The effect of phase conventions on vibration-rotation matrix elements

JOURNAL OF MOLECULAR SPECTROSCOPY

143, 111- 136 ( t 990)

The Effect of Phase Conventions on Vibration-Rotation Matrix Elements CARLODILAUROANDFRANCALATTANZI Dipartimento di Chimica Farmaceutica e Tossicologica, Universitci di Napoli, Via D. Montesano 49. 80131 Napoli, Italy

AND

GEORGES GRANER Laboratoire d’lnfrarouge, Associe’ au CNRS. Universite’ Paris-Sud, Bdtiment 350. 91405 Orsay Cedex. France

The transformationproperties of vibrational and rotational basis operators and functions under symmetry operations and time reversal are investigated, with emphasis on their dependence on normal coordinate orientation and phase conventions. The effect of phase conventions on the values of the off-diagonal vibration-rotation matrix elements is examined, and it is shown that if the molecular symmetry allows for an operation denoted W’, consisting of either a reflection through a plane containing the angular momentum quantization z-axis or a rotation about a binary axis normal to z, all vibration-rotation matrix elements of axially symmetric and asymmetric rotor molecules can be made real by appropriate vibrational and rotational phase conventions, which are discussed and recommended. Therefore vibration-rotation matrix elements of these molecules can be made all real for all molecular symmetry groups, with the exception of the groups containing separably degenerate E-species, C,, C, (no symmetry), and Cz, C,, and C,t, if the binary rotation axis is oriented along z and oh = bXY.It is shown that, with the same conventions which render all vibration-rotation matrix elements real, the matrix elements to be used in the calculation of electric dipole vibration-rotation intensities, when the M-degeneracy is not removed, are taken all real if 9’ is a reflection plane and all imaginary if 9’ is a binary rotation axis. Relative phases of rotational wavefunctions differing by the value of M have to be defined in order to determine the values of matrix elements with AM = + 1. 0 1990 Academicpress. hc. 1. INTRODUCTION

The study of vibration-rotation spectra nowadays normally requires the simultaneous analysis of interacting levels belonging to several different vibrational states, with the diagonalization of large Hamiltonian matrices where the interaction terms occur as off-diagonal elements. The values of the off-diagonal matrix elements depend on the conventions adopted about the relative phases of the involved basis functions, and in general are complex quantities. Since each off-diagonal matrix element can be made either pure real or pure imaginary, as will be shown, it is crucial to investigate which matrix elements are real and which are imaginary, under certain adopted conventions, and whether it is possible to make the off-diagonal matrix elements either all real or all imaginary. This is the main goal of this article, owing to the relevance 111

0022-2852190 $3.00 CopyrigJ~t0 1990 by Academic Press, Inc. All rightsof reproductionin any form reserved.

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of this matter to the calculation of eigenvalues and eigenvectors, which are related to vibration-rotation transition energies and intensities, respectively. We find that with appropriate phase conventions, it is possible to render all offdiagonal vibration-rotation matrix elements real, if the molecular symmetry allows for a symmetry operation .42’,consisting of either a binary rotation about the x or y axis or a reflection through a plane containing the z-direction, z being the angular momentum quantization axis in a molecule-fixed frame. In this case, the translations T,, T, (and the rotations R,, Ry) span different irreducible representations of the molecular subgroup (E, W’), and W’ allows a distinction between x-oriented and yoriented displacements or motions in a molecule-fixed frame. The transformation properties of wavefunctions and operators under the molecular symmetry operations provide restrictions on the allowed matrix elements and relations between nonvanishing matrix elements. The effect of time reversal on matrix elements provides additional relations involving complex conjugation. Therefore the study of the behavior of matrix elements under the effect of symmetry operations and time reversal allows investigation as to under which assumptions a given matrix element is real or imaginary. The explicit application of time reversal to this kind of problem, in spite of its wide use in the field of rovibronic spectroscopy, seems to have received little consideration by infrared spectroscopists. We adopt a notation particularly suitable to the problems of axially symmetric molecules, but all the following treatment can be applied to asymmetric top molecules with appropriate simplifications of both formalism and results. 2. PHASE ANGLES

AND THEIR NOTATION

The transformation properties under time reversal and certain molecular symmetry operations of the one- and two-dimensional harmonic oscillator wavefunctions, and those of the rotational wavefunctions, are discussed in Appendices I and II, respectively. The reading of this article requires knowledge of most of the content of both appendices, and in particular of the definitions of the phase angles which are listed below. (i) One-dimensional harmonic oscillator wavefunctions. Relative phases are determined by the value of a phase angle 6,, defined by Eqs. (I. la, b) in Appendix I. We identify nondegenerate oscillators by the subscript r, and in this case their wavefunctions and phase angles are denoted 1vr) and 6,. We shall also use the symbols ~o,~I~ and 6“r(n)to distinguish the phase angles of the wavefunctions of nondegenerate normal modes which are symmetric or antisymmetric with respect to a specified molecular symmetry operation. (ii) Two-dimensional harmonic oscillator wavefunctions. The values of two phase angles 6, and 61, defined by Eqs. (1.9a, b) in Appendix I, are required to determine relative phases. Degenerate oscillators are identified here by the subscript t, and in this case wavefunctions and phase angles are denoted I q, It), 6,,, and a,,. We shall use the notations V,and 6, to identify vibrational quantum numbers and phase angles which apply to both nondegenerate and degenerate vibrational modes. We recall that the q1 and q2 normal coordinate components of a given two-dimensional vibrational pair are oriented to transform as the translations TX and T, under an orienting operation B’ (either a rotation about a binary axis along x or y or a

PHASE

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CONVENTIONS

reflection through a plane containing the z-direction, z being the molecule-fixed angular momentum quantization axis) if this operation occurs in the molecular symmetry group, as discussed in the Appendix I. (iii) Rotational wavefunctions. Relative phases of wavefunctions differing by the value of k are determined by the value of a phase angle bk, defined by Eq. (II. 1) in Appendix II. When matrix elements off-diagonal in J occur, as in the case of the transition moment operator, the value of the additional phase angle q$, defined by Eq. (11.4) in Appendix II, is required as well. Although phase angles may assume any value, we find convenient to bind each of the mentioned angles 6,,, a,,, a,,, &, and 7: to assume one of the simple values 0, ?r, k1~/2, modulo 27~.With this choice the matrix elements (I. 1 ), (1.9), and (II. 1) and the reduced matrix elements (11.4) assume either real or imaginary values. 3. VIBRATIONAL

WAVE

FUNCTIONS

AND

NOTATION

The vibrational harmonic eigenfunctions of an axially symmetric molecule, with defined values of the vibrational angular momenta generated within each doubly degenerate normal mode, are given as usual by products of separate factors related to the independent oscillators. We use the notation IV, i> =

III~, I+, . . . , vi,

vi:,

. . .) =

n rEAorB

1~~) n

IV!),

(1)

IEE

where i and ? represent the ensembles of v- and I-quantum numbers associated with the separate vibrational components

. . . ) Of,Ill<,. . .) i = (lt,it/,. . .).

(2)

i = (v,, IQ,

(3)

The harmonic vibrational energies are h C, ( vo,+ d,/ 2 ) us, where d, is the degeneracy of the sth vibrational normal mode, and depend only on i. Thus we denote by “?system” the system of vibrational levels of given V, whose components are all degenerate in the harmonic approximation in a nonrotating molecule. Each particular component of a i-system is identified by its i ensemble, which defines the value of the vibrational angular momenta of each doubly degenerate normal mode. We denote by i = 6 the component of a t-system with all &-values equal to zero, which exists if all Q’S are even. A T-system consists of a single i = 6 component if no degenerate vibrational mode is excited. Asymmetric top molecules can be thought to have i = 6 always. A f-system where each degenerate mode is excited by an even number of quanta always has a component with i = 6. Given a nonvanishing 1 ensemble, the related -i ensemble is obtained by inverting the sign of each lt in Eq. (3). The [?,I) and I ?, -i) states differ by the sign of the vibrational angular momenta of all degenerate normal modes. 4. TOP AXIS

OPERATIONS

IN AXIALLY

SYMMETRIC

MOLECULES

Since all groups of axially symmetric molecules have at the least a rotation C, or a rotation-reflection S, about the top axis (with n larger than 2 for C, and n even and larger than 2 for S.), we first examine the behavior of vibrational wave functions

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under this operation. This symmetry behavior is easily treated by use of the vibrational Gquantum number defined by Hougen (I): however, we prefer to follow the molecular group classification of Henry and Amat (2,.3) which, by considering symmetry behavior and selection rules related to the inversion i and to the reflection through a ub plane normal to the quantization axis z separately, allows definition of the vibrational G quantum number by a single expression for all symmetries. This choice has also been considered by Hougen (see footnote 23 of Ref. ( 1)) and adopted by Mills (4). Thus, following Henry and Amat’s notation, we denote by L&?the single step rotation CA about the top axis for all groups, except DcnlZjdand S, with n / 2 even in both cases, where 99 is the single step rotation-reflection SA operation. The j7,i) basis functions transform under W as !???I?, i) = exp(-27riG/n)l?,

i)

(4)

with

G= 5 C

Vr +

C m,l,,

(5)

I

TEB

where t runs over all degenerate normal modes E,,, with all possible values of m, and m, = m. Thus the symmetry behavior of I?, 1) functions under B is determined by the values of G modulo n, and the symmetry species A, B, and E,,, correspond to G = 0 modulo n

A

G = (n/2) modulo Iz

B

G = m modulo n

E,(a)

G = -m modulo n

L(b),

(6)

where 0 < m c n/2. The ensembles $ and i determine the G-value of a 17, i) but not vice versa. Equation (6) also defines the symmetry labels a and b for the components of degenerate pairs of wavefunctions. Application of Eq. ( 5) to I f , i) and I F, -i) gives G(?, -i) = -G(t,

i)

modulo n

(7)

and comparison with Eq. (6) shows that each Ii, i)/l?, --I) pair spans A/A, B/B, or E,( a)/E,,,( b) symmetries. For I i, 1) pairs spanning E,-symmetry, we shall denote by I T, i) the partner with E,,,(a) symmetry and by I 5, -i) the partner with E,(b) symmetry. No such convention is practicable for A or B pairs. Only A/A and B/B pairs can be split by anharmonicity (vibrational l-doubling) since the vibrational Hamiltonian can only mix pairs of levels I Z, i) and I i, -i> with AG = 0 modulo n. The effect of multistep rotations or rotation-reflections about the top axis is obtained by repeated application of Eq. (4). Alternative symmetric top vibrational wavefunctions are I(?, iI&) = 2-1/2(l?, i> + IT, -i>).

(8)

PHASE

CONVENTIONS

115

If i = 6, the I(T, 1))) combination (8) vanishes and I(?, a),) = I?, 6). When I+, l), IF, -1) span E,(a) and E,(b) symmetries, then I(?‘, I)+) and I(?, l)_ ) transform under W as Wl(i, i)+> = cos +,I(%, i)+> - i sin @I(?, i)_) &I(?, i)_> = - i sin *I(?, iI+> + cos +,I(?, i)_>

(9)

with + = 2?rm/n. The functions (8) transform under W as the qt, and iq, normal coordinate components of an E,,, mode, if a positive rotation about z brings q[, into qtz through the smaller angle. In linear molecules, symmetry allows for rotations by any angle @ about the internuclear axis z, and Eq. (4) applies with 2a/n replaced by ‘3. The contributions to G in linear molecules are zero for nondegenerate normal modes and 1, for degenerate normal modes. In fact only II degenerate normal modes, corresponding to El, occur in linear molecules. Linear molecules can be treated as symmetric top molecules with k = C, Ztand all considerations and results of this article applying to symmetric tops apply to linear molecules as well. The symmetry of linear molecules does not allow for vibration-rotation operators which would involve different shifts of k and C, It, in agreement with the requirement for equality of these two quantities. Asymmetric top molecules contain at the most a binary rotation about z. If we want to treat asymmetric tops as a special case of symmetric tops without degenerate vibrations, then Eq. (4) formally holds with B = C 2Z,and in this case Eq. ( 5) applies with G equal either to zero or II /2 = 1. However, in the usual notation, some Bspecies of certain symmetry groups, such as B, of V and B1,, B1u of DZh, are symmetric under CZZ(5) and then would correspond to G = 0 modulo 12.All binary rotations and all reflections transform the asymmetric top vibrational wavefunctions in a similar way, involving either invariance or change of sign. The definition of an .!%?= CZZ operation in asymmetric top molecules is of no special utility, as will be shown later, and does not even help to establish the notation for A and B irreducible representations as shown shortly above. 5. OTHER

SYMMETRY

OPERATIONS

The transformations of wavefunctions ( 1) under symmetry operations can be easily worked out from the transformations of the separate terms of the product ( 1) , which are discussed in Appendix I. When the molecular geometry allows for additional symmetry elements other than W, in the case of axially symmetric molecules, it is convenient to separate all the symmetry operations into two categories: (i) symmetry operations whose effect on )i, i) functions depends only on 5. This category includes the inversion i and the reflection Qhthrough a plane normal to the z-axis. (ii) symmetry operations whose effect on 1T, i) functions depends on both i and i . This category includes rotations and rotation-reflections about the z-axis, reflections

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through a,, and uYZplanes, and rotations about binary CZ, or CQ axes. The above distinction is useless in asymmetric top molecules, where i is undefined. All i-components of a given V-system, or any linear combination thereof, have identical symmetry behavior with respect to the (i) type operations. Therefore a given T-system has a defined g or u character with respect to inversion i and/or ’ or ’ character with respect to bh for all its components, if the molecular symmetry includes these operations. The (ii) type operations, on the contrary, allow classification of the different Icomponents of a given i-system according to their different transformation behavior. In this category, we find the mentioned 99 operation, which allow classification of these components into the A, B, I?,( a), and E,(b) symmetry species according to the corresponding G-quantum number, as in Eq. (6). When the molecular symmetry of an axially symmetric molecule contains u, planes or dihedral C’, binary rotation axes, one of these operations can be used to further classify the i-components of a Vsystem according to their symmetry behavior (second orienting operation W’ in the notation of Henry and Amat). The J%?‘-operation interchanges the partners of a 1i, +i) doublet since angular momenta about the z-axis, as axial vectors, are reversed by all bV reflections and dihedral Cz rotations. From Eq. ( 1) and Eqs. (1.7, 13) of Appendix I, we obtain BfI:,

i> = (-i)~kli,

with p = parity of the excited nondegenerate antisymmetric with respect to W’

-i>

(10)

vibrational quanta in normal modes

T = parity of the excited vibrational quanta in degenerate normal modes t= lifB’=o,orCZx t = -1 if B?‘= u,,orCZy and (Y= exp(-2i

C 1,6,,).

(11)

1EE

If the phase angles 6,, are bound to assume one of the values 0, r, or +x/2, then CY can be either 1 or -1, and the I(?, i)*) wavefunctions defined in Eq. (8) transform consequently as B’I(V,

iI*> = +(-iytral(f, i)*>.

(12)

The vibrational wavefunctions I ?) of asymmetric top molecules, owing to the lack of degenerate normal modes, have cy= 1 and T =O. They are transformed into ( - 1)” I S) by any reflection or binary rotation, as already mentioned. A binary rotation of an asymmetric top molecule is conveniently treated as .?8, in analogy with axially symmetric molecules, only if there is another binary rotation or reflection to be chosen as LB’. Otherwise, the binary axis might be more conveniently treated as W’. Thus, the C,, C,, and C2h groups can be assumed to have W’ and no B, and with this choice, 9Z?’is missing in an asymmetric top molecule only for the Ci symmetry or in the absence of any symmetry element. The choice of 99’ as the binary rotation CzXor

PHASE CONVENTIONS

117

CzYof the C, or C2h groups or as the symmetry reflection 6, or uyz of the C, or CZh group does not minimize the off-diagonal rotational matrix elements if the “near-top axis” is along the binary axis or normal to the symmetry plane associated to &I?‘, because in this case the quantization z-axis would not coincide with the near-top axis. The orientation of the I (i, i)+) and I (i, i)-) components depends on phase conventions; we examine in particular the case when (Yis equal to unity, which occurs for instance when all &,-values are zero or r. In this case 1(T, i)+), ((i, I)_) transform as T,, T, under u, and CzXfor even p and under uYZand Cz,. for odd ( p + T) . On the other hand, I(T, i)+), I(?, i)_) t ransform as TY, T, if the above parities in the two cases are reversed. States corresponding to the single excitation of one degenerate normal mode correspond to p = 0, 7 = 1. Their associated I (S, i)+) and I (7, i)-) wavefunctions transform as T, and Ty under 9’ as the normal coordinate components 41 and q2. 6. TIME REVERSAL

AND

MATRIX

ELEMENTS

Equations (4,9, 10, 12) show that B does not mix or interchange the components of a ( T, +I) doublet, and W’ does not mix or interchange those of a I ( ? , i)*) doublet. Therefore the I i, i) and I ?, -i) partners of a pair spanning E,-symmetry appear to be bases of unidimensional irreducible representations, &(a) and E,(b) respectively, for those groups where the B’ symmetry element is missing ( C,,, C,,, , S,) . However the behavior of wavefunctions under time reversal should be taken into account as well. Time reversal consists of the inversion of the direction of all motions, including angular momenta, and brings I ?, i) into )i, -i) and vice versa. Therefore E,(a) and E,(b) vibrational partners always must be thought of as being related to degenerate pairs, even if they are not mixed or interchanged by any molecular symmetry operation (separably degenerate irreducible representations). The time reversal operator is denoted 0, and its properties have been appropriately discussed by Wigner (6). Here we recall that this operation transforms constant factors and matrix elements into their complex conjugates, and changes the signs of all operators which imply any type of motion, such as momenta and angular momenta. The effect of time reversal on the eigenfunctions of harmonic oscillators and on rotational wavefunctions is discussed in Appendices I and II, together with the transformations under symmetry operations. It can be easily understood that all angular momentum projection quantum numbers occurring in wavefunctions change sign under time reversal, as a consequence of the inversion of the direction of motion. Application of the results (1.4, 12) to Eq. ( 1) yields elf,

i> = pp, -i>

el(f, i)>&> = +flI(q, I)>&>

(13) (14)

with P = exp(2i C vJ,J,

(15)

where the summation runs over all normal modes, degenerate and nondegenerate.

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118

If the 6, phase angles are bound to assume one of the simple values 0, ?r, 7r/2, or -n/2, /3 can be either 1 or - 1. The vibrational term involving degenerate normal modes in axially symmetric molecules can be written by using either linear component operators, such as qr, , q12,and/or their conjugated momenta, or ladder operators such as qt+ = q[, f &, ptk = pi, + ipt2. We suppose that the rotational operators, related either to the total angular momentum operators or to direction cosine operators, are expressed in terms of Cartesian components, though typical ladder operators could be used as well. Using linear component operators in the degenerate normal modes, both the vibration-rotation Hamiltonian and the transition moment operator consist of sums of products of hermitian terms k,,h,h,, where h, involves only vibrational operators and h, involves only rotational operators. The factors h, and h, are each hermitian separately. We define 0, = k,,h,, 0, = h,. Owing to the transformation Bk,,h,h,B-’ = kzh,h,, the invariance of k,,h,h, under time reversai requires the coefficients kyTto be real, when linear component operators are used for degenerate normal modes.’ Using ladder operators for degenerate normal modes, a vibrational hermitian term involving degenerate normal coordinates in general has the form Co,= k,,h,, + kZrh,_ = O,, + O,-

with

coV& = (O,?)‘.

Thus the operator 0, can always be written as the sum of two operators, hermitian conjugate of each other and with opposite ladder properties. We note that all terms of the vibration-rotation Hamiltonian, including those accounting for interactions with an external static electric field, are such that their invariance under time reversal implies hermiticity and vice versa, since hermitian conjugation generates the same transformations of these operators as time reversal. This does not hold for the separate vibrational and rotational partners of these operators, and this is not a general property, since hermitian conjugation and time reversal are basically different quantum mechanical operations ( 7). The relations between time reversal and hermitian conjugation applied to the rotational Hamiltonian have been also shown by Watson (8). Vibration-rotation Hamiltonian terms accounting for interactions with external magnetic fields, and magnetic dipole transition moment operators, transform with opposite signs under time reversal and hermitian conjugation. In any case, 0, transforms under time reversal as M$9-’ = aeo’o,

(16)

with&= lor&= - 1, depending on whether the operators containing “motion,” namely momenta and angular momenta, occur in 0, with an even or odd power. Thus, using Eqs. ( 13, 15, 16)) we find that a general vibrational matrix element transforms under time reversal as e(;‘,

Plo,It, i> = (G’,iJlovI;,i>* = upppf*(v’,-irIo,I Ii, -i>.

(17)

’ Operators accounting for interactions with external magnetic fields and magnetic dipole transition moment operators change sign under time reversal but this does not affect the result that the coefficients are real with the adoption of linear components for degenerate modes.

119

PHASE CONVENTIONS

Therefore the matrix elements of 0, between the components of two 1i, -ti) and 1T’, 1-i’) pairs are

(S’, (7,

iqo,p, i> = +p*p’(t~, -iqo,I~, -i>* -iyo,Is, i> = &3*pf(7, P(co,~~, -i>*

= w = w’

(18)

In general, the matrix elements on the left and right hand sides of Eq. ( 18) are due to different ladder components of 0,; for instance the left part comes from O,+ and the right one from 0,_ or the reverse. The matrix elements ( 18) are subjected to the following restrictions: (a) When the molecular symmetry contains the .%?operation (which is always the case for axially symmetric molecules), owing to the symmetry with respect to W, W vanishes unless2 G(+‘, i’) - G(?, i) 7 G( &) = 0 modulo n and W’ vanishes unless -G( F’, 1’) - G( T, i) T G( 8,) = 0 modulo n. The two signs + correspond to O,+ and O,_ respectively. Thus Wand W’ can both be nonvanishing if the two above equations are simultaneously satisfied with any pairing of the signs of G( 0,). This requires that either If’, +i’) or IT, d) or both be A or B pairs.3 (b) When the i or (Thoperation occurs, additional trivial selection rules hold, which depend on 0, and on ? and V, but not on i and i’. Wand W’ have different orders of magnitude if they involve different values of C, ) Al{1. This is never the case for i = b or i’ = b. It is easily found from Eq. ( 18) that ifi = 6, W’ = &P*B’W*, and ifi’ = 6, W’ = w.

If both i and i’ are 6, which also corresponds to the case of asymmetric top molecules, then there is a single matrix element W = W’ = &P*B’ W* in Eq. ( 18) which is real or imaginary if &/3*/Y’ is equal to + 1 or - 1, respectively. Transformation of Eq. ( 18) to the vibrational basis (8) yields

iy, (cov I(?,iI*>= [w+ w’ + &~*P’(w+ w’)*]/2 ((~‘,j’)710VI(~,i)f)=[~qw’-&p*pywi w’)*J/~ (?I, 6ls,p,i),) = [wk &p*~~w*]~V3 ((v, if), Io,I?, S) = [Wk &*p’w*]/fi.

((St,

(19)

Equations ( 18)) ( 19) can also be applied to the special case of matrix elements within a ?-system, that is, i’ = f. In this case, p’ = /3 and /3*p = 1. 7. SPECIAL CASE OF MOLECULAR

SYMMETRY GROUPS WITHOUT

9’

When the molecular geometry of an axially symmetric molecule does not allow for the SP operation (which is relatively rare), there are no further conditions to be obeyed by the vibrational matrix elements ( 18 ) and ( 19 ) , except the restrictions due 2 The ladder operators O,+ and 8,- transform under W as Wd,+ W-’ = exp( +ZriG( @,)/n)Co+ and G( 0,) can easily be calculated from the behavior of the single operators occurring in O,*. 3In this case, either G(i’, i’) = -G(i’, i’) modulo n or G(i, i) = -G(i, i) module n or both.

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to symmetry with respect to W, i, and Qh,whichever occurs in the molecular symmetry group discussed in the preceding paragraph. Obviously, W is always present in axially symmetric molecules. The matrix elements ( 18) and ( 19) in general have complex values. The matrix elements ( 19 ) are either real or imaginary if up@*/3’ = AI1, as can be understood by inspection. In fact, in this case, these matrix elements assume the form either a + a* (real) or a - a* (imaginary). This can be achieved by adopting a vibrational phase convention such that p*/3’ = +-1, since a? IS . either + 1 or - 1. This is always the case, if all aVsin Eq. ( 15) assume values not different from 0, +?r/2, ?r. If for instance we assume that each 6, is either 0 or P, both p and p’ are equal to 1 (see Eq. ( 15)) and the matrix elements ( 19) of any &-operator which is invariant under time reversal ‘v = 1) are real for + * + and - - - combinations and imaginary for + ++ (a@ combinations. The opposite result holds, with this phase choice, for the matrix elements of any 8, operator which is inverted by time reversal (a? = - 1). States with i = d behave as + states. The real or imaginary nature of these matrix elements, under the mentioned phase conventions, can also be obtained from Eqs. ( 14)-( 16). This procedure, however, would not give the relations between the matrix elements ( 19) and the matrix elements Wand W’ of Eq. ( 18 ) . The matrix elements of 0, for asymmetric tops without any symmetry element or of Ci symmetry are real if UP/~*@’= + 1 and imaginary if a@*@’ = - 1. The same result apply to the nonvanishing vibrational matrix elements of CZ, C,, and CZ~ molecules, if the binary axis is z or/and the reflection plane is xy. We observe that the matrix elements ( 19) of a given 0, can be made all real for &?*/3 = 1 and all imaginary for &p*/3’ = - 1 if we multiply by +i or -i either all 1(i, i), ) or all 1(i, i)- ) functions in Eq. ( 8 ). In fact, the values of the matrix elements ( 19) depend both on the vibrational phase conventions relative to the basis ( 1) and on the coefficients of the transformation (8 ) . The operators 8, in general occur in O,O, terms and we are actually interested in the values of vibration-rotation matrix elements which include the contributions of the rotational operators as well. The values of the matrix elements of rotational operators are determined once a choice is made for the values of the angles & and ~13defined in Appendix II. Time reversal brings a ) J k M) wavefunction into (J -k -44) and therefore its application is not suitable to problems where A4 is constant. However, the effects of time reversal and symmetry operations on rotational wavefunctions have been included in Appendix II for the sake of completeness. When we consider a problem where several vibration-rotation Hamiltonian terms are simultaneously involved, in general it is not possible in axially symmetric molecules to make all of them have real or imaginary off-diagonal matrix elements by appropriate phase conventions. On the contrary, it will be shown in this article that the off-diagonal matrix elements of the vibration-rotation Hamiltonian can be made all real by appropriate phase conventions if the molecular symmetry allows for an operation W’ (see Section 8b). As an example, we mention the z-axis Coriolis coupling of two different _&-modes, t and t’, of a symmetric top molecule by the Hamiltonian terms

PHASE

121

CONVENTIONS

and

HyoR= C,t;(Qt,Pt; -

Qt;Pt,

-

QtJ't;

+ Q,;P,,)J,.

HFoR vanishes in the presence of 33’ owing to the isoorientation of the qt,, qr2/qt;, qt; pairs, which are chosen in such a way that qt, and qt; are invariant and q12and ql; change sign under ~39’(2). With the vibrational phase choice 6, = 6! = 0 for both modes, the matrix elements of ql, , qt; , pf2, and pt; are real and those of q12,qt;, pt, , and pt; are imaginary (see Appendix I), so that the matrix elements of HFoR are real (they occur even in the presence of W’), and those of HFoR are imaginary (they occur only in the absence of 9’). 8. MOLECULAR

SYMMETRY

GROUPS

CONTAINING

d’

a. Vibrational Matrix Elements and Phase Conventions We come back now to the more frequent case when the molecular symmetry allows for the 94?‘-operation. In this case, other relations are found by considering the invariance of the nonvanishing matrix elements under W’. The transformation of vibrational wavefunctions under W’ is given in Eqs. ( lo), ( 12), and the vibrational hermitian operators 0, transform as #(J&@-l

= &‘O,

(20)

with ~2~ = + 1 if the qt, and qf2components of degenerate normal modes are oriented to transform as T, and T, under B’. Thus, from Eqs. ( 18), ( 1 1 ), and (20) and the invariance of the matrix elements of a vibrational operator Co”under GP, we find

= (-l)*~~*r (T~raa~*(7,iq~V~~, -i> = w’.

l.21)

The values of LYand cy’depend on the particular i and i’ components involved, owing to Eq. ( 11). In particular, if a value (Yapplies to )if, I), then the value cy* applies to I i, -I), owing to Eq. (11). On the contrary, (5, i) and IV, -i), have identical P-values, according to Eq. ( 15 ). All symbols in Eq. (2 1) have already been defined in relation to Eqs. ( lo), ( 11)) ( 15 ) . Ap and AT are the total shifts (or the parities thereof) of the vibrational v-quanta excited in nondegenerate modes antisymmetric with respect to B’ and in all degenerate modes, respectively. If ( - 1) AP~ *‘asp&*/ &B*p ’ is equal to + 1 or - 1 and both numerator and denominator are real (which is always the case if the vibrational phase angles b,,, a,,, 61,are bound to assume one of the simple values 0. ?r, a/2, or -r/2, leading to the values 1 or - 1 of (Y,(Y’,,B, p’, and their complex conjugates), the matrix elements (2 1) are all real or all imaginary. The above expression can be rearranged (remember that 02, and a3 can be either 1 or - 1 and cy,LU’, ,L3,and p’ are unitary coefficients) to show

DI LAURO,

122

LATTANZI,

AND

GRANER

that if the vibrational phase angles assume the above mentioned simple values and ( - 1) AptA.‘aa’*pp ‘* = 1, then the matrix elements (21) are real if a? = ~2~ and imaginary if a? = -a>(. This statement has to be reversed if the above product assumes the value -1. The above expression is the product of two separate unitary factors belonging each to one of the two wavefunctions occurring in a matrix element, and it can be shown, by rearranging the expression, that if ( - 1)Pt’o$ = (- 1 )P’~T’cy’p’ = f 1, the matrix elements (2 1) with a? = a>~ are real and those with a3 = -a>! are imaginary. When all vibrational phase angles assume the simple values 0, T, +~r/2, the cyand 6 factors are real, and the above relation is satisfied by all matrix elements (2 1) if, for any 1i, 1) vibrational state, c@ = +(-l)pt’ or, in expanded form, exp{2i[ C r,(S&Sj + C u,~+L,~B,+ C (rJU, -

r(s)

r(a)

Mk,)lI = *(-lYt’,

(22)

IEE

where I(S) and r(a) run over the nondegenerate normal modes symmetric and antisymmetric with respect to W’, respectively, and t runs over the degenerate normal modes. The symbols p, E, and 7 have been defined in relation to Eq. ( 10). Thus the convenience of a phase choice leading to Eq. (22 ) with vibrational phase angles bound to assume the simple values 0, a, a/2, or -a/2 is related to the observation that if Eq. (22) holds with the same sign for both the wavefunctions in a matrix element, the off-diagonal matrix elements IV, IV’ of vibrational operatorsjhat transform in the same way under 8 and .%?I,i.e., a? = a>(, are real in the 1i, +l) bases, and the offdiagonal matrix elements of vibrational operators that transform oppositely under 0 and 99’, i.e., a? = -*>I, are imaginary. These conclusions have to be reversed if Eq. (22 ) is obeyed with different signs by the two wavefunctions in a matrix element. The above considerations hold for asymmetric molecules as well. In this case, Eq. (22) ass$“mes the simpler form @* = + (- 1)” for all I ?) states. Then the equation (-l)A%,,/ = 1 holds as well, to assure the invariance of nonvanishing matrix elements with respect to 8’. In Table I we examine the conditions to be fulfilled in order to satisfy Eq. (22)) when the phase angles assume the typical values 0, k7r/2, ?r modulo 2a. These conditions obviously depend on the particular type of operation B’. Owing the fact that ut and 1, always have the same parity, it turns out that only the combinations 6,, f 6/,matter for degenerate modes if the phase angles assume only the above mentioned values. To give an example of the use of Table 1, we assume for instance that 6,rca,and all 6, k 6,, are zero or P (first row of the table) and 6%’= (I,, or CzX. The off-diagonal matrix elements IV, W’ of an operator which transforms with a2 = a$( will be real for even Ap, since p must be either even or odd in both the involved vibrational states for Eq. (22) to be verified with the same sign relationship in both states, and imaginary for odd Ap. The opposite results hold for a? = -a%t. It is evident that the most convenient phase conventions are those leading to Eq. (22 ) with the same sign relationship for all values of p and 7, that is, for all )f, i) wavefunctions. Thus the most convenient associations of phase conventions and L4?’are those indicated in Table I

123

PHASE CONVENTIONS TABLE I Conditions under Which Vibrational Wavefunctions )i, i) Obey the Relation (22) for Several Typical Values of the Vibrational Phase Angles (See Text) 6vruj= 0, II always %fd

i*%

$6’= fJn CZ&(E a@(-17

e7

0.x

0,x

even p

o,n *It/2 flrn

krrl-2 0,x *x0

even (p+r) always even 7

up= -

= 1)

SL’=C+

CC Cq(E

(-1)P~3% aB=(-l)Q

OddP &f (P+7) odd7

even (p + 7) even p even ‘T always

= -1)

up= - (-1)PO

owp+r) OddP odd7

Note. The angular values are given modulo 27r. It is assumed that all 6,,,, are zero or K, and all 6,,,,, and all 6,, f 6,, assume the values quoted in the table. See Section 2 for notation.

by the word “always.” The simplest phase conventions leading to Eq. (22) with the + sign for all I?, 1) wavefunctions with the notation of paragraph 2 are Vibrational phase convention A (see section 2 for notation). or CzX:

It applies to .%” = u,,

6“r(a) = f*/2 6,, 5~ 61,= 0 or 7r.

Vibrational phase convention B (see section 2 for notation). or Cay:

It applies to LJ?’= uYz

6“r(a) = *?r/2

6”, It 61,= f7r/2. The two conventions differ only with respect to the degenerate modes; therefore they are identical for asymmetric tops. Since the orientation of the molecule-fixed Cartesian frame with respect to the molecular symmetry elements is arbitrary, the choice of u, or uYzand of CzXor CzY as W’ is a matter of convenience. We like better the choice g’ = u, or CzXwith convention A, which can lead to real off-diagonal vibrational matrix elements of all operators transforming in the same way under 13and LJi?’ and to imaginary off-diagonal vibrational matrix elements of all operators transforming with opposite signs under 0 and .G%?‘, with the largest number of phase angles equal to zero or r (e.g., all vibrational phase angles equal to 0 or 7r, except 60,(aj= +7r/2). We conclude from Table 1 that by an appropriate choice of phases, either A or B, all the matrix elements W, W’ of vibrational operators 0, are real if up = a?< and imaginary if up = -usr. In the basis (8), the matrix elements ( 19) are still valid for molecules with the W’ symmetry operations but with the further restrictions due to symmetry with respect to this operation. Matrix elements in the bases (8) are simply multiplied by a constant

124

DI LAURO, LATTANZI, AND

GRANER

TABLE II Real or Imaginary Nature of Vibrational Matrix Elements in the Bases 1(i, i),) under the Phase Conventions A and B (See Text) combinationsin matrixelements

Note. Matrix elements in parentheses occur only in absence of d’.

+a>t( - 1 )ApeATcz~‘* under @‘, where the upper and lower signs apply to f * -t combinations and to + tf - combinations, respectively. Invariance of matrix elements under .?@requires these factors be equal to unity. With both phase conventions A and B the above factors become + ( - 1) Ap~Lpv do so that vibrational matrix elements related to f t* + combinations are nonvanishing if ( - 1 )Ap~2r = 1 and those related to + - - combinations are nonvanishing if (- 1 )APa>t = - 1. Table II shows these selection rules, with the requirement that matrix elements be real if ~79 = a%~and imaginary if op = - a>~ with the above phase conventions. This last rule applies both in the 1i, i) and in the 1(T, i)& bases, since the basis transformations (8) are accomplished by all real coefficients. Noting that the quantities ufvP*P’ become ( - 1)Apu~ and that W and W’ are either real or imaginary with the above phase conventions, we obtain from Eq. ( 19) the simple expressions factor

it), lo,l(G,i)+>= wk w’ (7, &.zq(v,i)+>= filzw ((7, il)+jc?q?, 6) = E/z,

((7,

all nonvanishing if (-l)*“a>~ A,,> = -1, and

(23)

= 1, real if (-1 )APpap = 1, and imaginary

if

(-1)

Qs IovI(T,iI*>= WT w’ (7, @o,p, I)_>= tiw ((7, i~)_Ico,(t, 6) = fiw,

((7,

all nonvanishing (-1)

(24)

if (- 1)APa>f = - 1, real if (- 1)APap = -1, and imaginary if

*P&V = 1.

6. Vibration-Rotation

Matrix Elements and Phase Conventions

We have so far investigated the properties of vibrational matrix elements, which generally occur as vibrational factors of vibration-rotation matrix elements. Now we

PHASE CONVENTIONS

125

consider the vibration-rotation Hamiltonian and assume that all vibrational operators related to degenerate normal coordinates are written in terms of linear components, and the total angular momentum operators are written in terms of J,, J,, J,. With this choice, the single vibration-rotation terms kH,H, are not mixed by time reversal and by the g’ operation of the molecular symmetry group and must be invariant under them, and their coefficients must be real quantities as already mentioned in Section 6. Therefore the vibrational and rotational parts of each term of the vibrationrotation Hamiltonian must share the same symmetry species, I, = Ir, and exhibit the same behavior under 8. This can be verified on the expanded form of the vibrationrotation Hamiltonian, as reported for instance by Mills (9). Therefore we find it convenient to classify the vibrational operators occurring in the vibration-rotation Hamiltonian by the behavior of the rotational operators which they “multiply” in the operator sense. In Table III, we illustrate the behavior of pure vibrational matrix elements, and of vibrational operators which “multiply” rotational angular momentum operators J LII,J 2, and J 2 B. With vibrational phase convention A when 8’ = u, or CIX and B when GI!’= uYzor CzY,we find that the vibrational matrix elements are all real, except the following case: those elements whose operators form a vibrationrotation Hamiltonian term where the rotational part contains an odd power of J,, if ~3’ = u, or CIYand an odd power of J, if 8’ = CzXor uyzare imaginary. This situation is illustrated in Table III. Thus all vibration-rotation matrix elements can be made real by appropriate rotational phase conventions. In fact with & = 0 or x (which includes the most widespread convention of real and positive matrix elements of the operators J, k iJ,, followed among others by Hougen ( 1) and di Lauro and Mills (IO)), the matrix elements of J, are real and those of J, are imaginary, whereas with the choice 8k = +7r/2 (which includes the convention of real and positive matrix elements of J, * iJ, followed among others by King, Hainer and Cross (11, 12) and also by Oka (13)). the matrix elements of J., are imaginary and those of J, are real. Thus, all vibration-rotation matrix elements can be made real by the following recipe, also included in Table III: W’ = u,:

adopt vibrational phase convention A, and & = 0 or P

(25a)

94?’= G:

adopt vibrational phase convention A, and & = +?r/2

(25b)

Cz?l= I+:

adopt vibrational phase convention B, and & = t?r/2

(25c)

W’ = C&:

adopt vibrational phase convention B, and & = 0 or r.

(25d)

These conclusions apply even if terms of the vibration-rotation Hamiltonian with a power in the J-operators higher than two are considered. We consider for instance terms containing rotational operators J JgJ,. The operators JddJ, with CY= /3 # y and their associated vibrational operators behave as J, with respect to both W’ and B, and can be dealt with exactly as those terms containing J, as rotational operator. The vibrational operators associated with J,J,J, change sign under 8 and are invariant under any 8’; therefore they have imaginary off-diagonal matrix elements. However, the off-diagonal matrix elements of J,J,J, are imaginary for both & = 0

126

DI LAURO, LATTANZI, AND GRANER TABLE III Transformation Coefficients of the Vibrational Operators H, Occurring in the H,H, Terms of the Vibration-Rotation Hamiltonian under Time Reversal and the Symmetry Operation W’

Y

r,=r,

“” “r og =os

So/*= ox,

c 2X

=Yl.

C 2Y

Pure vib. lr

lr

lr

lr

Jz

UJ,,

-1

-1 r

-1 r

-1 r

-1 I

operat. t.s

1

Jx

UJ,)

-1

-1 *

li

li

-1 r

JY

uJy)

-1

li

-1 r

-1 r

li

JZ

t.s

1

lr

lr

lr

lr

J:

t.s

1

lr

lr

lr

lr

J:

t.s

1

11

lr

lr

lr

JXJ,

r(J,)

1

-1 i

-1 i

-1 i

-1 i

Jdz

UJ,)

lr

-1 i

-1 i

lr

J,JZ

nJ,)

1

-1 i

lr

1K

-1 i

vibrational

A

A

0.n

3X/2

suggested phase

convention which

B

B

make all vibration mtationmauix

rotational&=

3X/2

0.X

elemen& real

Note. The coefficient of a given vibration-rotation Hamiltonian term is included in H,. Totally symmetric behavior is indicated by t.s. Vibrational matrix elements with # = (12, are real and vibrational matrix elements with a? = -w% are imaginary in both IT, I) and l(i, I)+) bases, with the vibrational phase conventions A ( .W = o, or C,,) and B ( d’ = L+ or C&), as shown in the table. The suggested vibrational and rotational phase conventions that make all vibration-rotation matrix elements real (see text) are indicated on the bottom of the table.

or P (matrix elements real for J, and imaginary for J,,) and & = +a/ 2 (matrix elements imaginary for J, and real for J,,), leading to real vibration-rotation matrix elements. If the matrix elements of the vibration-rotation Hamiltonian are all real or all imaginary, it can be easily verified that they remain all real or imaginary even if vibrational operators related to degenerate coordinates and total angular momentum operators are written in the Hamiltonian in terms of ladder operators. The matrix elements of the involved operators containing linear components of degenerate normal modes will be real or imaginary as well, because they require real coefficients. If the vibrational operators are written in terms of ladder components for the degenerate modes, the real or imaginary character has to be applied to the vibrational operators inclusive of the coefficients, since the coefficients are not always real. Pure rotational matrix elements, including the off-diagonal matrix elements occurring in asymmetric rotors, are always real with bk = 0 or K or *r/2, since they are due to operators of the form J i.

PHASE CONVENTIONS

127

These results also apply to asymmetric top molecules, except those belonging to the C; symmetry groups and those without any symmetry elements. In the C2, C,, and CZ~molecular groups, a binary rotation axis or a reflection plane are conveniently oriented as C, or a,,, or as C,, or uYz, in order to have all real vibration-rotation matrix elements. However, such a choice is not favorable to the simplification of the vibration-rotation Hamiltonian matrix if it would not bring the quantization axis z into coincidence with a near symmetric top axis. This happens in C,, C,, and C;, molecules when the near-symmetric-top axis z is in the direction of the binary rotation axis or normal to the uh plane. In Cj the inversion i may be taken as the 99’ operation but this does not allow the vibration-rotation matrix elements to be made all real. For instance, the vibrational angular momentum components rX, 7rY,a, all change sign under B and are invariant under i, so that they can have all imaginary matrix elements. The rotational matrix elements of J,, J,, J, cannot be made all imaginary with any phase convention in order to generate all real matrix elements for all the Coriolis operators 7rI a. By similar arguments, we can show that it is not convenient to choose W’ to be a Czz or uXYoperation, although all binary rotations and reflections generate similar transformations of the vibrational wavefunctions of asymmetric top molecules. The nonvanishing vibrational matrix elemenA; ( i’ I CoyI i ) of asymmetric top molecules obviously must obey the relation ( - 1 )Yua,J = 1 with respect to any additional operation B”, other than 9?‘, occurring in the molecular symmetry group, where y = C, AU,,,, and the u,(,) are the vibrational quanta excited in the normal modes antisymmetric with respect to W”. c. Vibration-Rotation

Transition Moments

The electric dipole vibration-rotation transition moments are the matrix elements of the operators X,P,, where F is the space-fixed direction of oscillation of the electric vector of the exciting radiation and (Y = x, y, z. Normally F is assumed to be Z. These operators can be written in the form of spherical tensor components, and the matrix elements of the rotational part, calculated with the help of the Wigner-Eckart theorem, with F = Z (14), contain the phase factor exp[ i( Ak& + A&$)], as shown by di Lauro and Lattanzi ( 15). This phase factor is preserved when the X operators are expressed in terms of Xza components, and we have Ak = 0, and phase factors exp[ iAJv$] for Xzz, Ak = +l; phase factors exp[ i( +6, + AJv:)] for XzX, Ak = +I: and phase factors exp { i[ k( & + r/2) + A&$]} for X,. From these phase factors of the rotational transition moments and the real or imaginary nature of the vibrational transition moments reported in Table IV, we can show that all real or imaginary vibration-rotation transition moments can be obtained by the same conventions (Eqs. (25 a-d)) that make all vibration-rotation matrix elements real, as shown at the bottom of Table IV. This result is relevant with respect to the intensity calculation in a system of interacting vibration-rotation levels, involving the mixing of transition moments between zero-order basis states, since vibration-rotation intensities depend on the products of these resulting transition moments by their complex conjugates. Replacing all Xza of Table IV by XX, or XY, would introduce an additional phase factor aM in the transition moments due to the selection rule AM = &l of these operators. However, in absence of resolution of the M-structure, the vibration-rotation

128

DI LAURO, LATTANZI, AND GRANER TABLE IV

Transformation Coefficients of the Electric Dipole Moment Operators p, Occurring in the X,_p, Terms of the Transition-Moment Operator, under Time Reversal and the Symmetry Operation d’

~ZYZ

l-0-Z)

1

Ir

-Ii

lr

-1

hZ&X

UT,)

1

Ir

lr

-1 i

-1 i

IZYPY

WY)

1

-1 i

-1i

lr

lr

A

A

B

B

Sk=

0,r

til2

W2

0.x

vibration-rotation hansitionmoments:

real

suggested

phase

vibrational

conventions whichrender

1 rotation uansition moments

i

auz-plarizedvibralion-

eitherrealor imagimy

mtational :

l&l

;

imaginary 14

imaginary

Note. The matrix elements of vibrational transition moments with up = u$ are real and those with up = -& are imaginary with the vibrational phase conventions A and B as indicated by the labels r and i.

Hamiltonian matrices are independent of M, so that the M-dependent factors in the transition moments which are combined by the mixing of the zero-order wavefunctions have constant values that can be factored apart ( IO), and become always positive after multiplication of the resulting transition moments by their complex conjugates, regardless of the value of 6,,,,.Therefore, the results of Table IV apply in practice to any F = X, Y, 2 in the space-fixed Cartesian frame, if the M-degeneracy is not removed. 9. EXAMPLES

a. Anharmonic Potential Terms As an example, we consider three cubic potential terms occurring in axially symmetric molecules of any symmetry, klqnqnrqnv, k2qnq1+qr-,and k3qnqf+q+, where the subscripts n, n’, n” denote nondegenerate normal modes and t and t’ degenerate normal modes. The modes t and t’ related to the third term belong to the same E,,, species, in order to have G = 0 as required by the invariance of the Hamiltonian terms under L&?. The first two terms are both hermitian and invariant under time reversal; the third one is not hermitian and is transformed by time reversal into its hermitian conjugate. Since the Hamiltonian has to be hermitian, the third term must occur together with its hermitian conjugate in the sum k3q,,qt+q,t_ + k:q,,ql_qII+, which is both hermitian and invariant under time reversal. The qr, and qrZpartners of a degenerate normal coordinate are oriented to transform under 9’ as T, and T,, respectively, as assumed in Appendix I, and we assume a’ to be either u, or CzX.This is not a restriction, due to our freedom in orientating the

129

PHASE CONVENTIONS

reference molecule-fixed Cartesian system with respect to the molecular symmetry elements. The transformations of these operators under time reversal and 93’ are illustrated in the following scheme: Transformed

operators under 8 and W’

Operators

(- 1Yk,qnqna~

hmt+qr-

k:q,q,a,~~ khnqt-qt+

k3qnqr+qtl- + k:qnqt-Qtl+

k;qnqt_qtl+ + k3qnql+q+

(-l)Pk,qnqt-qt’+ + (- 1)Pk: qnqt+qtT- (26)

k,q,qna,~~

(-~Y~MwI~+

Invariance with respect to B requires k, and k2 to be real and does not impose any restriction on k3. Moreover, when the molecular symmetry contains B’, kl and k2 vanish if the related operators contain an odd power of nondegenerate normal coordinates antisymmetric with respect to W’ (odd p) due to the invariance of the Hamiltonian under B’. The invariance with respect to W’ requires k3 to be real for even p and imaginary for odd p. With the vibrational phase convention A, which is recommended with .94?’= u, or CzX,the matrix elements of all three operators are always real. In fact, the matrix elements of all normal coordinate operators are imaginary for nondegenerate coordinates antisymmetric with respect to 93’ and real in all the other cases, and the operators containing an odd power of these normal coordinates (odd p), which have imaginary matrix elements, occur with either imaginary or vanishing coefficient k. If the anharmonic Hamiltonian terms involving degenerate normal coordinates are written in terms of “linear” components qt, and qf2rather than in terms of “circular” components qIk = ql, + &, then all force constants k are real for any molecular symmetry owing to the invariance of vibration-rotation Hamiltonian terms with respect to time reversal. This applies to the coefficients of all terms of H,,, provided that they contain linear components of operators involving degenerate normal modes, as already mentioned in Section 5. Using linear components coordinates for the degenerate normal modes, the p exponent of the scheme (26) must include the powers of those linear components which change sign under W’ (qlz and qt; with W’ = uxz or C2, and the suggested orientation of the components of degenerate coordinates). Similar considerations could be applied to the a-polarized electric dipole vibrational transition moment operators containing the same normal coordinates as the previous three anharmonic potential terms. In this case k, , k2, and k3 should be replaced by appropriate third-order electric dipole moment derivatives with respect to normal coordinates, evaluated at the vibrational equilibrium, and one should remember that pL,transforms under B’ as the translation T,. 6. Coriolis Interactions in Ethylene Ethylene belongs to the D2h molecular symmetry group. The near top-axis z has the C= C direction and the x-axis is set normal to the molecular plane. With this

130

DI LAURO,

LATTANZI,

AND

GRANER

axis orientation, Smith and Mills (16) studied the Coriolis interactions in the system v4 (A,), u7 (B,,), uIo (I&) considering the interaction u4, vlo ( {~,ro = 0.90) and u7, ulo ( <$,ro = 0.44). They disregarded the interaction about x between u4 and u7, allowed by symmetry, because the relative Coriolis coefficient vanishes due to the planarity of the molecule. Smith and Mills followed the vibrational phase convention 6, = 0 for all normal modes, which corresponds to real matrix elements of the q-operators and imaginary matrix elements of the p-operators, with resulting imaginary matrix elements for the vibrational angular momenta, and the rotational phase convention of Allen and Cross ( 12) ( hK = r/2), with real matrix elements of J, and J, and imaginary matrix elements of J,. With this choice, their matrix elements, related to the operators ?T~J~,and rZJZ, are both imaginary. The matrix element of the Coriolis interaction between u4and u7is related to ?T_~ J, and would be real with the above phase convention if {i7 were not zero as in ethylene. With the vibrational phase conventions A or B (they are equivalent for asymmetric tops due to the lack of degenerate vibrations) and dK = 0 or ?r if 8’ is cXZor CZYand dK = +7r/2 if W’ is CZXor CZY(both types of operation exist in &), all three Coriolis matrix elements would become real. We illustrate this point first for the case of 8’ = uXZ.With this choice, u4 (A,) and ulo (&,) are antisymmetric and u7 (Br,) is symmetric with respect to @‘. Thus, with either convention A or B, the operators q7, p4, and plo have real matrix elements and p7, q4, and qlo have imaginary matrix elements. Therefore the matrix elements of the vibrational angular momentum operators 7ri.7, aS,ro, and a&o are real, real, and imaginary, respectively. With 15~= 0 (real matrix elements of J, and J, and imaginary matrix elements of Jy), it turns out that all three operators 7ri.7J.Y, a5,roJz, and r {.roJ, have real matrix elements. With the choice g’ = CZxr u4 (A,) and u7 (B,,) are symmetric and ulo (&) is antisymmetric with respect to 8’. Then, with convention A or B, q4, q7, and plo have real matrix elements and p4, p7, and qlo have imaginary matrix elements. Therefore, the matrix elements of 7ri.7, ?r;,ro, and lr:,ro are imaginary, real, and real, respectively. With the choice (Sk= t-7r/2, the rotational matrix elements of J, are imaginary and those of J, and J2 real, leading to all real Coriolis matrix elements again. Although it may be cumbersome to analyze how our recipe leads to all real vibrationrotation matrix elements in specific cases by looking at the behavior of the single operators and coefficients, its use is very simple and leads to the desired result in a straightforward way. 10. CONCLUSIONS

We conclude that for all axially symmetric and asymmetric molecules, with the exception of those with no symmetry elements and those belonging to the symmetry groups Ci, C,,, C,h with n > 2, and S,,, by appropriate phase conventions, it is possible to make all the matrix elements of the vibration-rotation Hamiltonian in the usual vibration-rotation bases real. When use of the ladder operators in the vibrational operators is made, it may happen that, under the appropriate phase conventions of those above, a vibration-rotation Hamiltonian term consists of an operator with imaginary matrix elements multiplied by an imaginary coefficient. We have mentioned the case of the cubic anharmonic term kjq,,qt+qlf_ + kzqnqr_qr,+, where q,, is anti-

PHASE CONVENTIONS

131

symmetric with respect to .?@I,and this is also the case of the l-resonance with Al = t-2, Ak = T 1, occurring for instance in C,, molecules if W’ is either CZXor uyZ. The vibrational phase conventions A and B are not uniquely defined since the vibrational phase angles are indicated to be 0, ?r, or +x/2, leaving free the sign of the related phase factors. The same holds for the rotational phase angle I&and the related phase factors. When a computer program is used to determine the values of the parameters occurring in the vibration-rotation matrix elements, generally by a least-squares procedure, one can only establish a priori that the matrix elements, inclusive of the respective parameters, are real, without any assumption about the signs. In practical calculations, when vibration-rotation parameters are adjusted by least-squares procedures, the signs of matrix elements are determined in general by the zero-order values of the parameters entered in the least-squares process. Use of different zeroorder values of vibration-rotation parameters can lead to different sets of final computed values, all differing by the signs of the parameters and all equivalent to the leastsquares fit of the data. The relations between the signs that these parameters assume in different equivalent sets must be compatible with the changes of sign of certain vibration-rotation wavefunctions in the Hamiltonian matrix. An example of comparison of values of vibration-rotation parameters determined under different conventions on the relative signs of the wavefunctions is discussed in Ref. ( 17). APPENDIX

I: TRANSFORMATION

PROPERTIES

OF VlBRATIONAL

WAVEFUNCTIONS

The present appendix deals with individual wavefunctions corresponding to a single vibrational mode. Dimensionless normal coordinate operators and their conjugated momenta are defined as q = 2?r (m/h) ‘12Q and p = -ialdq. a. Nondegenerate

Modes

The nonvanishing matrix elements of the ladder operators F’ = q T ip associated with a nondegenerate vibrational normal mode have the general form (V + 1 IF’lv)

= exp(i&)[2(v

+ 1)]‘/2

(V - 1 IF-(v)

= exp(-i6,)(2v)*‘2.

(Lla) (Ilb)

The phase angle 6, defines the relative phases of the 1u) functions, and the matrix elements of q and p are easily derived from (I. 1a,b) and the relations q = (F+ + F-)/2

(1.2a)

p = i(F+ - F-)/2.

(1.2b)

Thus the matrix elements with Au = f 1 of q and p contain the phase factors exp( +iS,) and exp[ +-i( 6, + r/2)], respectively. The vibrational wavefunction corresponding to the excitation of 2)quanta in a given nondegenerate vibrational normal mode can be generated by application of (q - ip)” to the ground wavefunction )O), and we obtain ,V) =

(q - ip)“lO) exp( iv&)( 2”v!)“2 ’

(1.3)

DI LAURO,

132

LATTANZI,

AND

GRANER

The ground state wavefunction (0) is invariant with respect to the molecular symmetry operations and can be chosen to be invariant with respect to time reversal 8 as well. The operators q T ip are invariant under time reversal since q is invariant and complex conjugation compensates for the change of sign of p and the denominator of Eq. (1.3) transforms into its complex conjugate. Thus application of 8 to both sides of Eq. (1.3) yields e(v) = exp(2iv&)]v).

(1.4)

The transformations under C, and S, have already been discussed in the text. The transformations of 1II) under molecular symmetry operations can be worked out by applying these operations to both sides of Eq. (1.3). For i and ch we find i]v) = (*l)“]v)

(1.5)

= eu”lq,

(I.61

An)

where the upper and lower signs apply to the wavefunctions related to normal coordinates which are symmetrical (g or ‘) and antisymmetrical ( u or “) with respect to the above operations. The orientating operation a’, which is either a binary rotation normal to the zaxis or a reflection through a plane containing the z-axis, transforms the above wavefunctions according to the relation @‘IV) = (*l)“lv),

(1.7)

where the upper and lower signs apply again to the wavefunctions related to normal coordinates symmetrical and antisymmetrical with respect to &?’respectively. b. Degenerate Modes

Twofold degenerate wavefunctions related to the normal mode pair ql, q2 are chosen to be simultaneous eigenfunctions of the Hamiltonian Hi + Hz and of the vibrational angular momentum operator q1p2 - pIq2, with eigenvalues defined by the quantum numbers v and I, and are denoted I v, 1) . Following Moffitt and Liehr (18), we define the ladder operators F+’ = (q, - ip,) + i(q2 - ip2)

(I.Ba)

F-’ = (ql + ipl) f i(q2 + ip,)

(1.8b)

whose matrix elements have the general form (v + 1, Z-t 1 ]F+‘]v, 1) = exp[i(& -t 6,)][2(v f I+ 2)]‘12

(1.9a)

(v - 1, 1+ 1 IF-‘Iv,

(1.9b)

r) = exp[i(-6,

+ &)][2(v + /)]“2.

From (1.8)) we obtain q1 = (F++ + F+- + F-+ + F--)/4 q2 = -i(F++

- F+- + F-+ - F--)/4

(I.lOa) (I.lOb)

133

PHASE CONVENTIONS

pl = i(F++ + F+- - F-+ - F--)/4

(I.lOc)

p2 = (F++ - F+- - F-+ + F--)/4.

(I.lOd)

The matrix elements of the operators ql, q2, pl , p2 in the I v, Z) basis can be calculated from Eqs. (1.8)-(1.10). The F-operators can be applied to generate 1II, 1) from the IO, 0) wavefunction and we find (F+-)(u-/)/Z(F++)(u+1)/2 (1.11)

Iv, 1) =

exp [ i( v6, + IS,)] - 2” - [ (!+)!(

Y$),]“’

‘O- Oh

The F-operators transform under time reversal as BF+‘K’ = F+’ and BF-‘K’ = F-‘, and application of B to both sides of Eq. (I. 11) yields eventually

e 1v, I) = exp( 2iv6,) I v, -1).

(1.12)

The transformations of 1v, 1) under i and oh depend only on the parity of v and on the symmetry behavior of the related normal coordinate with respect to this operation, and are given by Eqs. (1.5) and (IA), replacing 1v) by Iv, 1). can be worked out The transformation of ( v, 1) under the orienting operation L&I?’ from Eq. (I. 11) by the usual procedure. If the q1 and q2 normal coordinate components are oriented to transform as T, and T, under L?@‘, we obtain 9’1 v, I) = c”exp( -2il&) (v, -I)

(1.13)

with c = 1 if 9%?’= uxz or CzXand t = - 1 if W’ = uYzor CzY. Bra-functions under the mentioned operations transform with complex conjugate coefficients with respect to the corresponding ket-functions. APPENDIX

II: TRANSFORMATION

PROPERTIES

OF ROTATIONAL

WAVEFUNCTIONS

The simultaneous eigenfunctions of J2, J,, and Jz, where z and Z are Cartesian axes in the molecule-fixed and space-fixed frames, respectively, are denoted 1J, k, M) as usual. The wavefunction lOO0) is invariant under all molecular symmetry operations, and is chosen to be invariant under time reversal as well. We also use the shorter notation ) J, k) to deal with problems where A4 can be constant. We represent by K either a positive k-value or the absolute value of k. The matrix elements of J,, J,, and their combinations J7 = J, T iJ,, emphasizing phase factors, are (J,k&

llJ,+iJ,IJ,k)

= exp(+&)[J(J+l)--k(kf

(J, k + 1 lJ,( J, k) = l/2 exp(*&)[J(J+l)

(~,k+ll~,~~,k)=fexp[+~+~)][~(~+l)-k(ktl)P2.

1)1”2 - k(k f 1)]“2

(11.1) (11.2) (11.3)

DI LAURO,

134

LATTANZI,

AND

GRANER

The relative phases of rotational wavefunctions differing by the value of the Jquanturn number are defined by the values of the reduced matrix elements of the direction cosine operators arranged in forms of irreducible tensor operators (Id), as shown by di Lauro and Lattanzi ( 1.5): (J’llXjlJ)

= exp[i(J’-

J)q$][(2J+

1)/(2J

(11.4)

+ 1)1”2.

These operators behave as irreducible spherical tensor components acting in both molecule-fixed and space-fixed frames, and application of the Wigner-Eckart theorem (14) and of Eq. (11.4) yields the transformation &IJOO)=exp(iq$)[(25+ +exp(-iq:)[(2J+

1)/(2J+3)“2]C2(J1

J+

1)“2]C2(J1

1)/(2J-

J-

l:OOO)lJ+

l:OOO)]J-

100) 100).

(11.5)

The term with 1J 0 0) is missing in the above expansions since C(J 1 J; 0 0 0) is zero. The direction cosine operators are invariant under time reversal and we can write in general

f%olJ 00) =fJhdJOO)

(11.6)

8~,lJ+lOO)=f,,&,,,lJ+lOO).

(11.7)

Using Eqs. (11.6, 7), application of B to both sides of Eq. (II.5 ) gives

hblJoo)=f X I J+

1)/(2J+

J+lexp(-iv$)[(2J+ 100)

+fJ_,exp(io$)[(2J+

3)1’2]C2(J1

1)/(2J-

l)i’*]

XC2(J1

J-

J+

l,OOO)lJ-

l;OOO)

100)

(11.8)

It is easily seen that Eq. (11.8) is consistent with Eq. (11.5) if&*, = exp( +2iv$)f, and,withthechoicefo= 1 ore)OOO)= lOOO),weobtain 81 JOO)

= exp(2iJq:)l

JO 0).

(11.9)

The I J +K 0) wavefunction is generated by the K-iterated application of the operatorsJ+= J,TiJ,to IJOO), )J+KO)=

Jr] JOO) exp( tiKG,)S( J, K)

(11.10)

with S(J, K) = { [J(J + l)][J(J + 1) - 2][J(J + 1) - 61. - .[J(J + 1) - (K - l)K]}“*. Angular momentum operators, as all motions, change sign under time reversal, so that

eJ,e-1

= -J,

eJ:e-1

= (-~)KJ,.

(11.11)

135

PHASE CONVENTIONS

Application of 13to both sides of Eq. (II. 10) with the help of Eqs. (II. 1 ), (II.7 ), (II. 10) and after cancellation of S( J, K) factors and of phase factors containing aK yields eIJ*KO)

(11.12)

= (-l)kexp(2iJn:)lJTKO).

By a much similar procedure, we can generate the ) J k +M) wavefunctions by application of ( JX f iJy)M to I J k O), and the transformation of the ( J k M) wavefunction under time reversal is found to be = (-l)k’Mexp(2iJn$)l

81 JkM)

J-k

-M).

(11.13)

The X, direction cosine operator transform as the R, rotation about the top axis under symmetry operations in the molecule fixed frame. Then, operating by molecular symmetry operations on both sides of Eq. (11.5), we obtain G~JOO)=(~l)‘~JOO)

(11.14)

with the upper sign for G = C:, Sg, i, and u!, (which leave R; unchanged) and the lower sign for G = o,, , gv;, Cz.,, Cz,. which change the sign of RZ. Equation (II. 14) applies to 1J 0 M) with any M value as well since the result of operations related to molecular symmetry is not affected by the value of M. In a similar way, operating on Eq. (II. 10) by the usual procedure, we obtain ( 1, 13. 19) iI JkM) ahlJkM) &?I JkM)

= 1JkM)

(11.15)

= (-1)kIJkM)

(11.16)

= (fl)“exp(2wik/n)l

JkM)

(11.17)

with the upper sign for W = C, and the lower sign for W = S,; WI JkM)

= (-1)J-Y”exp(-2ik&)I

J-kM)

(11.18)

with y = 0 if @’ is uYzor CzXand y = 1 if 8’ is u.~=or Czl’. ACKNOWLEDGMENTS The authors are grateful to Mrs. A. Mont-Reynaud for the preparation of the manuscript. Financial support from the Italian CNR and Minister0 P.I. is gratefully acknowledged.

RECEIVED:

November

14, 1989 REFERENCES

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DI LALJRO. LATTANZI,

AND GRANER

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