations
333
eigenfunctions which are more nearly localized onto a single RE surface trajectory. For example, the following combinations of the highest (A1B2) pair of J = 10 levels in fig. 2.2 or 2.3 yields (2.15a) (2.15b) where the coefficients a10 = (Y_~~are nearly equal to one and all the other components cr* a, (Y* 6, (Y* 4; and so forth are much smaller. The odd coefficients (YT 9, (Ye 7, etc. vanish because of symmetry; only the states with local symmetry of 0, can contribute to (2.15). The positive combination (2.15a) corresponds to a wavefunction centered on the upper trajectory in fig. 2.2, and the negative combination (2.15b) is on the lower one. Similarily, the next highest pair of energy levels labeled (A,B,) in fig. 2.2 yields the following combinations of states with odd (12) local C,( 2) symmetry (2.16a)
( I A,)
-
I B,))/O
=
a.9
1’;)
+
ci_,I’$
+
* * * .
(2.16b)
ak5, etc., are small. The next pair (A,B,) has the same form as (2.15) except that (Y+ 8 are the dominate coefficients. These combinations correspond to wavefunctions centered on the K,(z) = 8 or the K2( z) = 8 trajectory, respectively in fig. 2.2. An approximate calculation of these states will be discussed in section 2.4~. In order to obtain wavefunctions that are localized on a single classical trajectory it is necessary to mix states of different D, symmetries as in eqs. (2.15, 16). In this way a wavepacket is made which has some given local symmetry, say, 0, or 1, from C,(z) by combining two species such as A, and B, or A, and B,, respectively, of global symmetry D,. Since the different species have different energies the resulting combinations are non-stationary states, i.e. the wavepackets move from one location to another. However, if the energy differences are small this motion can be very slow. This is the case for the states localized far from the separatrix, i.e., high-K states. In fig. 2.2, the K = 10 states on the x or z axes have energy differences of only 26 kHz which is very small compared to their 150 GHz precessional frequency. Here it is easy to distinguish faster precession from the much slower tunneling motion. For states in the neighborhood of the separatrix the wavefunctions are not so localized onto the classical paths. Instead, the wavefunctions tend to pile up around the saddle region where the classical trajectories slow down and turn around. A J-vector on a trajectory that is closer to the separatrix will take longer to turn around and come out of the saddle region. (If it is exactly on the separatrix it will take forever!) The sensitivity of a given eigenwave to the change in the term BJ; in (2.la) depends on the wave amplitude near the y axes. Waves B, and B, which have odd C,(y) symmetry must vanish at the saddle turning point and be generally smaller in that neighborhood than the A, and A, waves. Therefore it is always A-levels which rise more quickly as a function of B in the separatrix region of fig. 2.3.
Here ak7,
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
2.4. Semiclassical quantization A number of semiclassical methods are being developed for approximating quantum rotor eigensolutions directly from the classical rotational energy surface. Such methods might be applied without having to calculate an effective tensor Hamiltonian for a non-rigid rotor. So far, these techniques have only been used to study rotor eigensolutions where well defined effective Hamiltonians exist and can be compared to the approximations. Here three examples of semiclassical methods based upon rigid top RE surfaces are described briefly. The first method is just a very simple visualization aid based upon angular momentum cones which provides quick rough estimates of certain spectroscopic features. The second and third methods are based upon simple semiclassical path integration and wavepacket propagation, respectively.
(a) Angular momentum come quuntization The angular momentum cone method is quite accurate for nearly symmetric tops and for non-rigid rotor surfaces with local regions of C,, C,, or higher symmetry. As we will see it is not so accurate for the asymmetric top, but it is based upon the properties of a symmetric top for which it is exact. Hence it will be introduced here. The angular momentum cones are defined as the loci of the angular momentum vector J obtained from a “literal interpretation” of the symmetric top base state 1;) and its quantum eigenequations
JzIi) = K(IK) 7
The interpretation magnitude of
J’i;) = J(J+
1) k).
(2_17a,b)
of these equations is that J is constrained to have a z-projection of K and a
(2.18)
(JI =/m.
This constrains the J-vector to lie on a cone of altitude K and slant height 1J 1 = /m. The polar angle 0; of a quantizing cone is defined by the arccosine of the altitude to slant height ratio (2.19) The base of the cone provides a measure of the uncertainty (A J,)( A J,) of the two unquantized components of angular momentum. This uncertainty is minimum for the highest magnitude K values which are K = f J. In this sense a state 1;) (or I-“,)) is a minimum uncertainty state. The symmetric top Hamiltonian can be written as a function of the polar angle -p = 0 using (2.6). H(O)=
lJ12[(2B+C)+(C-B)(3~~~20-l)]/3.
Substituting quantum values (2.18) and (2.19) for magnitude and angle of J yields the correct eigenvalues
H(@;)=BJ(J+l)+(C-B)K’.
(2.20)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
335
This implies that the quantizing trajectories on the symmetric top RE surfaces in figs. 2.la and c are precisely the intersections of the angular momentum cones with the RE surface for J = 10 and K= 10, 9, 8 ,..., - 10. This provides a graphic relation between the angular momentum uncertainty and the semiclasscial precessional paths of the RE surface. The cone quantization will be a useful approximation in the neighborhood of maxima or minima of any RE surface which is locally nearly symmetric. The symmetric neighborhood is the region where the relative contributions of off-diagonal tensor terms TJ with 4 not equal to zero are small compared to those of the diagonal terms TL. An asymptotic expression for the Wigner-Eckart coupling coefficient in the (4 = 0)-tensor metrix element (2.7a) yields a Wigner D-function or a Legendre function of the cone angle
(;IT$) =D&(.@;.)) 1Jl’=P,(cos
@;) 1JI’
(Jx=- 1)).
(2.21)
Hence, the diagonal or isotropic part of the matrix is asymptotically equal to the RE surface altitude at the cone intersection. This is true since the RE surface is defined by replacing each multipole operator T; by its corresponding multiple function, that is, Ti + II,& ( - ,p, y) 1J 1r, according to (2.4), and only the (4 = O)-functions are nonzero in the neighborhood of p = 0. This approximation is better for higher rank (r > 2) tensors as we will see in the next section. For the asymmetric top in figs. 2.lb and 2.2 the majority of paths are “squeezed” by the anisotropy arising from the off-diagonal tensors Ti2 which take effect quickly. An angular momentum cone with 0: = 45 O which just reaches the separatrix corresponds to the following K-value K= \I[lool
cos 45” = 7.4.
From this one might conclude that only states with K = 10, 9, 8, and (with difficulty) 7, could fit in the available phase space between the separatrix circles. However, we have neglected the squeezing which makes it possible for paths labeled by K = 6 and 7 to fit easily in the quadrant of the RE surface shown in fig. 2.2. Of course, squeezing ruins the strict conservation of K and off-diagonal tensors T: z mix in states with K-values that differ by multiples of two.
(b) Semiclassical path integration For highly squeezed RE surface trajectories it is necessary to use more general forms of quantization. The Bohr-Sommerfeld action integral quantization formula is useful
#
p*dq=nh
(n=0,1,2
,... ).
For free rotors the three Euler angles and their conjugate momenta are the 4’s and p’s of the action integral (2.22)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
336
However, for a free rotor, the Euler angle cx is an ignorable coordinate and J, = ALAB is a constant integer which can be left out. Furthermore, for J-paths with bilateral xy symmetry the /3-integral will vanish identically. This leaves only a one-dimensional quantization condition involving the body z-component J, = JzBoDY 1 2TJYdy=K %e J
(unitsof
ti).
(2.23)
For the asymmetric top J, can be expressed in terms of H = E by solving (2.la) using (2.5) J, = Jy (z-axis)
(2.24a)
=
For paths that encircle the x-axis a change of the quantization coordinate is equivalent to simply exchanging A and C J, = J., (x-axis)
=
//*
(2.24b)
The integrals (2.24) are evaluated numerically for a sequence of energies E which converge onto a solution to the quantization condition (2.23) for each desired integer K. The resulting semiclassical E-values are accurate to between three and four significant figures for J = 10 and between four and five figures for J = 20 (see table 2.1). A slight modification of the path integral can be used to estimate tunneling rates which account for the superfine splittings in fig. 2.2. The necessary integral for the (K(z) = lO)-levels, for example, is along the dashed circle that connects the (K= lO)-path to the (K= - lO)-path and crosses the saddle point at the y-axis. The integral is evaluated between the closest approach points y_ and y + of the two z-paths at the correct energy value for the (K = lo)-level. Since the integral is around the x-axis the x-expression (2.24b) is used in the integral and limits are the points where J, = 0, i.e., the points where the z-path crosses the yz-plane. Between these limits J, is imaginary and the e-integral below is real.
e=i
‘+dyJ,=i
JY-
IJ12(Ccos2y+B
sin2y)-I&,,,,,
C cos2y + B sin’y - A
dv.
(2.25)
The tunneling amplitude is given by a standard semiclassical approximation S = (l/T)
e-*
(2.26)
where l/T is the classical precessional or “attempt” frequency on the z-path. The tunneling amplitude must be doubled since the two dashed tunneling paths connecting the z-paths in fig. 2.2. The tunneling matrix
(W=(& ‘E”)
(2.27)
W. G. Harter / Graphical approaches to molecular rotations and vibrations Table 2.1. Semi-classical Qua&ration
for Asymmetric Top (in cm-‘)
EK (QM)
J%(SC)
9 8 7 6
63.181 57.851 53.147 49.113 45.833
19 18 17 16 15 14 13 12 11
246.352 235.360 224.977 215.214 206.083 197.603 189.801 182.721 176.448 171.132
337
A = 0.2, B = 0.4, C = 0.6
SK(QM)
Si4C)
63.176 57.842 53.133 49.084 45.781
2.16x 1O-7 1.93 x 10-s 6.98 x 1O-4 1.27x10-* 1.11x10-’
1.96~10-~ 1.87 x 1O-5 6.91 x 1O-4 1.29x lo-* 1.20x10-’
246.347 235.354 224.970 215.205 206.071 197.586 189.777 182.684 176.382 171.029
2.84x lo-‘* 2.55 x10-” 1.42x10-* 5.33 x lo-’ 1.41 x1o-5 2.71 x 1O-4 3.76x1O-3 3.64x10-* 2.18 x10-’
1.25 x 1014 2.66x10-l2 2.49x10-” 1.40x 1o-8 5.29x lo-’ 1.41 x 1o-5 2.72~10-~ 3.80~10-~ 3.75 x lo-* 2.38x10-l
J=lO K=lO
J=20 K=20
QM - Quantum mechanical diagonalization. SC - Semi-classical treatment.
gives A and B eigensolutions EA=E+2S, EB=E-2S
(2.28)
which have a total superfine splitting of 4s. The semiclassical superfine splittings calculated using (2.25) thru (2.28) vary over six to twelve orders of magnitude yet are correct to one or two figures of accuracy for J = 10 or J = 20 (see table 2.1). (c) Wavepacket propagation and quantization Another method of semiclassical approximation yields approximate eigenfunctions as well as eigenvalues. This method is an adaptation to the rotational problem [33] of a vibrational quantization method developed by De Leon and Heller [34] which is described in this volume [l]. The wavepacket quantization methods often use a minimum uncertainty state ]MU(q, p)) or wavefunction ( ]MU( q, p)) which is localized around a certain point (q, p) in classical phase space. In the original formulation of this problem for vibrational mechanics, this state was chosen to be a (q, p)-translation of an oscillator ground state. In this case the corresponding wavefunction ( 1MU( q, p)) is a translated complex gaussian packet with minimum value (h) for the uncertainty (Aq)( Ap) for all degrees of freedom. The method uses classical equations of motion to derive the time evolution of the phase variables (p(t), q(t)). It starts with a judicious choice of initial values (p(O), q(0)) which have a
338
W.G. Harter / Graphical approaches to molecular rotations and vibrations
classical energy reasonably close to the eigenvalues that are desired. Then a path propagation state ) q(t))
= )MU(q(t),
p(t)))
e(i/h)sp(t)
(2.29)
is obtained by numerically integrating the classical equations to obtain the phase variables (q(t), and the classical action or Hamilton’s Principle Function S,.
p(t))
(2.30) Generally, one defines a classical translation operator CT which has the effect of changing the (q, p) parameters of the wavepacket to the correct classical values for each time t IMU(
p(t))>
= CT(q(t),
p(t):
q(O),
P(O))
IMU(
P(O)))-
(2.31)
The same CT operator can be used to set the initial wavepacket, as well. The initial wavepacket expectation value of this operator, including the action phase, is called the time autocorrelation function.
AC(t) =
PfU(dO)7P(O))
ICTkdt),
p(t): q(O),P(O)) IMU(q&%P(O))) e(“h)Sp(r) (2.32)
= ( W) I W)) *
This function oscillates or “beats” each time the evolving wavepacket returns on a classical path that is close enough to overlap with the initial packet. The energy Fourier transform of the autocorrelation FTAC( E) provides information which can be used to generate eigenfunctions. The values E = E, which are peaks of the following transform function FTAC( E) = &
can be used to obtain an approximate propagation state. 1EFT)
= &
/:dt
(2.33)
/TTdt e(‘lhjE’AC( t) semiclassical eigenfunction
e(i/h)ERr I ‘k( t ))
by a Fourier integral of the
.
Finally, the desired semiclassical energy is computed by an expectation Hamiltonian.
Esc=(E,lHIE,)/(E,lE,). This last step is necessary since, as we will see, the FTAC peaks {E,, slightly from the desired quantum energy values { Es,-, Ei,, . . . } .
(2.34) value of the quantum (2.35) EfT,. . . } may differ
W. G. Harter / Graphical approaches to molecular rotations and vibrations
339
One way to understand the wavepacket method is to study the conditions which would cause (2.33) to have a peak or to make (2.34) keep accumulating a stationary state over longer and longer integration times. Such a peak or accumulation implies that E has been chosen so as to obtain a phase coherence or constructive interference each time the evolving wavepacket returns to the neighborhood of the initial packet. This occurs when the action for a closed path satisfies (EBK) certain quantization conditions. In general, these are the Einstein-Brillouin-Keller quantization conditions applied to Hamilton’s Characteristic Function S, or “reduced action” S,=S,+Et=
/
(2.36)
p*dq.
For a closed path the EBK quantization
conditions are
p dq = (n + a)2nA LfJ l
(2.37)
where ar is called the Maslov constant. In terms of the classical action this gives Esc7 + S, = (n + a)2llft,
(2.38)
where 7 is a classical period for the closed path and Es, is the desired semiclassical approximation to the energy. This may be written EFr7 + SP = (n)2nA, where E, E,
(2.39)
is a peak value in the FTAC (2.33) given by = Es, + 2aaA/7.
(2.40)
If the Maslov constant is not zero, then the true value Es, is shifted. From (2.32) one sees that condition (2.39) makes the phase in (2.33) or (2.34) come out to a multiple of 2ai after classical period 7. Hence the wavepacket method amounts to a numerical resonance or spectroscopy experiment in which one finds stationary states by “tuning” E to get a peak in (2.33) and then “painting” the eigenfunction using (2.34). The resulting eigenfunction consists of a sum of a series of MU packets set out on a classical trajectory. As explained in refs. [33] and [34] the last step can be accomplished with relatively small number of time steps. It is also helpful to “tune” the initial conditions (q(O), p(0)) in (2.31) to get the strongest peaks. The rotational adaptation of wavepacket propagation involves a different type of minimum uncertainty state, phase variables, and classical translation operator. The latter is a rotation operator which affects the Euler angles { (upy }, and these angles are the rotational equivalent of phase variables. Recall that the angles -p, and -y are the RE surface coordinates. The wavepacket propagation can be viewed as taking place on the RE surface shown in fig. 2.lb.
340
W.G. Harter / Graphical approaches to molecular rotations and vibrations
A form for the rotational wavepacket is suggested by the discussion of angular momentum cones in section 2.4a. A choice for a rotational minimum uncertainty wavepacket is the wavefunction associated with the narrowest angular momentum cone, that is,
IMU(0,090))= 1;).
(2.41)
A good choice for an initial wavepacket is one centered on some part of a quantizing RE surface path. Assuming that the point (/3 = 0, y = 0) is the center of an RE surface hill or valley, then the Kth quantizing path would be near to the point { & = - Oi, y0 = 0). A suitable initial wavepacket is therefore obtained by the following y-rotation of initial state (2.41) ]MU(O, -O;,
0)) =R(O,
-O;,
(2.42)
O)];).
(Negative Euler angles are used since the rotation is defined with respect to the body fixed frame.) The choice of initial angles is usually not so critical, but a good initial guess can reduce computational time somewhat. The classically propagating wavepacket state (2.29) has the following form for the rotational problem. (2.43)
The Euler angles and the action are obtained Hamiltonian’s equations. . aH a=a~,,
.
aH
Jzi= --as,
a=a, P,
by numerical
Runge-Kutta
[35] solutions
of
(2.44a)
Y,
and (244b) Detailed examples of these equations are given in appendix C. The initial conditions (a(O), /3(O), y(O)) given by eq. (2.42) may be used along with S,(O) = 0 for the action. The autocorrelation function is given in terms of the initial and final Euler angles AC(t) = V(O) I W)) = (.$W#),
P(O), y(WWt)9
IW),
Y(NIJ,)
exp[(Vh)S,(t)].
(2.45a)
The rotation product is reduced and represented by a Wigner D-function AC(t) = (J”I+,,
&
Y,$)
ew[(i/h&(t)]
= S(~,,
BP9
Y,> exp[(i/h)~,(t)].
(2.45b)
341
W. G. Harter / Graphical approaches to molecular rotations and vibrations 60 000
A- 020 a*0.00
8- 0.40 &--30.00
c-
WO E,,,-
0 80 To’ 0.00
50.000
A-
0.20 0.00
B= 040 /3,--30.00
CP 0 60 ,p 0.00
Time
Fig. 2.5, Wave packet autocorrelation function and Fourier transform for rigid asymmetric top (a) Function shows beats (b) Function FAC( w) has peaks which correspond to eigensolutions.
AC(t)
The group product rule and the D-function formula are given in appendix A. An example of a rotor autocorrelation function and its Fourier transform (2.33) are plotted in figs. 2.5a and b, respectively. The beats due to the returning wavepacket are shown in fig. 2.5a. A peak in the transform at energy E,(K= 8) has been singled out in fig. 2Sb. Each of the larger peaks E,(K) indicates a possible eigensolution. According to eqs. (2.34) and (2.43) the following state is an approximate eigenstate associated with that peak
IJ%rW) = >ir
$/_rrdf
ex~[(i/h)(%r(K)
+ s,(t))]R(+),
B(t),
A(t)) 1;).
(2.46)
The representation of this state in the angular momentum basis reduces to a Fourier transform of D-functions
(2.47) It can be shown that the coefficients (2.47) are always real [33]. The coefficients chosen in fig. 4.5b are the following =
=
10, 0.118,
8, 0.929,
6, - 0.339,
4, 0.075,
2, -0.012,
for the example
0 0.001,
1. . ...
(2.48)
342
W. G. Harter / Graphical approaches to molecular rotations and vibrations
These are to be compared with the exact values for a localized state of the form (2.15) which are the following (ll& --0.129,
0.922,
-0.347,
Finally, the semiclassical according to eq. (2.35). &c(K=
8) = G%(8)
0.106,
-0.062,
0.018 ,....
energy value for the wavepacket
IH I ~%r(8))/(Gr(8)
I E&8))
generated
= 53.134.
(2.49) state (2.48) is given
(2.50)
This to be compared with the quantum mechanical values of 53.149 and 53.146 for the (K= 8)cluster. The accuracy is comparable to that of the elementary path integration result in table 2.1. This completes an example of the most elementary “bare bones” wavepacket propagation technique. More sophisticated treatments [33] give more accurate wave functions and energy values by taking tunneling into account.
3. RE surfaces for tensor Hamiltonians
of semi-rigid rotors
Some examples of generalizations of the RE surface techniques to semirigid rotors of higher symmetry will be considered in this section. Semirigid rotors are deformable bodies whose distortion energy is readily described by simple analytic tensor Hamiltonians. A brief discussion of the derivation of the tensors, their limitations, and some possible alternatives will be given in sections 5 and 6. Much of the development of the RE surfaces for semirigid rotors follows the development given for the rigid rotors in section 2. Analogies between the rigid and semirigid rotors will be emphasized as will be the points where they differ. 3.1. Fourth-rank octahedral tensor Hamiltonians The simplest non-trivial rotor Hamiltonians having tetrahedral (Td), octahedral (0,) or cubic symmetry is the fourth rank Hecht Hamiltonian [16]. This may be expressed in polynomial form
H = Hscalar+ lot,,
[ J,4 +
J; + J; - (3/5) J"] .
(3.la)
or in tensorial form (3.lb) where the scalar part contains the undistorted spherical top rotor kinetic energy B 1J ) 2 and a fourth order correction D 1J 14 due to isotropic distortion.
H scalar = BJ2 + DJ4.
(3.lc)
The detailed dynamics and spectroscopy of this type of rotor depends upon the anisotropic distortion effects which are described by tensor parts of Hamiltonian such as the t,,&erm in the
W. G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 3.1. Octahedrally
symmetric RE surfaces for Hecht Hamiltonian (eq. (3.2)) with H_,, (b) tw 1J4 1= - 1.0.
343
= 1.0, (a) tw 1J4 1= 1.0;
Hamiltonian (3_la, b). To help visualize these effects the tensor part of (3.1) is converted to an RE surface function using the substitution (2.4) which was quoted in the discussion of rigid tops. The RE surface function follows from (2.4) and (3.lb). (3.2a)
=
+tC14M4[(35 Hscalar
c0s4p
-
30 cos3p + 3) + (5 sin”p cos 4y)].
(3.2b)
This RE surface is drawn in fig. 3.1 for arbitrarily chosen values of HSCdar= 1.0 and 4t,, 1J I 4 = 1.0 in fig. 3.la. The form of the RE surface levels and the corresponding eigenfunctions are independent of the values chosen for either HSCdaror I,,.+,.The eigenvalue fine structure patterns belonging to each J value will be shifted by adjusting HScalar and expanded uniformly by increasing t,, . A positive value 4t 044) J 14 = 1.3 is used in fig. 3.la, while negative value 4t,, = - 1.0 is used to give fig. 3.lb. The positive value corresponds to an octahedral XY, molecule which is more susceptable to bending its radial bonds than stretching them. One could imagine the x, y, and z body axes to lie along the X-Y equilibrium radial bonds of an XY, molecule. The axes are centered on the six peaks of the RE surface in fig. 3.la (or in the corresponding six valleys in fig. 3.lb). The second term in (3.2b) gives an RE surface height of 4.0 units on the z-axis (/I = 0). These are units of t,, I J I 4 above Hdar. The rotational energy is the following. RE(0, y) = HSdar + t,,
I J ( 4(4.0).
(3.3)
The third term in (3.2b) does not contribute in the neighborhood of the z-axis (/3 = 0 mod IT), but it is mostly responsible for the four peaks along the x and y axes (@ = + n/2) in fig. 3.la.
344
W.G. Harter / Graphical approaches to molecular rotations and oibratrons
04
A,
I
A,
.
I
E T,
I I
I 2
T2j2
14
24
34
.
’
Fig. 3.2. Different choices of rotation axes for octahedral rotor corresponding to local symmetry (a) C,, (b) C,, and (c) C,. Tables correlate global octahedral symmetry species with the local ones.
The locations of topography lines indicate that all six peaks are equivalent so the surface has a global Oh symmetry. The local symmetry of each peak is indicated by the shapes of the topography curves. Near the tip the curves have a nearly circular (R,) symmetry while further down they change into a square or C, symmetry near the separatrix. Beyond the separatrix the curves assume a triangular or C, symmetry which becomes more nearly circular again at the valley bottoms. Trajectories near the C, valley bottoms of fig. 3.la correspond to an XY, molecular rotation that is similar to that which is shown in fig. 3.2a. On the other hand trajectories near the C, peaks correspond to an XY, rotation pictured in fig. 3.2~. In the latter (C,) case centrifugal force would have a tendency to stretch the spinning bonds while in the former (C,) case it would bend the bonds. If bending is easier than stretching then the motion which causes bending will have lower energy for a given magnitude ( J 1 of the angular momentum. Hence there will be energy valleys for the XY, RE surface on the C, symmetry axes, and the C, symmetry axes will be on hills as shown in fig. 3.la. This would correspond to positive to&. For tetrahedral (XY,) on cubic (XY, or X,Y,) structure this constant may be negative. Then the RE surface has a cubical shape as shown in fig. 3.lb. As for the rigid tops it often happens that the RE surface is shaped something like the molecules that it represents. The topography lines in fig. 3.1 represent possible classical trajectories of the lab-fixed-J-vector or lab z-axis. This is like the rigid top surfaces in fig. 2.1 with one very important difference: The XY, molecule is constantly changing shape as the J vector precesses around any of the topography lines in fig. 3.1. Each point on this RE surface corresponds to definite amounts of the vibrational coordinate distortions induced by the angular momentum J at that point. The effective Hamiltonian (3.1) is a useful description only if the rotational distortion follows the precessional motion of J udiabatically. That is, only “virtual” mode polarizations are allowed; a significant amount of “real” or resonant mode excitation by rotation signals a departure from the simple RE surface.
W.G. Harter / Graphical approaches to molecular rotations and vibrations
345
Note that these adiabatic requirements apply to the effective tensor Hamiltonian. The RE surfaces in fig. 3.1 are merely ways to visualize effective Hamiltonians and to clarify the adiabatic requirements. An analogous set of adiabatic requirements exist in order that a potential energy (PE) surface be a useful description of vibrational motion in the Born-Oppenheimer approximation. In that case the electronic orbital distortion must adiabatically follow the vibrational motions without undergoing real or resonant transitions. Ultimately, it is hoped that the RE surface can be used to carry the Born-Oppenheimer approximation to the next level of detail and treat rotational motion which is adiabatically driven by vibrational and electronic (or vibronic) distortions. This may involve RE surfaces obtained directly from generic Hamiltonians as discussed in section 5 and multiple RE surfaces which are introduced in section 6. We consider now how the RE surface mechanics is related to the eigensolutions of the fourth rank tensor Hamiltonian of the SF, molecule. The RE surface levels in fig. 3.la are the appropriate quantizing curves for angular momentum quantum number J = 30. (fig. 3.lb shows levels for J = 36.) The lowest levels in the C, valleys and the two or three highest levels on the C, hills are very close to the intersections of the RE surface with angular momentum cones. The angles of these cones are found using (2.19). They are given for J = 30 and a range of K-values as follows. K
=
9;
=
30,
29,
28,
10.3”,
18.0”
23.3”,
27, 27.7”,
26, 31.5”,
25, 34.9”,
...
...
(3.4)
Simple geometrical analysis shows that the separatrix in fig. 3.1 approaches to within 35.3” for the C, axes and to within 19.5O of the C, axes. Using this along with (3.4) one can estimate which K-states will be bound by each region. The C, valley regions can only hold K = 30 and 29, while the C, hills can hold K = 30 thru 25. This prediction is verified by the exact level spectrum for J = 30 which is displayed in fig. 3.3. This is drawn using the same format as the asymmetric top spectrum in fig. 2.2. The six magnifying circles on the righthand side show clusters of levels associated with K-values of 30 through 25 while the two circles on the lower lefthand show the clusters associated with K-values of 30 and 29. Furthermore, the cubic symmetry species contained with in each of these circles can be predicted using the selected columns of the correlation tables in fig. 3.2. The use of these tables is the same as for the rigid top tables in fig. 2.4. For example, the lowest cluster of levels in the C, valley is labeled by K, = 30 and has local symmetry 0 mod 3 or 0,. The O3column of the C, table predicts that one each of the A,, A,, T,, and T2 species would appear in the cluster. The K = 29 = 2 mod 3 cluster contains one each of T,, E, and T2 species which are exactly the ones listed in the 2, column. Note that A, E, and T levels are one, two, and three degenerate levels, respectively. Thus the (AITrTzAZ) and (TrET,) clusters each contain eight levels and this corresponds to having eight equivalent classical trajectories for each K-value. The clusters above the separatrix are each associated with six trajectories having C, local symmetry. The trajectories belonging to K4 = 28 are indicated by arrows which show the direction of precession in fig. 3.3 Since this K-value is 0 mod 4 it is correlated with the 0, column of the C, correlation table in fig. 3.2, and the species E, Tr and A,, account for the six quantum levels in this cluster. The other C, clusters also contain six levels each but with different species. Odd-K, clusters contain Tr and T,, while A,, T,, and E show up for K = 2 mod 4.
346
W G. Harter / Graphical approaches to molecular rotations and vibrations
VISUALIZING THE J = 30 LEVELS
OF A
SPHERICAL
c, Clusters -4)
,-31
,
-21
TOP
C4 Clusters
i :
_,I
,
7 I
,
II
, Relative
12
,
Units
13
of
,
(4
,
(5
10‘4cmS’=3.0MHz
84.77319
Fig. 3.3. J = 30 Octahedral rotor levels and related RE surfaces paths (after ref. [3]).
The six-hold and eight-fold near-degeneracies of the C, and C, clusters are analogous to the C,(x) and C,(y) clusters of the rigid asymmetric top as can be seen by comparing figs. 3.2 and 2.4. We noted that asymmetric top clusters could be traced back in fig. 2.3 to the K-doublets associated with the irreducible representatives of the symmetric top symmetry R,. However, this connection can be misleading. First, there is no obvious symmetry like R, which has six or eight dimensional irreducible representations. Second, the cluster degeneracies are not perfect but have exponentially small superfine splittings that are a function of a tunneling amplitude between semiclassical trajectories. This tunneling amplitude depends upon the RE surface geometry in the
W. G. Harter / Graphical approaches to molecular rotations and vibrations
saddle point regions of the separatrix and not necessarily K-conservation of a particular trajectory.
347
on the degree of symmetry
or
3.2. Symmetry properties of eigensolutions A more detailed picture of the eigensolutions on the RE surfaces in fig. 3.1 emerges if one takes account of the geometry of the tunneling mechanisms. An example involving the superfine level structure of the 0, type clusters should demonstrate this point. For an O&pe cluster such as the (K = 28)-levels on the right hand side of fig, 3.3 one can imagine six equivalent angular momentum cones touching the RE surface at angle of approximately 82: = 23.3” near the classical trajectories. Centered on each of these one could imagine a wavepacket consisting mostly of a symmetric top wavefunction with M = J = 30
(a, P Y(;t) =~%**(% P Y) defined with responect to the local axis of quantization. These six states may serve as a basis { II>, 12)Y.V 16)) of a tunneling matrix of the following form.
m =
H
0
0
H
-S
-H
-S -S -S .-S
-S -S -S -S
-S -S H
-S -S 0 0 H
H 0 -S
-S
-S -S
-S -S -S -S -S H
0
-S -S -S -S -S 0 H
11) 12) 13) 14)
(3.5)
I? 16)
Here the base states 11) and 12) are associated with the wavepackets on the +z and -z axes, respectively, while 13) and 14) are similarly related to +X axes, and 15) and 16) go with the +y axes. A tunneling amplitude - S is assumed to exist between nearest neighboring trajectories, i.e., from +z to +X or to + y but not from + z to -z. Eigenvectors of this are easily obtained analytically. The totally symmetric eigenvector is
(A,) = (1, 1,1,1,1,1>/6
(3.6a)
with eigenvalue ,?(A,) = H - 4s. Three antisymmetric
(3.6b)
eigenvectors
ITIn)= ((4% 1, -1, 0,0)/d% lT,y) = (0, CO, Cl, IT,z) =
-1)/d%
(1, -1, O,O, 0,
0)/d%
(3.7a)
348
W.G. Harter / Graphical approaches to molecular rotations and vibrations
each share the triply degenerate eigenvalue E(T,)
= H.
(3.7b)
Finally, two eigenvectors ]El) = (l,l,
-l/2,
]E2) = (0, 0, l/2,
-l/2, -l/2,
-l/2, l/2,
-l/2)/&
-l/2),
(3.8a)
each share the doubly degenerate eigenvalue E(E)=H+2S.
(3.8b)
These eigenvalues explain the ordering and spacing that is observed repeatedly for each ETA cluster in the magnifying circles of fig. 3.3. A similar analysis can be done for all superfine structures in that figure [8]. A wavefunction of type-O, localized onto a single trajectory of, say, the +z-state 11) would involve a non-stationary combination of symmetry-defined states analogous to the rigid top states (2.15) or (2.16). Inverting (3.6a) thru (3.7a) gives the following combination at time t = 0 1q(o))
= 11) =
k
IA,) + +
IT,4 + + I’%.
(3.9)
At a later time the state would evolve into a delocalized combination e-i(H-4S)t
IW))
=
fi
e-i(H)t
IAd + - fi
of the following form
e-i(H+2S)r
IW
+
0
El).
(3.10)
However, this combination would relocalize perfectly again and again. Perfect recurrance occurs with an angular frequency of w = 2s since the eigenvalue (3.6b) through (3.7b) are integral multiples of this values plus a constant H. The values of tunneling amplitudes S decrease quasi-exponentially as clusters move away from the separatrix. In the case of the SF, rotor levels with J = 30 shown in fig. 3.3, the total superfine splitting varies between 4.8 Hz on the highest ATE cluster to about 2.0 MHz near the separatrix. For higher J the superfine structure splittings cover an even larger spectroscopic range. We consider now numerical approximations for fine and superfine levels. 3.3. Semiclassical quantization The semiclassical techniques which were used to approximate the asymmetric top spectrum can also be applied to higher rank tensor eigensolutions. Here we discuss the angular momentum cone and action path integral approximations to the fourth rank tensor solutions. The maximum points on the RE surface in fig. 3.la are the [lOO] directions of the coordinate axes. Substituting J= (0, 0, I J I) into (3.la) or Engler angles (p = 0, y = 0) into (3.2b) gives E ,,_=E(O,
0) =4( J14
(3.11)
349
W. G. Harter / Graphical approaches to molecular rotations and vibrations
Table 3.1. Comparison of exact cluster centroid energies and tunneling amplitudes with classical RE surface and semiclassical approximations. (Units of [ J( f + l)j2 for tw = 1) (a) J= 30
E(exact diag.)
E(RES-cone)
E(Semi-class)
S(exact)
S(SC)
K=30 k = 29 kc=28 K=27 K=26 K=25
3.3829967 2.256160 1.302880 0.51132 -0.1272 - 0.616
3.373 2.246 1.292 0.495 -0.157 - 0.677
3.371 2.249 1.293 0.503 - 0.140 - 0.637
1.68E-07 6.083X- 06 9.758- 05 9.05E-04 .5.30E-03 1.49E - 02
1.59E-07 6.10E-06 9.94E - 05 9.378-04 5.51E-03 204E - 02
(b) J= 88
E(exact diag.)
E(RES-cone)
E(Serni-class)
S(exact)
S(SC)
1.55E-21 1.95E- 19 l.l8E- 17 4.60E - 16 1.29E - 14 2.80E- 13 4.86E- 12 6.93E- 11 8.26E - 10 8.36E - 09 7.22E - 08 5.38E-07 3.47E - 06 1.93E-05 9.348-05 3.80E - 04 1.29E-03
1.46E-21 1.91E- 19 1.17E - 17 4.59E- 16 1.29E- 14 2.81E- 13 4.89E - 12 6.98E - 11 8.348- 10 8.45E-09 7.32E-08 5.47E - 07 3.53E-06 1.98E-05 9.52E - 05 3.97E-04 1.40E-03
K=88
K=87 K=86 K=85 K=84 K=83 K=82 K=81 K=80 K=79 K=78 K=77 K=76 K=75 K=74 K=73 K=72
-
3.778691053 3.349362373 2.942069530 2.556379795 2.191699743 1.847493649 1.523240328 1.218437672 0.932~5088 0.665288448 0.416068852 0.18456690 0.0295167 0.226355 0.~6~7 0.56832 0.7104
-
3.777 3.348 2.941 2.555 2.190 1.846 1.521 1.215 0.928 0.659 0.408 0.174 0.044 0.246 0.433 0.604 0.762
-
3.778 3.348 2.940 2.555 2.190 1.846 1.522 1.217 0.932 0.664 0.415 0.183 0.031 0.228 0.408 0.571 0.717
where we set t,, = 1 and H,,, = 0. The ~~rnurn points are the [ill] directions. Setting J= (l/0)( 1J 1, 1J f, 1J 1) or (/3 = 54.7*, y = 4S”) gives the following energy for the lowest points on the fourth rank RE surface. =E(54.7”, enin
45’)
= -(8/3)
1J14=
-(2/3)E_.
(3.12)
The saddle points occur at the [llO] directions where J = (I/a)( I J I, 1J 1, 0) or (/3 = 90 O, y = 45 * ) and the energy is
E sadd,e=E(900,450)=
-lJ14=
-(l/4)&_.
(3.13)
This is the separatrix energy with divides the fine structure interval E,, - E_ into two parts. The upper three-quarters of the region is devoted to C, levels while the lower quarter of the fine structure pattern is devoted to C, levels. By substituting the J = 30 cone angles p = 8;’ from (3.4) into the RE expression (3.2b) with y = 0 one obtains the approximate values shown in the second column of table 3.la. These agree within a few percent with the exact values for the C, level clusters in units of t,, 1J I 4 with Hscalar= 0. Note that the highest cluster value 3.38 is well
350
W.G. Harter / Graphical approaches to molecular rotations and vibrations
below the classical maximum (E,, = 4.0). The difference is the rotational zero point energy associated with the angular momentum cone uncertainty angle. This zero point energy decreases as J increases. The highest value of the J = 88 level clusters is 3.78 as seen in table 3.lb. This is closer to the classical maximum. Also, the RE surface-cone approximation is more accurate for higher J. Semiclassical path integration formula (2.23) applies to the fourth rank RE surface paths, as well, and the numerical results are quoted for J = 30 and J = 88 in the third column of table 3.la, b. Some improvement in accuracy over the simple cone approximation is obtained for lower K-values. However, the main strength of the action path integration is the ability to estimate the tunneling amplitude or S-values in the superfine pattern formulas like (3.6b) thru (3.7b). The procedure used to give the semiclassical results in the S(SC) column of table 3.1 and analogous to the rigid top formulation in (2.25) thru (2.28). Details are given in ref. [3]. 3.4. Mixed-rank
octahedral tensor levels
So far we have considered only the lowest order rotational tensors which exhibit the symmetries D, of the rigid rotor and 0, of the semi-rigid cubic or octahedral rotor. We consider now the effect of the sixth-rank normalized octahedral tensor operator
PI=
(l/fi)[g-
(d7/Jz)(T46+P4)].
(3.14)
This will be added in varying amounts to the normalized fourth rank tensor (3.15) contained in (3.1). A sixth rank centrifugal distortion may be necessary in the presence of anharmonic and other higher order effects. The magnitude of the Tt6] contribution would vary according to a higher power of J than that of Tt4] and might be significant at higher J values. Here the magnitudes of their respective contributions are varied artificially through an angle parameter v in a combination which maintains the overall normalization T4,6( v) = Tt41 cos v + PI
(3.16)
sin v.
The exact quantum (J = 30)-eigenvalues for this mixed [4,6]-rank tensor operator are plotted as a function of the mixing angle v in fig. 3.4. The plot begins on the lefthand side (v = 0) with a scaled copy of the T [41level spectrum in fig. 3.3 and ends on the righthand side (v = V) with the same spectrum inverted. Between these limits the level clusters become completely reorganized [36,37]. Certain values of the v parameter in fig. 3.4 are marked (b), (c), (d), and (e). At these values the RE surface of the combination tensor (3.16) is drawn in fig. 3.5. The RE surface function used for Tt4] is as follows Et”(/3, y) = (7/12)“2(9/4+2(35
cos4b - 30 cos2p + 3 + 5 sin4p cos 4y)/8.
(3.17)
W.G. Harter / Graphical approaches to molecular rotations and vibrations
(4
09 00
10
(4 ..,“t,”
20
v III
40
351
(4 JO
60
of 7r/6
Fig. 3.4. J = 30 eigenvalues of varying mixtures of 4th and 6th rank tensors. (Y = 0) corresponds to levels in fig. 3.3.
For
TL6] the RE function is as follows
I?(p,
y) = (1/8)“2(13/4+2(231
co@
- 315 cos4/3
+105 cos3p - 5 - 21 sin4p(11 cos2p - 1) cos 4y)/16. The tensors
(3.18)
T[‘] and RE functions E [rl have a spherical harmonic normalization factor ([2r + 11/4a) ‘I2 that was not included in previous definitions (2.4) or (3.2). This factor is used here to slightly enhance the effect of the 6th rank tensor for this particular example. Also, the 1J 1r factors are deleted in (3.17) and (3.18) so that the higher rank tensor effects are not J dependent. The eigenlevels marked by (b) in fig. 3.4 correspond to the RE surface drawn in fig. 3.5b. The later shows that the separatrix has taken over the regions that formerly held C, symmetric trajectories, and only C, trajectories remain. (Note that these are equally spaced contours and are not quantized paths.) The result is the destruction of C, clusters in the spectrum which is composed almost entirely of C, clusters above the (b) point in fig. 3.4. Beyond this point a remarkable new type of cluster is formed. Just above the points marked (c) and (d) in fig. 3.4 lie two clusters which contain twelve levels each. These correspond to trajectories which encircle twelve equivalent valleys which lie on the C, symmetry axes in figs. 3.5~ and d. The symmetry species within each of these clusters are exactly the ones contained in the C, correlation table in the center of fig. 3.2. The lowest cluster in fig. 3.4 would correspond to K = 30 and hence to the even local symmetry or the 0, column of the table in fig. 3.2 which contains species A, E, T,, and 2T2. The next cluster has K = 29 and contains the five species A,, E, 2T2, and Tr listed in the odd column 1,. The superfine splittings between these five levels are
352
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 3.5. RE surfaces corresponding to selected v-values in fig. 3.4. (a) v = 0.0; (b) v = 0.4 (T/t); v = 4.0 (n/6); (e) v = 4.6 (q/6); (f) v = 5.0 (IT/~).
(c) v = 2.0 (~/6);
(d)
W. G. Harter / Graphical approaches to molecular rotations and vibrations
353
actually visible in the scale of fig. 3.4. As Y changes the levels as seem to change order within this cluster. This is result of competition between tunneling mechanisms. Between the (b) and (d) points in fig. 3.4 there is another phenomenon which occurs in the upper energy levels. There are a number of crossings or Fermi-like resonances between accidentally coinciding C, and C, clusters. This is because there are two kinds of mountains on the RE surfaces in figs. 3.5~ and d: the C, mountains which are shrinking and C, mountains which are growing with Y. For certain values of v quantizing paths on one type of mountain are bound bo te in resonance with different kinds of paths on the other. The result is an extraordinary kind of tunneling in which eigenfunctions are delocalized over both kinds of paths at once and a peculiar sort of hybrid superfine structure occurs in the eigenlevels. The spectral region containing the unusual fine structure is bounded on the righthand side by the (e) point in fig. 3.4 which corresponds to the RE surface in fig. 3.5e. At this point the eight C, mountains dominate the surface geometry entirely and the eigenlevels are composed entirely of very strong C, clusters of eight levels each. The final fig. 3.5f shows the situation at v = 5.0( ITS) where the C, trajectories begin to return. Now they are occupying the valleys. By v = 6.0( 76) the RE surface resembles the one shown in fig. 3.lb.
4. Quasi-spin RE surfaces for two-dimensional
vibrations
The following is a method for using RE surfaces to help visualize anharmonic vibrations [38]. The method is based upon an analogy between rotor motion, two-dimensional oscillation, and quantum evolution of a spin-l/2 or spinor system. This analogy has a long history, some of which will be mentioned in section 4.1. 4.1. Spinors, rotors, and coupled oscillators
(a) Spin and rotors The quantum mechanics of spin-l/2 operators is related to the algebra of the special unitary group SU(2) and to the three dimensional rotation group R, which is homomorphic to SU(2). The evolution of spin-l/2 or general two-level systems is described by the Rabi-Ramsey-Schwinger [39] or Feynman-Vernon-Hellwarth [40] formulations. These formulations treat the two-state Schrijdinger equation
@/at) I+> = H I SV
(for A = 1)
(4.1)
by interpreting the solution I +W>
= cHf
I WN
(for constant H)
as a rotation. If the representation
(4.2)
of (4.1) has the following form
(4.3)
354
W. G. Harter / Graphical approaches to molecular rotations and vibrations
then the Hamiltonian operators as follows.
matrix may be expanded in terms of Pauli spin angular momentum
+%4
+w$
32
(4.4)
The first component w. = (A + D)/2 is th e angular velocity of the overall phase. The remaining three components can be viewed as Cartesian components of an angular velocity vector w = (o,, u,,, wz) which can be written in terms of a set of polar coordinates { o, 8, +} as follows. w, = w cos t#~sin 9
wr = w sin $3 sin 8
= 2B,
o, = o cog e
= 2c,
(4.5)
=A-D.
The time evolution in eq. (4.2) is viewed as a rotation combined with an overall phase change. = e-i(wo-oeJ)t I*(O)).
1*k(t))
(4.6)
A similar three-dimensional vector description is given for the state 1\k) or for the two-level system or for the density operator which represents that state. Each state can be written as an Euler angle rotation of the first base state 11)
A spin-l/2
representation of this construction has the following form (see appendix A.) e-‘(” +y)/2cos( B/2)
(I ( *I
‘k2 =
e1(a-u)/2 sin( B/2)
- e-1(a-v)‘2
sin( B/2)
e’(* +7)/2 cos( B/2)
1 i!i 0
=
eeia12 Cos( B/2) i ela12 sin( B/2) I
e_iy,2 .
(4.8) The expectation value of the spin angular momentum vector J gives an expected spin vector S.
~x=Pl4l~) =+ cosasinp,
~y=(~lJ,lW =- i sinasinp,
~z=(~l4IW =- : cos P,
(4.9)
This spin vector can be viewed as the main axis of a rotor which will be acted upon by the evolution operator in eq. (4.6). The third Euler angle ( y) is the “ twist” of this rotor, and it is proportional to the overall phase of the spin state (4.8). The rotor rotates and twists around the w-vector given by eq. (4.5) as the spin state evolves according to eq. (4.6).
355
W. G. Harter / Graphical approaches to molecular rotations and vibrations
(b) Spinors and oscillators Spinor state evolution can also be related to that of a two Dimensions oscillator. The real and imaginary parts of the spinor amplitudes { JI,, ttz ) can viewed as oscillator phase space variables h P17 x27 P2 1. These in turn can be related to the Euler angle definition in eq. (4.8). G1 = x1 + ipr
(4.10a)
+,=x,+ipz
= JFe-‘(or+y)‘z cos( /3,/2),
= Jie-‘(y-“f/2
(4.10b)
sin( b/2).
Here an intensity variable I or normalization factor fi has been included so that the Euler angle definition will have a total of four variables which is the same number as the oscillator phase space dimension. We now consider the following coupled oscillator Hamiltonian H HO = (A/2)(p:
+x:>
+ (o/2)(
p; + x2’) + B(P,P,
+ Xl%)
+ C(XlP2
- X2PJ
(4.11)
This Hamiltonian yields classical equations of motion that are identical to the quantum equations that result from substituting eq. (4.10a) into the SchrGdinger equation (4.3). Either (4.3) or (4.11) yield the following . Xl
. x
=hK3
=-=Ap1fBp2-Cxz, 8P1 %iO
2 =-=BpI+Dpz+Cx,, 3P1
a&i0
-A
=
-+Ax,+BxZ+Cp2, ax, (4.12)
-j2
=
a%i0
-=BxI+Dx2-CpI_ 3x2
In general, an n-level quantum system has the same equations of motion as an n-dimensional classical harmonic oscillator. This idea goes back to the Planck and DeBroglie hypotheses at the beginning of quantum mechanics. However, it is only recently that this idea has been used as a visualization aid for molecular vibrations 1381. The idea of spinorial ch~acte~zation of two dimensional oscillators goes back even farther. Over a century ago, Stokes developed a theory of electromagnetic polarization ellipsometry [41]. In this development the complex electric field amplitudes E, and EY correspond to spinor amplitudes #i and #2, respectively. The spin expectation value 5’ corresponds to the Stoke’s vector which defines the state of elliptical polarization. In fig. 4.1. the three-dimensional Stokes vector space is shown on the righthand side. The Stokes “quasi-spin” vector components (S,, S’, S,) are labeled (B, C, A) and they characterize the shape of a polarization ellipse shown on the lefthand side of fig. 4.1.. The ellipse is a plot of the real parts x = x1 and y = x2 of the spinor amplitudes in eq. (4.10a) for a state of constant S. One characterization of the ellipse is its major to minor axis ratio b/a = tan JI and the inclination angle cp of the major axis. These two angles determine the orientation of the Stokes vector relative to the C-axis. The latitude and longitude of S relative to the C-axis are GM& these angles, that is 2lc/ and 2+, respectively. If the C-axis is chosen as the z = axis of quantization in the rotor-spin analogy, then the polar angles in eq. (4.9) would be ((Y = ~QYJ, p = 7~/2 - 2$). The C-axis would be a convenient choice if the oscillator was to be described in terms of left and right handed circular polarization states. In the molecular vibrational model
356
W.G. Harter / Graphical approaches to molecular rotations and vibrations
a
Fig. 4.1. Ellipsometric characterization of oscillator trajectories for spin-rotor analogy. (a) Two-dimensional spinor or SU(2) oscillator space; (b) three-dimensional vector or R(3) rotor space.
this would be an appropriate choice in the presence of a strong Coriolis term, that is, a large value of C in the Hamiltonian H,, in eq. (4.11). The choice of z-axis for the present description is the A-axis. This is a convenient choice if the oscillator is to be described in terms of x- and y-polarization states. In the molecular vibration model this would be an appropriate choice when ) x) and 1y) correspond to normal modes, and the value of (A - D) is large compared to B or C in H,,. The polar angles of S relative to the A-axis in fig. 4.1 are ((Y = 29, p = 2~). This choice corresponds to another way to characterize the polarization ellipse on the lefthand side of fig. 4.1. Here the ratio X/Y = tan v of maximum x and y amplitudes and the angle 9 determine the ellipse. The geometrical interpretation of the angle 19 in the two dimensional oscillator picture is tricky, and will be discussed shortly. However, the azimuthal angle 26 is easily visualized in the three-dimensional part of fig. 4.1. Also, the algebraic interpretation of 29 is simple; it is the relative phase of 1x) and 1y) in the following state 1‘k) = (X e-”
1x) + X e-”
1y)) = (cos v e-”
I x)
+ sin v eis 1y))
e-‘ofi.
(4.13)
The third Euler angle y = 20 is proportional to the overall phase as in eq. (4.8). Also, the amplitude factor J? is necessary since a classical oscillator can have arbitrary amplitude. The geometric interpretation of fi and I as the radius of a circle and a sphere, respectively, is shown in fig. 4.1. The expectation values for Pauli spin angular momentum operators in the state (4.13) can be expressed in terms of the spin-rotor parameters 9, v, and I, or else in terms of the oscillator phase space variables xi, x2, pi, and p2
~x=cwJ~k)
=xlxz+PlP2
=
(1/2) sin(2v) cos(29),
(4.14a)
S,=(‘k
IJYIW
=xlP,-x,P,
=
(1/2) sin(2v) sin(26 )9
(4.14b)
S,=(!P
MIW
=(P:+xx:
JO=(+
luwk)
=
(P:+
- p2’ x; +pz’+
x22)/2 = (1/2) cos(2v),
(4.14c)
x,2)/2 = (I/2).
(4.14d)
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Then the oscillator Hamiltonian quasi-spin vector S. H HO =
[CA+ D)/21JO+ ‘&
(4.11) can be expressed in terms of the components
+ cs,
357
of the
(4.15a)
=
woJo
(4.15b)
=
woJo
(4.1%)
This appears to have the form _ZQ,J, of an action angle Hamiltonian, neither the classical Poisson brackets such as (S,, S,,) nor the quantum commutators such as [J,, J,] will vanish. One may use Euler angle variables to obtain a legitimate action angle form as discussed in appendix B. However, the form of eqs. (4.15) is useful for comparing the angular rates of the oscillator models. It is seen that the angular rates w, of the three-dimensional spin vector or rotor are double the correspondence rates 1(2, of the two-dimensional oscillator. This doubling was also seen when comparing angles between the two parts of fig. 4.1. Roughly speaking, this is because spin “up” and “down” are only 90 o apart in the two-dimensional space but are 180 o apart in three dimensions. The angular doubling is an important part of the correspondence between the rotor and oscillator dynamics. Some examples which compare the two ways to visualize spinor or oscillator motion are shown in fig. 4.1. Two archetypical cases are used. The first case is shown in fig. 4.2a, and it corresponds to isotropic (A = D), uncoupled (B = 0), oscillators under the influence of a Coriolis force. The resulting motion is called a Foucault Pendulum precession in mechanics, or Faraday rotation of the polarization in optics. The w-vector for this case points along the C-axis as shown in the three-dimensional side of fig. 4.2a, and this is consistent with eq. (4.5) and our choice of C as the y-axis in (4.15). S-vectors which are parallel or anti-parallel to the C-axis correspond to the left or right circularly polarized optical states, respectively. These vectors would not precess, and circular orbits would remain circular. However, the speed of the circular orbits depends upon whether the orbit is right or left handed. All other states correspond to elliptically polarized orbits which precess. Their S-vectors go around at an angular rate of w,=2Cin{A, B,C}spacewhileanellipsegoesaroundatarateofonly~~=Cin{ lx), Iv)} space. The second archetypical case is shown in fig. 4.2b, and its corresponds to an anisotropic (A # D), uncoupled (B = 0), and Coriolis-free (C = 0) two dimensional oscillator, or to two one-dimensional oscillators in which 1x) and 1y) represent symmetric and antisynunetric normal modes. The resulting motion is called beating in mechanics and electronics, or polarization birefringence in optics. The w-vector for this case points along the A-axis as shown in the three-dimensional side of fig. 4.2b, and this is consistent with eq. (4.5) and our choice of A as the z-axis in (4.15). S-vectors which are parallel or anti-parallel to the A-axis correspond to x and y linearly polarized optical states, respectively. These vectors would not precess, and linear orbits would remain linear. However, the frequency of the linear oscillations would depend upon whether they were x or y polarized; the former is A and the latter is D. All other states which are mixtures of 1x) and 1y) will exhibit beating and generate a Lissajous pattern in the
358
W.G. Harter / Graphical approaches to molecular rotations and vibrations (a ) Faraday Rotation
fb 1 Birefringence
Fig. 4.2. Two diiensional
oscillator trajectories and corresponding three dimensional rotations. (a) C-axial rotation or Faraday precession; (b) A-axial rotation or birefringence.
two-dimensional space as shown in fig. 4.2b. At the same time the S-vector will precess around the A-axis at an angular rate of w, = (A - D) which is commonly called the beat frequency. This is the frequency of the beat intensity, but the beat amplitude oscillates at half this frequency: s2, = (A - 0)/2, in agreement with eq. (4.15). The visualization of beating as a rotation is a useful concept in discussing models of molecular vibrations. Suppose the 1x) and 1y) represent symmetric and antisymmetric combinations of localized models I&) and l (f2) I4
= IA,) = (10
where 15,) and I&) molecular system IEd =(I4
+ ltz))/fi,
IY) = I&)
= (It,>
- I&%‘~~
(4.16a)
represent motions confined to atom-l or atom-2, respectively, in a
+ IYWfi~
152) ‘(IX)
- IYW.
(4.16b)
The I &) and I(,) correspond to optical polarization at 3145 * to the fast and slow axes. The S-vectors for these states lie along the f B axes. The rotation around the A-axis of such a vector causes the atoms to take turns vibrating or “beating” as the S-vector rotates from + B through
359
W.G. Harter / Graphical approaches to molecular rotations and vibrations CORIOLIS
HARMONIC
OSCILLATOR
NH=2
CORIOLIS
NL=l A
HARMONIC
OSCILLATOR
NH=2
NL=l
A
Fig. 4.3. Phase space trajectory on torus in {x, y, px} space for a mixed (ABC) rotation, (Stereoptic figure.)
+ C to -B an then back to + B after passing - C. As the S-vector passes the + C axes each time the atoms will be vibrating with equal amplitude but differing in phase by *a/2. These are the points where the resonant transfer of energy between the atoms is maximum. Anharmonic perturbations can spoil this resonance and trap the S-vector near one side of the B-axis. This corresponds to local mode trapping which is discussed below. Computer animation of the beating motion provides an alternative view to a time trace such as the one given in fig. 4.2b. A properly animated view is seen immediately as a rotation, albeit a rather curious one. It is rotation of the angle 6 with angle Y held constant around 45”. What is curious is that the viewer sees it as an optical illusion which rotates around either the x-axis or the y-axis of the two-dimensional screen! This is an interesting observation since the three dimensional {A, B, C} picture of this motion is a simple rotation about the A-axis, but that is the axis which contains the x and y polarizations. The three-dimensional picture is always simpler, but the two-dimensional one has more information. The latter information exists in a subtle form which the human visual system can quickly interpret given a little practice. The &rotation is one example in which animated views of the spinor or two-dimensional oscillator picture are effective. Another way to enhance the oscillator picture is to display more of the oscillator phase space. An example of this is shown by a stereo plot of three of the four phase variables xi, x2, and p1 in fig. 4.3. This figure may be viewed with a standard stereopticon, or by simply relaxing one’s eyes so that the left (right) image is seen by the left (right) eye. The example is a Hamiltonian for which the w-vector has components on all three axes as shown by the {A, B, C} plot of the w and S vectors in the upper righthand corner of fig. 4.3. In stereo one can clearly see an invariant torus for this Hamiltonian and a closed phase path on that surface. A careful inspection of the torus shows that it is not just a “doughnut”. More details of this type phase picture are given in ref. [38].
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W. G. Harter / Graphical approaches to molecular rotations and vibrations
4.2. Quasi-spin RE surfaces (a) Harmonic oscillator surfaces The analogy between rotor dynamics and vibrational motion suggests that the RE surface picture used for rotors in sections 2 and 3 might also be of some value for describing vibrational dynamics and spectra. The Harmonic oscillator Hamiltonian (4.15) yields the simplest type of RE surface since it is linear in the quasi-spin components S,. By replacing these components with their polar coordinate form given in eq. 4.14a-c, one obtains the following expression for an RE surface E HO=(1/4)[(A+D)+2BcoscrsinB+2CsincysinB+(A-D)cosB].
(4.17)
Here the Euler angles (Y= 29 and B = 2v are used as the azimuthal and polar angles, respectively. The surface which results is a quasi-spherical or balloon-shaped object which is displaced and stretched in the direction of the w-vector (2B, 2C, A - 0) as shown in fig. 4.4a. The level curves on the surface are possible trajectories for the quasi-spin vector S, and these are seen to be a set of circles around the w-vector. This is consistent with the {A, B, C} pictures given in figs. 4.2 and 4.3. The S-orbit corresponding to birefringence or beating in fig. 4.2b is one of the circles in fig. 4.4a. The orbits on the harmonic oscillator surface all move in the same direction with the same angular frequency. The poles of the surface are the only fixed points. Note that orbits move counterclockwise when viewed from above a maximum point which is the positive A-axis in fig. 4.2b or 4.4a. This is opposite to the convention used for describing body fixed RE surfaces in sections 2 and 3. Here we are using lab fixed polar angles { (Y, B } instead of { - y, - B } . Note, also, that the vibrational RE surfaces do not necessarily have to have in inversion symmetry since their may be odd powers of the quasi-spin components in the oscillator Hamiltonian. (b) Anharmonic oscillator surfaces The effects of anharmonic perturbations can be visualized using the energy surface construction. For example, the following Hamiltonian will be examined as a function of the anharmonicity parameter a, (4.18) The quasi-spin RE surface function is the following E A0
=
E,,
+ a, ( 12/4)
cos*a sin*B.
(4.19)
Hamiltonians of this form have been introduced in refs. [24-281 as models for local mode behavior in quantum systems. The classical variable defined in eqs. (4.10) and (4.14) can be changed to quantum operators by replacing the phasor variables 4, = x, + ip, and J/,* = x, - ip, by boson operators a, = x, + ip, and ai = x, - ip,, respectively. The energy surfaces for a, = 1.0 and 3.0 are shown in figs. 4.4b and c, respectively. The harmonic constants are fixed to the values A = 4.0, B = 0.0 = C, and D = 2.0 for each case in
W.G. Harter / Graphical approaches to molecular rotations and vibrations
361
Fig. 4.4. Vibration quasi-spin RE surfaces and level paths drawn using eq. (4.17) with: J = 1, A = 4.0, B = 0 = C, d = 2.0. (a) Harmonic case: a, = 0.0; (b) Threshold anharmonicity: a, = 1.0; (c) Local mode due to large anharmonicity: a, = 3.0.
fig. 4.4. The RE surface for the anharmonicity constant a, = 1.0 is shown in fig. 4.4b. There is some squeezing of the circles that were near the positive A-axis in the harmonic case of fig. 4.4a, but the topology of the orbits is unchanged. However, as shown in fig. 4.4c, higher values of a, yield a new pair of fixed points surrounded by figure-eight separatrices which grow out of the positive A-axis as a, increases. The new fixed points approach the f B axes as a, becomes large. We recall that the f B axes are in the neighborhoods of the local mode state I&) or ) t2) given in eqs. (4.16). The ellipsometry embodied in fig. 4.1 allows one to precisely visualize the shape of the oscillator trajectory in {x1, x2 }-space corresponding to any given fixed point in { A, B,
362
W.G. Harter / Graphical approaches to molecular rotations and vibrations
C }-space. From eqs. (4.17-19) it follows that the extrema are located on the A&plane (a=29=Omod 71) at an angle (& = 21) to the A-axis for which the following derivative is zero W40
- ysin O= w a-o=
/3+ aXGsin
p cos p
i
$+_L!+.
(4.20a)
In general, there are the following four zeros for the derivative p. = 0,
7,
kcos-1%.
(4.20b)
x
The first two solutions p = 0 and p = IT correspond to the normal modes A, and A,, respectively, in eq. (4.16a). The second pair of solutions are valid only if a, is greater than the ratio a/I of the true normal mode beat frequency L? and the intensity of the oscillation. For the example shown in fig. 4.4 this threshold occurs at a, = 1.0, which is the value used in fig. 4.4b. Beyond this value the A, normal mode becomes unstable, and the new fixed points approach p = f 71/2 which belong to the local modes. The A, mode corresponding to p = IT remains stable, but the paths surrounding the negative A-axis become more and more squeezed as a, increases and the local modes occupy more of the RE surface. The shape of the RE surface and its level paths provide an indication of the form of the quantum level spectrum for each value of the oscillator intensity I or the principal quantum number N(afa, + a$a2) = (24).
(4.21)
In the case of fig. 4.4a the vibrational levels are nondegenerate and uniformally spaced. In the case of fig. 4.4~~the lower levels levels corresponding to the A, mode are still singly degenerate, but they become more crowded near the separatrix region in the middle of the level pattern. The upper levels corresponding to the pairs of local mode loop paths are nearly degenerate doublet cluster similar to the asymmetric top clusters seen in fig. 2.2. 4.3. Higher dimensional oscillators A number of anharmonic oscillators models have been investigated for local mode behavior. Many of them such as the well known Henon-Heiles [42] or Barbannis models [43] are two-dimensional, but do not necessarily conserve the principal quantum number given in eq. (4.21). Jaffe and Rheinhardt [42] have used Birkhoff-Gustaveson transformations to derive effective action functions of the Henon-Heiles Hamiltonian whose RE surface equivalent is like the symmetric top surface in fig. 2.la but with three local mode valleys around the equator. Patterson [44] has constructed vibrational energy (VE) surfaces for the anharmonic three dimensional model of SF, vibrations which resemble the RE surface in fig. 3.lb. This model also exhibits chaotic classical motion. It is possible that a graphical analysis of multiple N-surfaces might prove worthwhile for visualizing the irregular or chaotic motions found in these models. Elementary examples of multiple surfaces and the resulting irregular motions are discussed in section 6.
W. G. Harter / GraphicaI approaches to molecular rotations and vibrations
5. RE surfaces for generic Hamiltonians
363
of non-rigid rotors
The tensor Hamiltonians for semi-rigid rotors are effective operators that are derived from a fundamental or generic Hamiltonian of the form H=KE+PE.
61)
The effective operators such as the fourth order Hamiltonian (3.1) or the sixth order tensor (3.14) are just the first two terms in a series expansion of (5.1) involving arbitrarily high powers of the rotor angular momentum J. If higher order terms contribtite negligibly to the distortion of the rotor then its rotational dynamics and fine structure energy levels will be be well described by just the fourth order tensors. However, if many higher order terms are necessary, the algebra needed to derive them and their subsequent numerical treatment becomes increasingly laborious. As an alternative approach, one might derive the rotational distortion energy numerically as a function of the rotor momentum vector. This approach would avoid the algebra of the tensor expansion while, in effect, carrying the expansion to infinite order. Furthermore, it would work just as well for cases in which the potential energy of the Hamiltonians (5.1) was highly anharmonic; in fact the potential could be given numerically. For non-rigid or highly excited molecules, it is unlikely that any usable analytic forms will approximate their potentials. An example of this approach is shown in this section for an elementary model of the simplest polyatomic symmetric rotor X,. The numerical approach will be compared to the fourth order expansion for a harmonic model of the X, rotor, and RE surface plots will be shown for harmonic and anharmonic models. As shown in sections 2 and 3, it is possible to approximate rotational energy levels by quantizing paths on the RE surface. So far this has not been done for any of the numerically generated RE surfaces. However, since the RE quantization is already a numerical process, it will not matter if the input is given numerically instead of by a formula that has to be numerically evaluated anyway. A completely numerical quantization example should be done for a non-trivial model which can be solved by other means in order to compare the different approaches. A number of possibilities can be suggested. One example is the well known Hougen-Bunker-Johns non-rigid bender model [45] and it numerical implementation by per Jensen [30]. Another example is the stretch bender model which have been studied classically by Fredrick, McClelland, and Brummer [46] and Eua [47] and semi-classically by Frederick and McClelland [48]. It is likely that these models will involve multiple RE surfaces of the type that will be discussed in section 6. 5. I. The generic Hamiltonian
in the body frame
The present discussion is just meant as a preliminary study of the generic RE surface as an aid to visualizing the properties of numerical rotor models. The approach is based upon using a body-fixed version of the generic Hamiltonian such as the classical Wilson-Howard Hamiltonian. [49]. This is given below in a form similar to the one described by Papousek and Aliev [50] H vib--rot= ;R,&.R
+ Hcb.
(5.4
W.G. Harter / Graphical approaches to molecular rotations and vibratrons
364
Here the rotor momentum is R=
J-Ltib,
(5.3)
where J and Ltib are the total and vibrational angular momenta, respectively. The latter is defined in terms of Coriolis S coefficients of the first kind and vibrational normal coordinates and momenta which will be defined further below (Note that indices which are repeated on only one side of an equation are to be summed.) (5.4) The normal coordinates are linearly related to body fixed Cartesian displacement ~,Jv) {k=x, y, z; v=l...N} ofeachof Natoms ‘b =
Bb,k(v)dk(v)9 dj(
v,=B-l I(Y)*aSa
coordinates
(5.5)
’
The vibrational part of the Hamiltonian contains the Born-Oppenheimer approximate potential and an inverse G-matrix kinetic term that depends upon the B-coefficients in the normal coordinate definition Htib = G;;s,sb
+ PE( S) .
(5 -6)
The G-matrix is defined by
where m(v) is the mass of the v th atom. The B and G coefficients of the two kinds of Coriolis coefficients l and 6 as follows
also appear in the definition
where cink is a standard c-tensor. These in turn are used to define the canonical vibrational momenta (5 -9) where w is the instantaneous angular velocity of the rotor frame. Finally, tensor components $ involve both kinds of Coriolis coefficients
the inverse inertial
(5.10) where the nuclear inertia tensor is written as follows
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
in terms of body frame defined Cartesian coordinates x,(y)
=a&>
(5.12)
+ d,(v).
These include the positions of static equilibrium a,(v) as well as the displacements d,(v). The definition of the generic Wilson-Howard Hamiltonian is based upon having a subset of normal coordinates which are “genuine” vibrations, that is, they contain no components of translation or rotation. This leads to the first and second Eckart conditions on the molecular displacements m(v)di(v)
= 0,
m(v)+a,(v)dk(
v) = 0,
(5.13a)
(i = x, Y, 2).
The center of mass is understood to be the origin, that is ??2(v)a,(v)
(5.13b)
= 0.
The object of the following discussion will be to determine the Eckart-allowed displacements for a given angular moments .I. This involves finding a set of dynamic equi~b~um points for the genuine normal coordinates s, (a = 1 . . .3N - 6.) which, in general, differ from the static (J = 0) equilibrium points s, = 0. It should be noted here than Watson [Sl] has shown that the quantum mechanical form of the Wilson-Howard Hamiltonian is formally identical except for a small correction term E = -( tt2/8) Trace pR. Since we are here interested in semiclassical approximations it will not be necessary to invoke this fully quantum mechanical formalism. 5.2. X, centrifugal distortion energy A simple model for a C,, symmetric X, rotor will be considered here. The genuine normal coordinates s1 = sc, s2 = s:, and s3 = SF are defined by the Cartesian displa~ments in the first three columns of the B-‘-matrix.
Wl) d2(1)
d,(2) = d2 (2) dlW \d2c%
l/O 0 -J;5/6 l/2 -o/6 - l/2
l/G
0
0
l/a
0
0
-l/G
1/a
0
l/G
-O/6
-l/2
-l/2
- l/2
o/6
--G/6
-l/2
G/6
l/2
G/6
l/J?-
\ / Sl \
s2
0
s3
0
l/G
s4
l/2
l/0
0
s5
-J?-/6
0
l/0
I
\
(5.14)
%f
This relates all the in-plane coordinates shown in fig. 5.1. The out of plane motions are a rigid z-translation and two rigid rotations which are not genuine vibrations. The other non-genuine coordinates s4, sg, and $6 are a rigid rotation and two translations, respectively, in the molecular plane.
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W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 5.1. Normal coordinates of an X, molecule labeled by C,” symmetry.
The vibrational Hamiltonian will be constructed to have the C,, symmetry of the assumed equilibrium positions which lie on the vertices of an equilateral triangle (5.15) The harmonic frequency an is assumed to be the same for the two genuine modes s2 and sj that belong to the E species of C,,. The C,, symmetric anharmonic potential is assumed to be in the form of the Henon-Hieles cubic function. (a) Harmonic case The desired dynamic equilibrium values for the genuine normal coordinates si, s2, and s3 may Hamiltonian (5.2) and (5.15) for a range be found by minimizin g the generic rotation-vibration of values of the J-vector with zero vibrational angular momentum (L = 0). It is instructive to compare the minimum values obtained numerically with a fourth order approximation. This is easily done in the harmonic case with A,, = 0 in (5.15). The minimum energy distortion values are found by including (5.15) in (5.2) and setting to zero the derivative with respect to coordinate s,
+
w,zs, .
(5.16)
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W. G. Harter / Graphical approaches to molecular rotations and vibrations
The resulting coordinate displacements
for dynamic equilibrium are then given
-1 a/-%, s, = -JJ.2a,2 ’ J as, * The change in the inverse inertia tensor can be expanded in the displacement
(5.17)
coordinates (5.18)
If second or higher order terms in this expansion are negligible then the Hamiltonian the following
becomes
(5.19a) Using eq. (5.17) gives the following H = :J,JIpiJ(0)
+
This result has the form of a rigid rotor plus a fourth degree J-polynomial Hamiltonian (3.la).)
(5.19b)
(Recall the Hecht
(5.19c)
It can be written in terms of the harmonic potential energy of the centrifugal mode distortions using eq. (5.17) H = :JQqj(0)
- &a&,2.
(5.20)
There is a gain in potential energy due to stretching according to the third term in eq. (5.19b). However, in this approximation, the rotor loses twice that much kinetic energy due to the expansion according to the second term. The result is net loss in total energy equal in magnitude to the gain in the potential energy as stated in eq. (5.20). The displacements (5.17) can be rewritten in terms of the derivatives of the inertia tensor I = p-l (5.21) In the approximation
which neglects quadratic or higher terms in the inverse inertia (5.18), it is
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W. G. Harter / Graphical approaches to molecular rotations and vibrations
only necessary to evaluate the inertial quantities in eq. (5.21) at static equilibrium (s, = 0). This yields a simple approximate expression for the X, centrifugal distortions 5
“=s,=(J,‘+J,‘+~J,*)k/~~,
(5.22a)
sf = s* = (J,2+ JyZ)k/L&
(5.22b)
SF = sj = (-2J,J,)k/&.
(5.22~)
The constant factor depends upon molecular mass (m) and equilibrium radius (a) k = 2(3m)-3’2a-3.
(5.22d)
For a numerical example, let us choose the mass and molecular radius to be a unity. (m = 1, u = 1). Let us also pick a unit angular momentum vector, and direct it along the (x = x,)-axis. (J, = 1, Jy = 0 = J,). Then the centrifugal distortions of the normal coordinates beyond static equilibrium will be calculated for two different sets of the vibrational normal mode frequencies. A comparison will be made between a “semirigid” rotor for which wA= 50 and wE = 30, and a “ semifloppy” rotor for which wA= 5 and wr = 3. The approximate eqs. (5.22) give normal mode distortion values for the floppy rotor that are exactly a hundred times those for the rigid rotor. This follows from eq. (5.22) since their frequencies differ by a factor of ten. The approximate values given in the top two rows of table 5.1 are to be compared with the corresponding numerical values given in the bottom two rows. The coordinate SF is identically zero by symmetry for this choice of J-vector, and this provides a first check for the numerical routine. It is seen that the approximate equations (5.22) are accurate to three places for the semirigid rotor but are considerably off for the semifloppy rotor. The accuracy of eqs. (5.22) deteriorates further in the presence of higher J or with the inclusion of anharmonicity. Note that eq. (5.22) overestimates the distortion since equilibrium values of I- ’ were taken in eq. (5.21). In fact, I is generally larger, and so 1-l is generally smaller when the correct expanded distortions are used. The numerical results were calculated using standard Powell-Brent minimization routines [35] of the Hamiltonian given by eqs. (5.2) and (5.15). These routines do rapid calculations of global
Table 5.1. X, Distortion values due to angular momentum J = (LO, 0) Semirigid rotor (WA= 50, wa = 30)
Semifloppy rotor 5, WE = 30) cwA=
Approximation Distortion value (from eq. (5.22))
sA = 1.5396 x 1O-4
sA =1.5396x10-’
sl” = 4.2767 x 1O-4
~1”= 4.2767 x 1O-2
Exact numerical Distortion values
sA =1.5381 x1O-4 SF = 4.2721 x 1O-4
~~=1.4062~10-~ s1”= 3.9011 x 1o-2
W. G. Harter / Graphical approaches to molecular rotations and vibrations
369
b
a
, Fig. 5.2. Equatorial (xy) section of numerically generated X, RE surface for centrifugal distortion model. mA = 5, tiE = 3. J-vector is of constant magnitude ( 1J 1= 3/2) and variable direction in xy plane. Distortions are magnified six times. (a) Harmonic model: X Haron= 0; (b) anharmonic model: XHenon= 1.5. minima, but are compact enough to easily be used on a Macintosh or other personal computer. They are also rapid enough that the J-vector can be swept over a circle or sphere to quickly produce RE surfaces. The speed of the numerical minimization is usually not affected by the inclusion of anharmonicity, and so it is easy to show the effects of anharmonic forces on rotations. The two plots in fig. 5.2 show the scaled equatorial sections of an X, RE surface for harmonic nenon = 0) and anharmonic (hn_, = 1.0) Hamiltonians of the form given in eqs. (5.2) and (A (5.15). These include scaled drawings of the actual rotor distortions belonging to select points on the RE contour. The scale of the distortion is magnified sixfold for each figure. The harmonic RE surface is circular, but the anharmonic RE surface develops hexagonal anisotropy in the (J, J,)-plane. Note that a triangular RE surface is impossible for the rotor since the Hamiltonian must be invariant to angular momentum inversion (J + -J) or time reversal. The scale of the hexagonal anisotropy in fig. 5.2 is exaggerated compared to the scale of the anisotropy of the oblate planar symmetric top. The complete RE surface corresponding to fig. 5.2 is shown in fig. 5.3a. This resembles the surface for the oblate top in fig. 3.lc, except for the beginnings of six small resonance islands that distort the equatorial contours. By increasing the magnitude of J, the islands are seen to grow in figs. 5.3b, c until they occupy a significant fraction of the surface. Also, as J increases, the angular momentum cone angles decrease while the angular size of the islands increases. At some critical value of J three will be room for one or more @-cones (recall eq. (2.19).) inside the equatorial separatrices of the surfaces in fig. 5.3. At or shortly below this critical J-value, the rotor will cease to behave like an oblate symmetric top. It will then be possible for it to become stuck rotating like a spinning coin around one of the six hills on its edge or equator. The symmetric top rotational spectrum will become extremely perturbed, and the simple doublets at the top of each J-manifold will be replaced by clusters of six levels. The C,, species in those sextets will be in the form (A,EEA,). The splittings are likely to resemble the projections of a hexagon according to the simplest tunneling models [lO,ll].
370
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 5.3. Numerically generated anharmonic model X, RE surfaces and level paths showing the effects of increasing the total angular momentum. Harmonic constants are U* = 5, aE = 3. Anharmonic constant is XHenon= 1.0. (a) J = 2; (b) J= 5; (c) J=lO.
The semiclassical methods described in section 2 can be developed to derive approximate energy levels and eigenfunctions directly from the numerical RE surfaces. The same graphics routines which draw contour paths can be modified to satisfy quantization conditions as well. Classically propagated wavepackets can be used to approximate eigenfunctions in the neighborhood of quantizing trajectories. 5.3.
RE surfaces for vibronically excited states
The minimization methods which gave the RE surfaces in figs. 3.3 were chosen to deal with the special case of a rotor in a vibronic ground state. In the first try of a generic approach to RE surfaces one needed to have a rapid search for global minima more than a general algorithm for treating excited states. Now we speculate on alternative methods which might yield usable RE surfaces in excited states, as well. One alternative is to run classical equations of vibrational motion while changing the J-vector adiabatically. This would involve procedures related to those discussed by Skodje [2]. One could pick a quantized vibrational action and slowly turn on the J-vector as well as the anharmonicities. One would obtain a quantized vibrational state polarized by the frozen rotation. Then the J-vector direction could be changed slowly enough to trace out an adiabatic RE surface. It would also be possible to determine to what extent the true J-vector precession was adiabatic. Another alternative is to solve quantum vibronic equations to obtain effective adiabatic RE surfaces. The RE surface or surfaces would be plots of quantum vibrational eigenvalues as a function of J-vector directions. As mentioned previously, this is analogous to the derivation of Born-Oppenheimer vibrational PE surfaces from quantum electronic eigenvalues. In the following section, we consider some examples of rotational dynamics driven by vibrational or vibronic eigenvalues. 6. Multiple RE surfaces and Coriolis coupling effects This section contains a discussion of some simple models for Coriolis coupling between rotors and their internal degrees of freedom. Much of molecular physics involves an analysis of coupled
W.G. Harter / Graphical approaches to molecular rotations and oibrations
371
systems involving a rotor and some passengers such as orbiting electrons, electron spins, nuclear spins, vibrational angular momentum, or combinations of any of these. Here we will discuss ways to visualize coupling effects using RE surfaces. It is a common practice to analyze coupled angular momentum systems in terms of special cases in which certain components of rotor or passenger momenta are conserved quantities or good quantum numbers. Certain types of coupling operators may conserve certain quantum numbers; and their eigensolutions are classified as archetypical cases. For example, there are the LS and jj coupling cases for multielectron problems in which spin-orbit coupling is weak or strong, respectively. There are the Hunds coupling cases (a) through (d) for levels of diatomic or linear molecules. These depend on which type rotor, spin, or orbit coupling is dominant. [52] The connection between energy levels of two archetypical cases are often shown using eigenvalue correlation diagrams. [53] This is particularly useful if the desired eigensolutions are close to one of the archetypical cases. For polyatomic molecular problems, the classification or archetypical cases is more difficult. Polyatomic molecules do not generally have the infinite R, symmetry or the associated quantum numbers found in symmetric diatomic or linear rotors. At best they have a finite permutational symmetry, but these symmetries are labeled by a finite number of ‘“quantum letters” such as A,, A,, T,, etc., instead of quantum numbers. These letters usually cannot label states uniquely since each label must be used over and over again. Associated with an inadequate labeling system is an increased difficulty to visualize the eigensolutions and to identify what, if anything, is being conserved or nearly conserved. Also most finite symmetry Hamiltonians yield spectra which must be classified region by region or even level by level. Most of the levels diagrams in sections 2 and 3 exhibited two or more archetypes in the same spectrum, and simple quantum letters or numbers do not give enough info~ation by themselves. In sections 2 and 3 the labeling and visualization problem was solved for finite symmetry rotors by using RE surface trajectories and their associated semiclassical quantum numbers or quasi-conserved momenta. The effects of perturbations on level correlations could be visualized continuously as the perturbation strength increased, not just when the perturbation was very strong or very weak. A generalization of RE surface visualization methods is introduced in this section for describing Coriolis coupling between the angular momenta of a rotor and its passengers. Again, it is sometimes possible to visualize the level correlations continuously between archetypical cases of strong or weak couplings. However, this generalization involves the use of multiple or overlapp~g RE surfaces, and it leads to more complicated semiclassical dynamics. These ideas are introduced in the following section 6.1. Then examples are treated in sections 6.2 and 6.3. 6.1. Semiclassical mechanics on multiple surfaces In preceeding sections the rotational has been described using a single rotational energy surface. The analogy has been made between a rotational energy (RF) surface and a vibrational potential energy (PE) surface. A PE surface is a measure of electronic and electrostatic energy, and it depends upon having an electronic state which follows the vibrational motion adiabatically. In the case of pure vibrational excitation this would be the ground electronic state. Electronic adiabaticity is necessary in order to have an effective potential energy as a function of
372
W. G. Harter / Graphical approaches to molecular rotations and vibrations
the classical vibrational coordinates q = ( ql, q2, . . . ). In analogous fashion, an RE surface depends upon the adiabatic following of rotational motion by a vibrational or vibronic state. In the case of a deformable rotor, this would be the ground vibronic state. Vibrational adiabaticity is necessary in order than an RE surface can accurately described the total effective rovibronic energy as a function of the direction of the classical rotational momentum vector J = ( J1, J2, J3). This section contains a description of models which involve two or more neighboring RE surfaces. Multiple RE surfaces are needed if two or more vibrational or vibronic states are degenerate in the absence of rotation (J = 0). This is analogous to well known Jahn-Teller or Renner models [54] which involve two or more neighboring PE surfaces. Multiple PE surfaces are needed if two or more electronic states are degenerate in the absence of vibrational distortion (4 = 0). The multiple Jahn-Teller or Renner PE surfaces are each plots of different electronic eigenvalues as a function of vibrational coordinates q. By analogy, the multiple RE surfaces are each plots of different vibrational or vibronic eigenvalues as a function of the direction of total angular momentum J. Each set of eigenvalues gives one point on each PE (or RE) surface, and these are evaluated in the adiabatic approximation at constant q (or J). However, each PE (or RE) surface determines a possible classical path of motion for q (or J) that is, q (or J) are usually not constant, after all. Each PE (or RE) surface belongs to a different electronic (ro vibronic) eigenstate, and it provides a different set of marching orders for q (or J). The adiabatic approximation is valid if each electronic (or vibronic) eigenstate can remain nearly an eigenstate as q (or J) follow their matching orders. Then the electronic (or vibronic) quantum states are said to be adiabatically following the q (or J) motions and the configuration is said to reside on a single surface as far as its own dynamics are concerned. However, the presence of nearly coupled PE (or RE) surfaces belonging to different electronic (or vibronic) eigenstates allows for more general types of motion. It is possible to consider PE (or RE) surfaces belonging to mixtures of electronic (or vibronic) eigenstates as well as the base surfaces belonging to eigenstates. With two or more electronic (or vibronic) states, there will be a continuum of possible different mixtures of states and a corresponding continuum of possible different PE (or RE) surfaces. Each mixture of eigenstates corresponds to another surface which lies between the base surfaces belonging to eigenstates. If adiabatic following occurs on each of the PE (or RE) surfaces belonging to electronic (or vibronic) eigenstates, then it may be possible for adiabatic following to occur on surfaces belonging to mixed states. Eigenstates are not necessarily the only states capable of adiabatic following. At this point, it is necessary to temporarily abandon the analogy between PE and RE surfaces in order to reemphasize their differences. A single adiabatic RE surface is a two-dimensional manifold, and its intersection with the constant energy sphere defines a one-dimensional classical J-trajectory. In contrast, a PE surface is a (3N - 6)-dimensional manifold, and its intersection with the constant-energy (hyper) sphere defines only the outside boundary of classical motion. The classical rotation J-motion or precession is determined directly from RE surface topography lines, but the determination of vibrational motion on a PE surface requires a detailed (generally numerical) solution Hamiltonian’s equation of motion. Motion of on a single RE surface is simple integrable and periodic, while PE surface motion may be irregular or chaotic even for a single two-dimensional surface. However, rotational motion also becomes complicated if an eigenstate cannot follow J-motion
W. G. Harter / Graphical approaches to molecular rotations and vibrations
313
adiabatically. Then the J-vector will no longer be constrained to lie along the intersection of the energy sphere and a single constant RE surface. This will happen if an original RE surface path changes the J-vector too rapidly for the original eigenstate to follow, and a mixture of eigenstates results. As the state mixture evolves, so does the RE surface, and the precessional mechanics is no longer a simple one-dimensional problem. The energy sphere is constant but the RE surface deforms and vibrates through a continuum of possible shapes. The paths of J-motion can still be thought of as the energy sphere-RE surface intersections, but these paths can change dramatically as the surfaces change even a little. As a result, the J-vector is not confined to any particular region of J-space, and there is a possibility for classically chaotic rotational motion. Nevertheless, the RE surfaces can still provide useful information even when parts of them show a breakdown of the adiabatic approximation. Each RE surface has regions surrounding stable fixed points. These are separated by separatrix regions that connect unstable fixed points. There may be little or no tendency for the J-vector to move near the center of a fixed point region. The vibronic state corresponding to that part of the RE surface may be able to keep up with the J-vector if it is close enough to the fixed point. Hence, one might expect those portions of RE surfaces to contain regular motion and provide a useful semiclassical characterization of a subset of rovibronic eigensolutions. In separatrix regions, on the other hand, the single constant surface is most likely to fail and irregular motion may occur. Still, the simple RE surface calculation is useful for locating the irregular regions. Some examples will now be given to demonstrate these ideas. 6.2. Asymmetrically
coupled spin rotor systems
One of the simplest polyatomic models for Coriolis coupling is an asymmetric rotor coupled to a spin-l/2. This is a model which involves just two RE surfaces, and it could serve as a model for coupling of an electronic or nuclear spin S to a rotor. It could also be a model for a two-dimensional vibrational Coriolis coupling in which the vibration was represented by a quasi-spin S as described in section 4. Here the model Hamiltonian will be restricted to have just first and second order terms in the coupled spin momentum and total momentum J=R+S
(6.1)
where R is the angular momentum of the rotor by itself. The body defined components of J satisfy the usual commutation relations with the spin
1J& I =
-kzjkJk,
[J,9 S,] =O,
[&S,]
=ic,jkSk.
(6.2)
The general second order Hamiltonian has the following form (6.3) Here we will consider special cases of the Hamiltonian in which the spin is assumed to have zero rotational inertia (E,, = 0). Only the diagonal inertial and coupling terms which have D2
374
W.G. Harter / Graphical approaches to molecular rotations and vibrations
symmetry will be kept. For an example of an asymmetric spin coupled rotor, only the quadratic terms are kept in order to preserve time-reversal symmetry. This special case is the following: HR= C,,J,‘+
C,,J;
+ C,,J,‘+
D,,J,S, +DyyJ,Sy + Dz,JzSz.
However, the simplest nontrivial spin Hamiltonian following example, which will be discussed first.
(6.4
which requires two RE surfaces is the
(6.5)
H, = A, J, + B,S, + D,, J,S,
This represents two precessing zero-inertia rotors or spins asymmetrically coupled along one component. A form like this was studied by Feingold and Perez [55] as an example of classical chaos. Their objective was to study the correspondence between exact quantum eigensolutions and classical dynamics the exhibited chaos. The following describes the constriction RE surfaces which help to visualize both the classical and quantum dynamics. (a) Asymmetric coupled spins As usual, the total angular momentum vector components (J,, JY, J,) will be treated as classical functions (2.5) of Euler direction coordinates for RE surfaces. The spin components (S,, SY, S,) will be treated as quantum mechanical operators whose eigenvalues determine points on each RE surface along the direction of a given J-vector. This is accomplished by first replacing each spin operator in the Hamiltonian by its two-by-two Pauli matrix representation. For the Hamiltonian (6.5) the result is the following matrix
’ (Hs) =
A, 1J 1cos p + Bz2ISI -D
\
xx “J I2I ’
-D XX ]J]]S]sinpcos 2
I sin /3cos p
A, I J 1cos p -
ISI Bz2
(6.6a)
Here the spin magnitudes are I J I = \i[J(J+l>l and ( S ] = \i[s(sm. The resulting Hamiltonian can be viewed as a two-state Hamiltonian that is a function of the polar angles (- /3) and (- y) of the J-vector. For each J there is a precession vector Ic2 that is analogous to the w-vector derived for the general two-state Hamiltonian in eqs. (4.4) through (4.6). This derivation gives the following D vector for eq. (6.6a). 52,= -D,,I
JllSl
sin /3cos y,
Q,,,=O,
Q2,=BzIS(
(6.6b)
This determines the angular velocity vector for the expectation value of the spin vectors. The two eigenstates of (H,) will have spin vectors that are aligned “up” or “down” along the L&vector, and these are the only directions for which S is stationary relative to 0. However, the Q-vector is a function of J, and both will move according to the Hamiltonian H,. The possible J-motion can be determined approximately by plotting the “up” and “down” eigenvalues of the (H,) matrix in (6.6) as a function of polar angles ( -p) and (- y) of J. Some examples of the resulting pairs of surfaces are shown for a range of values of the coupling
W. G. Harter / Graphical approaches to molecular rotations and vibrations
ig. 6.1. RE
surfaces for asymmetricallycoupled spins J = 5/2
and S =1/2.
(a) D,, =l.O;
375
(b) D,, = 3.0; (c)
D,, = 9.0.
parameter D,, in figs. 6.la-c. If the S vector could follow the motion of 52 and J, then the latter would be confined to the equal-energy contours which are drawn on the surfaces. The contours on each of the two RE eigensurfaces start as a set of parallel circles. Then as the coupling constant DXX increases, the circles around one pole on each surface become very squeezed. This is shown happening at the north pole of the outer (high energy) surface and the south pole of the inner (low energy) surface in fig. 6.la. Then the squeezed pole of each surface becomes an unstable fixed point or saddle point at the center of a figure-eight separatrix which surrounds a pair of nearly circular loops. One loop of each pair is visible on each surface in fig. 6.lb. The contours on each surface resemble those of the local mode paths in the anharmonic oscillator of fig. 4.5~. The pole opposite to the saddle point remains a stable fixed point, but as D,, increases, the paths around the stable pole get squeezed by the figure-eight loops which are approaching the equator on each surface as shown in fig. 6.la-c. For high D,,, the figure eight centers on both surfaces approach the x-axis, and symmetric top surface like figs. 2.la,c emerge. The stable pole for the high energy RE surface is above the unstable pole for the low energy one, and visa versa. Also in fig. 6.lb, the stable figure eight loops of the high energy surface lie above the separatrix region of the low energy surface. A J-vector that is initially inside one of the top loops in fig. 6.lb will have much less tendency to move rapidly if the initial spin vector is “up” than if it is “down”. To test these ideas the exact classical equations derived from the Hamiltonian Hs were solved numerically. The resulting trajectories for the vectors J, S, and 1(2 were ploted in 3D stereograms in fig. 6.2. The stereo pair in fig. 6.2a show the small amplitude precessional motion of J for the case that the S-vector is initially “up” along 9. For the stereo pair in fig. 6.2b, the initial conditions are the same except the S-vector is “down” along Sz. The result is irregular motion with large amplitude precession of J. It appears that J takes turns going around curves that are close to one or the other of the figure-eight loops on the spin-down surface in fig. 6.lb. Since the S cannot follow the rapidly moving J and 9, it is possible for the system to classically jump from one loop to the other. Such a classical jump would be impossible on a single RE surface, but which resonance between two surfaces the J-vector will sometimes find the saddle point pass “open” while other times the pass will be “closed”. The preceeding example shows that while the J-vector does not stick precisely to RE eigensurface paths, one can still use the paths to visualize qualitatively what the resulting motion
376
W G. Harter / Graphical approaches to molecular rotations and vibrations a
J= 1.0S,= 1.00
A,= 000 A,= 0.00 A,= 1.00 B,= 0.00 By= 0.00 B,= 1.00 c = 0.00 c,= 000 c = 0.00 D",=3.00 D,= 0 00 d",=0.00 aO= 0.00 &= 45.00yO= 000
a J= 1.0S,= 1.00 Emiti.lly= 3.052315 *to EO p"te=3.052315
A,=
0.00
A,=
0 00
A,=
, 00
B,= 0.00 B,= 0.00 B,= 1.00 c,= 0.00 c,= 0.00 c = 0.00 D,= 3.00 D,= 0.00 I$,-0.0~ ao= 000 &= 4500 -yO=000
II 0
b J= 1.0S,=-100 E =-1636101 inithlly
AX=000 A,= 000 A,= 100 000 000 100 c B,= = 000 c,= By= 000 c= B,= 000 D",=300 D,= 000 dz= 000 aO= 000 &= 4500 yO= 000
b J= 10 S,=-100
A,= B,= c,= D,= aO=
000 000 100 000 A,= El,=0.00 A,= B,= 100 000 c,= 0.00 c = 000 3.00 D,= 000 o",=0.00 000 p,= 4500 To= 000
Fig. 6.2. Examples of classical motion for J near an upper surface fixed point in fig. 6.lb. (Stereoptic figures.) (a) Initial spin is “up” along &?-vector; (b) initial spin “down” results in irregular motion.
will be. It is also possible to obtain qualitative information about the quantum energy levels. One can see that the high D,, surfaces (figs. 6.lb, c) contain two pairs of equivalent figure-eight loops which become more localized at very high or very low energies. These loops should correspond to level cluster pairs similar to asymmetric top clusters. The clusters should become more nearly degenerate in the higher and lower regions of the spectrum. This clustering is seen in the quantum energy level correlations shown in fig. 6.3. The energy levels in the center correspond to irregular near-separatrix motion and are unclustered.
317
W. G. Harter / Graphical approaches to molecular rotations and vibrations
(b) Asymmetrically coupled spin-rotor Some examples of RE surfaces and level correlations will be shown for the elementary asymmetric rotor coupled to spin - l/2 through the Hamiltonian HR in eq. (6.4). The spinor representation of HR is the following h+D
‘J”s’cms~
(HR)=
-(D,,
cm y -*:D,,,,:in
u)ysin
-(D,,
/3
cm y+iDy,,
I J I I s I sin p
sin y)--~_
h - Dz,~cos
/3
(6.7a) ’
Here h is the free asymmetric top energy which is similar to eq. (2.1) h = 1J 1*(Cxx sin*/3cos*y + C,, sin*P sin*y + C,, cos’p).
(6.7b)
The eigenvalues of the matrix in eq. (6.7a) are shown by the RE surfaces in figs. 6.4a-c. The figures represent a range of coupling parameters in which the ratio D,, = 3D,, and D,,,, = 2 D,, are maintained as 0,. increases, while the inertial parameters C,, = 2.0, C,, = 4.0, and C,, = 6.0 are kept constant with similar ratios. These are also similar to the values used for the simple asymmetric top in section 2. This choice of ratios yields a peculiar set of trajectories on the inner (low energy) RE surface. The outer surface is very similar to the simple asymmetric top RE surface in fig. 2.1, but the
Fig. 6.3. J = 5/2
energy levels for asymmetrically couples spins plotted as a function of coupling constant Dxx. Clustering is most pronounced for the highest and lowest levels.
W. G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 6.4. RE surfaces for asymmetric spin coupled rotor. (C,, = 2.0, C,,y = 4.0, C,, = 6.0; Dyy = 2D,,, (a) D,, = 30.0; (b) Dxx = 40.0; (c) D,, = 50.0.
and D,, = 3D,,)
inner surface becomes more nearly spherical with a much more complex topography. For one thing there are sets of four equivalent paths in the form of two figure eight valleys. These appear on the x-axis and move toward the z-axis as OX, increases. This indicates the possibility of an extraordinary level cluster made of four D, species {A,A,B,B,}. Such a cluster is formed in the lowest level of the J = 21/2 spectrum in the middle of fig. 6.5. Above and to the right or left are levels which resemble the J = 10 asymmetric top pattern of figs. 2.2 and 2.3. The upper pattern corresponds to the outer surface in each of figs. 6.4 and is very much like a single asymmetric top. As OX, becomes very large, the lower surface must also approach a simple asymmetric top. For very large OX,, the rotor momentum quantum numbers R = 10 and R = 11 become good quantum numbers for the upper and lower RE surfaces, respectively. However, for intermediate values (30.0 < 0,. c 50.0) in fig. 6.5, there is a clustering of two ordinary cluster doublets into a fourfold near-degeneracy in the lowest levels. This is the region described by the RE surfaces in fig. 6.4. In addition to the unusual clustering, there is a considerable crowding of all the levels in the lower energy branch of fig. 6.5. This is related to the low hills and shallow valley of the inner surface of fig. 6.4. Also, the inner surface has a complicated separatrix region. Together these properties tend to make the classical motion irregular or chaotic. However, if J is large compared to S, it cannot be upset as easily as it was in figs. 6.2 or 6.3 where they were given comparable magnitudes. Hence, the levels for J = 21/2 and S = l/2 are fairly orderly. 6.3. Octahedral scalar and tensor Coriolis coupling The fundamental vector vibrational modes vj and vq of SF, are described by a triplet of base states { &, &, #- 1> which have a total vibrational angular momentum L = 1 and cubic symmetry label T1. The coupling of the momentum L with an octahedral rotor of momentum R yields states of total momentum J = R + L as in eq. (6.1). However, the three L = 1 states will give rise to a triplet of RE surfaces for each J-value. Here a set of surfaces will be derived from a semiclassical analysis of the following effective Hamiltonian due to Hecht [16] H, = v + BJ* + 2B{J*L
+ t,,[v*(rotation)
8 u*(vibration)]i,.
(6-g)
379
W. G. Hurter / Graphical approaches to molecular rotations and vibrations A,= 0.00
B,= 0.00
A,=
0 .OO
A,=
0.00
B,= 0 .OO B = 0.00 4.00 ‘cm= 6.00
R= 11 I
0
8
0.0
10.0
0
20.0
30.0
I
40.0
50.0
Dxx
60.0
70.0
80.0
90.0
10
With D,,=2Dxx and D,,=3D,,
Fig. 6.5. J= 21/2 energy levels for asymmetric spin-coupled rotor plotted as a function of Ox,. The levels in the crowded region in the lower branch correspond to the innermost RE surfaces shown in fig. 6.4. The lowest two clusters in that region contain four levels each and correspond to a quartet of equivalent paths on the RE surfaces.
The first two terms are constants for a given J which give the vibration and scalar rotational energy. We shall be concerned with the scalar and tensor vibration-rotation coupling in the third and fourth terms, respectively. The tensor operator is a fourth rank combination of vibrational and rotational quadrupole operators u2(vib) and u2(rot). A brief description of tensor construction and notation is given in appendix D [ u2(rot) @ u’(vib)]i,
= V% [ u2 @ U’]: + 5([ u2 QDZI’]: + [u2 c3 d]‘q.
(6.9)
Except for an overall factor this is the same combination of [ 1: tensor terms as found in the T,” expression in eq. (3.15). The vibrational quadrupole tensors are quadratic functions of vector boson creation and destruction operators
380
W.G. Harter / Graphical approaches to molecular rotations and vibrations
The rotational quadrupole quadrupole tensors quadratic function of J-vector form given in eqs. (2.2). ui (rot) =
operators of the
Tq2
The necessary Clebsch-Gordon
[T*@v*];=&(
products of the rotational and vibrational tensors are as follows
T,2?,+
T:,u;)+
@(
T,2&+
T~l~~)+~(T~v;),
[T*@u*]~~= T:,u;,. Assembly of these terms in eq. (6.9) yields the desired operator = &Tt( alal- 2&z,,
[ v*(rot) @ U*(vib)]i,
+ T2,( &_,
+ u~~Q)
+ 5u;u_,)
+ 2fiT!,(
&_r
- a$~,) (6.10)
+ C.C.
*+= (- l)qT,* (q # 0)are included in where the complex conjugate of the off-diagonal tensors T_, the c.c.-term to make the operator self-conjugate. The details for construction of quantum mechanical matrix elements of the Hamiltonian Hv are given in ref. [56]. Here we shall only consider the semiclassical matrix whose eigenvalues give the RE surfaces. A semiclassical matrix follows by replacing boson operators in eq. (6.10) by their fundamental 3 x 3 representations, L by its vector representations, and the tensor operators by their Euler angle functions according to eq. (2.4). The result is the three-dimensional matrix which is the representation in the basis of body-fixed vibrational states which are labeled using the standard Z and II notation IW
= I&>
IX) = I&o)
IW
= Ih>*
The matrix in this basis is the following 1
0
0
0
1
0
0
0
1
3 cos*p - 1 - J8elY sin p cm /I si&?(6 cm 2y - i4 sin 2~)
-\/8e-‘Y sin ,9 cm p -2(3 cos*/?- 1)
sin*j3(6cos 2y + i4 sin 2y) J8e-‘Y sin B cos /3
fielv sin fi cos p
3 cd/3 - 1
1.
(6.11)
381
W.G. Harter / Graphical approaches to molecular rotations and vibrations
If the I: and II states are eigenstates, it means that the projection of the vibrational angular momentum on the body axis of quantization is conserved and (L,) = A = 0, & 1 is a good quantum number. For a very strong scalar Coriolis constant (B3 B tzz4 1J I) it may be advantageous to use bases in which the rotor momentum R is conserved and the good quantum numbers are R = J + 1, J or J - 1. These bases are labeled 1P), 1Q), and 1R) respectively, after the corresponding branches of the infrared spectrum to which they belong. In this basis, the scalar Coriolis operator is diagonal
(%)
PQR =(v+BIJ/2)
(6.12a)
The off-diagonal basis
tensor components in the PQR basis are more complicated
Hro = 5 sin /I[ 7 cos2@ - 3 cos fl-
sin2/3(cos p cos 4y + i sin 4y)] /2fi
than in the XII
= H& ,
HPR = 5[ - 7 COS”~+ 8 cos2/3+ (1- cos4/3) cos 4y + 2i cos B sin”/3sin 4y - l] /4 = H&, . (6.12b) However, the diagonal tensor components in the PRQ basis have precisely the same form as the simple RE surface expressions in eq. (3.2b) HPP = HRR = (35 cos4p - 30 cos2/3-t 5 sin4p cos 4y + 3)/4 = -J&&2.
(6.12~)
This means that the P and R branch RE surfaces for high B{ and positive t, (< B{ will be two nearly perfect copies of the single surface in fig. 3.la. The Q branch surface will resemble the surface in fig. 3.lb. It will have twice the amplitude of the P and R branch surfaces and lie between them. The high-( B{) case is the one usually encountered. Nevertheless, it is instructive to consider RE surfaces in the neighborhood of B{ = 0. The eigenvalues of either representation (6.11) or (6.12) yield the RE surfaces shown in fig. 6.6a-d. Each set of three eigensurfaces are shown as stereo pairs (a), (b), (c), and fd). These correspond to B{ = 0.0, 0.5, 1.0, and 1.5, respectively, with t,,, = 1. The four sets of surfaces are to be compared with corresponding points labeled (a), (b), (c), and (d) in the exact (J = 60) eigenvalue correlations in figs. 6.7a, b. A magnified view of the low-( B{) region (BS I 1.5) is shown in fig. 6.7b. In the center of this region the Z and II labels apply since the Coriolis force is weak and does not cause the L-vector to deviate rapidly from its
382
W.G. Harter / Graphical approaches to molecular rotations and vibrations
Fig. 6.6. RE surfaces for Coriolis coupled octahedral rotor-vibrator in the low B{ region. (t,,, 1J 12 = 1.0). (a) B{ = 0.0; (b) BP = 0.5; (c) Bl = 1.0; (d) B3 = 1.5.
precession around body-fixed axes. The lowest ten or eleven levels in fig. 6.7b at Bl = 0 each contain six levels in clusters of the 4-fold (C,) symmetry type which were shown in fig. 3.3. These clusters can be associated with the six sets of C,-symmetric valley paths on the innermost RE surface “cube” in fig. 6.6a. It can be shown that the L-vector precession is mostly normal to the instantaneous axis of quantization for the C, states, and so they are labeled 4-fold Z-clusters in fig. 6.7b. Above the C4-C levels is a separatrix region containing a few unclustered levels that create a dark band in fig. 6.7b at B{ = 0. The five or six level clusters just above the separatrix band each contain eight nearly degenerate levels and are associated with equivalent paths on the eight C,-symmetric comer hills of the innermost cube in fig. 6.6a. Note that the level curves in figs. 6.6 are equally spaced topography lines and are not actual quantizing paths for J = 60 levels. Hence, the number of paths in figs. 6.6 does not necessarily correspond to the number of levels in fig. 6.7. The cube comers are sharp points which are very different from the rounded hills on the simple fourth-rank RE surface in fig. 3.lb. Furthermore, the corner hills of the inner surface actually meet comer valleys of the middle RE surface to make a conical point. It can be shown
W.G. Harter / Graphical approaches to molecular rotations and vibrations
383
Fig. 6.6 (continued).
[56] that the paths on these corners correspond to states made mostly from combinations of ( II+K,) and 1IIR + 3) where K, is the rotor momentum component (R,) along the C,-axis. There is one state 1n-59) whose mixing partner 1l-I-62) does not exist since its total J,-component ( KJ = A + K, = 61) exceeds J. Its lone level cluster belongs to the straight line crossing the middle of the 3-fold II * cluster region in fig. 6.7b. This “lone state” is a quantum mechanical entity that does not follow the usual RE surface mechanics. All the other Q-II, levels correspond to paths on the cones in fig. 6.6a. These levels appear to “repel” each other in fig. 6.7 because their angular uncertainty angles prevent their paths from approaching the conical intersection. The middle surface for (B[ = 0) in fig. 6.6a has an octahedral shape. Its C, symmetry axes lie in valleys that form eight conical intersections with the lowest energy surface. The C, symmetry axes lie on hills that form six oscullating (i.e., “kissing”) intersections with the outermost and highest energy surface. The C, intersections are surrounded by a very peculiar arrangement of paths belonging to II-states. A careful inspection of fig. 6.6a reveals that the middle surface has three square shaped stable path around each C, axis while the outer surface has no stable C,
384
W G. Harm / Graphicalapproachesto molecularrotationsand vibrations
(a)
r-,
EXACT ,
DIAGONALIZATION ,
,
,
J = 60
lb),
EXACT
DIAGONALIZATION
J=60
Fig. 6.7. J = 60 energy levels of octahedral rotor plotted as a function of B[ corresponding to RE surfaces of fig. 6.6. (a) Higher B{ values yield normal PQR level patterns similar to those in fig. 3.4; (b) low B[ values yield extraordinary XII level patterns and two-fold symmetry clusters (see fig. 6.6).
paths at all. There are only C, symmetric paths on the hills of the outermost surface while a separatrix goes through the C, axes. Another way to plot multiple RE surfaces is shown in figs. 6.8a-d, where the energy is plotted vertically instead of radially. The upper hemisphere of each surface is plotted as a function of projected polar coordinates. The cubic symmetry is not so obvious in these plots, but the surface topography is easier to see. For example, the four C, hills are clearly visible on the top surface of fig. 6.8a. (This corresponds to the outer surface in fig. 6.6a.) The outer surface has only C, paths and only C,-type clusters show up in the highest levels of the (B[ = O)-spectrum in fig. 6.7b. These are Z-type 3-fold symmetry levels which go along with the (C,, II) pairs of levels discussed previously. Directly below the (C,, II)-levels he the (C,-II) levels. However, unlike the (C,-II) stack, the (C,-II) levels only contain II+ level from each IT-pair. The II_ state has no stable paths and no band of quantum levels. Only the band of levels corresponding to III-states belong to stable paths. This is similar to the spin-rotor effects seen in figs. 6.1, 2 for which spin “up” is stable and spin “down” is unstable. The stability is very sensitive to the value of Bc. At point (b) in fig. 6.7 where B{ = 0.5, the upper levels change dramatically. The middle and upper RE surfaces in figs. 6.6b or 6.8b show how the changes come about. The outermost surface has &-paths replaced by Cd-paths which lie above C,-paths. The latter are difficult to see in fig. 6.6b, since much of the outer surface is cut away. The C,-paths are easier to see in fig. 6.8b, and they correspond to 2-fold clusters indicated on the bottom of the upper set of levels above point (b) in fig. 6.7b. Note also, that the 3-fold Z-clusters have changed into a new set of 4-fold symmetry clusters above the 2-fold clusters. Also, the middle surface has a region of instability near the Q-axes and all stable paths surround the C,-axes. At this value of coupling (BS = 0.5) the 2-fold clusters begin to form on the middle surface, they become very strong at B[ = 1.0 as seen by examining figs. 6.6~ and 6.8~ or the (c) point in fig. 6.7b.
W. G. Harter / Graphical approaches to molecular rotations and vibrations
385
a
b
b
Fig. 6.8. Vertical projection RE surfaces for octahedral rotor-vibrator in the low B{ B{ = 0.0; (b) Bl = 0.5; (c) B{ = 1.0; (d) B{ = 1.3.
region. (tZZ41J 1’ =l.O) (a)
It is remarkable to have stable 2-fold or C, symmetry paths on the twelve axes that normally contain saddle points on a fourth rank tensor RE surface. In each case a near degeneracy of order twelve appears in the exact eigenvalue spectrum just as happened for sixth rank tensors in fig. 4.6. Here it happens when the scalar Coriolis coefficient BS is comparable to the fourth tensor coefficient t,,. Coriolis stability on a saddle point is analogous to ion cyclotron stability in a quadrupole field. For values near Bl = 1.5 (see fig. 6.6d or 6.8d), the lower surface becomes nearly spherical and develops stable C, symmetric paths as well as the usual C, and C, trajectories. This corresponds to an extremely crowded band containing all three types of clusters in the lower part of fig. 6.7b. This band crowding was also seen in fig. 6.5 for the spin rotor. As B{ increases further one eventually recovers the normal situation which can be described
386
d
W.G. Harter / Graphical approaches to molecular rotations and vibrations
d
Fig. 6.8 (continued).
accurately by single independent RE surfaces. In fig. 6.7a, one can see the recovery of three separate P, Q, and R bands of the type discussed in section 3. There it was shown that roughly 25% of a band is devoted to C3 clusters and the remaining 75% belongs to the C, clusters with a single C, separatrix in between. This recovery of standard bands is expected, given the form of the representation (6.12). It is worth noting that a revealing RE surface picture is possible even when the level structure is not so simple! 7. Conclusion The adiabatic rotational energy surface has been shown to be a useful visualization and approximation device for semiclassical treatments of diverse rotational and vibrational phenom-
387
W.G. Harter / Graphical approaches to molecular rotations and vibrations
ena. Presently, it complements the usual numerical tour de force diagonalizations on a large basis sets by providing insights into dynamics and spectral phenomena that might otherwise be unexplained or missed entirely. It does this by providing phase portraits of the eigenvector properties as well as those of the eigenvalues. By combining these techniques with wavepacket dynamics, it is now possible to calculate both eigenvalues and eigenfunctions semiclassically. An introduction has been given to some generalizations of the energy concept including vibrational quasi-spin surfaces, generic or ab initio numerically generated surfaces, and multiple Coriolis coupling surfaces. With the concepts, there is possibility for visualization and approximate numerical solution of problems involving highly fluxional molecular or ionic complexes for which the standard basis set analysis may be impossibly complicated.
Acknowledgements I would like to thank Dr. Phil Bunker for inviting me to begin this article and to Dr. Chris Patterson for telling me to get it finished. I would also like to thank Drs. Chris Patterson, Daniel Huber, Nelson De Leon, and Eric Heller for past and present collaborations and many helpful suggestions. For typing and editorial assistance I would like to thank Margot Harter and Lea Hermann. This work is supported in part by the Atomic and Molecular Theory Division of the National Science Foundation (Grant PHY-86-96052), and by Los Alamos National Laboratory Theoretical Division T-12 through a grant from the Department of the Army. Graphics equipment used in the later stages of this project were provided by a grant from the Arkansas Science and Technology Authority (ASTA).
Appendix A. Computation
of rotation matrices and products
The Euler angler parameters are defined here according to conventions found in most texts on rotation groups [57]. A pictorial view of the standard definition is contained in refs. [3] and [38]. All rotations are products of a Z-rotation by y, followed by a Y-rotation by p, and finally, another Z-rotation by cr.
R(% P, Y) = R(% O,~)NO,P?wm 0, VI.
(A-1)
Always the Y and Z are lab-fixed axes. The fundamental SU(2) representation is defined using the same parameters, however the volume of the SU(2) parameter space is twice that of the R(3) space. The fundamental or spinor representation is e-ia/2
@‘2(%8, Y) =
o
i
where
co4P/2)
0
e i42
i sin( b/2) ii
-i
e-iy’2
sin( p/2) cos(P/2)
i(
0
0 eir/2
1
(A-2)
388
W. G. Harter / Graphical approaches to molecular rotations and vibrations
This was used in section 4 to derive the general spinor state in eq. (4.8). For R(3) it is only necessary to consider P-values between 0 and 7~. A binomial expansion [57] of the spinor representation yields the general W-matrix for all integer and half integer values of j. e-_i(ma+my)
(j+m-k)!k!(j-n-k)!(n-m+k)!
k=O
(A-3)
x(COS(j3/2))2’+m-n-2k(COS(~/2))n-m+2k. In computing D ‘, the limits of the sum over k are set first. They are determined requirement that the arguments of the factorials must be non-negative. The limits are max(O, m-n)
by the
j-n).
(A-4)
The resulting sum is all that is needed for the RE surfaces in sections 2. and 3. The wave packet minimum uncertainty function is section 2.4 is
DA( c@y) = e-iJ(u+Y)(cos /3/2)2”.
(A.5)
Other D-functions in the wavepacket calculation follow easily from (A.3). An expansion of the spin-l/2 rotation matrix (A.2) in terms of Pauli spin matrices provides a convenient way to compute rotation (or SU(2)) group products. The desired expansion is the following.
W2(a, j3,
y) = exp( T)
cos( t)(
i
-isin
sin(f)(~
-i
cos( t)(
sin(T)
y) k)-ices(y)
t
sirr($
-d)
_!I)).
If one wishes to combine two rotation operators R( a,a,a,)
(A.6) and R( b,b,b,)
into a product
then the expansion (A.6) of each operator has the following forms. R(u,u,u,)=A,I-iA*a=A,I-i(A,u,+A,u,+A,u,),
(A.8a)
R(b,b,b,)
= BoI - i(BXuX + BYuY+ B,u,),
(A.8b)
R( c1c2c3) = Co1 - iC* u = Co1 - i( C,u, + CYuY+ C,u,).
(A.%)
= B,I - iB*u
389
W.G. Harter / Graphical approaches to molecular rotations and vibrations
The coefficients identity.
in the third expansion are computed from those of the first two using Pauli’s
[A*a][B*a]
=A*B+i(A
XB)=a.
(A-9)
The results are as follows. c, = cos[ c/2]
cos[(c3+c&2]
(A.lOa)
=A,B,-A,B,-A,B,-A,B,,
C, = sin[ c,/2] sin[(c,-c,)/2]
=A,Bo+AoB,+A,B,-A,By,
(A.lOb)
C, = sin[ c/2]
COS[($-cr)/2]
=A,B,+A,B,-A,B,+A,B,,
(A.lOc)
c, = cos[ c/2]
sin[(c,+c,)/2]
=A,B,+A,B,+A,B,-A,B,.
(A.lOd)
The relation between the Euler angles and the four-vector (C,,, C) follows from eq. (A.6), and applies to the A and B quantities as well. The inverse of this supplies the desired Euler angles in terms of the four-vector components. c2 = 2 cos-l[
cr=c+-c_,
C,/sin
c,] ,
c3 =
c,
+
(A.lOe)
c_
Here c, and c_ are expressed in terms of a four-quadrant FORTRAN ATAN ~+=(c,+cr)/2=tan-~[C,/C~],
arctangent
function such as the
C-=(c3-c1)/2=tan-‘[C,/C,].
Appendix B. Body frame transformations
(A.lOf)
and momenta
The angular momentum vector operators may be expressed in terms of lab-fixed components { JxJyJz} or in terms of body-fixed components { J, Jy J,}. The two systems are related by Euler rotation given by eq. (A.l) using the Cartesian vector representation. cos (Y (R(a,B,y))=
(
sina 0
-sin a cosa 0
0 0 1 Ii
msP 0 -sin/3
1 0
x
= x
cosacos~cosy-sinasiny
Y sinacos/3cosy+cosasiny Z i
-cm
y sin /I
sin /3 0 cos/3 ii
cos y
-siny
0
sin y 0
cos y
0
0
1i Z
Y -cosacos/3sin~-sinacosy
cm d: sin /3
-sinacosfisiny+cosacosy
sin a sin /3
sinysinp
cos /!3
’
(B.1)
i
leads to the following relation [3] between the lab-fixed components of angular velocity ( wx, wY, wz) and Euler angle velocities { ai, p, i, } . This
-sin
(Y
*X
0
WY =
0
cos (Y
cos (Y sin p sin (Ysin p
OZ
1
0
cos j3
III
& fi 11 9 1
.
(B-2)
390
W. G. Harter / Graphical approaches to molecular rotations and vibrations
The body-fixed components of o involve the internal polar angles ( - /3) and ( - y) only *X WV =
- cos y sin p
sin y
0
d!
sin y sin fl
cos y
0
#8 .
cos p
0
ij(@z
(B-3)
1 )i 9 i
These matrices contain Jacobian transformation components. The inverse matrices coefficients for relating Cartesian angular momenta to Euler momenta.
contain
The lab-fixed momenta depend on Euler angles a and p I-cosacotp
‘Jx
4
-sin
\A
\
a cot /I 1
- sin a
cos a/sin /3 i J,
cos a
sin a/sin
0
/3
Jp _
0
)I
(B.9
J-r1
The body-fixed momenta depend on the body-defined polar coordinates ( - y) and ( - p) 1 - cos y/sin /3
Jy JZ I*)
sin y/sin /_I
= \
0
sin p
cos ycot p
cos y
-sin
0
(J,
y cot /3 1
Js .
(B.6)
I J-7I
The commutation relations are different for each set of momenta. [ J,J,]
= iJ, etc,
[ JxJy] = -iJ,
etc.,
[J,,
Jo] = 0 etc,
(B.7)
The Euler momenta are completely mutually commuting. Euler coordinates are holonomic and canonically conjugate to the momenta { J, JaJy }. There are no holonomic coordinates conjugate to Cartesian momenta.
Appendix C. Classical equations of rotational motion Numerical solution of classical rotational equations is generally easier to accomplish using Euler coordinates ( a/3y} and momenta { J, Ja Jy } . The relations in appendix B are used to convert Cartesian angular velocities, and momenta into Euler quantities. In some cases the analytic expressions are more revealing, too. The Hamiltons equations for Euler angles S = a, p, y are as follows
W. G. Harter / Graphical approaches to molecular rotations and vibrations
For the rigid symmetric top Hamiltonian aH
2B(Ja
-Jy
(2.1) with A = B # C this become the following
~0s P)
sin*P
&=aJ,=
(C.2a) ’
. aH p=aJ,=2BJa,
aH
391
(C.2b)
-2B(Ja-J,cosP)cosP
‘=aJ,=
sin*p
+2ccos J
Y’
(C2a)
The equations simplify considerably if the J vector is assumed to lie along the lab axis. If we let J,=J,=J,
Ja=O,
J,=J,=Jcosp,
(C-3)
then the coordinate equations yield ai = 2BJ,
,d=O,
jl=2(C-B)Jcosp.
(C.4)
The y-equation describes the classical precessional motion discussed after eq. (2.10). The motion of azimuth ( - y) is negative (clockwise) if C > B and J cos j? > 0. The equations for Euler momenta and the action are used in eq. (2.44)
The righthand expression is valid only if the Hamiltonian is an explicit function of body components of Cartesian momenta only. This is the case for rotors free of external torques. External fields may introduce an explicit coordinate dependence. This requires H to be re-expressed in terms of Euler coordinates and momenta before the coordinate derivitive in eq. (C.5) is evaluated. Numerical solutions for general free rotors using eqs. (C.l) and (C-5) are easier to program if the Cartesian A-derivatives of the Hamiltonian are evaluated separately in a numerical subroutine. This makes it easy to change the Hamiltonian or to derive the classical mechanics. It also facilitates the use of general Hamiltonians to obtain classical equations of motion directly from an ab initio RE surface. However, analytic expressions such as were given in the examples by eqs. (C.2) will usually run faster.
Appendix D. Tensor operator construction The members of a set of r th rank irreducible tensor components { c, T,‘_i, . . . , Tl,} are each composed of certain polynomial combinations of r th degree monomial products V;:,V;.,. . . V;.r of vectors. Generally, a vector set { V,, VY, VY} or, equivalently, a rank-l tensor set { V:, Vi, I? 1} is
W.G. Harter / Graphical approaches to molecular rotations and vibrations
392
the starting point for the development of higher rank tensors. It is first necessary to specify the relation between Cartesian or “linearly polarized” {x, y, z}-components and the angular or “circularily polarized” { 1, 0, - 1}-components as follows
The phase convention is the standard Condon-Shortley convention for spherical harmonics YA = { Yt, Y,‘, Yyl}. Examples of vectors V include operators of spatial position r = {x, y, z}, angular momentum J = { J,, Jy, J,}, or three-dimensional vibrational boson creation operators a+={,+ .+ a! }. The fundamental symmetry properties of rank-r tensor operators are that they tral;lsfo’rm according to the following D-matrix form
Rank-2 tensor operators T2 may be made from Clebsch-Gordan vector operators V(a) and V(b) as follows T4’= [V’(a)
Q V’(b)];=
~C;,~2,V&z)V,‘2(b).
products of any pair of
(D-2)
..’ “’ are the standard Clebsch-Gordan angular momentum coupling Here the coefficients C~J,,j,~f coefficients for coupling outer products of states 1;) and I,$) into a state I( j X j’);:!) of definite total angular momentum j”.
Examples of rank-2 tensor products are the following. T2 = J7’JT1 2
The 4 = 0 also used operators n = C&z, vibrational T4’=
1
1,
T2, = V:,V’,,
and 4 = + 2 terms are proportional to the tensors in eq. (2.2). The rank-2 tensors are in eq. (6.9). For the latter application one needs the fact that boson destruction al, transform like a!,( - 1)“. This is necessary in order that the number operator is a scalar operator. This implies that the diagonal (number-preserving) rank-2 tensor operators are constructed using the following form of eq. (D.2)
[u’+@ul]Z,=cc~~~,,u~:U’,,(-1)42.
tw
W. G. Harter / Graphical approaches to molecular rotations and vibrations
393
Higher rank tensors are derived in a similar manner. For example, rank-4 tensors could be made from products of any two rank-2 factors as follows
(D-6) This type of tensor product is used explicitly in eqs. (6.8) to (6.10). This structure is implicit in the simple rank-4 tensors which are described in section 3. Other examples are described by Hecht [ 161.
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