On pseudorotation

Chemical Physics 349 (2008) 9–31

Contents lists available at ScienceDirect

Chemical Physics journal homepage: www.elsevier.com/locate/chemphys

On pseudorotation Alexander F. Sax * Institut für Chemie, Karl-Franzens-Universität Graz, Strassoldogasse 10, A-8010 Graz, Austria

a r t i c l e

i n f o

Article history: Received 1 December 2007 Accepted 6 February 2008 Available online 13 March 2008 Dedicated to Hans Lischka on the occasion of his 65th birthday. Keywords: Pseudorotation Jahn–Teller Ring systems Cage systems

a b s t r a c t Pseudorotation was introduced as a molecular motion related to degenerate ring deformation modes in ring systems. This motion is related to the motion of a particle on a ring and is, therefore, called a onedimensional pseudorotation. This motion is related either to ring puckering or to the fluctuation of weak bonds. It is shown that there are systems where bond fluctuation occurs on the surface of polyhedra which are related to the motion of a particle on a sphere. We call these bond fluctuations two-dimensional pseudorotation and give the energy expressions for a free two-dimensional pseudorotation. One-dimensional pseudorotation is thoroughly discussed and shown how one can calculate the symmetry number of pseudorotation. We discuss systems where two-dimensional pseudorotation can be expected such as bullvalene or vacancies in solids with strong bond fluctuations. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Pseudorotation (PR) is a special form of anharmonicity of molecular vibrations. The phenomenon was first described in 1947 by Kilpatrick, Pitzer and Spitzer (KPS) [1] who wanted to give an explanation for the diffuseness of the symmetric C–C stretching mode in the vibrational spectrum of cyclopentane and why the experimental entropy and specific heat of cyclopentane was smaller than predicted by the harmonic oscillator-rigid rotator model. The difference for both quantities was consistent with the loss of one degree of freedom in the vibrational contributions to the partition function. KPS gave the following explanation: one of the two degenerate pucker modes in cyclopentane mutates from a vibration to a nuclear motion which has the properties of a rotation. This is because the planar D5h structure of cyclopentane is not a local minimum but a second-order saddle point, ring puckering along these two modes yields several energetically equivalent low lying ring conformations. Force field calculations gave evidence that the change between these conformations caused by a sequence of deformations of bond lengths and angles as described above can be compared to a nearly free rotation of a particle on a ring. Therefore, this nuclear motion was dubbed pseudorotation. This usage of the term pseudorotation refers to both, the description of structure changes due to a sequence of deformations of bond lengths and bond angles in ring systems which give the impression of a rotation of the whole molecule, and the thermo* Tel.: +43 316 380 5513. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2008.02.060

chemical and spectroscopical effects that can be experimentally measured. Since the publication of the KPS paper the concept of pseudorotation was used in investigations on other cycloalkanes or heterocyclic organic molecules like sugars [2], later it was also used for planar molecules such as the alkali trimers (see, e.g. Ref. [3] or [4]) or the benzene cation [5,6]. These molecules are all planar Jahn–Teller (JT) systems. Whereas for stable molecules like cyclopentane it is possible to measure not only spectroscopical data but also thermodynamic properties like the specific heat or thermodynamical potentials which depend on the entropy, most of the mentioned JT systems are so reactive or unstable that only spectroscopic data are available. But all molecules to which the concept of pseudorotation was applied have in common that they are floppy meaning that there are internal nuclear motions which are not small vibrations about one equilibrium structure but large amplitude motions connecting several isoenergetic or nearly isoenergetic equilibrium structures which are separated by very small saddle points, or where saddle points do not exist at all. Such a situation is according to Bersuker [3] typical for systems where symmetry breaking due to the coupling of electron and nuclear motions (vibronic coupling) leads to structures of lower spatial symmetry. That this can happen there must be a molecular structure for which the electronic state (in the Born–Oppenheimer approximation) is degenerate due to a high spatial symmetry of this structure which is labeled HSS (highly symmetric structure) in this paper. The distorted structures can sometimes be observed but frequently they are connected by internal motions so that the

10

A.F. Sax / Chemical Physics 349 (2008) 9–31

molecular structure is not static but dominated by dynamic effects. Such internal motions are called pseudorotations. Besides cyclic JT systems with planar HSS there are cage-like molecules for which JT effects are well known like, e.g. the Cþ 60 cation [7]. Therefore PR should also be observed in such molecules. Another example for PR in cage molecules was given by Manz [8] who described a rotating CO molecule in a solid argon matrix where the Ar atoms forming the cage perform synchronous pseudorotations. Treatment of the JT effect is dominated by symmetry considerations of the HSS and of the nuclear motions which are responsible for the vibronic coupling and for the number and the type of the equilibrium structures. This is relatively easy for cyclic molecules but not at all trivial for three-dimensional molecules. Therefore, most of this paper deals with PR in cyclic molecules. The paper is organized as follows:  In the first part the vibrational problem is discussed for the twoand three-dimensional isotropic double minimum oscillators which serve as a model for certain pseudo Jahn–Teller systems (PJT).  In the second part vibrations of a regular N-sided polygon are discussed, which is a model for any N-membered ring molecule.  Finally, an outlook to pseudorotations in three-dimensional systems is given.

The Born–Oppenheimer approximation by which electronic and nuclear motions in a molecule are separated is the basis for a qualitative and quantitative treatment of nuclear motions in a molecule. Solving the electronic problem for fixed nuclear configurations gives the electronic energies of the different electronic states as a function of the nuclear configurations which are also called potential energy surfaces of the electronic states and which act as the potential energy for the nuclear motions in the respective electronic state. A HSS is described by position vectors ðR1 ; . . . ; RN Þ in a space fixed cartesian coordinate system, and the description of distorted nuclear configurations needs displacement vectors ðn1 ; . . . ; dN Þ of the vertices which may be represented by cartesian coordinates ðd1 ; . . . ; d3N Þ ¼ ðx1 ; y1 ; z1 ; . . . ; xN ; yN ; zN Þ with respect to a body fixed coordinate system with basis vectors ðe1 ; . . . ; e3N Þ ¼ ðex;1 ; ey;1 ; ez;1 ; . . . ; ex;N ; ey;N ; ez;N Þ. The Hamiltonian for the nuclear motions about the HSS ! 2 N h X  1 o2 o2 o2 b þ Vðx1 ; y1 ; z1 ; . . . ; xN ; yN ; zN Þ ð1Þ þ þ H¼ 2 I¼1 mI ox2I oy2I oz2I is in general not separable. For small vibrations the potential which is a general quadratic form in the displacement coordinates (harmonic potential) vij di dj

3N X o2 i¼1

ð2Þ

i;j

can be transformed into a sum of purely quadratic terms by the introduction of new coordinates which describe collective displacements of all atoms. This is done by simultaneous diagonalization of the coefficient matrices of the kinetic and potential energies which gives the eigencolumns Q ¼ ðq1 ; . . . ; q3N Þ with which we construct the normalized mass weighted vibrational eigenvectors qi ; i ¼ 1; 2; . . . ; 3N as new basis vectors in the 3N-dimensional vector

þ 2

oQ i

3N X

V i ðQ i Þ ¼

i¼1

3N X

bi H

ð3Þ

i¼1

The fi are the harmonic force constants with respect to the normal coordinates Q i . If the symmetry group of the molecular structure at the HSS contains at least the cyclic group CN ; N P 3 as a subgroup there will be doubly degenerate normal modes; if the symmetry group of the molecule has at least the cubic group T as a subgroup there will be triply degenerate normal modes. Systems with only doubly degenerate vibrations can be thought of being regular N-sided polygons, systems with triply degenerate vibrations may be regular polyhedra with cubic symmetry. We assume that the HSS has a symmetry lower than icosahedral, therefore the harmonic vibrational Hamiltonian can be written as a sum of Hamiltonians for non-degenerate, as well as doubly degenerate and triply degenerate isotropic oscillators, in the following also called the subsystems, 1-deg X

bs þ H

s

2.1. Implications of the Born–Oppenheimer approximation

3N X

2

 b ¼ h H 2

b ¼ H

2. The vibrational Hamiltonian

Vðd1 ; . . . ; d3N Þ ¼

P space, qi ¼ j qji ej in which we represent normal modes Q i ¼ Q i qi by means of normal coordinates Q i ; i ¼ 1; 2; . . . ; 3N. In the ith normal mode all atoms perform in-phase vibrations with the same frequency mi along straight lines Q i ¼ Q i qi ei2pmi t : If degenerate normal vibrations superimpose with a different phase one gets nuclear motions where the atoms do no longer move along straight lines. P In the basis of qi the potential is V ¼ 3N i¼1 V i ðQ i Þ with 2 V i ðQ i Þ ¼ fi Q i and the Hamiltonian is separable

2-deg X

bs þ H

s

3-deg X

bs H

ð4Þ

s

where the index s labels the different oscillators in each class. The vibrational eigenvectors of the doubly and triply degenerate oscillaðaÞ ðbÞ ðaÞ ðbÞ ðcÞ tors will be labelled qs ; qs , and qs ; qs ; qs , respectively. The potential is the sum V¼

1-deg X

V s ðQ s Þ þ

2-deg X

s

ðbÞ V s ðQ ðaÞ s ; Qs Þ þ

3-deg X

s

ðbÞ ðcÞ V s ðQ ðaÞ s ; Qs ; Qs Þ

ð5Þ

s

where ðbÞ ðaÞ2 V s ðQ ðaÞ þ Q sðbÞ2 Þ s ; Q s Þ ¼ fs ðQ s

ð6Þ

or ðbÞ ðcÞ ðaÞ2 V s ðQ ðaÞ þ Q ðbÞ2 þ Q ðcÞ2 Þ: s ; Q s ; Q s Þ ¼ fs ðQ s s s

ð7Þ

It is always possible to introduce in the two- and three-dimensional subspaces curvilinear coordinates such as polar coordinates in the two-dimensional spaces or spherical coordinates in the threedimensional spaces. If the two-dimensional space is spanned by the vibrational eigenvectors qðaÞ and qðbÞ the relation to the coordinates r and / is Q ðaÞ ¼ r cos /

Q ðbÞ ¼ r sin /

ð8Þ

r describes the overall deviation from the HSS during a normal mode along a line with direction vector qðaÞ cos / þ qðbÞ sin /, / describes the rotation of this line about the origin of the coordinate system (passive transformation) or the change of the molecular shape during the degenerate normal mode with constant r (active transformation). In the three-dimensional space of normal coordinates we introduce spherical polar coordinates by Q ðaÞ ¼ r sin h cos /

Q ðbÞ ¼ r sin h sin /

Q ðcÞ ¼ r cos h:

ð9Þ

r gives again the deviation from the shape of the HSS during vibrations along a line with direction vector qðaÞ sin h cos / þqðbÞ sin h sin / þ qðcÞ cos h, while the angles h and / describe the rotation of this line about the origin in the coordinate system, or the change of the molecular shape during the degenerate normal

A.F. Sax / Chemical Physics 349 (2008) 9–31

mode. The potential for doubly degenerate vibrations is in general anisotropic, it may depend explicitly and implicitly on the polar angle, V ¼ Vðrð/Þ; /Þ but because of the degeneracy of the vibration there must be a CN symmetry with N P 3 and, consequently, the potential must be periodic in the angle /   2p ¼ Vðr; /Þ; n ¼ 0; 1; 2; . . . ð10Þ V r; / þ n N with period 2p . The potential becomes exactly isotropic, V ¼ VðrÞ, N when N goes to infinity, that means when the polygon becomes a circle. For finite N the potential may be approximately isotropic. The eigenfunctions of the Hamiltonian of such an isotropic twodimensional oscillator have the form wðr; /Þ ¼ RðrÞeim/ ; m ¼ 0; 1; 2; . . ., the radial part RðrÞ depends on the potential. 2 Only for the isotropic harmonic oscillator with VðrÞ ¼ kr =2 the radial part RðrÞ can be given in closed form. The anisotropic potential for triply degenerate vibrations depends implicitly and explicitly on the angles Vðrðh; /Þ; h; /Þ, the potential must be periodic with respect to each of the symmetry axes, and their number and order depends on the symmetry group of the polyhedron. The potential is exactly isotropic only when the regular polyhedron becomes a sphere that is again for N ! 1. Again, for finite N the potential may be approximately isotropic. For an isotropic potential V ¼ VðrÞ the eigenfunction has product form wðr; h; /Þ ¼ RðrÞY m l ðh; /Þ; l ¼ 0; 1; 2; . . . ; m ¼ 0; 1; 2; . . ., only for the harmonic oscillator potential the radial part can be given in closed form. Any potential at the HSS which is sufficiently often differentiable can be expanded into a Taylor series. Vðd1 ; . . . ; d3N Þ ¼

3N 3N X X o2 V o3 V di dj þ di dj dj þ    odi odj odi odj odj i;j i;j;k

ð11Þ

Since a harmonic potential is the basis of the concept of normal coordinates, one might assume that any kind of anharmonicity destroys the separability of the potential and thus the reduction of the full Hamiltonian to a sum of low dimensional Hamiltonians. Nevertheless, experience shows that the anharmonicity in the potential can frequently be a treated as anharmonicities of the subsystems, so the potential can be approximated by a power expansion with respect to the normal coordinates 3N X i;j;k

3N X o3 V o4 V di dj dk þ di dj dk dl    odi odj odk od od i j odk odl i;j;k;l



3N X 4 ðaQ 3i þ bQ i þ   Þ

ð12Þ

i¼1

2.2. Vibronic coupling, Jahn–Teller, pseudo Jahn–Teller A special form of anharmonicity results from vibronic coupling. The separation of nuclear and electronic motions is always only approximately valid, in principle they are always coupled. In the Born–Oppenheimer approximation one says that two or more electronic states may be coupled by normal modes. The energies of the new states yield potential energy surfaces which are qualitatively different from those of the original Born–Oppenheimer surfaces. Which electronic states are coupled by which normal modes is determined by their spatial symmetries at the HSS. The order of the vibronic coupling depends on the power of the normal mode in the coupling terms. One speaks of a Jahn–Teller system when the electronic state at the nuclear configuration of the HSS is f -fold degenerate, the corresponding f Born–Oppenheimer potential energy surfaces intersect then in a single point. The symmetry of the normal modes which couple the components of the degenerate electronic state,

11

called the JT active mode, is determined by the requirement ðCelec Þ2  Cvib that the irreducible representation of the normal modes Cvib is contained in the product representation of the degenerate electronic state Celec . For such a JT system the notation Celec  Cvib is used. The f new potential energy surfaces show a conical intersection at the nuclear configuration of the HSS so they are not stationary at this point. Local minima are always at nuclear configurations with a lower spatial symmetry than the HSS. The most simple and important case is the E  e JT system, for linear coupling (linear JT) the potential energy surfaces are isotropic. With polar coordinates we get V  ðrÞ ¼

x 2 r  kr 2

ð13Þ

If k ¼ 0 one has an isotropic harmonic oscillator with frequency x. These potentials are at the HSS not stationary. Only when quadratic or higher order coupling terms are used the potential V depends on the polar angle / [3], e.g. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 V  ðr; /Þ ¼ r2  r k þ c2 r2 þ 2kcr cos N/ ð14Þ 2 If at the HSS two or more electronic states which are not degenerate are coupled by normal modes, vibronic coupling gives new potential energy surfaces without conical (but with glancing) intersection. This kind of vibronic coupling is typical for pseudo Jahn– Teller systems. The symmetry requirement for the coupling of electronic states A and B by a normal mode is CðAÞ  CðBÞ  Cvib . For PJT systems the notation ðCðAÞ þ CðBÞÞ  Cvib is used. The new potential energy surfaces are stationary at the HSS. In contrast to JT systems the HSS in PJT systems is not automatically destabilized, but it depends on the strength of the coupling and the energy gap between the electronic states [4]. For a linear PJT system ðA þ EÞ  e, where a non-degenerate ground state of symmetry A and a degenerate excited state ðEÞ are coupled by a degenerate normal mode e, the energies are x 2 r þ E 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r x E þ A E  A 2  þ k2 r 2 V  ðrÞ ¼ r2 þ 2 2 2

V 0 ðrÞ ¼

ð15Þ ð16Þ

In this case the lower energy surface V  coincides for r ¼ 0 with the non-degenerate surface A , the higher surface V þ coincides with the degenerate surface E and is degenerate with V 0 . Both V  and V þ depend on the origin quadratically on r. Only when V  has at the HSS (r ¼ 0) a local maximum the system will have more stable structures with lower symmetry than the HSS. The condition is k2 > xðE  A Þ=2. A typical example for such PJT systems is cyclopentane [3] which in the planar conformation with D5h symmetry (HSS) has a non-degenerate electronic ground state and several degenerate excited states. The dependence of V on r is shown for JT and PJT system in Fig. 1.

Fig. 1. The radial dependence of the lower potential energy surface for Jahn–Teller and pseudo Jahn–Teller systems at a highly symmetric structure.

12

A.F. Sax / Chemical Physics 349 (2008) 9–31

can be regarded as composed of two independent systems, a harmonic oscillator with radial vibrations about r 0 and a particle rotating on a ring with radius r 0 ! 2 2 h  o2 1 o k h b2  2 b þ ðr  r þ Þ ð18Þ H¼ L  0 2l or2 r or 2 2lr 2 / Inserting the wave function wðr; /Þ ¼ RðrÞeim/ into the Schrödinger equation we get a differential equation for RðrÞ ! ! 2 2 2 h d RðrÞ 1 dR  h  m2 k 2 þ ðr  r R ¼ ER ð19Þ  þ þ Þ 0 dr2 r dr 2l 2lr 2 2 by using the ansatz RðrÞ ¼ r 1=2 SðrÞ this transforms into a differential equation for SðrÞ  2 2  h  1  h 1 k S þ ðr  r0 Þ2 S ¼ ES m2   S00 þ 2 ð20Þ r 2l 4 2 2l With x ¼ r  r 0 and d=dr ¼ d=dx we get ! ! 2 2 h d SðxÞ  m2 k 2 x  þ þ SðxÞ ¼ ESðxÞ dx2 2l 2ðr0 þ xÞ2 2

Fig. 2. Potential energy surface for an anisotropic two-dimensional double minimums oscillator showing the local maximum at the origin of the coordinate system and the equivalent local minima separated by saddle points (top) and the bottom line of the moat (bottom).

For higher order coupling the energy depends also on the polar angle and the potential energy surface is of the form shown in Fig. 2. 2.3. Isotropic anharmonic oscillators We consider now only that kind of anharmonicity that is represented by a double minimum potential and that is found for PJT systems as discussed above. 2.3.1. The two-dimensional double minimum oscillator The potential VðrÞ for the isotropic oscillator has a local maximum at r ¼ 0, the bottom line of the moat (Fig. 2) is a circle and the potential at the bottom is constant. A convenient representation of the r-dependence of the potential is a quadratic–quartic polynomial 4

VðrÞ ¼ ar 2 þ br ;

a; b; > 0

ð17Þ

a two-dimensional isotropic oscillator with such an r-dependence of the potential was dubbed an isotropic double minimum oscillator [9]. Since any power of r higher than two in the potential destroys the unitary symmetry of the underlying isotropic harmonic oscillator, coordinates as well as momenta can only be transformed among themselves. Consequently the degeneracy of the energy levels will be lower. Eigenvalues of anharmonic oscillator must in general be calculated with approximate methods, a variational treatment with the eigenfunctions of the two-dimensional isotropic harmonic oscillator as basis was, e.g. done by Harris et al. [9,10] and Davidson and Warsop [11]. However, for high central barriers at r ¼ 0 the system

ð21Þ

which is the Schrödinger equation of a perturbed one-dimensional harmonic oscillator. The unperturbed energies are   1 ; n ¼ 0; 1; 2; . . . ð22Þ En ¼ hm n þ 2 qffiffi with 2pm ¼ lk. 1 1 x Taylor expansion of the perturbation ðr þxÞ 2 ¼ r 2 ð1  2 r þ   Þ 0 0 0 and taking only the constant term as perturbation yields the first-order energy correction 2    2 1 h  h  S0  2 S0 ¼ ð23Þ 2l 2I r0 where I ¼ lr20 is the moment of inertia of the pseudorotation. With B ¼ 8ph2 I we get for the approximate energies [1]     1 1 þ hB m2  ; n ¼ 0; 1; 2; . . . ; En;m ¼ hm n þ 2 4 m ¼ 0; 1; 2; . . .

ð24Þ

The vibrational energy levels are equally spaced, and the pseudorotational levels are quadratically spaced and doubly degenerate for all m 6¼ 0. With these energies the partition function of the isotropic two-dimensional double minimum oscillator is approximated by the product of those of a one-dimensional harmonic oscillator and of a modified particle on a ring rffiffiffiffiffiffiffiffi 1 pkT 1 hB=4kT ð25Þ q ¼ qHO1D  qPR  e 1  ehm=kT hB r The motion of the particle on the ring is a one-dimensional pseudorotation (1D-PR) and r is the symmetry number. For the isotropic oscillator r is 1. Compared with a two-dimensional isotropic harmonic oscillator the energy of the system in the high temperature limit hEHO1D i þ hEPR i 

3kT 2

ð26Þ

shows a loss of kT=2 or the contribution of one degree of freedom and the entropy of the system is     kT k pkT þ 1 þ log SðTÞ ¼ k 1 þ log ð27Þ hm 2 hBr2 From all system-dependent quantities in the partition function the pseudorotational constant B is most difficult to calculate because it contains the reduced mass of the pseudorotation and the displacement r 0 from the shape of the regular polygon, which are both un-

13

A.F. Sax / Chemical Physics 349 (2008) 9–31

known. The smaller the ring system is, the less is the assumption of isotropy of the potential justified. Therefore, for small ring systems direct determination of the pseudorotational constant, as it was done by Katzer and Sax [12], is preferable. One may wonder if pseudorotation in JT and PJT systems is different. For E  e JT systems one can derive an approximate energy expression   1 2 þ hBm ; n ¼ 0; 1; 2; . . . ; En;m ¼ hm n þ 2 m ¼ 1=2; 3=2; 5=2; . . .

ð28Þ

where the second quantum number has not integer but half integer values which is the result of explicit accounting for the Berry phase [3]. All pseudorotational levels are doubly degenerate and the spacing is different from those in PJT systems. The thermodynamic properties in the high temperature limit obtained with both energy expressions are, however, the same and there should be no differences between the thermodynamic effects caused by pseudorotation.

tem for which the masses of all ring members are equal so that they need not be explicitly taken into account but can be set to unity. The side length of the polygon is l, vertices are numbered in positive sense and have position vectors RI ¼ Rðcos aI; sin aI; 0Þ; I ¼ 0; . . . ; N  1, where a ¼ 2p and R ¼ 2l sin 2a. N Two local right handed cartesian coordinate systems (Fig. 3) are frequently used to describe the displacements dI of the vertices. In CS1 the basis vectors ðiI ; jI ; kI Þ; I ¼ 0; . . . ; N  1 are parallel to the basis vectors of the space fixed coordinate system, the coordinates of the displacements are dI ¼ ðxI ; yI ; yI ÞT ; I ¼ 0; 1 . . . ; N  1. In CS2 each vertex has its own frame with unit vectors ðrI ; tI ; vI Þ; I ¼ 0; . . . ; N  1 and the displacement coordinates are nI ¼ ðrI ; t I ; vI ÞT ; I ¼ 0; 1 . . . ; N  1. The basis vectors normal to the polygon are in both coordinates systems the same, vI ¼ kI . The transformation from CS1 to CS2 is done with matrix T T ¼ diagðTð0Þ; Tð1Þ; . . . ; TðN  1ÞÞ where  TðIÞ ¼

2.3.2. The three-dimensional double minimum oscillator One can treat the three-dimensional problem in an analogous manner. Here, we treat only free pseudorotation on a sphere and harmonic oscillations about r 0 . The Hamiltonian for this system is ! 2 2  o2 2 o k h b2  b ¼h þ ð29Þ þ ðr  r0 Þ2  L H 2 r or 2 2l or 2lr2 With the ansatz wðr; /Þ ¼ RðrÞY m l ðh; /Þ for the wave function we get the Schrödinger equation for RðrÞ ! ! 2 2 2 h d RðrÞ 2 dR  h lðl þ 1Þ k  2 þ ð30Þ  þ þ ðr  r 0 Þ R ¼ ER dr 2 r dr 2lr2 2 2l and with the ansatz RðrÞ ¼ r 1 SðrÞ we finally get a differential equation for SðrÞ 2



2

 00 1 h h  k S þ 2 lðl þ 1ÞS þ ðr  r 0 Þ2 S ¼ ES r 2l 2 2l

With the energies   2 1 h  þ lðl þ 1Þ; En;l ¼ hm n þ 2 2I

n ¼ 0; 1; 2; . . . ;

ð31Þ

l ¼ 0; 1; 2; . . . ð32Þ

the partition function is the product of that of a one-dimensional harmonic oscillator and that of a particle on a sphere HO1D

q¼q

PS

q

1 kT ¼ 1  ehm=kT hBr

ð33Þ

The motion of the particle on the sphere is a two-dimensional pseudorotation (2D-PR) and r is the symmetry number. For the isotropic oscillator r is 1. The energy is in the high temperature limit hEHO1D i þ hEPS i ¼

hm þ kT þ kT  2kT 2

ð34Þ

ð36Þ

DðIÞ

0

0

1

 ;

I ¼ 0; 1 . . . ; N  1

ð37Þ

where DðIÞ is the rotation matrix   cos aI  sin aI DðIÞ ¼ ; I ¼ 0; 1 . . . ; N  1 sin aI cos aI

ð38Þ

therefore, the basis transformation at the Ith vertex is ðrI ; tI ; vI Þ ¼ ðiI ; jI ; kI ÞTðIÞ

ð39Þ

and the transformation of the displacement coordinates is done by nI ¼ TT ðIÞdI

ð40Þ k

CS2 has the advantage that any cyclic permutation P of the polygon Pk RI ¼ RIþk (indices are calculated modulo N) permutes only like basis vectors, P k rI ¼ rIþk , P k tI ¼ tIþk and Pk vI ¼ vIþk which form, therefore, three sets of equivalent objects. In the following, CS2 will be used throughout. Given a purely quadratic potential in the 3N cartesian displacement coordinates ni ; i ¼ 1; 2; . . . ; 3N, where ni can be any of ðr I ; tI ; vI Þ, I ¼ 0; . . . ; N  1, and Hn is the ð3N; 3NÞ Hessian matrix of the second derivatives with respect to ni , the potential can be written as ! 3N 1X o2 V n n ¼ nT Hn n ð41Þ V¼ 2 i;j¼1 oni onj i j In the case of small vibrations of the ring system, one can assume that out-of-plane or vertical vibrations, also called ring pucker vibrations, are decoupled from in-plane vibrations. So the ð3N; 3NÞ Hessian matrix Hn can be rearranged to a ð2N; 2NÞ block matrix Hip for the in-plane displacements and an ðN; NÞ block matrix Hop for the vertical displacements, ! 0 Hip ð42Þ Hn ¼ 0 Hop

which means a loss of kT, that is of two degrees of freedom, compared to the three-dimensional isotropic oscillator. The entropy of system is     kT kT þ k 1 þ log ð35Þ SðTÞ ¼ k 1 þ log hm hBr

3. Normal modes of a regular N-sided polygon We discuss now harmonic vibrations of a regular N-sided polygon which is a model for any homonuclear N-membered ring sys-

Fig. 3. (Left) Local coordinate system CS1 with unit vectors i; j parallel to the laboratory coordinate system. (Right) Local coordinate system CS2 with radial and tangential unit vectors r and t.

14

A.F. Sax / Chemical Physics 349 (2008) 9–31

Other ð3N; 3NÞ matrices related to the displacement coordinates can be analogously partitioned, for example the transformation matrix T ! 0 Tip T¼ ð43Þ 0 Top ip

Molecules like cyclohexane or cyclopentane have puckered structures, therefore, the planar HSSs which correspond to regular N-sided polygons must have Hessians with at least one negative eigenvalue k, tor

str

tor

str

kðs; k ; k Þ ¼ kstr ðsÞ þ ktor ðsÞ ¼ k AðsÞ þ

k

l

2

BðsÞ;

where T is a block diagonal matrix with the rotation matrices DðIÞ as blocks

s ¼ 2; . . . ; N  1

Tip ¼ diagðDð0Þ; Dð1Þ; . . . ; DðN  1ÞÞ

The eigenvalues k depend on the index s and on the force constants of the bond stretch and torsion potentials. Since both AðsÞ P 0 and BðsÞ P 0 (see Appendix B), it is obvious that pure bond stretch or str pure torsional model potential with positive force constants k or tor k has only positive eigenvalues. If the Hessian is to have both, positive and negative eigenvalues, the model potential must be a sum of bond stretch and bond torsion contributions and the force constants must have different signs. For physical reasons the bond stretch force constant of a stable molecule must be positive, str k > 0, so a negative torsional force constant is necessary for the HSS corresponding to a saddle point. This suggests the use of reduced eigenvalues

ð44Þ

Every N-sided polygon has 3N  6 genuine and six non-genuine vibrational eigenvectors, three of them are eigenvectors of Hip and Hop , respectively. 3.1. The out-of-plane modes Any out-of-plane displacement of a ring atom can be described by both, stretching of the two adjacent bonds and torsion of three adjacent bonds. In Appendix B we discuss the two potentials which can be used to describe such vertical motions. For both potentials, the Hessian matrix Hop is a symmetric, real circulant matrix. According to Appendix A its eigenvectors are the columns of the real Fourier matrix R ðaÞ

2

2

ð45Þ

for odd N the last eigenvector must be omitted. The matrix Hop has only real eigenvalues, for odd N there are one non-degenerate and d ¼ ðN  1Þ=2 doubly degenerate eigenvalues, for even N two nondegenerate and d ¼ ðN  2Þ=2 doubly degenerate eigenvalues. Index s runs from s ¼ 0 to s ¼ q with q ¼ ðN  1Þ=2 for odd N and q ¼ N=2 for even N. The three non-genuine vibrations are the translation in z-direction and two degenerate rotations about the x and y axes. To be a translation the coordinates of a vibrational eigenvector must be equal and not zero. This is true only for eigenvector q0 ¼ c0 . A vibrational eigenvector q ¼ ðq0 ; q1 ; . . . ; qN1 Þ describes a rotation if the sum of coordinates of the angular momentum vector L is not zero. N1 N 1 X d X RI qI / RI qI dt I¼0 I¼0 ðaÞ

str

k

tor

¼ AðsÞ þ ðsignk ÞnBðsÞ

ð50Þ

tor

ðbÞ

Q ¼ ðq0 ; q1 ; q1 ; . . . ; qN Þ ¼ ðc0 ; c1 ; s1 ; . . . ; cN Þ ¼ R

L¼

kðs; nÞ

ð49Þ

ð46Þ

ðbÞ

This gives for qs ¼ cs and qs ¼ ss 0 1 0 1 cos aI 0 N 1 X 1 B B C C ðaÞ Ls ¼ R@ sin aI A pffiffiffiffi @ 0 A N I¼0 0 cos aIs 0 PN1 1 I¼0 sin aI cos aIs R B PN1 C ¼ pffiffiffiffi @  I¼0 cos aI cos aIs A N 0 0 1 0 1 cos aI 0 N1 X 1 B C B C R@ sin aI A pffiffiffiffi @ 0 A LsðbÞ ¼ N I¼0 0 sin aIs 0 PN1 1 sin aI sin aIs I¼0 R B P C ¼ pffiffiffiffi @  N1 A I¼0 cos aI sin aIs N 0

ð47Þ

ð48Þ

where n ¼ j l2kkstr j is the ratio of the curvatures of the potential energy surface along the stretch and the torsional internal coordinates. tor k > 0 or n > 0 is sufficient for a local minimum at the HSS and tor k < 0 or n < 0 is necessary for a saddle point at the HSS. The change from a stable planar polygon to a stable non-planar polygon is described by the variation of n from positive to negative values. All eigenvalues are linear functions of n, the slope BðsÞ increases with increasing index s (Fig. 4). For positive n values the eigenvalues are in ascending order with respect to s so index s ¼ q has the highest eigenvalue. For very small negative n values we find the same ordering of the eigenvalues as for positive n values. When the n decreases eigenvalue kðqÞ becomes smaller then kðq  1Þ and finally negative. If n is sufficiently negative the ordering of all eigenvalues is reversed. In this case, the lowest eigenvalue has always the largest index s ¼ q which is for even N the non-degenerate pucker mode with q ¼ N=2, for odd N the index q ¼ ðN  1Þ=2 corresponds to a degenerate normal mode. The number of nodes of the degenerate normal modes, i.e. the degree of ring puckering, increases with increasing index s. For negative eigenvalues strong puckering is favored, for positive eigenvalues puckering is disfavored. Fig. 5 demonstrates this for the 7-sided polygon: the top row shows the less puckered higher normal mode with s ¼ 2, the bottom row shows the more puckered lower normal mode with s ¼ 3. From the condition kðqÞ < 0 one can derive the criterion jnj > AðqÞ BðqÞ that at least the eigenvalue with s ¼ q is negative. As Fig. 4 shows, the number of negative eigenvalues depends on the value of n. For the model potential used it is easy to formulate the condition for positive eigenvalues kð2Þ to kðjÞ and negative eigenvalues kðj þ 1Þ to kðqÞ, this gives the inequality for n,     Aðj þ 1Þ AðqÞ Að2Þ AðjÞ max ;...; < jnj < min ;...; ð51Þ Bðj þ 1Þ BðqÞ Bð2Þ BðjÞ 3.2. The in-plane modes

Only for s ¼ 1 these sums are not zero so the degenerate vibrational eigenvectors c1 ; s1 are the rotations about the x and y axes. The eigenvalues of the genuine vibrations start with index s ¼ 2, polygons with odd N have ðN  3Þ=2 degenerate pairs of genuine vibrations, polygons with even N have in addition to ðN  4Þ=2 degenerate pairs one non-degenerate genuine vibration with index s ¼ N=2, which is the ring pucker mode.

For the treatment of in-plane vibrations we use as model potential the sum of N bond stretch and N angle bend potentials. Again str we assume that the stretch force constant is positive, k > 0, whereas the angle bending may have negative force constants. Using group indexing (see Appendix A.4) the matrix H ip consists of ðN; NÞ circulant block matrices

A.F. Sax / Chemical Physics 349 (2008) 9–31

15

str

Fig. 4. Reduced eigenvalues kðsÞ=jk j for the vertical vibrations for N ¼ 4; 5; 6; 7; 8; 9 from top left to bottom right.

Hip ¼



Hrr

Hrt

Htr

Htt

 ð52Þ

 2   2  Hrr with elements oroI orV J , Htt with elements otoI otV J , and Hrt with ele 2  ments oroI otV J . Hrr and Htt are symmetric, and Htr is skew-symmetric.

16

A.F. Sax / Chemical Physics 349 (2008) 9–31

If matrix Hrt , which describes the coupling of r and t coordinates, would vanish, the eigenvectors of H ip are the complex or the real Fourier matrices Fð2Þ ¼



F

0

0

F



or

Rð2Þ ¼



R

0

0

R

 ð53Þ

and would transform Hip into diagonal form. Instead we get T

Rð2Þ Hip Rð2Þ ¼

RT Hrr R

RT Hrt R

RT Htr R

RT Htt R

! ð54Þ

and RT Hrt R 6¼ 0. So neither Fð2Þ nor Rð2Þ in any of its possible transformed forms are the eigenvectors of Hip . From the transformed matrix we see, however, that proper linear combinations of columns of either Fð2Þ or Rð2Þ will diagonalize Hip . 3.2.1. Transformation with Dð2Þ The three non-genuine normal modes are the doubly degenerate translations and the in-plane rotation which can easily be identified when symmetry matrix Dð2Þ for the displacement representation (Appendix A) is used to transform the Hessian. The columns of Dð2Þ are built from displacement vectors of the same length at each vertex. Fig. 6 shows the graphical representation of the columns of Dð2Þ for N ¼ 5. In coordinate system CS2 the displacement vectors of the inplane rotation must be purely tangential and identical for all vertiðbÞ ces, which is true for D0 . In-plane translations must have identical non-zero displacements at all vertices in coordinate system CS1 which holds for ðaÞ ðbÞ ðD0 ; D0 ÞCS1 . Back transformation to coordinate system CS2 T T ðaÞ ðbÞ ðaÞ ðbÞ ðD0 ; D0 ÞCS1 ¼ ðD1 ; D1 ÞCS2 identifies the degenerate pair with s ¼ 1 as in-plane translations. Omitting the three non-genuine modes from Dð2Þ gives the following basis for the genuine vibrations   ðaÞ ðaÞ ðbÞ ðaÞ ðbÞ ðaÞ ðbÞ ðbÞ ðaÞ ðbÞ D0 ; Dþ1 ; Dþ1 ; Dþ2 ; Dþ2 ; . . . ; DðaÞ s ; Ds ; Ds ; Ds ; . . . ; DN ; DN 2

ð55Þ

2

Fig. 6. Graphical representation of the columns of matrix Dð2Þ for N ¼ 5.

in which the Hessian has the form      N N ;s ; Dð2ÞT Hip Dð2Þ ¼ diag qð0Þ; Cð1Þ; . . . ; CðsÞ; . . . ; q 2 2 s ¼ 2; . . . ; d

ð56Þ

The last two columns and rows of Dð2ÞT Hip Dð2Þ must be omitted for odd N. The block matrices are

Fig. 5. (Top row) The higher pucker mode s ¼ 2 for cycloheptane. (Bottom row) the lower pucker mode s ¼ 3.

17

A.F. Sax / Chemical Physics 349 (2008) 9–31

 Cð1Þ ¼

qð1Þ þ sð1Þ þ 2ð1Þ



0

0 qð1Þ þ sð1Þ þ 2ð1Þ qðsÞ þ sðsÞ þ 2ðsÞ 0 qðsÞ  sðsÞ 0 qðsÞ þ sðsÞ þ 2ðsÞ 0 1B B CðsÞ ¼ B 2@ qðsÞ  sðsÞ 0 qðsÞ þ sðsÞ  2ðsÞ 0

0

0

ðqðsÞ  sðsÞÞ

CðsÞ ¼

qðsÞ þ sðsÞ  2ðsÞ

0

0

0

qðsÞ þ sðsÞ  2ðsÞ

ðqðsÞ  sðsÞÞ

0

0

ðqðsÞ  sðsÞÞ

qðsÞ þ sðsÞ  2ðsÞ

ðaÞ

ðaÞ

0

ð60Þ

ben

The reduced eigenvalues as functions of n ¼ k

2 str

=l k

ð61Þ ð62Þ

are

¼

ð64Þ

it is an affine function of n. Diagonalizing the (2,2,) block matrices of CðsÞ for s  2 gives two doubly degenerate eigenpairs and

ðbÞ ½kðsÞ; qðaÞ s ; qs ;

s ¼ 2; . . . ; d

ð65Þ

with ðaÞ ðaÞ qðaÞ s ¼ ðcos wðsÞDs þ sin wðsÞDs Þ

ð66Þ

qsðbÞ ¼ ðcos wðsÞDsðbÞ  sin wðsÞDðbÞ s Þ

ð67Þ

ðaÞ ðaÞ qðaÞ s ¼ ð sin wðsÞDs þ cos wðsÞDs Þ

¼

ðsin wðsÞDðaÞ s

þ

cos wðsÞDðaÞ s Þ

N ¼ F  G;

P ¼ LN þ 4EH

ð71Þ ð72Þ

ð68Þ

ð69Þ

ð73Þ ð74Þ

As can be seen in Fig. 7, all reduced eigenvalues are positive for positive n, and only for negative n some eigenvalues may be negative. For positive n the non-degenerate eigenvalue kð0Þ is asymptotically the lowest one, for negative n and even N it is the non-degenerate kb ðN2 ; nÞ and for odd N it is the degenerate eigenvalue kðd; nÞ. For even N the largest index s ¼ N=2 belongs to two non-degenerate states, the graph of the eigenvalues ka ðN2 ; nÞ and kb ðN2 ; nÞ is a pair of crossing straight lines; for the next larger polygon with N þ 1 the largest index s ¼ d ¼ N=2 belongs to two degenerate states of the same symmetry, so they interact and the curves of the eigenvalues cannot cross. The interaction of degenerate normal modes depends on the value of n. Since bond stretching force constants are about one order of magnitude larger than bond bending force constants the physically relevant values of n should be in the interval ð0:5; þ0:5Þ. The maximal and minimal values of w depend on the value of the ratio in Eq. (73) and thus on index s. (w ¼ p=4) occurs when the denominator is zero, this is for nd ðsÞ ¼ 

ðbÞ ¼ ðsin wðsÞDsðbÞ þ cos wðsÞDðbÞ qs s Þ ðbÞ ¼ ð sin wðsÞDðbÞ s þ cos wðsÞDs Þ

M ¼ F þ G;

Q ¼ L2 þ 4E2 ;

1 1 CðsÞ  DðsÞ þ nðFðsÞ  GðsÞÞ arctan 2 2 EðsÞ þ nHðsÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 str kðs; nÞ=k ¼ KðsÞ þ nMðsÞ  RðsÞn2 þ 2PðsÞn þ Q ðsÞ 2

1 ðCð1Þ þ 2Eð1Þ þ nðFð1Þ þ 2Hð1ÞÞÞ 2

ðbÞ ½kðsÞ; qðaÞ s ; qs

L ¼ C  D;

wðsÞ ¼

has the reduced eigenvalue str

qðjsjÞ  sðjsjÞ ð70Þ 2 When the angle w and the reduced eigenvalues are expressed by the quantities C; D; E; F; G; H of Appendix B.2 with

we get for the eigenvalues more compact expressions

¼C

because Fð0Þ ¼ 0 and, for even N, also CðN2 Þ ¼ 0 and GðN2 Þ ¼ 0. kð0Þ and kb ðN2 Þ are constant, whereas ka ðN2 ; nÞ is a linear function in n. The degenerate eigenpair for s ¼ þ1

 1 ðaÞ ðaÞ ðbÞ ðbÞ ð63Þ kðþ1Þ ¼ ðqð1Þ þ sð1Þ þ 2ð1ÞÞ; qþ1 ¼ Dþ1 ; qþ1 ¼ Dþ1 2

k

ð59Þ



R ¼ N 2 þ 4H2 ;

      N N N þ nF ¼ nF str 2 2 2 k N       kb 2 ; n N N N þ nG ¼D ¼D str 2 2 2 k

kðþ1; nÞ

C C C A

eigenvectors of the Hessian. For w ¼ p=4 the coupling is maximal and the eigenvectors are purely radial or tangential. wðsÞ is a skewsymmetric function of s. Using this symmetry of w the eigenvalues expressed by w are   qðjsjÞ þ sðjsjÞ þ ðsÞ kðsÞ ¼ cos2 wðjsjÞ 2   qðjsjÞ þ sðjsjÞ 2 þ sin wðjsjÞ  ðsÞ þ sin 2wðsÞ 2

K ¼ C þ D;

¼ Cð0Þ þ nFð0Þ ¼ Cð0Þ

ka 2 ; n

1

0

q0 ¼ D0

ð58Þ

The angle wðsÞ ¼ 12 arctan qðsÞsðsÞ describes the coupling of the basis 2ðsÞ ðaÞ ðbÞ vectors DsðaÞ and Ds , as well as DsðbÞ and Ds , respectively. For wðsÞ ¼ 0 there is no coupling of the basis vectors, which are then also

qðsÞ  sðsÞ

1B B B 2@

0

and for even N two additional non-degenerate eigenpairs  

   N N ðaÞ ðaÞ ¼q ; qN ¼ DN ka 2 2 2 2  

   N N ðbÞ ðbÞ ¼s ; qN ¼ DN kb 2 2 2 2

str

0

qðsÞ  sðsÞ

½kð0Þ ¼ qð0Þ;

k N

C C C A

qðsÞ þ sðsÞ þ 2ðsÞ

3.2.2. Eigenvalues of Hip Hip has for every N the non-degenerate eigenpair

kð0; nÞ

0 0 qðsÞ þ sðsÞ  2ðsÞ

With reordered columns in the four-dimensional subspaces, ðDsðaÞ ; ðaÞ ðbÞ Ds ; DsðbÞ ; Ds Þ, matrices CðsÞ with s > 1 have the following block structure 0

ð57Þ 1

EðsÞ HðsÞ

ð75Þ

nd ðsÞ is negative for positive s and positive for negative s. Because of Eq. (B.45) the corresponding nn ðsÞ have the opposite sign. This gives the interval IðsÞ ¼ ½jnd ðsÞj; jnd ðsÞj .

18

A.F. Sax / Chemical Physics 349 (2008) 9–31

str

Fig. 7. Reduced eigenvalues kðsÞ=jk j as a function of n for N ¼ 3 to N ¼ 7.

Fig. 8. (Left) tan 2wðsÞ for s ¼ 2 and s ¼ 3. (Right) wðsÞ for s ¼ 2 and s ¼ 3.

Some numerical values for nd show that the physically relevant interval ð0:5; þ0:5Þ covers indeed the maximal intervals for the individual s. For N ¼ 5, jnd ð2Þj ¼ 0:28 and for N ¼ 9, jnd ð2Þj ¼ 0:60, jnd ð3Þj ¼ 0:33 and jnd ð4Þj ¼ 0:26. In Fig. 8 tan 2wðsÞ and wðsÞ for N ¼ 7 are shown as a function of n. As can be seen from Fig. 8b w ¼ 0 is found only for positive n, that is, when the HSS is a minimum; w  p=4 is possible only for negative n that means when the HSS is a saddle point.

As for the out-of-plane modes we want to know if there are criteria for the existence of negative eigenvalues of the Hessian and their number. Eigenvalues ka ð0Þ and kb ðN2 ; nÞ are constant and positive, for eigenvalues kb ðN2 ; nÞ and ka ð1Þ which are affine functions of n it is easy to check for which n values they are negative. Also the signs of the eigenvalues kðsÞ for jsj  2 are rather easily determined. The discriminant is positive for all n 2 IðsÞ,

19

A.F. Sax / Chemical Physics 349 (2008) 9–31

therefore the sign of the eigenvalues depends on the sign of the sum and difference of the affine function g and the root function r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðs; nÞ ¼ KðsÞ þ nMðsÞ rðs; nÞ ¼ þ RðsÞn2 þ 2nPðsÞ þ Q ðsÞ ð76Þ

or one could mix a rotation matrix with a rotation–reflection matrix as eigenvectors. When the rotation matrix is used for the construction of the eigenvectors the degenerate eigenpairs of CðsÞ have the form

Positive eigenvalues result from gðs; nÞ > rðs; nÞ, which is found for all positive n, and negative eigenvalues need gðs; nÞ < rðs; nÞ, which is found for all negative n. In Fig. 9 theses inequalities are shown for s ¼ 2 and s ¼ 3 for a polygon with N ¼ 7.

ðbÞ ½kðþsÞ; qðaÞ s ; qs

where the vibrational eigenvectors for s ¼ 2; . . . ; d and x ¼ a; b ðxÞ are linear combinations of the basis vectors Ds , e.g. ðaÞ ðaÞ qðaÞ s ¼ cos wðsÞDs þ sin wðsÞDs

qðbÞ s

diagðO1 ðsÞ; O2 ðsÞÞ ¼ diagðDðwðsÞÞ; DðwðsÞÞÞ ð78Þ

ð79Þ



cos wðsÞ

 sin wðsÞ

 sin wðsÞ

 cos wðsÞ

ð83Þ

ðaÞ ðaÞ ~ðaÞ q s;J ¼ cos wðsÞqs;J þ sin wðsÞqs;J

ð84Þ

ðbÞ ðbÞ ðbÞ ~s;J ¼ cos wðsÞqs;J  sin wðsÞqs;J q

ð85Þ

ðbÞ ~s;J q

ðaÞ sin wðsÞqs;J

ð86Þ

ðbÞ ðbÞ ðbÞ ~s;J ¼ cos wðsÞqs;J  sin wðsÞqs;J q

ð87Þ

¼

ðaÞ cos wðsÞqs;J

þ

~ are in general not unit vectors, only The displacement vectors q when there is no coupling of degenerate modes (i.e. w=0) or for s=+1. For w ¼ p=4 (maximal coupling of radial and tangential distortion) the displacement vectors are either purely radial or tangential pffiffiffi cosða  s  JÞ  pffiffiffi  sinða  s  JÞ  ~ðbÞ ~ðaÞ 2 q 2 ð88Þ q s;J ¼ s;J ¼ 0 0     pffiffiffi pffiffiffi 0 0 ðbÞ ~ðaÞ ~s;J 2 q ¼ 2 ð89Þ q s;J ¼  sinða  s  JÞ cosða  s  JÞ

then O2 ðwðsÞÞ ¼ O1 ðwðsÞÞ ¼ O1 ðwðsÞÞ ¼

þ

ð82Þ

cos wðsÞDðbÞ s

When conventional indexing is used every basis vector is a colðxÞ lection of unit displacement vectors qs;J at the atoms J and, therefore, the vibrational eigenvectors are collections of linear ðxÞ combinations of the corresponding qs;J :

One can choose O1 ðwðsÞÞ as the rotation matrix DðwðsÞÞ that gives

or one can choose the rotation–reflection matrices   cos wðsÞ sin wðsÞ O1 ðwðsÞÞ ¼ DðwðsÞÞ  I1;1 ¼ sin wðsÞ  cos wðsÞ

¼

 sin wðsÞDðbÞ s

ðxÞ Ds

ð77Þ

¼ diagðDðwðsÞÞ; DT ðwðsÞÞÞ

ð81Þ ðxÞ qs

3.2.3. Eigenvectors of Hip The eigenvectors of CðsÞ, 2 6 jsj 6 d are simply block diagonal matrices composed of orthogonal (2, 2) matrices O1 ðsÞ and O2 ðsÞ. diagðO1 ðsÞ; O2 ðsÞÞ

ðbÞ ½kðsÞ; qðaÞ s ; qs

 ð80Þ

They have different lengths, and the eigenvectors resemble the pucker modes. For all other w values the displacement vectors are linear combinations of radial and tangential distortions. Fig. 10 shows how the normal modes change as a function of w. 3.3. Coupling of out-of-plane and in-plane modes The separation of vertical and in-plane modes in ring systems is only an approximation, which is reliable for small rings but not for large ones. For both, weakly and strongly coupled modes, the Hessian has the structure

Fig. 9. Plots of g, r and r for s ¼ 2 (left) and s ¼ 3 (right) as a function of n for a polygon with N ¼ 7.

ðaÞ

ðbÞ

Fig. 10. Graphical representation of qs , qs

ðaÞ

ðbÞ

and qs , qs (from left to right) for w ¼ p=4; p=8; 0 (from above).

20

A.F. Sax / Chemical Physics 349 (2008) 9–31

0

1

Hrr B H ¼ @ Htr

Hrt Htt

C Htv A

Hvr

Hvt

Hvv

Hrv

ð90Þ

where the additional block matrices Hrv and Htv are skew-symmetric and symmetric circulant matrices, respectively. Weak coupling of the coordinates r; t; v can be treated by perturbation theory, for strong coupling the separation of in- and out-of-plane motions is no longer valid. Coupling of vertical and in-plane modes is frequently found in excited electronic states where, e.g. in-plane modes such as JT modes are coupled to vertical PJT modes. 4. Hindered pseudorotation in regular polygons Hindered pseudorotation implies that the bottom line of the moat r0 ¼ r 0 ð/Þ and/or the values of the potential at the bottom line V ¼ Vðr0 ð/Þ; /Þ depend implicitly and/or explicitly on /, so the potential is anisotropic and has then a certain number of local minima separated by the same number of very low lying saddle points (Fig. 2), both types of stationary points are reached by the distortion of the HSS along the components of the degenerate vibrational eigenvectors. The number of stationary points and the form of the potential V depend on the type of normal modes and on the coupling of radial and tangential distortions. In real molecules there is the assumption that all vertices are equivalent not valid because of different substituents at the vertices or because the vertices themselves are different as in cyclohexene, which can be seen as a five-membered ring with four CH2 moieties and one CH@CH moiety. Similarly, sugar molecules in a cyclic configuration are other examples [2]. The molecular structures of low symmetry along the moat are obtained by following the degenerate vibrational eigenvectors, and the symmetry of the structures depends on the symmetry of the eigenvector which can be classified according to a rv plane. If a vibrational eigenvector is symmetric with respect to rv it is called an a-vector, in detail, it is vector þaJ, if the mirror plane goes through vertex J and if the displacement vector at vertex J points in the positive direction. If more than one vertex lie in the mirror plane the lowest index is used for labelling. For in-plane modes vector þaJ has only a radial displacement in positive direction at vertex J, and for out-of-plane modes a aJ vector has maximal displacement at vertex J in positive direction. Vector aJ is vector þaJ multiplied by 1. If the vibrational eigenvector is skew-symmetric with respect to rv it is called a b-vector, in detail, it is vector þbJ, if the mirror plane goes through vertex J. For in-plane modes vector þbJ has only a tangential displacement in positive direction at vertex J, and for out-of-plane modes a bJ vector has zero displacement at vertex J. Again vector pffiffiffibJ is obtained by multiplying þbJ by 1. Eigenvectors c and 1= 2ðc; sÞ are a-vectors, pffiffiffi eigenvectors s and 1= 2ðs; cÞ are a-vectors and b-vectors for outof-plane and in-plane modes, respectively. Following out-of-plane a-vectors leads to polylines of Cs symmetry, and the polylines obtained from following þaJ and aJ are mirror images of each other with respect to the rh -plane. Following inplane a-vectors leads to non-regular polygons of (at least) C2v symmetry, the polygons obtained from following þaJ and aJ are have different angles at vertex J so they are not even congruent. Following out-of-plane b-vectors leads to polylines of C2 symmetry, the polylines obtained from following þbJ and bJ are mirror images of each other with respect to the rh -plane. Following inplane b-vectors leads to non-regular polygons of Cs symmetry, which are mirror images of each other with respect to the rv . Another classification of a- and b-vectors is with respect to the arguments of the cosine pffiffiffi and sine functions, best seen for an inplane eigenvector 1= 2ðc; sÞ. If the argument at vertex J is a multiple mp=2 with m mod 4 ¼ 0, the cosine is positive and the sine is zero, so it gives a þaJ vector; an argument which is a multiple

Fig. 11. From left to right: the þa, a and þb polygons resulting from eigenvectors ðaÞ ðbÞ qs and qs for w ¼ p=4; p=8; 0 (from above).

mp=2 with m mod 4 ¼ 1 gives zero for the cosine and þ1 for the sine, so it gives a þbJ vector; analogously, the argument for aJ must be the multiple mp=2 with m mod 4 ¼ 2; and for bJ it must be the multiple mp=2 with m mod 4 ¼ 3. Besides symmetry, the strength of coupling radial and tangential displacements (measured by w) also determines the shape of the distorted polygons. If w ¼ 0 all vertices are displaced by the same amount however in different directions; for w ¼ p=4 vertex J (and equivalent vertices) is maximally displaced whereas displacement at the other vertices is smaller (Fig. 11). For symmetry reasons for every N-sided regular polygon there are N equivalent a-polygons þaJ, aJ, þbJ and bJ, respectively. The potential values for equivalent polygons must be equal, and because of the symmetry there must be pucker modes in-plane modes

VðþaJÞ ¼ VðaJÞ VðþaJÞ 6¼ VðaJÞ

VðþbJÞ ¼ VðbJÞ VðþbJÞ ¼ VðbJÞ

ð91Þ ð92Þ

According to the number and type of stationary points the polygons can be classified into three different types. Type 1 a-polygons þaJ, aJ correspond to equivalent local minima as do b-polygons þbJ and bJ. In addition to the 4N local minima other 4N saddle points are needed, giving together STP ¼ 8N stationary points. This type is only found for out-of-plane modes. Type 2 The 2N polygons þaJ, aJ correspond to local minima and the 2N polygons þbJ, bJ correspond to saddle points (or vice versa). The total number of stationary points is STP ¼ 4N. This type can be found for out-of-plane modes where the local minima þaJ and aJ are equivalent, but also for in-plane modes where the minima are in general different. The saddle points are always equivalent. Type 3 Only a-polygons þaJ correspond to local minima and apolygons aJ correspond to saddle points (or vice versa), altogether STP ¼ 2N stationary points. b-Polygons do not correspond to stationary points but to structures between a minimum and a saddle point. This type can only be found for in-plane modes.

21

A.F. Sax / Chemical Physics 349 (2008) 9–31

In Fig. 12 different potential energy surfaces for a 3-sided polygon are shown. The molecular structures at the local minima have lower symmetry than the HSS but we assume only small distortions from it, therefore the normal modes at the local minima are no longer exactly degenerate but they should be similar enough to allow the description of the motion in the moat by an orthogonal transformation of degenerate normal modes. 4.1. Transformation of the normal modes Rotation of the degenerate eigenvectors by multiples of p=2 transforms a-vectors into b-vectors and vice versa without changing the vertex with respect to which the vector is characterized. In detail þ2p

þ2p

þ2p

þ2p

þaJ ! bJ ! aJ ! þbJ ! þaJ

ð93Þ

Reading the sequence from right to left describes the transformation by p=2. On the other hand, in Appendix A.3.1 it is shown that rotation of the real eigenvectors of a circulant matrix by an angle b ¼ as; jsj ¼ 1; . . . ; d is equivalent to a cyclic permutation of the rows by one position. Therefore, rotation of the vibrational eigenvectors ðaÞ



ðbÞ qs ðbÞ; qsðbÞ ðbÞ ¼ qðaÞ DðbÞ ð94Þ s ; qs

The identical position is reached after rotation by Nb ¼ 2ps passing on this way STP stationary points 2ps ¼ NSTP x, so the angle between two stationary points is x ¼ 2ps=NSTP . ðaÞ ðbÞ Rotation of the vertical vibrational eigenvectors ðqs ; qs Þ causes at each vertex vertical oscillations between the maximal and the minimal values of the displacement, whereas rotation of the inðaÞ ðbÞ plane vibrational eigenvectors ðqs ; qs Þ with jsj  2 means that the displacement vectors at the vertices rotate and change their length so each arrow head follows an ellipsis. Only for the genuine in-plane mode with s ¼ þ1 each arrow head follows a circle. On the way from one local minimum to another one passes other stationary points corresponding to a-, þb- and b-vectors. To find out to which vertices these stationary points belong, one has to look at the arguments which must be multiples n of p=2 for some vertices asJ þ kx ¼ n

or

4sJ 4sk þ ¼ n; N N STP

¼ 0; 1; 2; . . . N STP  1

n 2 N;

k ð96Þ

whether an eigenvector becomes an a-vector or the second one becomes a b-vector depends on the value of n mod 4 and this depends on the three polygon types discussed above. We get the following relations between N; s; k; J: Type1

by b transforms a-vectors into a-vectors and b-vectors into b-vectors whereby the index of vertex J is changed by one.

Type2

ðaðJ þ 1Þ; bðJ þ 1ÞÞ ¼ ðaJ; bJÞDðbÞ

Type3

ð95Þ

p 2

  4s k Jþ mod 4 N 8   4s k Jþ mod 4 n¼ N 4   4s k Jþ mod 4 n¼ N 2

n¼

ð97Þ ð98Þ ð99Þ

Fig. 12. Potential energy surfaces for N ¼ 3 systems. The top row shows surfaces for realistic in-plane modes, the bottom row surfaces for fictitious out-of-plane modes. (Top) Type 3 (left) and Type 2 (right). (Bottom) Type 2 (left) and Type 1 (right).

22

A.F. Sax / Chemical Physics 349 (2008) 9–31

þ a0; U; b2; U; a1; U; þb0; U; þa2; U; b1; U; a0; U; þb2; U; þ a1; U; b0; U; a2; U; þb1

Table 1 Values of n for a Type 3 system with N ¼ 3 and s ¼ 1

k¼0

J¼0

J¼1

J¼2

0

4 3 6 ¼2 3 8 3 10 3 12 ¼0 3 14 3

8 3 10 3 12 ¼0 3 14 3 16 3 18 ¼2 3

2 3 4 3 6 ¼2 3 8 3 10 3

k¼1 k¼2 k¼3 k¼4 k¼5

Underlined is an integer value of n mod4. Table 2 Values of n for a Type 2 system with N ¼ 3 and s ¼ 1

k¼0 k¼1 k¼2 k¼3 k¼4 k¼5 k¼6 k¼7

J¼0

J¼1

J¼2

0

4 3 5 3 6 ¼2 3 7 3 8 3 9 ¼3 3 10 3 11 3

8 3 9 ¼3 3 10 3 11 3 12 ¼0 3 13 3 14 3 15 ¼1 3

1 3 2 3 3 ¼1 3 4 3 5 3 6 ¼2 3 7 3

Underlined is an integer value of n mod4.

In Tables 1 and 2 calculation of n is demonstrated for a Type 3 and a Type 2 system of a 3-polygon with s ¼ þ1.

4.2. The symmetry number of pseudorotations The symmetry number r of a molecule is ‘‘obtained by imagining all identical atoms to be labelled, and then counting the number of different but equivalent arrangements that can be obtained by rotating (but not reflecting) the molecule” [13]. Therefore, the symmetry number of planar cyclopentane is 10 ¼ 5 2 because there is a 5-fold symmetry axis and a set of equivalent 2-fold symmetry axes. Since in pseudorotation the molecule is not rotated as a whole we have to find out the number of equivalent arrangements in a different way. ðaÞ If a vibrational eigenvector, say qs , which corresponds to a stationary point, has a form so that only after rotation by 2ps the original vector is obtained, N equivalent stationary points are visited during the rotation, therefore the symmetry number rPR ¼ N. If, however, rotation by a smaller angle gives the original vector the coordinates for two or more vertices in the vector must be identical. That means that the arguments of the cosines and sines for these vertices must be multiples of mp=2 with the same m mod 4). By construction, for eigenvector þa0 the value for k in Eq. (96) is zero, so we just have to find out whether the arguments asJ for other vertices with J < N  1 in this vector are integer multiples of 2p, s asJ ¼ m2p () J¼N ð100Þ m so s ¼ ms is a divider of N. On the other hand is s ¼ ms, so s is also a divider of s. The number of vertices with equal arguments is the greatest common divider s of N and jsj. s ¼ gcdðN; jsjÞ

and the pseudorotational symmetry number rPR is also a divider of s. The number of vertices with equal coordinates, that is the symmetry number of pseudorotation, is the greatest common divider of N and s, and if it is 1 the symmetry number is N, rPR ¼

4.1.1. Type 3 For a three-membered ring of Type 3 there are N STP ¼ 6 stationary points, 3 local minima, corresponding to distortions along +avectors and 3 saddle points, corresponding to distortions along a-vectors, or vice versa. Distortions along b-vectors do not lead to stationary points. The result of the calculations in Table 1 is summarized in the sequence þa0; a1; þa2; a0; þa1; a2 which has to be read like this: the þa0-vector becomes after rotation by 2p=STP the a1-vector, after rotation by another 2p=STP it becomes the þa2-vector and so on. For Type 3 polygons no stationary points correspond to b-vectors. 4.1.2. Type 2 A three-membered ring of Type 2 has N STP ¼ 12 stationary points, 6 local minima, corresponding to distortions along a-vectors and 6 saddle points, corresponding to distortions along b-vectors, or vice versa. The result of the calculations in Table 2 is summarized in the sequenceþa0; b2; a1; þb0; þa2; b1; a0; þb2; þa1; b0; a2; þb1. 4.1.3. Type 1 A three-membered ring of Type 1 has N STP ¼ 24 stationary points, 12 local minima, corresponding to distortions along a-vectors and b-vectors, and 12 saddle points. For the saddle points the arguments of the cosines and sines are not integer multiples of p=2, that means they correspond to neither a- nor b-vectors. They are indicated by a U in the sequence,

ð101Þ

N s

ð102Þ

rPR is, therefore, for prime N always N, and for the 6-sided polygon and jsj ¼ 1 it is 6 whereas for jsj ¼ 2 it is 3. The greatest common divider s has a simple geometric meaning: the cyclic subgroup CN of the full symmetry group of the regular polygon has itself cyclic subgroups, at least C1 . s is the order of the cyclic subgroup which remains after the deformation of the regular polygon by the vibration. If N is prime, destroying the CN symmetry leaves only C1 . As an example, for N ¼ 9 and jsj ¼ 3 the distorted polygon has (D3h) symmetry (Fig. 13). In PR one considers no rigid rotations of the whole molecule so rPR should be N=s. But the rotational symmetry number of a molecule is introduced to correct the rotational contribution to the partition function, so that the theoretical thermodynamic data agree with the experimental ones and the same holds for the pseudorotational symmetry number. Therefore, the latter was frequently adapted according to the results in fitting procedures. For example,

Fig. 13. Symmetry of the distorted polygons. (Left) N ¼ 6; s ¼ 2, (Middle) N ¼ 6; s ¼ 1, and (Right) N ¼ 9; s ¼ 3.

A.F. Sax / Chemical Physics 349 (2008) 9–31

Kilpatrick et al. used a PR symmetry number of 5 for cyclopentane [1] whereas some years later another group used a value of 10 [14] which is the symmetry number of the rigid rotations of planar cyclopentane. This is certainly wrong, because pseudorotations about a 2-fold symmetry axis perpendicular to the N-fold axis are not possible, otherwise there were a phase angle / transforming all pairs of vertices so that asJ þ / ¼ asðN  JÞ holds for the arguments in the normal modes irrespective of the vertex number J. But this is not true. One reason why one cannot give a definite pseudorotational symmetry number is the lack of rigidity in pseudorotating systems and accordingly the determination of the ‘‘effective” symmetry of such floppy molecule. An interesting alternative seems to be the concept of a continuous symmetry measure as proposed by Avnir [15] which can be applied to time averaged structures of ring and cage molecules. 5. Systems showing one-dimensional pseudorotation Pseudorotation was first found in cyclic molecules undergoing out-of-plane vibrations. For these motions the torsional angles change considerably but the bond lengths do not. Pseudorotation caused by in-plane vibrations was found mainly in cyclic molecules which are JT systems. In four-membered rings and larger ones these motions change bond angles but again not the bond lengths. In three-membered rings, however, any change of angles implies considerable change of bond lengths which can be described as bond fluctuations. In p-systems like the cyclopropenyl radical the rbonds are only stretched but the p-bonds fluctuate, whereas in the alkali trimers the weak r-bonds fluctuate. In bulky molecules but especially in solids pseudorotation may also occur when atoms are removed from the bulk leaving the so-called vacancies. 5.1. Out-of-plane modes Cycloalkanes and cyclopolysilanes are the most prominent molecules where pseudorotation caused by pucker modes can be observed. These molecules are PJT systems, so their potential energy surfaces at the HSS correspond to saddle points of various orders. The modes with negative force constants are the ring puckering modes for which is of typical that the bond lengths of the ring skeleton remain nearly constant during vibration, but the torsional angles change considerably which is coupled to changes in the valence angles. Pseudorotation in these systems is connected to isomerization of conformers of puckered rings. That means that only dihedral angles but not bond lengths of the puckered ring are changed, and no rearrangement of bonding electrons occurs during such pseudorotations. The number of normal modes in cycloalkanes and cyclosilanes is about three times as large as that for an N-sided polygon; our arguments are valid only for the lowest modes of each symmetry which can be attributed to vibrations of the ring skeleton, the higher normal modes dominantly describe hydrogen motions. We discuss now only genuine modes of the three smallest ring systems with puckered equilibrium structures, namely the five-, six- and seven-membered ring. We find for all investigated planar cycloalkanes and cyclosilanes that the negative degenerate genuine modes are ordered with index s, the E2 mode is always the lowest one. In planar cyclobutane and cyclotetrasilane only the non-degenerate ring pucker mode has a negative force constant, and the vibration describes the oscillation between 2 equiv. puckered ring structures. The smallest cycloalkane with a genuine degenerate ring deformation mode is cyclopentane where the Hessian for the planar HSS has one negative doubly degenerate eigenvalue

23

with eigenvectors of E2 symmetry, therefore, the equilibrium geometry of cyclopentane is non-planar. Cyclopentane is a Type 1 molecule, that means there are 2N ¼ 10 equivalent puckered ring conformers of Cs symmetry which are reached by following the acomponent in positive direction and in negative direction and 2N ¼ 10 puckered ring conformers of C2 symmetry corresponding to distortions along the b-component in both directions, giving together 4N ¼ 20 local minima. These local minima are separated by 20 saddle points. A graphical representation of the modes leading to local minima is given in Fig. 14. The two types of local minima are nearly isoenergetic, the energy difference is 0:3 cm1 , and the energy barriers are of the same size [12]. So, cyclopentane is indeed nearly an ideal free pseudorotator. The same is true for cyclopentasilane Si5 H10 [12]. Planar cyclohexane has a Hessian with two negative eigenvalues, the lowest one is non-degenerate and the higher one is doubly degenerate. Following the non-degenerate mode leads to the chair conformations, following the þaJ and þbJ modes of the degenerate eigenvectors of E2 symmetry leads to the boat and twisted conformations of cyclohexane, respectively, which are both energetically higher than the chair conformations. The socalled boat conformations with C2v symmetry are indeed saddle point structures, only the twisted boat conformations with C2 symmetry correspond to local minima. In the E2 normal mode cyclohexane is a Type 2 molecule with 12 local minima and 12 saddle points and since the symmetry number is 2, the number of stationary points is halved, giving finally 6 minima and 6 saddle points. The saddle point in the boat conformation lies about 10 cm1 above the twisted local minima, therefore cyclohexane in the E2 mode is a hindered pseudorotator but nevertheless, rather floppy. Planar cycloheptane has a Hessian with two negative doubly degenerate eigenvalues, the eigenvectors are in increasing order of E3 and E2 symmetry. Because N ¼ 7 prime cycloheptane is a Type 1 molecule, that means we have in each mode 4N ¼ 28 local minima and the same number of saddle points. The E2 distortions (Fig. 5, top row) lead to less stable structures whereas the E3 distortions (Fig. 5, bottom row) yield more stable and more puckered structures. Chemists say that the reduction of Pitzer strain in ring molecules is the reason for ring puckering, and another explanation is that puckering provides additional covalent bonding [3]. 5.2. In-plane modes The best known molecules with in-plane pseudorotation are JTsystems with a conical intersection at the HSS where no vibrational spectrum can be calculated, but there is also planar PJT ring system for which normal modes can be calculated at the HSS. Example is Li3 which is a JT system whereas Na3 is a PJT system due to the coupling of the low lying A1 state with the E ground state. K3 is regarded to be very similar to Na3 . For small ring systems like 3rings and 4-rings, where only one degenerate mode is possible its symmetry is certain, but in higher rings with two or more degenerate modes one can speculate whether their ordering is in the same JT and PJT systems. The smallest ring systems where pseudorotation can be observed are three-membered rings and for them any in-plane deformation implies changes in the bond lengths. Well-known examples are the alkali trimers Li3 , Na3 and K3 , see, e.g. [16–19] but also the trimers Cu3 , Ag3 or Au3 [17,20–28]. All these molecules are of Type 3 with three local minima which correspond to the obtuse C2v structures which can exaggeratedly be described as a three electron-three center bond. These structures are reached by following the aJ modes. The acute C2v structures (following þaJ) correspond to saddle points, and these structures can again exaggeratedly be described as a singly bound dimer coupled to an atom. Calculated barriers between the local minima strongly

24

A.F. Sax / Chemical Physics 349 (2008) 9–31

Fig. 14. Graphical representation of the 20 local minima corresponding to the puckered configuration of c-C5 H10 . Gray is an atom in the molecular plane, black and white are atoms above and below the plane, respectively.

depend on whether a JT or a PJT model is used and the differences between the values can be substantial. For a discussion see [3]. In general pseudorotation in X3 systems is hindered, and nearly free pseudorotation is possible when very small barriers exist between the local minima but also when the zero point energy is larger than the barrier as was reported for Cu3 [3]. In all X3 molecules discussed there is a change of the bond orders in the molecule, or a fluctuation of r-bonds. The cyclopropenyl radical is a JT molecule with 3 p electrons, so it is a Type 3 molecule with obtuse local minima and acute saddle points, and the barrier is 116cm1 [12], so the pseudorotation in this molecule is hindered. In this molecule one observes a fluctuation of p-bonds. The cyclopentadienyl radical is a JT system with a 2 E001 electronic ground state with 5 equiv. local minima. Therefore it is a Type 3 molecule. For symmetry reasons the JT active mode must have E02 symmetry which is an in-plane mode. There must be two degenerate in-plane modes with indices s ¼ 2 and s ¼ þ2; for a PJT system the 2 mode is the one with the negative eigenvalue of the Hessian. According to Fig. 7 the degenerate mode with index s ¼ þ1 ðE01 Þ should be higher than the second E02 mode (index s ¼ þ2) for 0:5 < n < 0, that is for physically relevant n-values. Kiefer et al. calculated frequencies for the cyclopentadienyl radical at the HSS and found indeed for the low E0 modes the following ordering: E02 with a physically insignificant frequency of 6300i wave numbers, the second E02 with m ¼ 922 cm1 and E01 with m ¼ 1054 cm1 [29]. Kiefer et al. classified the E02 mode with imaginary frequency as ‘‘stretching” and the real one as ‘‘bending”. In Fig. 15 the deformations due to both E02 modes are shown; they were calculated for pure radial and tangential displacements. The sequence of the polygons for the mode with s ¼ 2 (in the bottom) shows that there are indeed polygons with stretched sides in con-

Fig. 15. Form of the local minima of the cyclopentadienyl radical caused by the E modes with s ¼ þ2 (left) and s ¼ 2 (right).

trast to the polygons corresponding to the mode with s ¼ þ2. The agreement of the calculated results with our model predictions is very satisfactory. More planar JT molecules are discussed by Bersuker [3]. We mention another type of JT system with one-dimensional pseudorotation which is equivalent to the metals trimers discussed above. Divacancies in crystalline silicon can be characterized as two adjacent holes (missing atoms), each surrounded by three silicon atoms with unpaired electrons. The local symmetry of each hole is C3v and D3d for the divacancy. When the deformations due to the JT active mode within C2h symmetry are investigated one finds either obtuse arrangements of the three silicon atoms at each hole, called resonant bond, or acute ones, called large pairing [30]. The acute structure lies about 10 meV or about 80 wave numbers above the obtuse one and without symmetry restriction the systems goes without barrier into the obtuse structure. There are 3 equivalent local minima of obtuse geometry separated by three saddle points of acute geometry, the system is therefore of Type 3. Due to the small barrier which will become even smaller when the zero point energy is accounted for the system will perform nearly free pseudorotation.

25

A.F. Sax / Chemical Physics 349 (2008) 9–31

6. Systems showing two-dimensional pseudorotation Two-dimensional pseudorotation can only occur in an essentially spherical system such as cage molecules or multivacancies or microvoids in solids. As regular polygons are the model systems for 1D-PR in anisotropic ring systems, regular polyhedra or objects derived from them serve as model systems for 2D-PR in anisotropic cage systems. If such a system has a HSS with cubic or higher symmetry and if the electronic ground state is triply degenerate, there are triply degenerate JT active normal modes which couple the components of the degenerate electronic state and lead to equilibrium structures of lower symmetry than the HSS. Again one has to distinguish between JT and PJT systems. For one-dimensional pseudorotation it was shown that if the order of the highest cyclic subgroup of a regular polygon is not prime the symmetry group of distorted structures may contain cyclic subgroups of lower order than N, while for prime N symmetry will always be reduced to C1 . Similarly, if the HSS of a cage system has a symmetry group with an inversion center such as the octahedral group Oh , the distorted polyhedra may still have an inversion center (high symmetry) or the inversion center may have been destroyed (low symmetry). Structures of the HSS having no inversion center like the tetrahedral group Td will always be reduced to low symmetry structures. To determine PR symmetry numbers of cage systems use of Avnir’s method seems to be the best choice. Treatment of PR in cage systems is difficult due to the rather complicated underlying JT or PJT problem. For triply degenerate electronic states of systems with tetrahedral or octahedral symmetry the T  ðe þ t2 Þ JT problem occurs frequently. If the coupling of the electronic state to the e mode is much stronger than to the t 2 mode, the T  e JT problem gives a 1D-PR connecting three tetragonal structures. If it is the other way round one gets a 2D-PR connecting four trigonal structures. Both types are shown in Fig. 16. An example for a cage system which is a T  e JT problem is a monovacancy in a diamond lattice where four atoms surrounding a hole carry an unpaired electron, the local symmetry is Td . In crystalline silicon there is a strong tendency to form two LP bonds in a tetragonal arrangement [31], there are three possibilities for doing that and, therefore, this bond reconstruction is of T  e type. If the surrounding bulk is stiff enough to prevent the formation of strong bonds, a hindered PR with fluctuating weak bonds will be the result. The corresponding T  t JT problem would lead to four equivalent structures with a three electron–three center bond weakly coupled to an unpaired electron. Such structures were, however, not found in silicon. If not one silicon atom is removed from the lattice but a Si5 cluster of Td symmetry a pentavacancy with 12 neighboring atoms is

created and there will be triply degenerate electronic states with several possibilities of bond reconstruction due to JT distortions of any of the mentioned types and, thus, 2D-PR can be expected to occur in such systems. A different type of 2D-PR related to cages in solids was described by Manz [8]. A CO molecule in a solid argon matrix is surrounded by 12 argon atoms, and due to the non-spherical shape of the CO molecule the 12 argon atoms form an ellipsoidal cage. The CO molecule rotates and concomitantly the shape of the argon cage changes which is clearly a pseudorotation because it is only caused by vibrational displacements of the argon atoms about their lattice sites. One may wonder whether there are also cage molecules showing pseudorotation. Since most pseudorotating molecules show bond fluctuations one can assume that this will be observed also in such molecules. The best known cage molecule with fluctuating bonds is bullvalene C10 H10 [32] (Fig. 17) with its 10!=3 isomers. The symmetry is C3v , and there are two sets of carbon atoms: 4 tetracoordinated and 6 tricoordinated ones. If this structure is obtained due to JT distortions there must be a HSS having at least tetrahedral symmetry. This structure can be derived from a tetrahedron with CH groups sitting at the centers of the six edges. Central projection of the latter on the sphere passing through the vertices gives HSS (Fig. 18). Since there are low lying electronic states in bullvalene at the HSS [33] bullvalene is probably a PJT system. The trigonal structure

Fig. 17. The bullvalen molecule.

Fig. 16. The tetragonal (top) and trigonal (bottom) distortions of a tetrahedron.

26

A.F. Sax / Chemical Physics 349 (2008) 9–31

Every circulant matrix can be written as a polynomial in the shift matrix C¼

N 1 N1 X Ni X ci P1 ¼ ci PNi i¼0

ðA:3Þ

i¼0

A.2. Complex eigenvectors and eigenvalues

Fig. 18. Construction of the high symmetric structure of bullvalen.

of the bullvalene molecule is certainly caused by a triply degenerate JT mode (Fig. 16), but the high number of isomers results from the fact that due to fluctuation of p and r bonds every tetragonal carbon atom can become a trigonal and vice versa. 7. Conclusion Pseudorotation is a kind of nuclear motion which is the result of the dynamic JT or PJT effect and can mostly be described as periodic bond weakening and bond strengthening frequently due to bond fluctuations. In ring molecules only one-dimensional pseudorotation is possible, in cage molecule one- and two-dimensional pseudorotation can occur. One-dimensional pseudorotation caused by JT and PJT yields the same thermodynamic properties in the high temperature limit. The eigenvalues of the Hessian of N-sided regular polygons were calculated assuming that the potential consists only of bond stretch and bond torsion terms for out-of-plane vibrations and bond stretch and angle bend for in-plane vibrations. It seems to be justified to assume that the ordering of normal modes for a PJT system is the same as for a JT system. Two-dimensional pseudorotation was discussed for cage systems in the form of the inner surface of multivacancies in solids or in the form of nearest neighbors of host molecules in a noble gas matrix, as well as for the cage molecule bullvalene. Acknowledgements Helpful discussions with Brian Sutcliffe, Horst Köppel and Bernd Thaller are gratefully acknowledged. Appendix A. Circulant matrices A.1. Definition

All circulant ðN; NÞ-matrices have the same eigenvectors which form the complex symmetric Fourier-matrix F, the elements of the kth column are the kth powers fkl ; l; k ¼ 0; 1; . . . ; N  1 of the Nth roots of unity fl ¼ eila with a ¼ 2p=N 0 0 1 f0 f10 f20 . . . fN2 fN1 0 0 B 0 C B f1 C f11 f21 . . . fN2 fN1 1 1 B C B f0 C f12 f22 . . . fN2 fN1 C 1 B 2 2 2 B C F ¼ pffiffiffiffi B . . C . .. . . ... ... N B .. C . . . B C B 0 N1 C f @ fN2 f1N2 f2N2 . . . fN2 N2 N2 A f0N1 f1N1 f2N1 0 1 1 1 ... 2 B1 f ... f B B 2 4 B 1 f f ... 1 B ¼ pffiffiffiffi B . .. . . .. .. . NB . B. B 2 4 @1 f f ...

1

f

2

f

...

fN2 fN1 N1 N1 1 1 f C C C f 2 C C .. C . C C C f2 A

... 1 f 2 f 4 .. . f4 f2

ðA:4Þ

f

F ¼ ðf 0 ; f 1 ; f 2 ; . . . ; f N2 ; f N1 Þ ¼ ðf 0 ; f 1 ; f 2 ; . . . ; f 2 ; f 1 Þ

ðA:5Þ

For every N column f 0 is real, (and for even N also column f N=2 ) and there are d pairs of complex conjugate columns ðf i ; f i Þ with d ¼ ðN  1Þ=2 for odd N and d ¼ ðN  2Þ=2 for even N. More convenient for the following considerations is the nonsymmetric form of F obtained by reordering of the columns. F ¼ ðf 0 ; . . . ; f s ; f s ; . . . ; f q ; f q Þ ¼

for odd N

ðf 0 ; . . . ; f s ; f s ; . . . ; f q ; f q ; f N=2 Þ

ðA:6Þ

for even N

ðA:7Þ

Index s runs from s ¼ 0 to s ¼ q where q ¼ N=2 for even N and q ¼ ðN  1Þ=2 for odd N. This form of F will be used throughout. The form of the eigenvectors is not unique, multiplying the ith column of F by a phase factor eibi yields again eigenvectors of the circulant matrices to the same eigenvalues. A.2.1. Cyclic permutation matrices The eigenvalues of P1 are the N/th roots of unity P1 ðf 0 ;f 1 ; f 1 ; .. .Þ ¼ ðf 0 ; .. .;f 1 ;f 2 ;f 2 ;f 1 ; .. .Þdiagð1;f 1 ;f1 ;f 2 ; f2 ; .. .Þ

The rows 0 c0 B B cN1 B B C ¼ B cN2 B . B . @ . c1

of a circulant ðN; NÞ-matrix (in short a circulant) [34] 1 c1 c2 . . . cN1 C c0 c1 . . . cN2 C C cN1 c0 . . . cN3 C ðA:1Þ C ¼ ððc0 ; c1 ; c2 ; . . . ; cN1 ÞÞ .. C .. C . A . c2 c3 . . . c0

are obtained from the N elements of the first row c0 ; c1 ; . . . ; cN1 by repeated right shifting. Real circulants may be symmetric, skewsymmetric or non-symmetric like the cyclic permutation matrices Pk , which are the powers of shift matrix P1 . The ðN; NÞ unit matrix I is equal to PN ¼ P0 : 0 1 0 0 0 ... 0 0 1 B1 0 0 ... 0 0 0C B C B C B0 1 0 ... 0 0 0C B C ¼ ðð0; 0; 0; . . . ; 0; 0; 1ÞÞ ðA:2Þ P1 ¼ B . . . . . . ... ... ... C B .. .. .. C B C B C @0 0 0 ... 1 0 0A 0

0

0

...

0

1

0

¼FK

ðA:8Þ kðN2 Þ ¼ 1, ðNsÞ

For every N eigenvalue kð0Þ ¼ 1, for even N eigenvalue the d pairs of eigenvalues kðsÞ ¼ f s and kðN  sÞ ¼ f ¼ f N  f ðsÞ ¼ fs are pairwise complex conjugate. For the general cyclic permutation matrix Pk the eigenvalues Kk are the kth powers of K kk ðsÞ ¼ f ks ;

s ¼ 0; 1; 2; . . . ; N  1

ðA:9Þ

and therefore cyclic row permutation of F is the same as scaling the columns by the corresponding eigenvalue Pk F ¼ FKk

ðA:10Þ

A.2.2. General circulant matrices Eq. (A.9) together with the representation of a circulant matrix as a polynomial, Eq. (A.3), gives the eigenvalues of a general circulant matrix kðsÞ ¼

N 1 X j¼0

cj fsj ;

s ¼ 0; 1; 2; . . . ; N  1

ðA:11Þ

27

A.F. Sax / Chemical Physics 349 (2008) 9–31

All eigenvalues of real symmetric circulant matrices are real kðsÞ ¼ kðN  sÞ ¼ c0 þ cN ð1Þs þ 2 2

d X

cj cos ajs

ðA:12Þ

j¼1

the d pairs of eigenvalues kðsÞ and kðN  sÞ are doubly degenerate, eigenvalues kð0Þ (and for even N also kðN2 Þ) are non-degenerate. For odd N the cN contribution must be omitted. 2 All eigenvalues of real skew-symmetric circulant matrices are purely imaginary kðsÞ ¼ kðN  sÞ ¼ 2i

d X

cj sin ajs

ðA:13Þ

j¼1

so the real eigenvalues kð0Þ (and for even N also kðN2 Þ) are zero. A.3. Real eigenvectors Each pair of complex conjugate vectors ðf s ; f s Þ; s ¼ 1; 2; . . . ; d spans a two-dimensional complex space Iðf s ; f s Þ which is invariant against complex conjugation which means that for any linear combination of the basis vectors also the complex conjugate is element of this space. In such a space it is always possible to obtain a real basis by applying the unitary transformation   1 i 1 u ¼ pffiffiffi ðA:14Þ 2 1 i to the complex basis pffiffiffi pffiffiffi ðf s ; f s Þu ¼ ðcs ; ss Þ ¼ ð 2Rf s ; 2If s Þ 80 1 0 19 cos 0as sin 0as > > > > > B C B C> > > > > >B cos 1as C B sin 1as C> rffiffiffiffi> > > B cos 2as C B C> < C B sin 2as C= 2 B B C B C ¼ ;B .. .. B C C N> . >B > . C B C> > > > B C B C> > > > @ cosðN  2Þas A @ sinðN  2Þas A> > > > > : ; cosðN  1Þas sinðN  1Þas 80 1 0 19 1 0 > > > > > > > B sin as C> cos as C > >B > B C B C> rffiffiffiffi> > > B cos 2as C B sin 2as C> < C B C= 2 B B C; B C ¼ .. .. B C B C > >B N> . . C B C> > > > B C B C> > > >@ cos 2as A @  sin 2as A> > > > > ; : cos as  sin as

ðA:15Þ

ðeib f s ; eib f s Þu 0

cosðbÞ

sinðbÞ

1

B rffiffiffiffiB 2B B ¼ B NB B @

cosðsa  bÞ cosð2sa  bÞ .. .

sinðas  bÞ sinð2as  bÞ .. .

C C C C C C C A

cosððn  1Þsa  bÞ sinððn  1Þas  bÞ  cos b  sin b ¼ ðcs ; ss ÞDðbÞ ¼ ð~cs ; ~ss Þ ¼ ðcs ; ss Þ sin b cos b

ðA:21Þ



ðA:22Þ

and the rotated eigenvectors ð~cs ; ~ss Þ are eigenvectors to the same eigenvalues as the ðcs ; ss Þ. If b is a multiple of as, e.g. b ¼ kas, the result of the rotation is the same as the cyclic permutation of the rows of the vectors, as if b is in addition shifted by p=2 the cosine are transformed into sines and vice versa. A.3.2. F and R as symmetry matrices Both F and R are symmetry matrices which can be used to reduce a reducible representation of cyclic point groups. Using F the irreducible representations are one-dimensional and in general complex, using R one gets real one- and two-dimensional irreducible representations. For every N there are d two-dimensional representations which for the cyclic groups CN are termed E1 ; . . . ; Ed , for odd N there is one and for even N there are two additional non-degenerate representations. The eigenvectors ðf s ; f s Þ and ðcs ; ss Þ, respectively, belong to the representation Es . A.4. Circulant block matrices

ðA:16Þ

So, applying the transformation matrix U with d diagonal block matrices u (for odd N the last row and column must be omitted) to the complex Fourier matrix gives the real Fourier matrix R FU ¼ Fdiagð1; u; . . . ; u; 1Þ ¼ R ¼ ðc0 ; c1 ; s1 ; . . . ; cd ; sd ; cN=2 Þ

ðA:17Þ

with c0 ¼ f 0 and cN=2 ¼ f N=2 for even N. The columns of R are eigenvectors of real symmetric circulant matrices C, for c0 and cN=2 this is obvious and for the pairs ðcs ; ss Þ; s ¼ 1; . . . ; d it follows from  s   s  k 0 0 k u ¼ ðcs ; ss ÞT Cðcs ; ss Þ ¼ uy ðf s ; f s Þy Cðf s ; f s Þu ¼ uy 0 ks 0 ks ðA:18Þ The columns of R are, however, not eigenvectors of real skew-symmetric circulant matrices C, because of Eq. (A.13)  s    ijk j 0 0 jks j ðcs ; ss ÞT Cðcs ; ss Þ ¼ uy u¼ ðA:19Þ s s 0 ijk j jk j 0 and not of cyclic permutation matrices Pk !  cos kas f ks 0 ðcs ; ss ÞT Pk ðcs ; ss Þ ¼ uy u¼ ks sin kas 0 f

A.3.1. Non-uniqueness of R Multiplication of the complex vectors ðf s ; f s Þ by phase factors ðeib ; eib Þ corresponds to a rotation of the real vectors with the rotation matrix DðbÞ

 sin kas cos kas

 ¼ DðkasÞ ðA:20Þ

Treatment of in-plane vibrations of an N-sided polygon involves two sets of equivalent objects such as the basis vectors ðr1 ; . . . ; rN Þ and ðt1 ; . . . ; tN Þ of CS2. We call this way of indexing the two sets group indexing (GI). This leads to block matrices of the form   A 0 ðA:23Þ Cð2Þ ¼ 0 B where A and B are ðN; NÞ circulants. In conventional indexing (CI), the two basis vectors at each vertex form an ordered pair ðrI ; tI Þ, the pairs are numbered from 0 to N  1 giving ðr0 ; t0 ; . . . ; rN1 ; tN1 Þ. The change between the two indexing schemes is merely a permutation, which applied to rows and columns gives the block matrix using CI. The eigenvectors of matrices Cð2Þ are the matrix Fð2Þ which is the Kronecker product of the ð2; 2Þ unit matrix I2 and F   F 0 ðA:24Þ Fð2Þ ¼ I2  F ¼ 0 F The following reordering of the columns of Fð2Þ is appropriate

ðA:25Þ

because the pairs of eigenvectors         0 f 0 fs ; s and ; fs 0 0 fs

ðA:26Þ

span each a 2-dimensional space T which is invariant to complex conjugation. The two pairs of eigenvectors transform as the irreduc-

28

A.F. Sax / Chemical Physics 349 (2008) 9–31

ible representation Es of point group CN . On the other hand belong the pairs         0 0 fs fs ; ; and ðA:27Þ fs 0 fs 0 to degenerate eigenvalues of the Kronecker matrices I2  Ps so the two pairs span a four-dimensional space Fs in which one can always construct real basis vectors of arbitrary shape with the help of unitary transformations. Any set of such eigenvectors are real symmetry matrices which can be used to reduce 2N-dimensional representations spanned by the two sets of equivalent objects, but only few are important, either because matrices transformed into these bases have convenient forms or because of a geometric interpretation of the real eigenvectors. A.4.1. Matrix Dð2Þ Transformation of the four-dimensional spaces Fs with the unitary matrix 0 1 1 0 1 0   B u 0 1 B 0 1 0 1 C C pffiffiffi B u3 ¼ ðA:28Þ C 0 u 0 1A 2@0 1 1

0

1

0

s ¼ 0; 1; . . . ; dðand

N for evenNÞ 2

ðA:33Þ

The local displacement vectors form a set of right handed orthogoðaÞ ðbÞ nal unit vectors. qs;j  qs;j ¼ dab . For odd N there is in each a-component at least one atom with a pure radial displacement vector and the same atoms for a pure tangential displacement in the b-component. Matrix Dð2Þ is a special symmetry matrix which is frequently used in vibrational spectroscopy to give the so-called displacement representation [35] of the potential energy. A recipe for the construction of the columns of Dð2Þ is given by Herzberg [36]. Like the real fourier matrix R the columns of Dð2Þ are not unique, rotations by multiples of a ¼ 2p=N give columns with row permutation of the displacement vectors q. ð2Þ Transformation of DCI from coordinate system CS2 to coordinate system CS1 is done with transformation matrix Tip (Eq. (43)   ðbÞ DðaÞ s ; Ds

CS1

  ðaÞ ðbÞ ¼ Tip Ds1 ; Ds1

CS2

ðA:34Þ

it shows that change from coordinate system CS2 to CS1 means an increase of the indices of corresponding vectors by one.

yields

ðA:29Þ

where by using the symmetry properties of sine and cosine the eigenvectors can be written in a more symmetric form

ðA:30Þ

If conventional indexing is used instead of group indexing, a simple geometrical interpretation of the basis vectors can be given: every column is a collection of unit displacement vectors of vertices qI ¼ ðr I ; t I Þ.     cs ss 1 ðbÞ DðaÞ ¼ pffiffiffi ffi ðA:31Þ s ; Ds GI 2 ss cs GI 0 1 cos s0a  sin s0a B C sin s0a cos s0a B C B C cos s1a  sin s1a B C C 1 B ðaÞ ðbÞ C sin s1a cos s1a ðDs ; Ds ÞCI ¼ pffiffiffiffi B C .. .. NB B C B C . . B C @ cos sðN  1Þa  sin sðN  1Þa A 0 B 1 B B ¼ pffiffiffiffi B NB @

sin sðN  1Þa ðaÞ qs;0 ðaÞ qs;1

ðbÞ qs;0 ðbÞ qs;1

.. .

.. .

ðaÞ qs;N1

ðbÞ qs;N1

1 C C C C C A

cos sðN  1Þa

Appendix B. Ring deformation potentials B.1. Vertical modes Vertical ring deformations can be described with different internal coordinates. B.1.1. Bond stretch Out-of-plane motions can be described by small upward displacement of the jth atom and small downward displacements of atom þ1, the internal coordinate is a bond stretch coordinate Sstr J ¼ ðvJ  vJþ1 Þ The harmonic deformation potential is the sum V str ¼

ðA:32Þ

ðB:1Þ

vertices X J

V str ðJÞ ¼

str k X str 2 ðSJ Þ 2 J

ðB:2Þ

and to each vertex J contribute two potential terms V str ¼ V str ðJ  1Þ þ V str ðJÞ

ðB:3Þ

29

A.F. Sax / Chemical Physics 349 (2008) 9–31 str which yields  2the contribution to the Hessian matrix Hvv with matrix V str . Hsvv is a circulant matrix defined by its first row elements ovo ov j

jþ1

str

Hstr vv : k ðð2; 1; 0; 0 . . . 0; 1ÞÞ:

ðB:4Þ

two adjacent polygon sides, direction vector dJ is the normalized vector from vertex J þ 1 to J, vector dNþJ is the normalized vector from vertex J to J  1. The corresponding normal vector to each direction vector dJ is nJ .

with eigenvalues str

str

kstr ðsÞ ¼ k AðsÞ ¼ k 2ð1  cos asÞ;

s ¼ 0; . . . ; N  1

ðB:5Þ

and AðsÞ > 0 for all s > 0 and Að0Þ ¼ 0. B.1.2. Bond torsion Out-of-plane motions can also be described by bond torsion, and the internal coordinate for bond torsion about the bond between atoms J þ 1 and J þ 2 is Stor J

1 ðvJ  vJþ1 þ vJþ2  vJþ3 Þ ¼ l sin a

ðB:6Þ

where l is the distance between two adjacent atoms. The harmonic deformation potential is the sum V tor

tor k X  tor 2 ¼ V tor ðJÞ ¼ SJ 2 J J  tor 

2 k 1 vJ  vJþ1 þ vJþ2  vJþ3 ¼ 2 l sin a bonds X

ðB:8Þ

ðB:9Þ

tor

BðsÞ P 0 for all s 8 2ð2  3 cos as þ 2 cos a2s  cos a3sÞ for N P 7 > >

> tor < 2 2  3 cos as þ 2 cos a2s  ð1Þs for N ¼ 6 k ¼ 2 2 > 2ð2  3 cos as þ cos 2asÞ for N ¼ 5 l sin a > >

: for N ¼ 4 4 1 þ ð1Þs  2 cos as l

2

V str ðJÞ ¼

ðB:11Þ ðB:12Þ ðB:13Þ

ðB:14Þ ðB:15Þ ðB:16Þ

str k X str 2 ðSJ Þ 2 J

ðB:17Þ str

The force constant must again be a positive quantity k vertex J contribute two potential terms V str ¼ V str ðJ  1Þ þ V str ðJÞ

The eigenvalues are k

bonds X J

which yields the contribution to the Hessian matrix 8 ðð4; 3; 2; 1; 0 . . . 0; 1; 2; 3ÞÞ for N P 7 > > > tor < ðð4; 3; 2; 2; 2; 3ÞÞ for N ¼ 6 k tor Hvv :¼ 2 2 ðð4; 3; 1; 1; 3ÞÞ for N ¼ 5 l sin a > > > : ðð4; 4; 4; 4ÞÞ for N ¼ 4

ktor ðsÞ ¼

a a sJ ¼ dJ ¼ sin rJ  cos tJ 2 2 a a sJþ1 ¼ dNþJþ1 ¼ sin rJþ1 þ cos tJþ1 2 2 Sstr J ¼ nJ  sJ þ nJþ1  sJþ1  a a  ¼ nJ  sin rJ  cos tJ 2 2   a a þ nJþ1  sin rJþ1 þ cos tJþ1 2 2 a a a a ¼ sin rJ  cos t J þ sin r Jþ1 þ cos tJþ1 2 2 2 2 a a ¼ ðr J þ rJþ1 Þ sin  ðtJ  t Jþ1 Þ cos 2 2 The harmonic deformation potential is the sum

ðB:7Þ

To each vertex J contribute four potential terms V tor ¼ V tor ðJ  3Þ þ V tor ðJ  2Þ þ V tor ðJ  1Þ þ V tor ðJÞ

B.2.1. Bond stretch The bond stretch coordinate Sstr between points J and J þ 1 is J defined as the projection of the displacements nJ and nJþ1 onto the vectors sJ and sJþ1 , respectively,

BðsÞ;

ðB:10Þ s ¼ 0; . . . ; N  1 B.2. In-plane modes Fig. B.1 shows the vectors needed for the definition of internal coordinates: each vertex J of an N-sided polygon is end point of

> 0, to each ðB:18Þ

which yields the following contributions to the circulant submatrices of the Hessian   str 2 a 2 a 2 a ; sin ; 0; 0; . . . ; 0; sin ðB:19Þ Hsrr :¼ k 2 sin 2 2 2   a a a str ðB:20Þ Hstt :¼ k 2 cos2 ;  cos2 ; 0; 0; . . . ; 0;  cos2 2 2 2  a a a a  str s ðB:21Þ Hrt :¼ k 0; sin cos ; 0; 0; . . . ; 0;  sin cos 2 2 2 2 The eigenvalues of the submatrices are a ð1 þ cos asÞ 2 str ¼ k ð1  cos aÞð1 þ cos asÞ; CðsÞ P 0 for all s a str str Hstt sstr ðsÞ ¼ k DðsÞ ¼ k 2 cos2 ð1  cos asÞ 2 str ¼ k ð1 þ cos aÞð1  cos asÞ; DðsÞ P 0 for all s a a str str Hsrt istr ðsÞ ¼ ik EðsÞ ¼ ik 2 sin cos sin as 2 2 str ¼ ik sin a sin as str

str

2

Hsrr qstr ðsÞ ¼ k CðsÞ ¼ k 2 sin

ðB:22Þ ðB:23Þ ðB:24Þ ðB:25Þ ðB:26Þ ðB:27Þ

We mention the symmetry properties qðsÞ ¼ qðsÞ, sðsÞ ¼ sðsÞ and ðsÞ ¼ ðsÞ. The three quantities C; D; E are not independent because 2 EðsÞ ¼ CðsÞDðsÞ; s ¼ 0; 1;ffi . . . ; N  1, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ðCðsÞ  DðsÞÞ2 þ 4E2 ðsÞ.

giving

CðsÞ þ DðsÞ ¼

B.2.2. Angle bend The angle bend coordinate Sben with respect to vertices J  1, J J and J þ 1 is defined by the projection of the displacements nJ1 , nJ and nJþ1 onto the vectors sJ1 , sJ and sJþ1 , respectively.

Fig. B.1. Definition of the vectors used for constructing the internal coordinates.

a a sJ1 ¼ nJ1 ¼ cos rJ1 þ sin tJ1 2 2 a a sJþ1 ¼ nJþNþ1 ¼ cos rJþ1  sin tJþ1 2 2 a sJ ¼ ðsJ1 þ sJþ1 Þ ¼ 2 cos rJ 2

ðB:28Þ ðB:29Þ ðB:30Þ

30

A.F. Sax / Chemical Physics 349 (2008) 9–31

Sben ¼ sJ1  nJ1 þ sJ  nJ þ sJþ1  nJþ1 J 1 a a a cos r J1 þ sin t J1  2 cos r J ¼ l 2 2  2 a a þ cos rJþ1  sin tJþ1 2 2  1 a a cos ðrJ1  2r J þ r Jþ1 Þ þ sin ðtJ1  t Jþ1 Þ ¼ l 2 2

ðB:31Þ

V ben ðJÞ ¼

ben vertices X

k

2

J

ðB:33Þ

¼

2l

ðB:34Þ to each vertex contribute three potential terms V ben ¼ V ben ðJ  1Þ þ V ben ðJÞ þ V ben ðJ þ 1Þ

ðB:35Þ

For polygons with N  5 the circulant submatrices of the Hessian ben

Hben rr :¼ Hben :¼ tt Hben rt

:¼

k

2

l ben k 2

l ben k 2

k

FðsÞ ¼

2

ben

k

l

GðsÞ ¼

2

k

l

2

ben

HðsÞ ¼ i

k

2

ðB:40Þ

ðB:42Þ

a 2 sin 2

with FðsÞGðsÞ ¼ H2 ðsÞ; s ¼ 0; . . . ; N  1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðFðsÞ  GðsÞÞ2 þ 4H2 ðsÞ and 26s6d

ðB:43Þ ðB:44Þ giving

FðsÞ þ GðsÞ ¼

ðB:45Þ

The expressions of the eigenvalues for 3- and 4-sided polygons are given below. Case N ¼ 4 The matrices ben

Hben rr :¼ Hben :¼ tt

k

2

cos2

2

sin

l ben k l ben k

2

a ðð6; 4; 2; 4ÞÞ 2

ðB:46Þ

a ðð2; 0; 2; 0ÞÞ 2

ðB:47Þ

a a sin cos ðð0; 2; 0; 2ÞÞ 2 2 2 l have eigenvalues

Hben :¼ rt

l

k

l

ðB:50Þ

ben

HðsÞ ¼ i

2

ð1  cos aÞð1  ð1Þs Þ

2

k

l

2

2 sin a sin as

ðB:51Þ

l

2

a ðð6; 3; 3ÞÞ 2

ðB:52Þ

sin

a ðð2; 1; 1ÞÞ 2

ðB:53Þ

sin

a a cos ðð0; 3; 3ÞÞ 2 2

ðB:54Þ

cos2

ben

Hben :¼ tt

k

l

2

2

ben

Hben :¼ rt

k

l

2

ben

ben

k

l

FðsÞ ¼

2

ben

ben Hben ðsÞ ¼ tt s

k

l

l

2

k

l

3ð1 þ cos aÞð1  cos asÞ

ðB:55Þ

ð1  cos aÞð1  cos asÞ

ðB:56Þ

ben

GðsÞ ¼

2

k k

l

2

2

ben

HðsÞ ¼ i

k

l

2

3 sin a sin as

ðB:57Þ

References

ðB:41Þ

l a cos ð2 sin as  sin 2asÞ 2 ben k ¼ i 2 sin að2 sin as  sin 2asÞ l

CðsÞ  DðsÞ EðsÞ ¼ ; FðsÞ  GðsÞ HðsÞ

k

ðB:39Þ

2

ben

k

GðsÞ ¼

2

The symmetry properties of the eigenvalues for all polygons are qðsÞ ¼ qðsÞ, sðsÞ ¼ sðsÞ and ðsÞ ¼ ðsÞ.

ben

l a ð1  cos 2asÞ ¼ 2 sin 2 ben k ¼ 2 ð1  cos aÞð1  cos 2asÞ; l for all s

ben ðsÞ ¼ i Hben rt i

l

ðB:49Þ

ben

2

2

GðsÞ  0

ðB:38Þ

k

l a ð3  4 cos as þ cos 2asÞ 2 cos 2 ben k ¼ 2 ð1 þ cos aÞð3  4 cos as þ cos 2asÞ; l for all s

ben ðsÞ ¼ Hben tt s

k

ben

2

FðsÞ P 0

k

ben Hben ðsÞ ¼ i rt i

ben

ð1 þ cos aÞð3 þ ð1Þs

2

ben

ben

Hben rr :¼

ðB:37Þ

l have eigenvalues l

ben ðsÞ ¼ Hben tt s

ben Hben ðsÞ ¼ rr q

a a sin cos ðð0; 2; 1; 0; . . . ; 1; 2ÞÞ 2 2

ben Hben ðsÞ ¼ rr q

ben

ðB:36Þ

a ðð2; 0; 1; 0; . . . ; 1; 0ÞÞ 2

2

sin

l

have eigenvalues

a ðð6; 4; 1; 0; . . . ; 1; 4ÞÞ 2

cos2

k

Case N ¼ 3 The matrices

ðSben Þ2 J 2 a a cos ðr J1  2r J þ r Jþ1 Þ þ sin ðt J1  tJþ1 Þ 2 2

J

FðsÞ ¼

2

ben

J

2

l

ben Hben ðsÞ ¼ i rt i

ben vertices X 

k

ben

k

 4 cos asÞ ðB:32Þ

The harmonic potential is vertices X

ben

ben Hben ðsÞ ¼ rr q

ðB:48Þ

[1] J.E. Kilpatrick, K.S. Pitzer, R. Spitzer, J. Am. Chem. Soc. 69 (1947) 2483. [2] D.G. Lister, J.N. Macdonald, N.L. Owen, Internal Rotation and Inversion, Academic Press, London, 1978. [3] I.B. Bersuker, Chem. Rev. 101 (2001) 1067. [4] W. Domcke, D.R. Yarkony, H. Köppel, Conical Intersections, Electronic Structure, Dynamics & Spectroscopy, World Scientific Publishing, 2004. [5] R. Lindner, K. Müller-Dethlefs, E. Wedum, K. Haber, E.R. Grant, Science 271 (1996) 1698. [6] K. Müller-Dethlefs, J. Barrie Peel, J. Chem. Phys. 111 (1999) 10550. [7] C.C. Chancy, M.C.M. O’Brien, The Jahn–Teller Effect in C60 and Other Icosahedral Complexes, Princeton University Press, 1997. [8] J. Manz, J. Am. Chem. Soc. 102 (1980) 1801. [9] D.O. Harris, G.G. Engerholm, C.A. Tolman, A.C. Luntz, R.A. Keller, H. Kim, W.D. Gwinn, J. Chem. Phys. 50 (1969) 2438. [10] G.G. Engerholm, A.C. Luntz, W.D. Gwinn, D.O. Harris, J. Chem. Phys. 50 (1969) 2446. [11] R. Davidson, P.A. Warsop, Faraday Trans. II 68 (1972) 1875. [12] G. Katzer, A.F. Sax, J. Chem. Phys. 117 (2002) 8219. [13] A.D. McNaught, A. Wilkinson, IUPAC Compendium of Chemical Terminology, Blackwell Science, 1997. [14] J.P. McCullough, R.E. Pennington, J.C. Smith, I.A. Hossenlopp, G. Waddington, J. Am. Chem. Soc. 81 (1959) 5880. [15] J. Breuer, D. Avnir, J. Chem. Phys. 122 (2005) 074110-1. [16] J.L. Martins, R. Car, J. Buttet, J. Chem. Phys. 78 (1983) 5646. [17] T.C. Thompson, D.G. Truhlar, C.A. Mead, J. Chem. Phys. 82 (1985) 2392. [18] M. Keil, H.-G. Kramer, A. Kudell, M.A. Baig, J. Zhu, W. Demtroder, W. Meyer, J. Chem. Phys. 113 (2000) 7414. [19] G. Delacretaz, E.R. Grant, R.L. Whetten, L. Woste, J.W. Zwanziger, Phys. Rev. Lett. 56 (1986) 2598. [20] S.P. Walch, B.C. Laskowski, J. Chem. Phys. 84 (1986) 2734. [21] S.P. Walch, C.W. Bauschlicher Jr., S.R. Langhoff, J. Chem. Phys. 85 (1986) 5900. [22] S.R. Langhoff, C.W. Bauschlicher Jr., S.P. Walch, B.C. Laskowski, J. Chem. Phys. 85 (1986) 7211. [23] E.E. Wedum, E.R. Grant, P.Y. Cheng, K.F. Wiley, M.A. Duncan, J. Chem. Phys. 100 (1994) 6312. [24] K. Balasubramanian, M.Z. Liao, Chem. Phys. 127 (1988) 313. [25] K. Balasubramanian, K.K. Das, Chem. Phys. Lett. 186 (1991) 577. [26] C.W. Bauschlicher, Chem. Phys. Lett. 156 (1989) 91. [27] G.A. Bishea, M.D. Morse, J. Chem. Phys. 95 (1991) 8779. [28] R. Wesendrup, T. Hunt, P. Schwerdtfeger, J. Chem. Phys. 112 (2000) 9356. [29] J.H. Kiefer, R.S. Tranter, H. Wang, A.F. Wagner, Int. J. Chem. Kinet. 33 (2001) 834.

A.F. Sax / Chemical Physics 349 (2008) 9–31 [30] [31] [32] [33] [34]

D.V. Makhov, L.J. Lewis, Phys. Rev. B 72 (2005) 073306. T. Krüger, A.F. Sax, Phys. Rev. B 64 (2001) 5201. W.v.E. Doering, W.R. Roth, Angew. Chem., Int. Ed. Engl. 2 (1963) 115. J. Kalcher, A.F. Sax, unpublished results. P.J. Davies, Circulant Matrices, Chelsea Publishing, VCH, 1994.

31

[35] R. McWeeny, Symmetry, An Introduction to Group Theory and its Applications, Dover Publications Inc., 2002. [36] G. Herzberg, Molecular Spectra and Molecular Structure, vol. II Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold Company, 1945.