Doubly degenerate vibrational levels of spherical top molecules

JOURNALOF MOLECULARSPECTROSCOPY 63, 227-240

Doubly Degenerate Definition

Vibrational

(1976)

Levels of Spherical Top Molecules

of Basis Functions as Irreducible

Tensors of O(3)

F. MICHELOT Laboratoire de Spectronvmie Molthlaire, au C.N.R.S.,

Bquipe

6 Bd Gabriel, 21000

de Recltercl~e AssociJe

Dz’jon, France

Although the vibrational wavefunctions for a doubly degenerate level of a spherical top molecule do not form the basis for an irreducible representation of the rotation group, we show that it is possible to define vibration-rotation basis functions as irreducible tensors of this group. We consider successively two important cases: the fundamental VP(E) and the sublevel G = 2 of the first overtone of a triply degenerate fundamental. From this we show that the method can be easily generalized.

I. INTRODUCTION When

the doubly

degenerate

rotation

energy

be made

using the tensor

considered

by Jahn (I)

However,

and Hecht

The

levels

the study

can be oriented

is

O(3). This

group

(Z), who used the standard

of O(3)

of the vibration-

group of which

of the full rotation

(3) to define

for which

indeed

for irreducible

Td or

Oh, can

method

representations

was

of O(3).

with regard to its cubic subgroups;

cubic representations

the doubly

the associated

it is possible considered

of O(3),

When

there

restricting The

(‘(H4 (6-8)

to use the tensor exists

vibration operators

of dimension

two-dimensional

well

adapted

to the

formalism

2 of the molecule representation

of O(3)

a vibrational

level

having

so far. Fox

(5) made

the study

it to the sublevel

FZwhich

he identified

other

investigations

the last study

presented

have

of which

a structure

is excited

must be considered

and wavefunctions of O(3).

provided

are the basis

symmetry

that

group,

Hilico

but

(4) showed

the vibrational

level

E.

is not of symmetry

been proposed

degenerate

elementary

E

representations

there exists no single-valued

state

excited

of such molecules.

separately;

level.

is not

tops, the symmetry

formalism

the representations

this led Moret-Bailly study

vibration

levels for spherical

been

to that

E, no general

with a triply

mainly

by Champion

comparable

symmetry

of the overtones devoted

method

2vR of XY4 degenerate

has

molecules

fundamental

to the fundamental

(9, 10) showed of the triply

degenerate

of

v2

that the levels

of this

fundamental

states. In the following results

we will consider

can be used for octahedral

results which

are required

3-j cubic symbols

only the case of tetrahedral ones.

in this paper;

and especially

In the first

Cowright

Q

IV76 by Academic

rights of reproduction

their connection

Preau. Inc.

in any form

reserved.

Xl’4

we will

this will lead us to specify

227 All

part

molecules, review

but our

some

basic

some properties

with the 3-j symbols

of the group

of Td.

F. MICHELOT

228

In the second part we will show that it is possible to define, as irreducible tensors of O(3), basis functions for the fundamental v2 and for the vibrational sublevel E of the first overtone of a triply degenerate fundamental. Finally, using these results we will show that a generalization is possible. II. TENSOR

1. Cubic Tensors and Coupling

FORMALISM’

Coeficients

The most often used representations of the full rotation group O(3) are the standard and contrastandard ones (II), corresponding to an orientation with regard to those of the rotation subgroup about the O.&axis. For an integral value of j, an irreducible spherical tensor is labeled by an additional index a! = u or g characterizing its behavior under inversion. The representations of O(3) can be oriented with regard to those of its (ja)G. Cubic tensors are so defined, the subgroup Td through a unitary transformation components of which are characterized by a triple index p = (C, n, u), where n is used to distinguish between different representations of identical symmetry C which appear in the reduction of the representation D VU) of O(3) in Td; u characterizes the different components of a tensor of symmetry C (c = 1,2 if C = E and g = x, y, z if C = Fr or F2). The covariant components of a cubic tensor are expressed in terms of its standard components by (3)2 T(&) = (j*)GFTfi) (1)

7

P

this relation showing that a tensor T(ja) oriented tensors T(javnnC)defined in Td (4). Its contravariant components are given by

with regard to Td is a direct sum of

I’?;;, = (j,)Upp’Tfi)

(2)

>

where (3) (&)uPP’ = (- l)i&pp’. The reduction of the tensor product of 3-j cubic symbols:

(3)

of two cubic tensors is then expressed in terms

(4)

[A (h,) x B(&z)]~) These symbols

= FT;, Tip,z)A

;‘+~Z~).

(5)

are different from zero if3

A(j,,

j2,

j,>

=

1;

Cl

x

c2

x

c3

3

Al;

aXPX-Y=g,

1 The main properties of irreducible tensors with respect to any group, and particularly those of cubic tensors, are given in Refs. (3, 16). The problem of the tensor connection from a group to a subgroup is considered in Ref. (IS). 2 Throughout this paper we will use the Einstein’s summation convention for all indices out of brackets. The symbol 8 with two indices is the Kronecker symbol. 3 A(j,, j2, j,) = 1 if j,, j,, j3 satisfy the triangular condition 1ji - j,l 6 js 6 ji f jz and is zero

otherwise.

E VIBRATIONAL

and are related by (3)

LEVELS

OF SPHERICAL

TOPS

229

:

(6)

The computation of these symbols depends on the choice made for the orientation; indeed the (ja)G matrices are well defined (within a phase factor) only if n is less than or equal to one. So as to remove this indeterminacy, J. Moret-Bailly (3) determines G matrices such that the symbols Fp?$‘= 1

0

for

p#p’.

(7)

The phase factors are chosen so as to have F(j’n jai?&) = Pl

PZ P3

FPI. p? pa = (3141 328 337)

(_

1) j~+h-kjaF~;,

where the bar stands for the complex conjugation. F(& ia6 jay) = p$ Pl

P2

$ ip) = (_

P%

piB V?,,

(8)

They also verify l)jl+j&jaF2

.$,

$)_

(9)

Tables of symbols Egl” iy p’ computed in the SO(3) group have been published (12,13) ; so as to deduce the symbols F$a 3 $$(a X p X y = g) defined in O(3) we must consider the reduction in Td of the representations Dog) and a>(ju) of O(3) (3, 14): (a) If C is not the E representation, then to a function !IJ$$ which transforms according to the irreducible representation C of Td is associated a function q,$$ which transforms according to the representation C’, the latter being obtained from C by exchange of the indices 1 and 2; i.e., F1 t+ Fz.

Al+-AZ, From Eq. (1) and the definition

(3) of the 3-j coupling coefficients we have

F;;‘I f ;;’ = FJp $7 g’ = F$u E 6;’ = . . .

,

the correspondence between the indices p and p’ being made as we indicated. (b) If C is the E representation, then taking into account the orientation (Table Ia) for the matrices of this representation we have the correspondence *t>; -&

!L$i

@& --+F$$

and

Making the choice El -+ E2 and E2 --+-El changes of sign on the 3-j symbols:

(10) chosen

(11)

we can deduce as before the corresponding

(1) there will be a change of sign if one makes, an odd number of times, the transformation $J = (tz, E2) -+p’ = (n, El) when the parity is changed; (2) the sign is unchanged for an even number of these transformations or if the change of parity leads to transformations of the type p = (n, El) --+ p’ = (a, E2). For example, We can summarize

Cases (a) and (b) by the relation F

(i& j@ jsl) =

PI’

where E = =t 1 is determined

Pll P3

as we indicated.

$

(ilo

is

PI

P2 P3

~JS) F

(12)

230

F. MICHELOT TABLE Matrix

Representation

Ia T/

for the Group generators

-t (3)4/z

- (3)v2 -4

a We chose the coordinate axes OX, OY, OZ to be fourfold axes of the tetrahedron. sional A representation the matrices coincide with the characters. b Nomenclature for the components of irreducible representations.

For the one-dimen-

2. Relation between 3-j Cubic Symbols and 3-j Symbols of Td Because

of the orientation

those of its subgroup n2C2, n&o

of the irreducible

Td, the 3-j

are proportional

representations of O(3) with regard to F,,h jt@d;’ for fixed values of nlCl,

cubic symbols

to the corresponding

3-j

symbols

of the subgroup

K(jln jzp jay) F(CI CZC3) I?%,,,, h:%,,, $zca = (nK* nzc*nrC,) 0, 62 S3 . We redefined

a matrix

representation

for Td, different

(1.5) (13)

from that given in Ref.

(16),

so that the orientation would be in agreement with that chosen for the computation of the (ja)Gr matrix elements (12, 17). In Tables Ia, b we give the matrix representation we used together From efficients

with the corresponding

the known properties through

Eq.

of the 3-j

(13). For example,

3-j symbols of T,+ symbols

we can deduce those of the K co-

they are invariant by an even permutation 1) h+jz+j3( - 1)C1+C2+Caby an odd permutation.

of the “columns” and multiplied by (The K coefficients for any (Y, /3,y can be deduced, as the 3-j symbols, from those for whichcr=P=y=

gby K &,

However, the determination

zc,, $&,, =

E'K$ycl ;Iycp,, ;/&.

(14)

of B’ is not as simple as before. Let us consider, for example,

the case when Q! = p = u and y = g (the other cases being easily deduced from this one). (a) If Cr, Cz, Ca are not E, then we established (Section 1.a) that in this case we have the correspondence

C’ = C X AZ for Cl and CZ, C3’ = CS, and c = +l

(Eq.

12);

that is, using Eq. (13), K;::“,,

$,,

$@;~I’

o”,o’ R’ = K ;h.$, kccl &~,,F;~l;;

The 3-j symbols for Td, for fixed Cl, Cz, Cp are determined possible to choose this phase factor so that (18) (Table F (cl’ cz’ ca) = F (cl oz c3) -1 #2 U33 S, -2 08 We then have, in all cases, E’ = +l.

;;‘.

within a phase factor.

(15) It is

Ib)

C’=CXAz.

(16)

E VIBRATIONAL

LEVELS

01; SPHERICAL

231

TOPS

(b) But if one or more of the Ci)s (i = 1, 2, 3) is E, it is not possible to choose all phase factors so that Eq. (16) is verified (Table Ib); e’ is then determined in each case taking into account (i) the rule established above, paragraph lb, (ii) the relation, Eq. (13), which defines the K coefficients, (iii) the table (Ia) of 3-j symbols for Td. 3. Elementary Tensor Operators and Vibrational Wave Fzcnctions An XY4 molecule has vibrations which are nondegenerate, vr(Ar); doubly degenerate, and triply degenerate, Q(F~) and Y~(FJ. To each oscillator we can associate two elementary tensor operators (3), an “annihilation” operator and a “creation” operator,

Q(E);

‘“‘C%J”)= qa, + wn>pw, cd&p = qs.3- Glh)P8,,

(17)

respectively. For the nondegenerate vibration they are of symmetry dr in Td and B(Og) in O(3). The operators associated to the triply degenerate vibrations are covariant components of irreducible tensors of symmetry FZ in Td or D(lu) in O(3). In contrast, those related to the doubly degenerate vibration of symmetry E are not the basis for irreducible representations of O(3); they are defined as tensors in Td and they must be coupled in that same group. The reduced matrix elements for the operators of Eq. (17) are given in Refs. (3, 15). In Table II we give the symmetries of the rotational and vibrational wavefunctions. We note that the wavefunction of a nonexcited oscillator is scalar; then we can consider by extension that it is a basis for an irreducible representation of symmetry SJ(O~)of O(3), (c=o)*A1 E (Y=O)\E(Og)

TABLE 3-j Symbols

cm :i

I

ii , $2

Y4, .‘l /
F(C’

CW1

GQ

A; Ea Bl F,p

Ai ECr E2 IJ,?

FIY Bl I<2 t;,z

F,? El El F,B

3 -l/34

FiP F&Y

F,p Fix

-)

Fiy

F,y

(rl

c2 Cz) bP

CJ

1 l/24 1/z* l/34 I:134 -+

l/2(3)4 +

(18)

Ib

for the Group

Tda cz CJ)

cl=1

cm

Gu3

F(CE

El Rl E2 R2 FJ FSX F,.r FXY Fly F28 Fez

FIX

F2.t

- *

Ply F,z F,fi F,3 F2y Fly F2Z Flz F2.x Fix

F2Y

f

(“22 F,@ F,Z FlZ Fzz F,*FIX

l/3& -l/2(3)+ -l/O -l/6+ -l/6& -l/6$ -l/6, -l/61 -l/6*

FIY FZY

c-1

02

c3

“--i= 1,2;cu= 1,2;p= x, y; Y = x, y, z. Symbols that are not given in this table are obtained C2 ('1)= F$p fi fi' = .,. from these through their symmetry properties: FCC’ a 69 w = (_ ~)C,+CS+CJF(C~ CL C? c7.3 c2 CI with(-l)C=lwhenCisA~,E,orF~and(-l)C=-lwhenCisAzorF,.

232

F. MICHELOT TABLE Quantum

numbers

II

Symmetry in O(3) or Td

Notations

n The wavefunctions for the doubly degenerate oscillator, expressed as tensors of Td, are given in Ref. (25). ba = g if vs is even; (Y = zc if vs is odd. c For transitions between different vibrational states we take, by convention (3), 01 = g in the ground state and OL= rc in the excited state.

We use this property to write the basis functions for a triply degenerate state in a way which is slightly different from the usual one: *@J) = [(“)I&C”U)X P

[(v2=OhpvJ,)

x

where s and s’ take the values 3 or 4. From now on we shall omit the scalar functions (U)@P(lU) = [oJz=O)*(O,) X we

(‘%)

(va=l)\kuuquuqP

(“l’o)gp4qD)

fundamental

(v,-o)qKb)

AI

,

09)

(Vl=O)q and (w~‘=o)*; setting (vs=l~\kuuqW,

(20)

will use Eq. (19) in the form *(en) = [(o)*VU) X P

III.

BASIS

FUNCTIONS

FOR

A DOUBLY

(v)cgluq~~.)~

DEGENERATE

(21)

FUNDAMENTAL

LEVEL

1. Vibrational Wavefunctions

Let us consider the annihilation operator (8)a(1ur p2) of a triply degenerate oscillator and the creation operator (2)(8(E) of the doubly degenerate oscillator. By coupling in Ta we obtain two tensor operators (E X Ft = Fl -I- Fz) (19) (y(R) = [(Z)@(E) x

b)~(h,F2)]@W,

pJ(F2)

b)@&FaqWz)

=

[W@(E)

x

(22) 7

which we can consider by extension as the covariant components of irreducible tensors, respectively, of symmetry ZD(‘~)and %J(‘u)of O(3). These operators acting on the vibrational wavefunctions of a triply degenerate fundamental state, Eq. (20), transform them then into vibrational wavefunctions for the doubly degenerate vibrational level; that is, 0 Pi) cv,+* 0

Ft) = &0

$[ (vl-l)\k(E) X (“,-o)\k(O.,.41)]~~),

where i = 1 or 2. The coefficients M, G)oP, are the matrix elements and are obtained using the Wigner-Eckart theorem (3),

of the operator

(23) 0:“)

E VIBRATIONAL

LEVELS

OF SPHERICAL

233

TOPS

y’ and y represent the quantum numbers labeling the vibrational states va = 1, 1~ = 1, E; v, = 0,l. = 0, A1 for y’ and 712= 0, l2 = 0, A r; v, = 1, 1, = 1, Fz for y. Let us now consider the functions [O(~r) X (v)~(IJJ($) Then taking into account [@ru) x (~)+J$~)

= FFI~ Fz0’N.+, (1.8FI) (v,@, (1D 1.1

P

c

Fz).

(25)

Eqs. (23) and (24) we have = (1/21)(r’;

Elj@‘F”ljy; F@;

:I;’ ~‘F;~l~:’

IfE! (.)@LB), (26)

where we have set (v)@jz) = [~~z=l)~(E) x

w.=O)\k(O..At)]~).

We use the proportionality relation, Eq. (13), to transform the summations on the various indices

(27)

Eq. (26) in which we clarify

P

where p = (C, 0). A priori k, can take the values O,, l,, or 2,; however, the unitarity relations between 3-j symbols (16) imply that the second member of Eq. (28) will be different from zero if and only if C = E and 6 = p therefore for k, = 2,. We then have [I

x WIJJ~;)

= (9/2)K[c ;: &‘;

Using the same method with the operator [@‘u’ x (~Q&)-J~;)

E/[W)lir;

Fz) ‘%I

%$(294

o(~u) we obtain

= (51/2)K $; ;; :I(“/‘; Ejjo’Fa’l/y; Fz) Wf’.

(29b)

Note. These results are in agreement with the fact that the vibrational wavefunctions (“2-r)*(E) can be indifferently considered as the covariant components of a tensor 5%) (respectively Vu)) of which the components of symmetry Fz (respectively F1) in Td are identically zero. 2. Vibration-Rotation

Basis Functions

So as to express these basis functions in a form which is convenient for the computation of matrix elements let us consider the functions obtained by coupling in O(3), of an operator O(la) (01 = u or g) with basis functions of a triply degenerate fundamental state, Eq. (21). Through a recoupling transformation it is possible to relate them to the functions previously defined : [@la)

x

[VU~VU)

x

(~)$&L)](WJ~~~)

x {ff

=

J+R+L+1[(2R + k?12(-1) - ,I

~)[CO’ipC’“~x [@7’

where 2 = u (resp. g) if (Y= g (resp. u). From the results of vibrational wave&n&m k = 2 are not identically zero; then

1)(2k + I)]”

x (V)f@Wpiq(~~), P

(30)

we know that only the terms for which

[oum’ x [co)pc~u) x W@(~U)](%)];~~) = (-l)J+fi+‘[5(2R x {f$ ;}[(o)*(J,)

+ l)]*

x [flC’-) x (++YJ(2a)];KQ).

(31)

234

F. MICHELOT

For each value of R (R = J f 1, J), K could take the values R f 1, R. We show in Appendix I that we obtain a complete system of functions if we choose, for

R=J+l,

K=J+2,

for

R=J--1,

K=J-2,

and

and opposite parities. (From now on we will choose a! = g for K = Jf2 and (Y= u for K = J - 2.) We can then expand the second member of Eq. (31) using Eq. (29) and we obtain, for K, = (J + 2)g, 1 FPo Er (J+% ~KI~~,E~(EIIo(~~)IIF~)(~J + 3) 4{1Ji-2 ii-1 J) (J” 2”) p 0 ” ”

(o)q$uY)

(32a)

(v)@LE),

and for K, = (J - 2),, $K{:;;;

l)i{:-”

~;(EI10(F~)jjF~)(2J -

;-’

:}FTJu“ST$+,

(O)qj$-, W;?

(32b)

Using these expressions we show in Appendix II that the functions so defined are orthogonal. The correct basis functions we will use in the computation of matrix elements for the Hamiltonian operators can be written @;xa’ = (l/X)[O

(LX)x [ (o)*((J,) x (vk@XJ (R’)];K$

(33)

with (R, K) = (J + 1, J + 2) when (Yis g and (R, K) = (J - 1, J - 2) when (Yis u. 3t is a normalization factor the determination of which we postpone until Section V. IV. BASIS FUNCTIONS FOR THE SUBLEVEL G = 2 OF THE FIRST OVERTONE OF A TRIPLY DEGENERATE FUNDAMENTAL

The vibrational wavefunctions of this level are the basis for an irreducible tion .DQg) of O(3) (Table II) the reduction of which in Td is E + Fz. Various coupling schemes can be used to determine the basis functions.

representa-

It may be desirable to isolate the Fz vibrational sublevel; considering by extension (v)~(2~,F*) have the symmetry a)(lu), that is that the vibrational wavefunctions (v)*&,F2) G C”,@CL’ , we can couple them in O(3) with the rotational functions \k@,) = [(o,*(Ju,

x

(Y)cpbq@8),

(34)

wavefunctions

so as to obtain

A(J, 1, 6) = 1,

basis (35)

similar to those of a triply degenerate fundamental state and allowing an interpretation of the Fz sublevel (5, 20). Another possiblity is to determine the vibration-rotation basis functions by the usual coupling scheme (5, Zl), *I(&) = [CO)*E(J,I x (v)@%)](Ru), The first coupling E and Fz vibrational

A(J, 2, R) = 1.

scheme is obviously the one to be used when the separation sublevels is large.

(36)

of the

E VIBRATIONAL

LEVELS TABLE

Coefficients

o We write

the reduced

matrix

elements

OF SPHERICAL

TOPS

2.~5

III

I’:$ 3” and X&l F a

(21~= 21, = 2; C~]Q@‘JIIC,= 21, = 2; C’) = (Cjl~(Fi)llC’).

We will prove that it is possible to determine basis functions expressed as irreducible tensors of O(3), allowing a simultaneous study of the E and FZ sublevels, the functions related to the Fz sublevel being given by Eq. (35). For this, let us consider an interaction operator between the E and Fz sublevels; it may have nonzero matrix elements if it is of symmetry FI or Fz (E X Fz = FL + Fz). If fi2(F11and ficFz) are two such operators, purely vibrational, we can consider by extension that they are, respectively, of symmetry D(‘u) and D(‘v). Then the functions obtained by the coupling in O(3), of these operators with the functions XP(*g),Eq. (3.9, can be written

We now have to determine the values of 6 and K, as well as the conditions to impose on the operators fi so that the functions obtained will be basis functions for the E sublevel, i.e., so that the coefficients YTf!$ of Eq. (37) will be identically zero.4 1. Expressions for the Coejicients

Y and X

These are easily obtained by means of the subgroup by expanding of Eq. (37), the coefficients being given by the scalar products5

the first member

y;i7; = (CO)\kJ$’ (ZI)\kJ~2)I[Q2(la)x \k&)]jG)), xl;,;,

= (CO’\E~“’C”)*F) 1[Qua’ x \kw]~K’).

(38)

In Table III we give the values of these coefficients. 2. Discussion

We first values J f deduce the (Appendix

note that in any case Eq. (37) is defined only if A(l, 6, K) = 1; 6 taking the 1, J we have a discussion similar to that of Section 111.2, from which we choice of the values J f 2 for K (then 6 = J f 1) and of opposite parities I).

*We assume here that the operator 0 cFz) has a reduced matrix element (Aljjn(Fz)IjF~) equal to zero, namely, that it is a pure interaction operator between the E and Fs sublevels. This assumption is not necessary; if suppressed the corresponding terms in Eq. (37) cancel (see Section 111.2). 6 A method similar to that of Section II can also be used; however, it is less interesting in that case than the one we give here.

236

I;. MICHELOT

In the case a! is u it appears that the Y coefficients will be identically zero if and only if the reduced matrix element (F2llVp)llF.J of the operator ~(~2) is zero, that is if it is of the same type as the operator 0(=2) considered for the level v2 = 1. When (II is g, the 3-j symbols F& 2 $*2u) b eing zero, there is no need to suppose that the reduced matrix (F211Q(Fr)IIFa) is zero. We can then write explicitly the functions we obtain, Eq. (37). For K, = (J + 2),,

gK!f’l + 3)*{:+, f+” {}FFu 2 y+2)g (We) I p” ~~(E/JQ(Fl)llF~)(2J ”

(~)\k,!~), (39a)

and for K, = (J - 2),, ~KI~~~~~~(EII~~(~~)IJF~)(~J -

:-” ;}FT$ R, f-2)*

l)“(;_r

(“hv~-, W+?

(39b)

The comparison of these expressions with those of Eqs. (32a) and (32b), obtained for the 2~2= 1 fundamental level, lets us foresee that a very large generalization is possible; it allows at first the simultaneous determination of the normalization factors of the basis functions we obtain. This has to be done so that the transformations, Eqs. (32)-(39), will be unitary. V. NORMALIZED

BASIS FUNCTIONS

We prove in Appendix II that the functions qf’2’” and XP$-2’U we determined are orthogonal; calculating the normalization coefficients we will simultaneously show that functions related to a given value of K, and different values of the index p = (C, n, u) are also orthogonal. For this we write in the unique form *((J+s)u = [2Y’r, x [(OXINU) x (V)~(lu)](~+l),]~+2)u P

=

J'&&~+~'~

(40)

Co,qd$) W\k;EE'

either Eq. (32a) or (39a) and we carry out the scalar product (J+% 1q+9, s = QPp’ S = I &n12F$;~ 2)

(41) F@ ~0 (J+2)~((oh$JJ~ ?;+2jg (2,~u)p

co)g~))((di#$)I

Taking into account the fact that the rotational wavefunctions the vibrational level considered are orthonormal we obtain S = 1gu, In Appendix

amounts

to

(4)

coefficients Fy: i+2g) $+2), being zero for p # p’, Eq. (6), we for p # p’ are orthogonal. the functions YX$+2)u and @J+2)u P S =

we deduce the expressions @(J+s), = [I/X P

(43)

+ S)*Fy; ;+20’~;+2j,].

S = 1J=(J) I”[@$ + 3(6/5)+{4, ;+2 ;+“}(2J

(42)

as well as those of

1“Fg; 2’ $+,,,F$u 2, g+2k

III we show that this matrix product

The Clebsch-Gordan see from Eq. (44) that Setting

(~J)*;JQ).

I GfLo

I 2N(J,P)

for the normalized (J(P)][Tus)

x

=

GJ,P),

(45)

basis functions

[(O)qNd

x

c~)~clu)]cJ+l),]g+2),_

(464

E VIBRATIONAL

Likewise from Eqs. (32b)-(39b) basis functions for K, = (J - 2),, @(J--z)* = [1/$(,,,,][~(G P

2.‘37

LEVELS OF SPHERICAL TOPS

we establish

the expression

x [(O)@(JU) x

for the normalized

(v)Q,(lu)](J-l)r]~-2)u.

(46b)

VI. GENERALIZATION

The vibrational levels of an XYd molecule can be classified in two large categories according to whether the doubly degenerate oscillator is excited or not. In any case the study of a vibrational level depends on the choice of a coupling scheme for the wavefunctions (that of the operators being then a copy of the former); it is then interesting to know the different theoretically possible coupling schemes, the choice of a particular scheme being in general determined by the experimental spectrum aspect. 1. Levels for Which v2 = 0 For these levels, the elementary operators and wavefunctions are defined as tensors of O(3); the basis functions are in general defined by coupling step by step the various vibrational angular momenta and the total angular momentum (3). As we saw, this coupling scheme does not seem to be the one to be used in the study of a level as 2~3 of CHI, and the method we propose in Section III offers an alternative; more generally it can be considered for the study of any vibrational interaction of the E-F type. 2. Levelsfbr Which

~2 #

0

i\ll overtone levels of the fundamental v2 have vibrational symmetry Al, AZ, or E. similar in the one established in Section III can be used. Let us consider, for example, the overtone 2~2. So as to study the E sublevel one only needs to consider the operators obtained by the tensor product, in Td, of two creation operators (2)~(E) with the annihilation operator (*)~Z(lu,~z). There are several possible coupling schemes but in any case the total operator will have symmetry F1 or F2 assimilable, respectively, to ~(~0) and a>(lu). Finally, for combination levels for which the doubly and triply degenerate vibrations are excited, the methods we propose enable us to express, as tensors of O(3), the wavefunctions of levels having a vibrational symmetry of type E and to consider various possible coupling schemes for levels having any vibrational symmetry. .A method

VII. CONCLUSIOK’

The study we made shows that it is possible to define vibration-rotation basis functions as irreducible tensors of O(3), for levels having a vibrational symmetry of type E (whether the doubly degenerate oscillator is excited or not). In a subsequent paper we will give the method allowing the computation of matrix elements of the vibrationrotation Hamiltonian. APPENDIX So as to

determine

the suitable

I

values for K,, Eqs. (31)-(37),

conditions: (a) the system of functions

must be of dimension

2(2_7 + 1);

we make use of two

F. MICHELOT

238

(b) the values taken by K, must be such that aD(Ju) X E = C aD(K.). K The decomposition in Td of the representations (3, 14). If we set J = 12~ + q, then zD(Ju) = pr,,

(1)

acJs) and a>cJu) is given in Refs.

+ Wu),

(2)

where rres = AI + AZ + 2E + 3Fr + 3Fz is the regular Td. We then have

representation

of the group

B(Ju) X E = PI’,, X E + %(*u) X E. Knowing

the reduction

of product

representations

for Td we have

LDcJu)X E = 2@‘,,, + D(Q=) X E. From Condition (a) we deduce that the possible also Eqs. (31)-(37)] J -

1, J -I- 1

or

(3)

J, J,

(4)

values for K in Eq.

(1) are [see

J - 2, J + 2,

or

the parities being arbitrary at this point. From Eq. (2) one easily shows that in general we have acJd

+

9cJa)

=

2pr,,

+

9cQd

~oCJ+‘)m +

D(J+“b

=

2pr,,,

+

D(Q-h

+

a)(Q+l)8,

a(J--%

a)(J+W

= 2pr,,,

+

a)(Q-2),

+

a)(4+2M.

+

+

a)(QH,

(54 (5b)

(54

b) Comparing Eq. (5) with Eq. (4), we see that these will be identical (Condition for any J value if they are for the values of J from zero to eleven. The investigation of are not identical to Eq. (4) for all these J the tables shows that Eqs. (Sa) and (Sb) values (for example, for J = 4), and that in the case of Eq. (SC) we must have opposite parities. We then have as possible choices for K, (except for J = 0 and l), (J - 2L

and

(J + 21,

(J - 218

and

(J + 21,.

or APPENDIX

Orthogodity

II

of Basis Functions

We saw that Eqs. (32a) and (39a) can be written ,$J+% P

= J’~~,F$~~

y+%

in the form [Eq. (4O)] (0)$,(&d

(v’\E$

(1)

In the same way we can write Eqs. (32b) and (39b), g(J-2)~ P

= &JjFg

R, y-2)~

(o,,$$“’

(v’i&,;EE’.

(2)

We consider then the scalar product S’ = (9,#(J--2)~ 1q$J+%).

(3)

239

E VIBRATIONAL LEVELS OF SPHERICAL TOPS

Using a method similar to that of Section V, we obtain 5” = df(J, $“bFg; 2’ $_,,UF~~ 2, r+2k

(4)

So as to show that S’ is identically zero, we will use a relation which exists between the 6-j and 3-j symbols of the group SO(3) (22). In the special case of 3-j cubic symbols one has (23)

In this expression and from now on we indicate explicitly the various summations. Equation (5) can be used in O(3) if one adds the indices u or g in a coherent way (see Section I). We now use Eq. (5) for the values II = J,, 12 = 2,, 13 = 2,, j, = (J + 2), and j, = (J - 2),. We have

We must have A(2, 2, JI) = 1 and A(Jr, J + 2, J - 2) = 1, therefore the only possible value is Ji = 4. When q3 = qz = Ep then pi is El or AI; but a)c4=) = A2 + E + F1 + Fz, so pl is El. Equation (6) then becomes

SO as to obtain S’ [in terms of 3-j symbols, Section II, Eq. (6)] we have to sum both members of Eq. (7) over the index P. In the first member we then find the factor C

F$;2P$;

= K$$‘$j

p=1,2 (see Table

C

I$=?)

= 0

(8)

p=l,Z

Ib); hence S’ is zero. .4PPENDIX

Coejjicients N(J,

p)

Let us consider we write it as c

III

Eq. (5) of Appendix

II with 12 = 13 = 2; simplifying

our notations

(2k + 1) {; ; ;‘} F$ f $“F;f ,“?3 = ( - 1).J C Fp’;r ; 2p)aF;f t2f?.

k.R

RI

(1)

When q2 = q;( = &.L, 6 is El or Al, then K can only take the values 0, 2, and 4. We then sum both members of Eq. (1) over the index p. The second member is simply given by [see Section II, Eq. (6)]: (--1)J

C F;;r;;;F;f&,;:) lJ.PI

= (-1y/[(2R

+ 1)(2R’ + l)]” C F~~sj;R,R’F~;,~~~‘;l’~. (2) P,Pl

F. MICHELOT

240 The first member

becomes

C (2k + l){:~~‘}F~~~~~)K~~~~~ k0=$;4

C FLE;;‘. Ir

(3)

For the same reason as in Appendix II, when Cu is El the corresponding terms in Eq. (3) are zero; then those terms which contribute to the sum are those for which C is Al, therefore, for K = 0 and k = 4. From 3-j symbols tables (12, 13) we obtain C FL; ;,, “A,= 2/S+ N-1,2 In addition,

and

C

F&;,,“d,

= 2/30*.

p=l,Z

we have (3)

Ffl ; F’ = (-1)%~&,,~/(2R We then obtain,

+ l)t

and

{“J;;}

= (-1)“+“+“/[5(2R

+ l)]‘.

when R’ is equal to R,

C F$sj;B)F&;/&j P.41 The coefficients N(J, setR= J+2(resp.R=

= $8; + 3(6/5)*{4,f:}(-l)J+R(2R p) [resp. J-2).

N’(J,

+ l)*Fy,,4)&.

(4)

p)] are then given by Eq. (4), in which one

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10. 11. 12. 13.

H. JAHN, Proc. Roy. Sot. Ser. A 255, 469 (1938). K. T. HECHT, J. Mol. Spectrosc. 5, 35.5 (1960). J. MORET-BAILLY, Thesis Paris, 1961; Cuhier Phys. 15, 237 (1961). J. C. HILICO, Thesis Dijon A. 0. 3666, 1969. K. Fox, J. Mol. Sfmtrosc. 9, 381 (1962). J. HERRANZAND G. THYAGARAJAN,J. Mol. Spectrosc. 19, 247 (1966). M. DANG NHU, Thesis Paris A. 0. 2360, 1968. H. BERGER,J. Mol. Spectrosc. 55, 48 (1975). J. P. CHAMPION,Thesis Dijon, 1974. J. P. CHAMPION,J. Phys. Paris 36, 141 (1975). U. FANO AND G. RACAH, “Irreducible Tensorial Sets,” Academic Press, New York, 1959. J. MORET-BAILLY,L. GAUTIER,AND J. MONTAGUTELLI, J. Mol. Spectrosc. 15, 3.55 (1965). J. C. HILICO, M. DANG NHU, J. Phys. 35, 527 (1974); B. BOBIN ANDJ. C. HILICO, J. Phys. Paris

36, 22.5 (1975). 14. K. Fox, J. Chem. Phys. 52, 5044 (1970). 15. J. C. HILICO, 1. Phys. Paris 31, 15 (1970). 16. J. C. HILICO, Cahier Phys. 19, 328 (1965). 17. J. P. CHAMPION, G. PIERRE, J. MORET-BAILLY,AND F. MICHELOT,to be published. 18. J. S. GRIFFITH, “The Irreducible Tensor Method for Molecular Symmetry groups,” Prentice-Hall, Englewood Cliffs, N. J., 1962. 19. J. MORET-BAILLYAND F. MICHELOT,C. R. Acad. Sci. Paris B, to be published. 20. B. BOBIN, J. Phys. Paris 33, 345 (1972). 21. G. PIERRE ANDK. Fox, to be published. 22. A. R. EDMONDS,“Angular Momentum in Quantum Mechanics, ” Princeton Univ. Press, Princeton, N. J., 1957. 23. E. PASCAUD,Thesis Dijon A. 0. 6853, 1972.