Vibration-internal rotation-rotation Hamiltonian of an asymmetric top molecule with C3v internal rotor

JOURNAL

OF MOLECULAR

SPECTROSCOPY

109, 374-387

(1985)

Vibration-Internal Rotation-Rotation Hamiltonian of an Asymmetric Top Molecule with CSvinternal Rotor KRISHNA

MITRA AND PRADIP N. GHOSH

Department of Physics, University of Calcutta, 92 A.P.C. Road, Calcutta 700009, India An expression for the kinetic energy part of the vibration-torsion-rotation Hamiltonian of an asymmetric top molecule containing a C,, internal rotor has been derived. The terms for various interactions in the molecule, viz. Coriolis interaction between rotation (both overall and internal rotation) and vibration, centrifugal distortion and anharmonicity of molecular vibrations induced by the internal, and overall rotation of the molecule, have been formulated. For a planar molecule with C, symmetry we have obtained the vibrationally averaged rotationinternal rotation Hamiltonian. Diagonalization of this Hamiltonian for a particular vibrational state will yield the rotation-internal rotation energy levels and hence the transition frequencies. These data will be useful for analysis of high-resolution infrared spectra obtained by laser or Fourier transform spectroscopy of nonrigid molecules with internal rotor. We also present a set of quartic centrifugal distortion coefficients associated with rotation and internal rotation. These data will be helpful for evaluation of vibrational potential constants of the orthorhombic asymmetric top molecules. Q 1985 Academic PXSS, Inc. 1. INTRODUCTION

Satisfactory interpretation of the fine structure in the ir gas spectra of nonrigid molecules requires detailed consideration of the interactions between various types of motion in the molecule. This has become particularly expedient because of the advent of the tunable infrared laser spectrometers which yield data of very high resolution. A number of papers (1-6) dealing with this problem have appeared in the literature. These papers were concerned with interactions of different types of motion in polyatomic asymmetric top molecules. Interactions between overall rotation and small-amplitude vibration in polyatomic molecules have been analyzed by Nielsen (1). Lin and Swalen (2) considered the effect of internal rotation on overall rotation in symmetric and asymmetric top molecules. Kirtman’s (3) paper dealt with the interactions between rotation, vibration, and internal rotation by the internal axis method (IAM). This paper emphasized the effect of these interactions on the microwave spectra. Recently, Ganguly and Ghosh (4) and Ghosh and Ganguly (5) have analyzed the effects of vibration-internal rotation interactions on the normal modes of a nonrigid molecule containing a nearly free internal rotor. In the present paper we derive the expression for the kinetic energy part of the vibration-internal rotation-rotation Hamiltonian of an asymmetric top molecule containing a symmetric internal rotor by the principal axis method (PAM). The effects of Coriolis interactions due to rotation and internal rotation, the change of moment of inertia with the normal coordinates, and the torsional angle dependence of the normal coordinates have been included. Finally, the interactions are interpreted 0022-2852185

$3.00

Copyright Q 1985 by Academic Press, Inc. All righrs of reproduction in any form reserved.

374

VIB-INT

ROT-ROT

HAMILTONIAN

315

in terms of Coriolis coupling, centrifugal distortion, and anharmonic coupling. The quartic centrifugal distortion coefficients in the rotation-internal rotation Hamiltonian have also been formulated. 2. KINETIC

ENERGY

PART OF THE HAMILTONIAN

To derive the expression for the classical kinetic energy, T, of the molecule the following sets of Cartesian coordinate axes are introduced: (i) the space-fixed XYZ axis with origin at S (say); (ii) the frame-fixed Xf Y, 2~ axis with origin taken at 0, the instantaneous center of mass of the entire molecule and unit axis _%?;,Y;-, Z; defined by the principal axes of the entire molecule; and (iii) the top-fixed X,Y,Z, axes with origin at 0’, the instantaneous center of mass of the top (2) (Fig. 1). Denoting z and q as the position vectors of the ith and jth atoms of the frame and the top, respectively, in XYZ axes, and mi, mj as their respective masses, the expression for T in XYZ axes may be written as -1, 2T= C miRi*Ri + C mjG*z.. (1) i

i

We define 0 and w’ as the angular velocity vectors of the molecule and the top, respectively; J; and 5 as the corresponding position vectors of the ith and jth atoms of the frame and the top in X, Y, Z, axes; Uj as the position vectors of the jth atom in X,Y,Z, axes; and q = S - 2, as the angular velocity of the top relative to the frame; and u, as the velocity of any arbitrary atom belonging to both the frame and the top relative to the Xl Y,Z,axes. Thus, using the Eckart condition, 2 Vii< + C

Wljc

L

FIGURE 1

=

0,

(2)

MITRA AND GHOSH

376 we obtain, from Eqn. (1) (5), 2T=

;m,t$+ s=l

;m,(GXFJ’+ A-=1

~m,(;X~)2+2~m,(WX~).(iXO;) j=l

j=l

+2;r.~m,(~X~)+2;‘~m,(~X~),

s=l

(3) j=l

where N is the total number of atoms in the entire molecule, and NI is the total number of atoms in the top. In Eq. (3) the first term represents the vibrational energy, the second term represents the overall rotational energy, the third term represents energy due to internal rotation of the top, the fourth term is the coupling term between overall rotation and internal rotation, the fifth term represents the interaction between overall rotation and vibration, and the sixth term represents interaction between internal rotation and vibration. The principal moments of inertia, Z,, of the entire molecule are defined by 5 m,(r; x ?,)2 = s=l

c

I,&

(4)

C?=X.y,Z

and 5 m,(; x i+

= zr?,

(5)

J=I

where Zr is the moment of inertia of the top about the ; axis which, for convenience, has been taken along the % direction of the X,, Y,, 2, axes. The rotation internal rotation interaction term is 2 2 WZj(Z, X 5) * (T X q) = 2;Zr

2

AaWe,

(6)

a=x,y,r

i

where the A, are the direction cosines of the ; axis with the axes of the XJ, Y,, 2, system, and use has been made of the center of mass condition ; T?ljs= 0.

(7)

j=l

The Sayvetz conditions are 2 mj(Z$X j

and

i$*P = 0.

2 mi(G X 6) + 2 rn,(G X 3) = 0, i

-A

(8) (9)

i

where roi, roj are the equilibrium values of < and < in X,, Y,, 2’ axes, 6 is the equilibrium value of a, in X,, Yt, 2, axes. The above Eqs. (8) and (9) imply that neither the top nor the entire molecule has zeroth-order vibrational angular momentum about the corresponding axis of internal rotation or overall rotation. Using the Sayvetz conditions, we get (3)

VIB-INT

ROT-ROT

377

HAMILTONlAN

225’Cm,(~X~)+2~.Cmj(slX~) s j =2w’Cm,(dr,X~)+2~‘Cmj(daJXTj;)+2(~X3)’Cm,(~X~), , J

(10) i

-

where dr,, d?, and daj are the small-amplitude displacement vectors corresponding to c, 5, and 5. On introducing the normal coordinates QK (K = 1, 2, . . . . , 3N - 7) through the relations 3N-7 dr,,

=

C

(11)

m;‘/2b,a,d!~,

R=l

where bsu,K is a function of 7: the expression for T is obtained from Eqs. (3) to (10) as 3N-7

2T=

c &+ K=l + 2

c I,&+ a=x.)‘.z C

S&Q&

G(C

a=x.>.=

IT+‘+ 2;Z7

c &o, a=n-,_V._

+ +Ma) + WC

&IQ&

(12)

+ f C c,lQ&h

KJ

K-1

K./

with the following notations: Cb = : (bs&Jsr,/ - b,,,&,c), S=l

(13)

M* = C 71%QKQl + AYQA

(14)

K.1 N 61

=

c

(b,Kb:,.l -

(15)

b,,,Kb:,d

_?=I Nl

Nt

ATQK) = A,[ c m~i2bja,KuoJoQrij=l

c

m,?2bjp,K~oJ,QK]

J=I

+

h[?

m~‘2bja.e%,& ,=

2 a

GE/

Z (bjp.KbJ,,l - JJJy,Kbj@,I)t ,=I

cK/ =

2

n and

f?

b%/

m]‘2bjy,K’JoJaQd,

(

16)

J=i

BKI =

=

LGb

-

1

+

c b:cxKbsn.r, .s.a

+ ; c

s,a

&,,&a.,,

(17)

(18)

(19)

MITRA

378

AND GHOSH

where cy, /3, and y represent cyclic permutations of x, y, and z. Here & is the Coriolis coupling term between vibration and overall rotation, G:, is the Coriolis coupling term between vibration and internal rotation, and 7;t-land N% arise from the torsional angle derivatives of the normal coordinates b&. The other terms originate from the finite angle of the internal rotation axis with the principal axes. On defining the frame-fixed component J, as the total angular momentum, J, as the momentum conjugate to ;, P, as the momentum conjugate to Q,, p, as the vibrational angular momentum, pT as the vibrational-torsional angular momentum, viz. J_=$;

J*=$;

a

pm = C GIQKPI;

fl = c

KJ we

BKI(~KPI,

(21)

KJ

can write Eq. (6) as 3N-7 2T

=

(Ja - P,)cL,s(J, -pa> + C P:,

c

(22)

I=1

a,a=x,y,z,+

where [pap] is the inverse of the matrix [Ias] that expresses J, in terms of w,; pa6 is a complicated function of ; and QK. The explicit form of the elements of [P,& has been presented in Appendix 1, for a planar molecule with C, symmetry which has its internal rotational axis ; in the X, Y, plane of the X, Y,Z, axis system. The quantum mechanical equivalent of T in Eq. (22) may be written as (6) T zz ;

p1/4

(J, - ~&L~&“~(J~ - pa)/~“~+ i j~“~ C PK~-“*PKP~‘~, (23)

C

K

d=x,YA~

where p is the determinant of [p,J and J,‘s are linear combinations of operators ps = -ih(a/ae); p4 = -ih(ald4); and px = -ih(a/ax) = (0, 4, and x being the Euler angle) and J, = -ih(a/&); pa = -ih c,, lb Q,dd/aQd; p, = -ifi c,,

BKIQKab%ih;

ami

PK

=

-ifL ai@K.

3. INTERPRETATION

OF THE TERMS

IN THE HAMILTONIAN

On expanding paS in terms of normal coordinates QK about the equilibrium configuration of the atomic nuclei, we get the kinetic energy T as T = C T,, s -I- t < 2, r.s,t

(24)

where the first subscript, r, is the degree in the vibrational operators (coordinate and momenta); the second subscript, S, is the degree in the components of the total angular momentum operator J, (a = x, y, z); and the third subscript, t, is the degree in the torsional angular momentum operator J,. Thus in Eq. (24) the terms T0209 Too2, and ToI, represent the rotation-internal rotation operator; T200 is the

VIB-INT

ROT-ROT

379

HAMILTONIAN

harmonic oscillator operator; TlzO and TzzOare operators of centrifugal distortion induced by overall rotation; T1oz and Tzo2 are operators of centrifugal distortion induced by internal rotation; Till and T2,, are operators of centrifugal distortion induced by both the overall rotation and internal rotation; TzLOis the Coriolis interaction between overall rotation and vibration; T20, is the Coriolis interaction between internal rotation and vibration; and Tdoo is the anharmonicity of the molecular vibrations. 4. ROTATION-INTERNAL

ROTATION HAMILTONIAN VIBRATIONAL LEVELS

FOR SOME PARTICULAR

On defining (J, - pa) [a = x, y, z] as an angular momentum (7) operator. we can effect vibrational averaging [Van Vleck-type perturbation] of the symmetric matrix [Jo] so as to get the rotation-internal rotation Hamiltonian for some particular vibrational state. Diagonalization of this averaged Hamiltonian would yield the corresponding energy levels so that the transition frequencies between different vibration rotation-internal rotation states can be found and, hence, analysis of the fine structure in the vibrational spectrum in the infrared region observed with high resolution can be accomplished. Explicit expressions of the elements of the averaged Hamiltonian have been shown in Appendix 2 for a planar molecule with C’, symmetry having its internal rotational axes in the X, Y, plane of the X, Y,Z,. axis system. In order to diagonalize the Hamiltonian for a particular vibrational state one has to choose an appropriate basis function depending on the height of the potential barrier, I’, . In case of low barrier we can choose the basis function Ic/“(x,6, 4, 7) = & which diagonalizes the unperturbed 5. QUARTIC

Ic/O(x,0, $)eimr,

rotation-free-internal

CENTRIFUGAL

DISTORTION

(25)

rotation Hamiltonian. COEFFICIENTS

Terms for quartic centrifugal distortion in the molecule can be formulated by performing a contact transformation (8) on the terms like T,, [r = 1; s = 0 or 2; t = 0 or 21 of the expanded kinetic energy operator T,$, of Eq. (24). The usual form of the quartic distortion term, Tqcd,of a molecule is written as

where h,, is some coefficient, and commutation rules between different Jo’s have been taken into account (J, commutes with all of .JX, Jy, J,). On this Tqcd of Eq. (26) we shall impose the condition that Tqcdmust be invariant to the operation of Hermitian conjugation (9) and time reversal (IO), i.e., Tqcd = Tficd= TTqcdT-’ = ( TTq,Tp’)+.

(27)

Using the following relations J = J+;

TJT-’ = -J;

hi,

= h&s;

Th,,.sT-’ = h&y

(28)

MITRA AND GHOSH

380

it follows that hPqrSin Eq. (26) must be real and p, q, r, s must each be even for an orthorhombic molecule. Thus, the form of the rotational Hamiltonian, I&, which includes rigid rotor terms and quartic centrifugal distortion terms, may be written as

=

C

; (2 + fiJ” + {g - ; (it? + +:

+ + FJZ - M,J,J,

- A,J4 - A,,J’Jz

(,f - fi - &J2 - S,vJ; - M,J,J,

- AKJ:!

1

+ (2 - fi - 65J2 - 6KJ2=] (J”.‘(- J2) Y]

+ [R, J:’ + RJ, J2Jf + rT(Jz - JG)Jf + r,,JiJz],

(29)

where the rotational constants 2, f, and 2, and the quartic centrifugal distortion constants, A,, A,,, AK, 6., and 15~associated with overall rotation have been defined by Watson (8, II). The additional terms present in the Hamiltonian of Eq. (29) arise from the effect of internal rotation and are defined as follows:

R = + K:,,,, RJ,

rK,

K&&a

=

;

W:m

+

&A

(30)

(31)

(32)

=

r, = i VG,,, - &A

(33)

= x, y, z) = -2 c BFBTw~‘,

(34)

K

and KL

= -2 2 BzB;lwi’,

(35)

K

where By [a, /3 can take up values x, y, z] are the coefficients in the term 7’,,, defined as (36) and B;I is the coefficient in the term r,,, defined as T 102

=

2 K

&%K.t;

(37)

VIE-INT

ROT-ROT

HAMILTONIAN

381

and wK = UJC, where vK is the Kth normal mode frequency. Explicit expressions for BF, Bg, SF, and B;(’ are given in Appendix 3. Thus Eq. (29) shows that, because of internal rotation or torsion in the molecule, there are four extra independent parameters besides the five usual constants occurring in the quartic centrifugal distortion terms of H,,, of an orthorhombic asymmetric top molecule without any internal rotor (8). 6. DISCUSSION We have formulated the complete vibration rotation internal rotation Hamiltonian for an asymmetric top molecule with a C,,-type internal rotor. Since the vibrational energy level spacing for the infrared transitions is much larger than the spacing between the rotation-internal rotation energy levels, we present the rotation-internal rotation Hamiltonian for each vibrational state. This Hamiltonian contains all possible interaction terms between the different kinds of motion. Thus, the Hamiltonian presented in Appendix 2 is the most general Hamiltonian for a planar molecule with a C3” internal rotor. Depending on the structural data and the barrier height and the quality of data that one has to analyze, one can make a suitable choice of the basis function and omit or retain a few of the terms given. If highresolution measurements are carried out for a molecule of this type by means of a laser spectrometer, it may be possible to identify individual transition frequencies. For the purpose of actual assignment one needs to consider the different interaction terms presented in this paper. In the case of observations with lower resolution, instead of the line frequencies one observes band envelopes. Nevertheless, the simulation of band envelopes demands that one pay proper attention to the internal rotation. Our experience in the case of Puns-methyl nitrite showed that the simulated band envelope looked completely different from that observed with a conventional spectrometer, although the same computational program could adequately explain the observed band envelope in the cis conformer of the same molecule which has a higher barrier (12). This is obviously due to our total disregard of the low internal rotation barrier (- 10 cm-r) in the lrans conformer. In fact, nearly free internal rotation in such molecules affords a special status to this low-lying normal mode of vibration. This mode lying at the boundary between the rotational and the vibrational motions crosses the border and adds an extra degree of freedom to the rotational motion at the cost of one normal mode of vibration from the usual (3N - 6) vibrational degrees of freedom. This, however, is only apparent because the torsional motion is still one of the normal modes. The torsional mode is shifted to higher frequency in the matrix phase. In the gase phase the band envelope of a particular infrared transition is constituted by the rotation-internal rotation structure, which is completely different from the rotational structure. There are not many records in the literature on band envelope studies by high-resolution instruments (e.g., diode laser spectrometer or FTIR spectrometers) of asymmetric top molecules with low barriers to internal rotation. However, microwave spectroscopic studies have been carried out in detail for a large number of such molecules. The rotationinternal rotation Hamiltonian has been formulated in detail for analysis of the microwave data. These analyses were concerned mostly with the ground vibrational state, and the effects of vibrational motion are not taken care of. For analysis of the

MITRA

382

AND

GHOSH

infrared data one needs a complete Hamiltonian, which we present here. One may use this Hamiltonian for numerical analysis of band envelopes. APPENDIX

I

Explicit expressions for the elements of the symmetric matrix [P,~] are presented below. P( 11) = A, - &A: - S,,r2Z$M~Mfi - S,M: + 2rZTAxMxMy R: + 2A,M,R,

+ 2rZTM~M$,,

(Al)

p( 12) = rZTMxM,, - SxrZTMxMyA, - SyrZTMxM,,A, - S,M,M, + R

642)

p(13)=R;$+RIM,M,~+S,,$ z p( 14) = -M,

+ S,A,M,

643) + S,,rZTM,M; + S,FM,

- RL[i ~(22) = A, - Sxr2Z$M:M$

+ 2rZlMiMy]

- Rx15 + 2Mi)

(A4)

+ 2RyMyA,,

(A5)

- S,Ac - S,M$ + 2R:rZTM.,M,Ay + 2R,rZ,M,Mj’

~(23) = R:

- 2R,M,M,,

MxMy‘-IT+R++S,+

~(24) = -My

z

z

z

CW

z

+ SxrZTM:My + S,M,A,

+ &FM,

MX - R: - + 2rZTMxM$ >

ZY

-2RxMxMJ,-R,(;+2Mt),

p(33) = ; - $) z

~(34) = -R; F

.?

(A7)

648) - R; 9

Z

~(44) = F - S,M: - S,M;

- S,, ;,

(A9)

z + 2R:M,M,

+ 2R,FM,

+ 2R,FM,

- F=S,.

(AlO)

and r(C_G = P(i, 0.

(Al 1)

VIB-INT

ROT-ROT

HAMILTONIAN

383

The following notations are used: & = c @?“QK- 2 E/GQ&,

K

ST=

cai&K+&,;

a = x, I’_ z;

(A13

K.r.l

(A13)

K

(A141 (A151

F=-$ T

(A161

(A17)

6418) 6419)

(AN 6421)

(A22)

(A23) (A241

(A251 (A261

384

MITRA

AND

GHOSH

and B smKl

=

(~27)

&,Kb,a,r.

APPENDIX

2

Explicit expressions for the elements of the averaged [cL,~] matrix [(&J presented below:

2rZ,MiM,,[ z C’H& - 2

+

K

pad

&K&l

-

2

CKNK

-

C

-

C&%

c

K

c

2

cK&&l

Gb?d2

db:,,K)2

-

2

c

C&K&KI

4B.K.l

& +2r’Z%f!M:)

+

My+

7

+

~M:MJZT

>

x MX 7 + Y

c CK&&,CTKI] + SAK,l

12

+

2rz&f&fc

Mx 7

[C

2

CL&14) = -M, - MA,

C

cKGd2

-

W&K

K

c W% K.1

-

2

>

[c

c&K

2 :&K&i K.1

-

rZhf&fz

+

c&Kl!%

KJ

C

CKGKIGI

KJ

K.1 c CKNK K

%‘&K

K

-

+ FMxP

1)

XY

K,l

-

2 W,K

W&K,I

x

+

K.I

cKBsaK,B&

(B

KJ

K,l

M,M,[2

CK&&~K/I;

SAKJ

12) = rZTM,M, + rZTMxM,& c c,&%)* + rl,M,MA, -

2

K.1

are

2 CKCS%~~ K.1

z: Cd&,K)2 S&K

-

2 CK~~&K~~; S,B,K,l

-

C s.0,k.l

CKEIB~~KII;

(W

U33)

VIB-INT ROT-ROT

-

385

HAMILTONIAN

2 IZ

cKGKIB~,vKIs,ic.K,I

2 CKBS~K/&~K/]; S.LLiS.K.i

035)

386

MITRA AND GHOSH

/&W(i,A= cLLW(_A 0.

(Bl1)

The following notations have been used:

and NK = A,NX,, + h,+N&,,

@13)

where h is Planck’s constant and UKis the vibrational quantum number associated with the normal mode frequency vK. APPENDIX

Explicit expressions following: Bz = dd-A;@

of the different

- r21$M$M$ag + 2A,M,{

Bf

= d,[-r21$-MzMza$x

&a;

3

terms mentioned

in Section 4 are the

- MsaKT + AX(QK)} + 2rl,MiM,{

&a$ + AYQKI}I;

(Cl)

+ AX(Q~))I;

(C2)

- A;a$? - a;M;

+ 2AyMy{ X,a$ + A’(QK)} + 2rZ,M,M;{

La;

(C3) Bj;’ = d&-M_:@

- M:aF

+ 2FM.,{ &a:

+ A”(QK)) + 2FM,(X,a~

+ A’(QJ}

- F*&l;

(C4)

VIB-INT ROT-ROT

HAMILTONIAN

387

and

w ACKNOWLEDGMENT Financial support by the University Grants Commission, Government of India is gratefully acknowledged. RECEIVED:

June 23, 1984 REFERENCES

I. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Il. 12.

H. H. NIELSON,Rev. Mod. Phys. 23, 90-136 (1951). C. C. LIN AND J. D. SWALEN,Rev. Mod. Phys. 31,841-892 (1959). B. KIRTMAN,J. Chem. Phys. 31, 2516-2539 (1962). S. K. GANGULY AND P. N. GHOSH, Chem. Phys. Lezr. 90, 140-144 (1982). P. N. GHOSH AND S. K. GANGULY, J. Mol. Spectrosc. 104, l-l 1 (1984). D. PAP~LJSEK AND M. R. ALIEV,“Molecular Vibrational-Rotational Spectra,” Elsevier, Amsterdam/ New York, 1982. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS,“Molecular Vibrations,” MC Graw-Hill, New York, 1955. J. K. G. WATSON,J. Chem. Phys. 46, 1935-1949 (1967). L. D. LANDAU AND E. M. LIFSHITZ, “Quantum Mechanics-Non Relativistic Theory,” 2nd ed. Pergamon, Oxford, 1965. E. P. WIGNER,“Group Theory,” Academic Press, New York, 1959. J. K. G. WATSON,in “Vibrational Spectra and Structure” (J. R. Durig, ed.). vol. 6, p. 1, Elsevier, Amsterdam/New York, 1977. P. N. GHOSH ANDHS. H. GUNTHARD,Spectrochim. Acta. A 37, 347-363 (1981).