Correlation between energy levels of linear and bent X2Y2 molecules

JOURNAL

OF MOLECULAR

Correlation

SPECTROSCOPY

between KOICHI

51,399-421 (1974)

Energy Levels of Linear

YAMADA,~

TORU

NAKAGAWA,

and Bent X,Y, Molecules

AND Kozo

KUCHITSU

Department of Chemistry, Faculty of Science, The University of Tokyo, Hongo, Bunkyo-knl, Tokyo 113, Japan

The permutation-inversion group developed by Longuet-Higgins is extended to a classification of the vibronic, torsional, and rotational wavefunctions of a nonrigid XZYZ molecule by introducing a symmetry operation p, which rotates the top half of the molecule by 2?r and, accordingly, the molecule-fixed x axis by ?r. Since the energy levels of linear (D-n) and bent (CZ~, C2hr and CZ) forms of XZY~ are classified according to a set of common symmetry operations of this extended permutation-inversion group, their energy levels can be correlated, including those of nonrigid forms such as a quasilinear system or a free internal rotor. Nuclear spin weights and selection rules are derived.

I. INTRODUCTION

The large-amplitude bending vibration and K-type rotation of triatomic molecules was investigated by Thorson and Nakagawa (1) and by Dixon (Z), and a quasilinear formalism was set up. Correlation of the energy levels of a bent triatomic molecule with those of a linear molecule was also discussed in detail by Johns (3) and by Herzberg (4). For four-atomic molecules, however, correlation between linear and nonlinear forms has never been studied in full, although the internal rotation in an XzYz molecule was discussed by Massay and Bianco (5), Hirota (6), and Hunt et al. (7); a symmetry classification of torsional and rotational wavefunctions was discussed by Hirota. Recently, Dyke et al. (8) analyzed the vibrational-rotational energy of the HF dimer as a nonrigid four-atomic molecule2 by use of the permutation-inversion group defined by Longuet-Higgins (9). The purpose of the present study is to make a group-theoretical treatment linking linear and nonlinear symmetric four-atomic molecules and to correlate the energy levels of the various forms that can be transformed into one another by way of bending displacements and internal rotation. A symmetric four-atomic molecule XsYz has LI_,h symmetry in the linear form, CZ* or Czh in planar forms, and CZ in a nonplanar form. As the valence angles of a nonlinear form increase to r, the energy levels vary continuously to those of the linear form. Similarly, the energy levels vary continuously as the dihedral angle increases from 0 1Present address: Institut fiir physikalische Chemie, Universimt Kiel, 23 Kiel, Germany. 2 The HF dimer has a hydrogen bond (HF. * . HF) , which is loose enough to allow tunneling of the end and the middle protons due to the antisymmetric bending vibration. Since it has an unsymmetric (C,, or C,) structure at its potential minima, this molecule is not discussed in the present study. 399 Copyright Q 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

400

YAMADA,

NAKAGA~A

AND RUCNITSU

to r (from CZVto CZh) passing through the CB form. Consequently, the energy levels of these different forms should be classified in terms of a set of common symmetry operations. Such a symmetry classification not only correlates the energy levels of the abovementioned forms with one another but also characterizes the symmetry properties of the energy levels of nonrigid forms such as a quasilinear system or a free internal rotor. In the following sections, the pe~utation-inversion group formulated by LonguetIIiggins (9) is extended for application to the present problem in a way parallel to the work by Hougen for the internal-rotation problem of dimethylacet~~iene (10). Since the vibrational-rotational wavefunctions of a linear molecule have been classified by Bunker and Papougek (If) by the use of an extended permutation-inversion group, the present treatment is concerned mainly with nonlinear forms. II. CLASSIFICATION

A. Extension of Permutation-Inversion

OF ENERGY

LEVELS

Group

The permutation-inve~ion group (PI group) introduced by Lon~et-Higgins (9) is applicable to a cl~ification of the rovibronic wavefunctions of a nonrigid molecule, since it does not assume a particular equilibrium structure. According to his theory, there are four fearible operations for an XZYz molecule : the identity operation 8, the permutation of the equivalent nuclei P = (12) (aa), the inversion of all the coordinates 8*, and the product (12) (ah@*, or in short, P*, It should be emphasized, however, that the above four operations are insufficient for a classification of the vibronic and rotational wavefunctions separately, even though they are sufficient for a classification of the rovibronic wavefunctions. In order to clarify this point, one needs to define a set of molecule-fixed coordinates (a rotating system). The present definition of the axes is illustrated in Fig. 1. (i) The line connecting the centers of mass of the XY atom pairs is defined as the z axis, in the direction from the XzYb pair to the XiY, pair. (ii) The x axis is defined to lie at the azimuthal angle of X = (Xl + X2)/2, where Xl and X2 are the azimuthal angles of the nuclei Xr and XZ,

FIG. 1. Definition of the coordinates. N shows the nodal line.

401

equiv.

vib. sym.

FIG. 2. Symmetry operations (A) (12) (a@, (B) J!?*, and (C) 9, and their equivalent rotational and equivalent vibrational symmetries.

respectively.3 (iii) The y axis is so chosen as to make the (x, y, z) axes a right-hand system. Thus, these axes are the principal axes of the moment-of-inertia tensor, and the x axis is always a twofold axis of symmetry. The angle of internal rotation (or torsion) is defined by 7 = XI - XZ. Now let us introduce the operation Z? which rotates only top half of the molecule (the X,Y, nuclei pair) about the z axis by 2~. This operation never changes the space8By the use of these axes and angles, the rotational and torsional parts of the wavefunctions can be separated in a good approximation, as discussed in Section VB.

402

YAMADA, Table

NAKAGAWA

Symmetry

I.

AND

RUCHITSU

Transformations

of

n

,.A

Operationa

^E

j

^E*

P*

PT*

x

x

x

x

x

-*

Y

Y

‘Y

-Y

z

z

-Z

@ w

gb

Y

z

-Z

-Y

Y

-Z

2

-X

Y

-Y

-z

z

Q

%

n-8

K-8

a

X

n- x

-X

“+X

2n-r

2s+T

2n+T

T-X

‘%

ll+x

*-X u

T-X,

T+Xu

L.

Schawlow,

Xb

x

T-X

-x

*x

T=

T

-T

-T

xg

xg

-xg

n+xg

“%I

-x

-X

n+Q

n-f3

%I

T

n-rql

r-8

-xu

h

PT

e

8

g

..n

@

-sb

T-X

“T*

n+$

a

T

Coordinates.

2r-r

-xu xu

xg r+x,

g

g

.%

P = (12) (ab)

a) b)

Eulerian

c)

coordinates

direction

of

the

see

Spectroscopy,

Torsional

angle

x,

fixed

Angles;

Microwave

are

of

r is

period

of the nuclei

27;

C. H. Townes

and A.

McGraw-Hill, of

period

see

4n,

New York even

@,

x, xg,

and

text.

and leaves

the Hamiltoni~

x and

{see

molecule-fined

though

(1955).

y axes

invarient.,

but

it reverses

the

Fig. 2). If one operates ? twice in succession, one obtains the identity 2. Consequently, the operation f is also a symmetry operation. The direct product of the PI group and the group (2, p) gives a complete representation of the rotational and vibronic wavefunctions for a nonlinear form. Since this double group is isomorphous to the point group L&h, it is called D2h* hereafter. The operations (12) (ab), &*, and p are illustrated in Fig. 2, and the transformations of the variables are listed in Table I.4 As is discussed in Section VA, the permutation-inversion operation may be considered as a product of partial symmetry operations, which operate on the translational, rotational, vibrational, and electronic coordinates respectively. Such partial operations, which may be called equivalent rotation (more strictly, equivalent rotational symmetry operation) and equivalent vibrational symmetry operation, etc., are listed in Table II for the present case. The equivalent vibrational symmetry operations of D2h* contain point-group symmetry operations for both CZ* (2, es”, 6,,, and 8,, according to the present choice of the axes) and CM (2, 6$, f, and 4,,). Even though the Czv and C2h point groups, as well as the PI group, contain only four symmetry operations for each, these operations cannot be correlated directly with one another with the change in 4The concept of permutation and symmetry operations wit1 be discussed

inversion operations again in Section V.

and

their

relationship

to the point-group

the ~uilibrium torsional angle rs. In this context, introduction of the operation I?, which results in a total of eight symmetry operations in the I&h* group, is necessary. The notation for the symmetry species is chosen to be compatible with the conventional notation for the point groups C,,, C 2h,and CZ. (i) For the operation (12)(ab), A is symmetric and 3 is antisymmetric. (ii) For the operation J??, suffix 1 is symmetric and 2 is autisymmetr~~. (iii) For the operation (12) (a&}?*, suffix g is symmetric and 26, is a~~symmetr~~. The character table is shown in Table II. 2% C~~~~~~~Q~

ofT~~~Q~~~-~Q~~~~Q~~W~~~~~~~~~Q~

Since the torsional motion is closeXy coupled with the rotation about the z axis, it is essential to treat the two motions simultaneously. The torsional-rotational wavefunctions may be written on the basis of the wavefunctions of a rigid symmetric rotor and a free internal rotor; i.e., *J,&,pa= ~~~~~~~~~~~~~~~~~

(1)

where X = (xx + x&Z and 7 = XI - XZ after the preceding definitions. The periodic boundary condition requires that K and n in Eq. (1) are integers and have the same parity. Wang’s linear ~Ornb~~~~iQusof the rotational ~~vefu~~t~~~s are extended to the Table

II.

Equivalent;

Katational

Symmetry Operations Group DZh* , and Their

and VibrationaL for

the Extended

Characters,

g* ;*

EPX operationa

;Ei I;

equivalent

E

CzX

C2y

B

C2x

0

equiv. notation

a}

rotation

vib.

symmetry

of species,

p”= ClZ){ab)

(A,B)

PX:

xz

(I,21

c2= OXY

g*

-g* ;;

$

E

CzX

c2y

CzZ

3

by2

C2y

C2=

(g tu)

YAMADA,

404 Table

NAKAGAWA

AND KUCHITSU

Symmetry Classification

III.

(i)

(ii)

(iii)

Electronic

Vibrational

Torsional (v,)b

of Wavefunctionsa

(iv) Rotational

(v) Rovibronic

J=even J=odd

J=even J=odd

c Vl

*a

%g

v2

*2u

v3

x

I

B2g

B2g *1kz

Bl”

A2u

Bl”

B2g

%”

*2u

v5

\

‘6

a.

Boundary conditions

b.

Symbols

pairs

the torsional-rotational

exE and 0x0 but not the pairs e and o represent

quntum number n; nonrigid is

allow

forms

equivalent

torsional-rotational

in Figs. to es,

6 and 7.

\kXJ,K*t.N+

=

of

of

torsional

the

internal

states

for

Our classification

ea, OS and oa defined

wavefunctions

of

the

ex0 and oxE.

even and odd levels

see the relations

wavefunctions

e+,

by Hirota

(z),

rotation

various

e-,

O+

and o‘

respectively.

:

T’,@J,K(e){eiK~

f

&K~}{eiNrlz

f

e--iN7/2},

(2)

where K = (k / and N = (n ( . These wavefunctions can be classified into the irreducible representations of the D,h* group. The symmetries of the rotational wavefunctions are as was originally shown by King, Hainer, and Cross distinguished by E* and O*, (12) ;5 they are summarized in Column (iii) of Table III. Similarly, the torsional wavefunctions are classified by e5 and of by use of the parity of N;‘j the symmetries in the group Dzh* are shown in Column (iv) of Table III. The representations of the torsionalrotational wavefunctions must follow the boundary condition, which allows the products e X E and o X 0 but not e X 0 and o X E. C. Classification of Vibrational ad

Electronic Wavejumtions

With the rotational and torsional coordinates defined above, the three bond lengths and two valence angles specify the positions of the nuclei. Variations of the bond lengths and valence angles from their reference values, which may be functions of the torsional angle 7, are taken to express the vibrational motions. Besides the torsional motion, there are five vibrational modes as shown in Table IV. By the permutation operation (12) (ab), such valence coordinates are interchanged. The operations p and 8*, on the other hand, do not affect the valence coordinates. Hence, only the AI, and 5Phase factors are arbitrary in the symmetric-top wavefunctions for + and x. They are chosen to be consistent with Nielsen’s treatment (13) but not with King et al. (12). BOur notation e*, 63 - , o+, and o- corresponds to Hirota’s es, ea, OS,and oa (6).

LINEAR-BENT Table

bent

mode

Vl v2 V3 V4 “s “6

sym. XY str. YY str. sym. bend. torsion

IV.

Vibrational

XY str.

antisym.

bend.

405

Modes of XzYz Molecule

forms

fl

linear

general

cis

trans

gauche

D2h*

5?v

‘Zh

‘2

Alg

Al

Ag

A

v1

sym.

AQ

A1

Ag

A

v2

YY

Alg AZu

antisym.

CORRELATION

Al

A

Ag

A2

Au

Blu

Bl

%I

Blu

B1

%l

form linear D -h

mode

str.

XY

c+ g z+ g

str.

v3

antisym

XY str.

A

~4

gerade

B

vS

ungerade

bend. bend,

C Tlg xu

B

B1, species appear as vibrational modes, excluding torsion, as listed in Table IV and in Column (ii) of Table III. The electronic wavefunctions can be of AX,, Bsg, _J&, or B1, symmetry only, because they are symmetric under the operation rr”. They are listed in Column (i) of Table III.

With all the possible symmetry properties for each part of the wavefunctions at hand, as summarized in Table III, one can determine symmetries of the overall coordinate wavefunctions (or rooibronic wavefunctions) as products of the electronic, vibrational, torsional, and rotational wavefunctions : The rovibronic symmetry can be one of AI,, Bt,, A zu, and BI,, as listed in Column (v) of Table III. All these symmetry species have the character 1 for the operation p; this resuit is reasonable because the overall wavefunctions may be expressed in terms of the space-fixed coordinates, on which the operation i” has no effect. Table III also shows that all the eight symmetry species are possible for the electronicvibrational-torsional (or ~~~~o~~~, hereafter) wavefunctions. Each of the vibronic states, however, can take only E* or only O* rotational wavefunctions because of the boundary conditions. Such relations and the symmetries of the resultant rovibronic wavefunctions are tabulated in Table V. For a classification of the rovibronic wavefunctions, the simple permutation-inversion group is more convenient: The rovibronic species AI,, BQ, Agu, and Br, may be called +s, --a, -s, and +a, respectively, where s and a represent the symmetric and antisymmetric species with respect to the permutation (12)(d), respectively, while + and - represent the even and odd parity with respect to the inversion a*, respectively.7 E. ~~~G~eaySpin Weights The characters of the nuclear spin wavefunctions and of the overafl coordinate wavefunctions for a symmetry operation G are given by Landau and Lifshitz (15) as xspin(Q

=

II

a

(zica

+

11,

(3)

7 This system is well known in the classification of the wavefunctions of a diatomic molecule (1-I)

406

YAMADA,

NARAGAWA

AND KUCHITSU

and x,,(6)

= n (2i, + 1)(-1)2&o@), a

(4)

where i, is the spin of the nuclei which are permuted by the operation c?, re, is the number of nuclei which belong to the ath permutation set, and the products are taken over all the nuclear permutation sets affected by the operation. The operations I?* and F cause no nuclear permutation. Accordingly, the spin wavefun~tions can be of symmet~ AI, {or +s) or Bz, (or +a) according to the symmetry for the operation (12)(4. The characters and the irreducible representations of the spin and coordinate wavefunctions are listed in Table VI. F. Linear Case For the linear pressed as

XaUz molecule,

the bending-rotational

wavefunction

may

be ex-

9 (Vet,, r,l@, E, J) = Fe,,, (per ~~)O~(B)ei’ofXptX)eit,(Xu+X) =

where the bending Table

displacements V.

of Symmetries

Bent and Linear

rovibronic

i

(5)

of ~avefu~ct~ons

Forms

symmetry

J=even

+s (Alg)

-a fs,,)

-s fA,,l

+a U&J

J=odd

-a

+s (A$

+a (Blu)

-s

(BZg)

Bent forms

(rotational E+

Ak3

i

~~)O~(B)e~lgX’e~~uxUe~~X,

of the IIS and 11, modes are expressed by the cylin-

Correlations for

z

Fygvu(pg,

levels

6)’

EE+

%l Of

A2g

%g

u

B2u

E‘

0+

0-

E-

E*

0-

0*

E+

0‘

Big

E-

0-

yI %u ‘2 % B171 f

(Azu)

0+

* Linear k-0 R-1 R=2

form

(vibrational 5: =g *g

+

substates) “g

z”-

?_i+

5%

=u-

?I+

*g

A””

*Il+

LINEAR-BEAT

Table

Symmetry

VI.

of

Spin

spin

and

character

Coordinate

Wavefunctions

of

coordinate

wavefunction

general

407

CORRELATION

wavefunction

general

D2G2

X2Y2

D2Q2

fora m2

4

m2

Iml

2

m

A

3A

(m2*m)/2

B

+B

; ; = (12) (ab)

4 -2

Irreducible (m2*~m~)/2

b

representations

*(m’-InI)/ 2iX a)

m = (2iX

b)

Full

notations

A = AIg A = lAIg (or

+ 1)(-l)

and

in

rovibronic)

Group

B = Big

or A2u)

* (2iY + 1)(-I)

and

for

+(m2.m)/2 2iy

A

A

B

+3B

.

D2h*: the

spin

B = (Blu

or

wavefunctions; B2g ) for

the

coordinate

wavefunctions.

drical coordinates co,, X,) and (&, X,), respectively, and the rotational azimuthal angle by X. The vibrational quantum numbers are denoted by ~,_,~a and D,,~v,while the quantum number of the z-axis projection of the total angular momentum is given by 1 = 2, + I,, where I,, I,, and 1 can take 0, fl, 62, . . . . The function Fmplu(pg, pJ isa scalar and symmetric under any symmetry operation. The bending azimuthal angle measured from the nodal line, i.e., X, i- X (or X, + X), is definite, whereas the angle X, (or X,) itself, which is measured from the molecular x axis, is arbitrary unless the rotational azimuthal angle X is defined in the linear case. As was done by Bunker and Papougek (11) more generally for the extended PI group Dmh, the angle in the present work is so taken as to follow the symmetry transformations defined in Table I for Dt,,*. Then the transformation properties of X, and X, are derived as shown on the last two lines of Table I. The vibrational states (zJ~~~, z’,~u,E) and (~@-~g,ZJ~-~=,--I) are doubly degenerate, but are split into the Z-type doubling components due to the interaction with the rotational motion. By the introduction of the linear combinations ‘Ik(QZ,, GZ=, JI / *, J) = 2-i[*

(Z1$Z,, G&l*,E, J) f

!I+-+,

ZL-l=, --I, J)],

one can obtain the irreducible representations of the vibration-rotation wavefunctions for the linear form. By use of the transformation properties listed in Table I, the rovibronic symmetry of the linear form is classified into four species: +s, --a, -s, and this confirms the +a (or AI,, &, Au, and Blu, respectively, in the Dzh* notation); results obtained by Bunker and PapouSek (11). The results are listed in Table V and are compared with those derived for the nonlinear forms. In the case of I # 0, it is impossible to separate the wavefunctions into the bending and the rotational parts, but the &type doubling components may be regarded as if

408

YAMADA,

NAKAGAWA

AND KUCHITSU

zs, 1EI*) ; such vibrational substates may they belong to the vibralional substates (v, zg, 2,1L be denoted as II*, A&, etc., corresponding to 111 = 1,2, etc. These + and - vibrational substrates are sometimes labeled c and d levels. III. CORRELATION

OF ENERGY

LEVELS

A. Correlation Between D*h and CZ,,Forms (a) Ge~ey~ ~e~ayks and no~a~~o~s. The correlation of the vibration-rotation energy levels of various X,U, molecules, linear or nonlinear, and semirigid or nonrigid, can be obtained by use of Table V: The rovibronic symmetry is kept unchanged (namely, the symmetry remains in the same column in Table V>, and the z component of the total angular momentum, denoted as 2 for a linear form and as k for a nonlinear form, is conserved. (We assume that k, or K, in the asymmetric-top notation, is a sufficiently good quantum number for distinguishing the rotational levels of a nonlinear form.) Bond stretching vibrations are omitted, since they are not essential in the following discussion. The vibrational states are labeled by (~~~0,efU~~)lQ*for the linear form; this is equivalent to (84z4,glsz6)/~l*in the standard notation of the bending vibrations of a symmetric Iinear molecule. For a nonlinear form, on the other hand, the bendingtorsional-rotational states are denoted as (Q, 86, uJx” in terms of the vibrational modes shown in Table IV. In the following, the rovibronic symmetry is denoted as fs or fa, whereas the vibronic symmetry is denoted in terms of the Ds* or Dwhspecies. In Fig. 3, the correlation between the energy levels of the D-h and Czv forms is illustrated schematically. As shown in the following, the degenerate vs(IIU) vibration of the linear form turns into the K-type rotation and the symmetric bending vibration ~3(-4 1) of the Csv form and that the P,(&,) vibration of Dmh splits into the torsional mot,ion Ye and the antis~metric bending vibration v,J(B~) of Cty. (b) R s~r~~~ye of the Czp ground v~~a~~o~aZ state. The ground vibrational state of the state of the ground linear form, (P, S)O &+, corresponds to the R = 0 rotational vibrational state of the Czu form, (0, 0, O)O41. The rovibronic symmetries of these states are +s for even J, and --s for odd J. (For the sake of simplicity, only the even .I levels are mentioned hereafter.) The K-type rotational structure of the ground state of the CZ. form is found in Table V on the row of the vibronic states 41, (for even K) and A lu (for odd K) ; one should recall that g/u discrimination is no longer applicable to the CZ” symmetry. Thus, the K = l* levels have rovibronic symmetry of --s and + a, which correspond

in turn to the II,

and II,+ vibronic

states of a linear form. These

II states are the t-type doubling components of the (00, l*l)lIL state. Similarly, the R = 24, 3+, . . . , states for the ground vibronic states of Czv correspond to the (O”, 2*2)2A,, (@, 3*3)3@is, . . . , states of D-h. Thus, the ungerade bending vibration &Iu) for Dmh turns into the rotational motion around the z axis in the case of Czv. The next lowest Z (I = 0) vibronic states of D,h are the overtones and combination tones of the bending vibrations. They are (00, 2O)OZ,+, (2O,00)02,+, (l*r 9 lIfl)oI;U+, and (l*‘, lF1)O Z p- , where the last two states are the symmetric and antisymmetric combinations of the (l+l, l+)O and (l-l, l+‘)O wavefunctions. For Czv, these states correspond to the (1, 0, O)O Al, (0, 0, 2)OAI, (0, I, O)OBI and (0, 0, 1)” A2 vibrational states, as described in the following.

Limur

c&&’

Baf’t c&b,

Fro, 3. Energy level correlation between the D,n and Ct. forms. Thor vibrational energy levels for the 1h) I11 ; bent form energy levels are labeled by ZJand K. The broken lines linear form are labeled by (va14 , t6 correlate ~nme of the Coriolis coupled levels. (cf K ~~~~~~~~ of &FL~ Y&&) G& Y&&) ~~~~~~~~ &.& of G+ Let US first consider the set of ~m~rnat~on states (PI, l”l)Q 8” of D& These X states for k = 0 correspond states of Cs,. From Table V the &* state to the X = 0 levels of certain vibrational bending vibration Y8 must have BI symmetry for Csn, to which the antis~metric or (0, 1, O)* 31 befongs. Similarly, the Z, state of DDohis uniquely related to the torsional vibration ~4 or (Q, 0, 1)” A$. The next problem is to find how the K structures of these vibrational states of CzS are eorreMed. Such K structures CM be obtained by a successive excitation of the PE,(II,) vibration of IL in accordance with the preceding rule, For the K = I = 1 states, there are (l&r, 00)’ II,* and (l*r, 2’Fz)rIi,* states. The K-type or d-type doubling in these states being taken into account, the II,+ states correspond to the K = t- fevet of Pi and the X = I* level of ~a@& while the XI@--states correspond to the X = I+ Ievel of ~~(3~) and the K = I- fevel of Y&J%& Since the pair of the levels X = Ii &) and X = fff~4) are eonphzd by an a-type Corioiis interaction, as described in the following, their wavefunctions are composed of the two wavefunctions of the II,* states of the linear form. In a similar manner, the K = 2 states for ~0 and y4 of CzVare related to the (l*l, l*r)* A,* and (l*l, 3F3)2 A$ vibronic states of L&A* In the above correlation, the 1t4fundamental vibrational state (IfI, @)I 11, of the Iinear form does SW~turn into a pure vibrationa state of a bent form because of its nonzero angular momentum (K = E = I). The normal ordering in the k” structures of ~6 and ~4 states of a bent form is no longer retained in the corresponding b structures of the linear form (see Fig, 3). This anomalous behavior can be inte~ret~ m terms of

410

YA~ADA,

~A~AGAWA

AND

KUCH~~SU

a large u-type Coriolis interaction between the ~6 (&) and ~4 (AZ) vibrations. This interaction couples the Ek levels of vg with the K” levels of v+ Since the interaction term for bent form is expressed by -2AS_&K, the K = 0 levels are not perturbed whereas higher K levels are separated more and more from each other. As the equilibrium valence angle increases, this Coriolis interaction has increasing importance sincethe rotational constant tl increases and the vibrational energy difference vg - iv4 decreases. In the limit of the linear form, the vibrational energies of ($*I, I”“)* 2,+ and (l+r, l%r)@&- are essentially degenerate, except for a very small difference on account of the vibrational E-type doubling- Consequently, the K > 0 levels are perturbed drastically for a linear or quasilinear system: The lower levels of X = 1 = 1, namely (liE, O”)l HP*, are pushed down even below the K = E = 0 levels. The Corioliscoupled levels have nearly equal separations of 2~s in the linear limit for any 1 (or R) value. (d) Overtolzes and combinations of va and V6 vibrations oj Csy. Let us next consider correlation of the (2@,00)* ZZ,+ level. In the Czy case, this level belongs to ~$1 and is related to the overtone of the torsional mode, (0, 0,2)O.* According to the above discussion, strong a-type Coriolis interactions among the states (0, 0, 2fc _4r, (0, 1, l)* BL, and (0,2,0>@ Ar are expected. The last two vibrational states may be related to the (2Bz 213)o 2, and (2ti, 21f”)”2,+ of the linear form, respectively. The above three vibrational states form a set of K (or E) structures coupled by the a-type Coriolis interaction. The K = 1 =t 2 levels, for example, correspond to the (2”, 2*2)2 Aphtrl; (2*2, 0°)2 As*, and (2ha, 4F4)s A,* states of D+ As an extension of the above discussion, the vibrational states of the linear form can be systematically related to the K = 0, 1,2, . . . , levels of the (Q = 0, ~6, ~4) vibrational states of the C% form. In Fig. 4, such a correlation is ~il~st~a.ted schematically. (e) z’s (&) &zG&~ s&&3 0j C&* Following the above discussion, the (@,2°)0 X,+ state of the linear form corresponds to the (eta = 1, 0, O)li AI ~bration~ state of the Gs form. Similarly, the (8,4a.Y &*, (PI @)OZ,+, . . . , states are related to the I, . . . , states of G,. The K’ structures and the (Q, Q.) structures (2,0,WA 1, (3, 0,OFA for these ve-excited vibrational states can be obtained in a manner similar to that discussed above. Some of the results are shown in Fig. 4. In this way, correlations among all the bending states (~~~0,~~~~~~~~ of th e 1’ mear form and all the bending-torsionalrotational states (83,06, ~4)~ for the bent Cz, form are derived. (j) S~rn~a?~ of Doe~-C~V~~~~~~~~0~. (i) The v5 (XT,) degenerate bending vibration of the linear form turns into the K-type rotational motion and the symmetric bending vibration vz (A I) of the Css form. The (O@,& 5= 06)’states of L)ochcompose the K’ structure of the ground vibrational state of C s8. The (0@,2~~~)~Z,+ states of D_,hcorrespond to the K. = 0 level of the vibratory states (s3 = na, 0, O)O A I of C2u. (ii) The vq (II,) degenerate bending vibration of &,,h splits into the torsional motion v4 (Ad and the antisymmetric bending vibration vs (&) of CzV. The combination states (I*‘, lF1)O 2,- and (l*“, lF1)oZ,+ of D oohcorrespond to the K = 0 level of the fundamental vibrational states of (0, 0, ~4 = 1)OAn and (0, u6 = 1,O)O El of Cz,, respectively.

LINEAR-BENT

CORRELATION

411

FIG. 4. Correlation of the D,h vibrational states to the K structure of the vibrational states for Czv and C:h. Vibrational levels for D,h are schematically shown with the labels WJS and K = 1.The K structures of a vibrational state for Crv are connected to one another by full lines, while those for Crh by broken lines. Vibrational levels for CZ, and Crh are labeled by (ww~).

(iii) The u-type Coriolis interaction plays an essential role in the correlation causes anomalous shifts in the energy system.

between the v4 (A,) and v6 (Br) vibrations between Doohand Cz,, forms. This interaction levels especially in the case of a quasilinear

B. Correlation between DWhand C2,, Forms In a similar manner, one can derive energy-level correlations between D-J, and C2h the role of the bending forms. In comparison with the above D,h-C 2” correlation, vibrations v4 (II,) and v5 (II,) of D-h is interchanged.

6)

The v4 (II,) degenerate bending vibration, the K-type rotational motion and the of C2h. (ii) The y5 (II,) vibration of DDohsplits into antisymmetric bending vibration v6 (B,) (iii) The u-type Coriolis interaction between linear-bent correlation of energy levels. in Fig. 4. C. Correlation among Nonlinear

instead of v5 (II,), of D-h turns into symmetric bending vibration va (A,) the torsional motion v4 (A,) and the of Czh. VJ and v6 causes a large effect on the These correlations are also illustrated

Forms and Internal Rotors

In addition to the linear-bent correlations discussed above, correlations among nonlinear forms can be accounted for in the present scheme. This problem has two aspects: (i) correlation following the change in the equilibrium torsional angle 7e, which is zero for Cz,,, a for Czh, and in between for Cs; and (ii) correlation following the change in the barrier hindering the internal rotation, i.e., correlation among rigid

412

YAMADA,

free int. rotor

NAKAGAWA

AND

KUCHITSU

vds

FIG. 5. Relations of nonlinear forms of XZYZ on the map of V trans vs Vcis. The equilibrium torsional angle changes from 0 to ?r. Marks a, b, c, and d show the typical cases used in the following illustrations.

rotors (of symmetry CZ,,, c2h, and C,), hindered internal rotors and a free internal rotor. Since the extended pe~utation-inversion group BP&* derived in the present study is applicable to all the forms in question, such correlations can be derived by tracing the energy levels with the same symmetry in this group and with the same angular-momentum projection K. The potential function hindering the internal rotation in various forms of nonlinear XzYp may roughly be characterized by three parameters : re and the potential energies llcir and Vtrans, at r = 0 and s, respectively, measured from the energy at r6. If the Vcis and Vlransvalues are given, one can roughly locate the equilibrium angle TV.Thus, various nonlinear forms may be mapped on a two-dimensional plane with Vois and V trans,-rebeing taken as a dependent parameter, as is illustrated schematically in Fig. 5. Energy schemes for hindered internal rotors with a high barrier (Fig. Sa, b, and c) and for a free internal rotor (Fig. 5d) will be shown as typical cases in the following illustrations. Figure 6 illustrates correlation of the torsional energy levels (K = 0). As shown in Table IV, the torsional motion belongs to A xUin the Dzh* group, which corresponds to AZ, A, and A, in the Czv, Cz, and C’zhsubgroups, respectively. In the Ct case the torsional states, which are doubly degenerate in a high barrier limit, are split with a decrease in Vcis and/or Vi,,,, as a result of tunneling. Therefore, the torsional quantum (0) C2”

“4

“4

(b)

k)

(d)

C2

‘2h

O:h

q4

n

FIG. 6. Correlation of torsional energy levels for hindered internal rotors with (a) Cz,, (b) Cz, and (c) C2Astructures and for (d) free internal rotor. The difference in the torsional (or out-of-plane vibrational) quantum numbers is shown.

LINEAR-BENT

(a)

(b)

C2Y

G2

----_ 2 --23 -a

I

-8 *o

Id)

Gl

+I

.~”

vg=0

413

CORRELATION

>I2

FIG. 7. Correlation of torsionai-rotational energy levels for (a, b, c) hindered and (d) free internal rotors. The quantum number K and the rovibronic symmetry for even J are labeled. The odd K levels of the ground and the excited torsional states for CI, and Ca are interchanged.

number ‘~4for CZ is not equal to v4 for Czv or C~J, (see Fig. 6). In the case of a free internal rotor, the torsional energy levels are roughly proportional to n2 and are doubly degenerate for n > 0, where n is defined in Eq. (1). The correlation of the rotational X structures of the ground and torsional excited states for the nonlinear forms are illustrated in Fig. 7. The rotational structures for the Cz,, [Fig. 7(a)] and C2h [Fig. 7(c)] f orms are those for the ordinary asymmetric tops; their rovibronic symmetries are obtained with reference to Table V, as was done in Section IIIA. One should note that the K = odd levels for the Cze and C2,, forms have different rovibronic symmetries even though the K = even levels have the same symmetries for the two forms. This means that the K = odd levels of the ground vibrational state of Cz,, should be correlated to the R = odd levels of the torsional excited state for C2h, and that the K = odd levels of the torsional excited state of Czv are correlated to the K = odd ground state of C H,. In Fig. 7(b) the K structures for the Cz form with 7. = ?r/2 are shown; the energy pattern is taken from Ref. (6) and the symmetry properties are determined with reference to Table III and Table V. The ground and the first excited torsional states have the ordinary rotational structures for the Ii: = even levels. The K = odd level energies, on the contrary, show a peculiar behavior with the change in TV; in the case of re = 7r/2 the molecule is an accidentally symmetric top and its K = odd levels are accidentally fourfold degenerate (6). This degeneracy of the four levels (&:s and z&) may be understood in Fig. 7 as a result of the energy crossing with the change in r e from the Cs,, form to the CZ~ form. Figure 7(d) illustrates the energy levels in the case of a free internal rotor. The energy levels may roughly be expressed by the sum of the free internal rotor energy and the rigid rotor energy E ,,,k = Fn2 + Ak3,

(6)

YAMADA,

414

~AKA~A~A

AND KUCHITSU

wheren=O,~l,f2,,..,andK=O,fl,fZ ,..., with the restriction that ft and k have the same parity. The energy levels are fourfold degenerate for the cases of n; # 0 and k # 0. By use of Table III and Table V, the rovibronic symmetries can be determined as shown in Fig. 7(d). As illustrated in Fig. 5, the free internal rotor may be correlated to any of the nonlinear forms of C2*, CZ, and C2h symmetry. By the use of the conservation rules of the angular momentum projection K and of the rovibronic symmetry, correlations shown in Fig, 7 are derived. IV. ~~RR~~ATIUN

OF TRANSITIONS

The electric dipole moment belongs to the rovibronic symmetry --s (or A 2U),because it is invariant under the operations (12) (ab) and ?” but is reversed by 8*. Therefore, the dipole selection rule is obtained as +s++

--s

and +a++

-a,

(7)

or in the I&* notation, A r. * A 2% and

RI, t+ Bzg

for the overalE (or ro&bronic) species. The selection rules for the ZJ&~P&Cspecies can be obtained in the usual manner by use of the symmetry species of the dipole-moment projections on the molecule-fixed axes. s According to Table I, the components pa, 1~~) and pz belong to Aru, Bgu, and Blu, respectively. Hence the vibronic selection rules are: z type:

AI, ++ A ru, A 2g+-+A 2u,31, ++ Bh, or 32g ++ B2u,

y type:

A~++i32~, A~t-)f&,

23type:

A lg +-+&PI AI, ++ Bz,, AZ* ++ Bb, or Azti ti I&=

A2,++&,,

or Aau+-+&,,

(81

The ~~#~~~~~a~selection rules for vibronic transitions can be obtained from the ~~~~~~~~ and the vibronic selection rules listed above. Let us consider the correlation of vibration-rotation transitions for nonlinear forms. Figure 8A illustrates the vibration-rotation transitions to the symmetric vibrational states (AI,) together with their torsional hot-band structures for the Czu and CzlLforms. The energy levels for the vibrational, torsional, and K-type rotational states are shown schematically, and their vibronic and rovibronic symmetries are labeled (for even J only, as before) with reference to Table V and Fig. 7. A symmetric vibrational mode, say ~1, is of symmetry A fg in the Dz$ group; the x component of the dipole moment pz (AnJ is responsible for the transitions related to this vibrational state. In the case of the Cfv form, this p2 component induces the fundamental band ~1 and the torsional hot band YI+ ~4 - ~4, which are of the b type with the AK = f 1 selection rule in a symmetric top approximation. With the aid of the rovibronic selection rules, Eq. (7), the allowed Q-branch transitions are drawn in Fig. 8A. (The P- and R-branch transitions have the same K structures as the Q-branch transitions but may have different pattern concerning the K-type doubling components.)

LINEAR-BENT

2

415

CORRELATION

%h

c2v

-5

I

illlAp,

0 2

(IO)Afg

i 0

(0 I) AOU

(001 Alg b,v,)

fund.

K

PX (4

hot.

K

b type

comb.

diff.

PX (A,,)

c type C2h

c2v

(vsv,l

K

fund.

hot

Pz (4”) a type. Au*0

comb.

diff.

ry (f32u) c type. Awf

K

fund. PY (4,)

I

hot

o type. aK= 0

fund.

hof

Py (B2”) b type. d
FIG. 8. Correlation of vibration-rotation transitions for Cza and &A. (A) Symmetric (Al,) band and its torsional band structure. (B) Antisymmetric (&) band and its torsional band structure. Rovibronic symmetry for even J levels and Q-branch transitions are shown.

The combination band YI+ vq and the difference band VI - ~4 are forbidden for Czy. In the case of the Cz,, form, on the other hand, the J.L~component induces the combination band VI+ v4 and the difference band v1 - v4, but the fundamental band VI and its torsional hot band are forbidden. The combination and the difference bands are

416

YAMADA,

NAKAGAWA

AND

KUCHITSU

not type b but type c bands in this case, having the selection rule AK = f 1. In parallel with the correlation shown in Fig. 7, where the odd K levels of the ground vibrational state and those of the torsional excited states for Czv and C2,, are interchanged, the fundamental ~1 (Al) band for CZv turns not into the fundamental (A,) but into the combination and the difference bands for C2h with the increase in 7e. In the above discussion, it is clear that the symmetry forbidden transitions such as the vr band of C2h never get allowed even by a large-amplitude vibration related to a low internal-rotation barrier nor by the change of the equilibrium torsional form into Cp. or Cz.‘O The understanding of the correlation of transitions as shown in Fig. 8A should be essential in the interpretation of the microwave spectra and high resolution infrared spectra of low barrier internal rotors. Figure 8B illustrates the transitions related to the antisymmetric B1, bands in a manner similar to Fig. 8A. Two components of the dipole moment, pZ (Br,) and cc, (Bw), are responsible for this system. The pr component induces the Br, fundamental band, say v5, and its torsional hot band, v6 + v4 - ~4, in both cases of Czv and CU. These bands are of the a type and have the selection rule of AK = 0.” Figure 8B shows that the interchange of the K = odd levels with the change from the CZ” to Czh forms interchanges the fundamental and the torsional hot bands but the resultant band structures (parallel) are not affected. Another dipole component pl/ (Bz,) induces the combination band V& + v4 and difference band v5 - v4 for the CzUform; these bands are of the c type and have the selection rule AK = f 1 in a symmetric top approximation. For the CD, form, on the other hand, this pr component induces a (perpendicular) component of the fundamental band v5 and its torsional hot band us + v4 - vq; these are b-type bands and have the selection rule of AK = fl. The c-type bands (vg + v4 and vb - VJ) for CZv and the b-type bands (IQ and vg + v4 - VJ for C2h are correlated in a manner similar to that illustrated in Fig. 8A. B. Raman Spectra In analogy with the preceding discussion on the dipole transitions, selection rules for Raman spectra can be derived. The polarizability tensor elements, which are responsible for the Raman transitions, can be classified according to the Dzh* species as listed in Table VII. The rotational selection rules AK are also listed under the assumption that the quantum number K is well-defined. Figure 9 illustrates the Raman selection rules for the Czzl and C~I, forms. The vibrational-torsional levels are shown in the figure, whereas the rotational structures are not specified but would be the same as in Fig. 8. The allowed Raman transitions are shown in the figure by arrows, together with the associated polarizability tensor elements and the AK selection rules given at the bottom. The energy levels for the Czv and CM forms are corrected by horizontal (for even K levels) and oblique (for odd K levels) broken lines in Fig. 9. For the Raman bands with AK = 0 and 4~2, the fundamental Raman band for CZv turns into the fundamental and the torsional hot bands for Czh, lo In the case of the Cr form, all the transitions shown in Fig. 8A belong to the fzmdamental band according to the ordinary definition of the v4 quantum number as shown in Fig. 6. ii Since Fig. 8B shows the Q-branch transitions, the transitions from K = 0 to K’ = 0 are forbidden in the a-type bands. However, P- and R-branch transitions from K” = 0 to K’ = 0 are allowed because of different rovibronic symmetries for even J and odd J.

LINEAR-BENT

Table VII.

palarizability tensor elements a

xx

+a

yy

+a

22

a toL -& xx ry @xx-“yy %Y a x2 %

Raman Selection

rotational selection AK

417

CORRELATION

Rules

vibronic

symmetry

rules

AJ

D2h*

c2v

‘2h

D-h

=%

c* g

Ag

%+ A B

0

0

%g

%

0

0,+1,*2

%g

Al

*z

0,*1,rz

%g

Al

Ag

$2

0,+1,rt2

%g

%

+I

0,+1,i2

%g

*I

Bg B g

+I.

0,*1,t2

%g

A2

hg

n

g

and vise versa. In the case of the c+ and azs Raman bands with AK = 3; 1, on the other hand, the fund~ental band for C’L. (or CZ~) splits into the combination band and the difference band for CILb(or G), just like the infrared bands shown in Fig. 8.

The selection rules derived above are also applicable to the transitions between two electronic states having different equilibrium structures, such as bent in the upper state and linear in the lower electronic state, as discussed extensively by Herzberg (4). The symmetry considerations introduced above allow systematic descriptions, because the upper and lower energy levels are classified within a common set of symmetry operations.

FIG. 9. Selection rufes for Raman spectra. Horizontal and oblique broken lines show the energy level correlation for even K and odd K, respectively.

418

YAMADA,

NAKAGAWA

AND

KUCHITSU

FIG. 10. Bent-linear transitions in the case of i?‘A,(&) - RQ,+(L?,h) branches are shown in full lines and PR branches in broken lines.

system

of acetylene

Q.

For example, the ;i ‘A, - 8 r2,+ band system of acetylene is illustrated in Fig. IO. This band system was studied by IGng and Ingold (16) and by Innes (17). The molecule is found to have a planar tray form (C,,) in the excited state, though it is linear (Doah) in the ground state. In the ground electronic state, t,here are two fundamental states of degenerate bending vibrations II, and III,, in which the angular momentum projection is K = 1 = 1. In the upper state, the K-type rotational energy levels of the vibrationless state (A, X A, = A, for the vibronic symmetry) and those of the torsional excited state (A, X A, = A,) are schematically shown in Fig. 10. The rovibronic symmetry for both electronic states are labeled in terms of the pe~utation-inversion group as fs or fa,r* which is applicable to any form of the XzYz molecule. (The rovibronic symmetry for the odd J levels is shown in parentheses.) The electronic transitions are dipole allowed due to the pz component (A,) with the rotational selection rule AK = f 1 in the approximation that the upper state is a near symmetric top, where the quantum number K is well-defined. According to the overall selection rule [Eq. (7)], the Q-branch transitions of the O-O band take place from the vibronic ground state to the lower K-doubling components of the K = 1 levels of the ~4’ = 0 upper state as shown by a thick full line in Fig. 10, while the P and R branches take pIace to the upper K-doubling ~oInponents as shown by a thick broken line. The effect of asymmetry in the upper state may induce AK = f 3, 15, . . . , transitions, though they are too weak to be observed in this band system of acetylene. Since the molecule changes its equilibrium form in the direction of the vq” (II,) or vg’ (A #) vibration, long progressions involving ~4” and ZJ~’are observed as a result of vibronic interactions. In Fig. 10 is shown an example of such a band, from the vq” = 1 (n,) state I2In Fig. 81 of Ref. (4), Her&erg classified the rotational levels in terms of the parity, + or -, for the inversion and the property, A or B, for the rotational subgroup CZ. These (A, B) labels in Fig. 81 correspond to the (s, a) labels in the present notation, though the basic ideas are different. For a rigid C2 molecule, one should leave the (A, B) or (s, a) labels and put + and - labels instead of either + or in Fig. 81.

LINEAR-BENT

CORR~ATIO~

419

to the vi = 0 state. This band has two subbands: one from K” = 1 to K’ = 0 involving the P, Q, and R branches, and the other from K” = 1 to K’ = 2 involving pairs of PQR branches. To the upper torsional excited state 84‘ = 1, a hot band is possible from the ws” (II,,) state, though it has not yet been assigned. This hot band also consists of twosubbandsK=O+-landK=2+-1. In the O-O band of this system, an extra series of lines were observed and assigned to Q-branch transitions to the K’ = 0 levels. Hougen and Watson (18) interpreted this phenomenon in terms of rzzis ~~~~~~g. They recognized that the Q branch is allowed but the corresponding PR branches are forbidden according to the rovibronic selection rules. The fact that the rotational constant A in the upper electronic state increases drastically with the increase in 03 is characteristic of a large-amplitude bending motion in a quasilinear system. V. DISCUSSION

It seems worthwhile to clarify some fundamental concepts of the permutationinversion operations and their relations to the point-group symmetry operations. A. PI O~e~ut~o~sand Fo~~t-~~u~ O~erat~~~s The permutation operation zi is simply defined as the interchange of the numbering of identical nuclei in the space-fixed coordinates, while the inversion operation _8* is defined as reversing the space-fixed coordinates of all the particles, including nuclei and electrons, at the same time (9). These operations are originally defined in terms of the space-fixed coordinates but can be interpreted in terms of the molecule-fixed coordinates, namely rotational, vibrational, and electronic coordinates, if the latter coordinates are uniquely defined. Any PI operation drr is expressed by _ * _a_ * GPI

=

Gf,.,.G=~t.Gvib*Geleet.GPPin,

where the partial symmetry operations on rotational or vibrational symmetry oflerations Equation (9) corresponds to Hougen’s operation 6,~ (19) *

69

the right-hand side will be called equinralelzt and so on. expression of the molecular point-group

_ GMP = etran*‘G,ot.dvib’~e*eot,

(10)

by which Hougen extended the definition of the point-group symmetry operation so as to cover not only the vibrational but also the rotational coordinates and to make the resultant C&p equivalent to the PI operation ~$1. The point-group operations, however, are applicable only to rigid molecules. On the other hand, the PI operations are applicable to nonrigid molecules as well. Originally, Longuet-Higgins explained that the PI operation times epui~alent rotation corresponds to an operation on the vibrational coordinates similar to a point-group operation, which he called an associated symmetry: &r ’ (6*&-l)

= Gvib.

(11)

Since the concept of PI operations is more fundamental than that of ordinary point group, we prefer the expression (9) to Eq. (11). It should be mentioned that the effect of A* on the rotational coordinates is in general a simple rotation, because the molecule-axe axes are always chosen to be a right-

420

YAMADA,

NAKAGAWA

AND

KUCHITSU

handed system, and that the effect of _& on the vibrational coordinates is not always identical to the inversion operation f in the ordinary point group, as in the present case of XZYs shown in Fig. 2. B. Necessity of Axis Dejinition

In order to understand the PI operation effect on the rotational and vibrational coordinates, molecule-fixed axes must be defined unambiguously: For a nonrigid molecule, a slowly varying large-amplitude coordinate need to be first defined to specify the reference configurations. Then the molecule-fixed coordinates must be introduced for any instantaneous reference configuration. By use of these definitions, constraints such as the center of mass, Eckart and Sayvetz conditions can determine the orientations of the molecule-fixed axes for any instantaneous nuclear positions. Though the above procedure looks almost obvious, a number of confusions originated in previous studies from inadequate definitions of molecule-fixed axes. Sometimes figures were displayed without explicit definition of their xyz axes. Such drawings are often ambiguous when nonrigid molecules are concerned. There are only two degrees of freedom to specify the xyz axes, because the third axis is always chosen to form a right-hand system. Therefore, in a larger molecule some nuclear positions are involved in the axis definition while others are not. Unless nuclear numbers used in the definition are clearly specified, a discussion of symmetry properties may be misleading since another equally valid axis convention is practicable. C. Extended PI Operations

Hougen (10) was the first who extended the PI group to classification of the vibrational and rotational wavefunctions separately in the case of dimethylacetylene. Double groups have been applied to various coaxial internal rotors by Hougen, Bunker, PapouSek et al. (21). In their papers the effect of a PI operation (say Hougen’s (123) in Ref. 10) on the rotational coordinate X (and therefore on the torsional coordinate 7) was regarded as undefined because of the double-valuedness, and extended PI operations (say A^ and A4 in Ref. 10) were introduced so that their effects on X were defined as either X’ or X’ + ?r. In the present study, as shown in Fig. 2 and in Table I, the PI operations are regarded as err

= GPILO

(12)

according to the notation of the angle Eby Bunker and PapouSek (11), and the operation F introduced in Section IIA corresponds to (.&.=~ concerning the transformation properties of X. For nonlinear (or noncoaxial) molecules and rigid coaxial molecules, the angle e is always zero and Eq. (12) holds. As the bond angles are opened to ?r or as the potential barrier hindering internal rotation decreases, the PI operations should be smoothly correlated to those in linear or nonrigid coaxial molecules. Therefore, &r in Eq. (12) is a natural definition in the treatment of extended PI groups. ACKNOWLEDGMENTS The authors the manuscript.

wish to thank

RECEIVED: September

Dr. B. P. Winnewisser

5, 1973

and Dr. P. R. Bunker

for their critical

reading

of

LINEAR-BENT

CORRELATION

421

REFERENCES 1. 2. 3. 4.

W. R. THORSON AND I. NAKACAWA, J. Chem.Phys. 33,994 (1960). R. N. DIXON,Trams.FaroduySot. 60, 1364 (1964). J. W. C. JOHNS, Can. J. Phys. 45, 2639 (1%7). G. HERZBERG, Electronic Spectra and Electronic Structureof Polyatomic Molecules, in “Molecular

Spectra and Molecular Structure,” Vol. III, Van Nostrand, Princeton, NJ, 1966. 5. J. T. MASSAYANDD. R. BIANCO, J. Chcwe.Pkys. 22,442 (1954). 6. E. HIR~TA,J. Chm. Pkys. 28, 839 (1958). 7. R. H. HUNT,R. A. LEACOCK, C. W. PETERS,ANDR. T. HECHT,J. Chem. Phys. 42, 1931 (i%S). 8. T. R. DYXE, B. J. HOWARD, ANDW. KLE~ERER,J. Chem. Phys. 56,2442 (1972). 9. H. C. LONGUET-HIGGINS, 3601.Pkys. 6,445 (1963). 10. J. T. HOUGEN, Cam. J. Phys. 42, 1920 (1964). II. I?. R. BUNKER ANDD. PAPOU~EX, J. Mol. Spec~rosc.32, 419 (1969). 12. G. W. KING,R. M. HAINER, ANDP. C. CROSS, J. Chem. Phys. 11,27 (1943). 13. H. H. NIELSEN, Rev. Mod. Phys. 23, 90 (1951); Handbuch der Physik 31, 173 (1959). Id. G. HERZBERG, Spectra of Diatomic Molecules, in “Molecular Spectra and Molecular Structure,” Vol. I, 2nd ed., Van Nostrand, Princeton, NJ, 1950. 15. L. D. LANDAU ANDE. M. LIPSHITZ,Quantum Mechanics, Pergamon, London, 1958. 16. C. K. INGOLDAND G. W. KING, J. Ckem. SW. 2702 (1953). 17. IL K. INNES,J. C&m. Phys. 22,863 (1954). 18. J. T. HOUGEN ANDJ. K. G. WATSON, Can. J. Phys. 43,298 (l%S). 19. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations,” McGraw-Hill, New York, 195.5. 20. J. T. HOUGEN, J. Chem. Phys. 39,358 (1963). 21. For example, P. R. BUNKER, J. Chem. Phys. 47, 718 (1967); 48, 2832 (1968); D. PAPOUSEK,K. SARKA, V. SPIRKO,ANDB. JORDANOV,Coil. Czech. Chem. Comm. 36,890 (1971).