Internal rotation in molecules with two asymmetric rotors. The symmetry of the Hamiltonian

Journal of Molecular Structure (Theochem), 166 ( 1988 ) 261-266

261

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

INTERNAL

ROTATION

SYMMETRY

OF THE HAMILTONIAN

P. G o m e z I , M.

IN M O L E C U L E S

Fernandez I

WITH

, L.

TWO ASYMMETRIC

Fernandez

(I) Dep. Qulmlca Flslca I. F. Qulmxcas. •

(2) Dep. Qulmlca General

•

~

•

y Bloqulmlca.

ROTORS.

THE

Pacios 2 U.Complutense.Madrid

E.T.S.

28040

Montes U.Politecnica

Madrid 28040 ABSTRACT

The kinetic energy operator for the internal rotation Hamiltonian in molecules with two asymmetric rotors has been derived in the framework of a semirigid model. As an example both potential and kinetic terms for the asymmetric deuterated derivatives of dimethyl ether have been calculated. The symmetry shown by the internal rotation Hamiltonians corresponding to these molecules is discussed and compared with the nonrigid molecule symmetry group obtained from the chemical formula. CALCULATIONS

The molecular model used in the present work to study internal rotation

in molecules

with two symmetric or asymmetric

rotors,

is the so called Semirigid Molecular Model

(see ref. 1-2 and

quotations

internal

freedom

therein

,different

equilibrium molecular for

~i

of internal

values.

Therefore

fixed axis system

and

~2

energy surface 1 and

). This model considers

corresponding

~2,which

The generalized rotation

describe

the orientation

coordinates

of internal

rotors

,thus

k=l ..... N

inertia matrix contains

, and two non square blocks

(akX%o

(i)

a 3x3 block describing

, a 2x2 block corresponding

giP= k ~ m k

in a

axis system

to a minimum of the potential

to the internal

(2x3) corresponding

teractions between overall and internal have the following expression:

0166-1280/88/$03.50

of the nuclei

( for example principal

rk = ak( ~ i' ~ 2 )

rotation

the positions

degrees of

,frozen in its

) ,depend only on the large amplitude

respect to the framework

overall

rotation ones

rotation.

ak ) i P

© 1988 Elsevier Science Publishers B.V.

to in-

Coupling

terms

(2)

262

and internal rotation terms can be w r i t t e n as:

gij

ak =~-mkZ__ ( ...... k

~

i

~ ak ) ( ...... ~

The internal r o t a t i o n H a m i l t o n i a n

) ; k=l, .... N

j

(3)

i,j= 1,2

for two-rotor m o l e c u l e s

when interactions with overall r o t a t i o n is n e g l e c t e d

,

, can be

w r i t t e n as :

~2

......

H = - -~- i,j

vk=

~52 ~

(4)

is similar to these in the vibra-

(see for example ref.

% g 13 % in g + g i j ( ~ 2 1 n g

U •

+ Vk + V ~ ~ j

where V k for internal rotation tion - rotation p r o b l e m

g 13

~i

3) 1

~ in g ~ In g)

J

i (5)

As shown by (5) V k does not contain any d e r i v a t i v e of the wavefunction and can be c o n s i d e r e d as a kinetic o r i g i n e d potential term. Elements gij are the internal r o t a t i o n ones in ~ -i matrix, being ~

the g e n e r a l i z e d inertia m a t r i x and g= det I~ 1 . In general

both g and g 13 are p e r i o d i c

functions of

be e x p r e s s e d as Fourier series only a constant

term remains

in the series.

g and g 13 depends on the m o l e c u l a r model used in the H a m i l t o n i a n ' s

~ 1 and

~2

, and can

,but when the rotors are symmetric

derivation

The e x p r e s s i o n of

and on the axis system

(ref.4).

In the present work V k and gij have been c a l c u l a t e d for CH3OCH 3 and its d e u t e r a t e d derivatives.

The kinetic o r i g i n e d po-

tential V k was found n e g l i g i b l e c o m p a r e d with the p o t e n t i a l V in every case.

In table 1 the kinetic terms o b t a i n e d for several

d e u t e r a t e d d e r i v a t i v e s , c o n t a i n i n g m a i n l y asymmetric rotors summarized.

Potential

,are

and kinetic terms for symmetric d e u t e r a t e d

derivatives of CH3OCH 3 can be found in the well known paper by Groner and Durig

(ref.

5).

The internal r o t a t i o n p o t e n t i a l has been c a l c u l a t e d w h i t h i n the B o r n - O p p e n h e i m e r a p p r o x i m a t i o n f r a m e w o r k by using an a b - i n i t i o m e t h o d with a double zeta plus p o l a r i z a t i o n basis set. A c c o r d i n g to the model used in the d e r i v a t i o n of T ,relaxation of g e o m e t r i cal p a r a m e t e r s was not allowed

. Table 2 shows the c o e f f i c i e n t s

of the Fourier series standing for the a b - i n i t i o potential•

In

263

the n e x t

section

the

symmetry

C H 3 O C H 3 as n o n r i g i d symmetry present

of the

of the d e u t e r a t e d

molecules

internal

is s t u d i e d

rotation

derivatives

and c o m p a r e d

Hamiltonians

with

derived

of the

in the

work.

Table i cm -I

B II CC

B 12 SS

CC

B 22 SS

CC

b0( - 4 . 3 9 5 7 8

1.07865

-4.39578

b0]

1.563C~i0 -I

-1.2826x10 -I

1.5630xi0 -I

]-D20C~-D2 bl£ -1.2820x10 -1 b0~ 2.451x10 - 2

-1.763xi0 -2

b2£ bl]

SS

-1.763xi0 -2 l~g~xlO -2

2.436x10 -2

2.451xi0 -2 -3,;7623x10- 2

bOO ~.55257

2.436xi0 -2 -5.38493

1.20760

2.0107xi0-I

~.3022xi0 q

b01

blO bO2-3.916xlO -2

2.740xi0 -2

-3.316xi0 -2

bOO -6.39672

1.12688

-3.97508

D300D3

bOO -3.74896

1.00718

-3.74896

~30CM3

bOO -6.62914

1.2534

-6.62914

_

~ij

Kinetic factors. ~

2 ~ ij :- ~-- g . CC = cos iB 1 cos JB2, SS= sin i81

sin j8 2

Factors lower than i0 -~° cm_ 1 do not appear in the table 813= X(bij

Cos iS 1 Cos jS 2 +

6ij Sin i8 1 sin j S 2

Table 2 V03

V30

1126.7

V06

V60

-9.66

v33 161.74

Potential Coefficientes in cm

V36

V63 -6.52

V33 -131.2

%

% 6 •66

v& -i. 88

-i

V(B I, B2)= X{Vio/2 (1-Cos iB I) + Voj/2 (l-cesJB2) + Vij/2 (Cos i B 1 cosJB2-1)+ t

+ Vij/2 sin ig I sin jB 2 }

264

SYMMETRY The symmetry of the n o n r i g i d m o l e c u l e s can be s u m m a r i z e d in the group of feasible p e r m u t a t i o n - i n v e s i o n of identical nuclei. That group

,for the d e u t e r a t e d derivatives of CH3OCH 3 has been

found to have a semldirect product structure that permit to write G = (R 1 ~ R 2 ) ~

S

(6)

where R 1 and R 2 are rotational groups of the rotors and S stands for a isomorphic group to the framework one. F r a m e w o r k is c o n s i d e r e d to contain the center of masses of the rotors. The feasible p e r m u t a t i o n - i n v e r s i o n

group for every m o l e c u l e

in the family is o b t a i n e d by means of (6~ taking into account whether the rotors are symmetric or not

,and the e q u i v a l e n c e of

them. Thus Using the n o t a t i o n of p e r m u t a t i o n (ref.

inversion groups

6 ) if the two rotors are symmetric:

~i =~ ~ 0

(123) • (132) I : c 3

R2 = ~ E ~

(456) ~

(7) (465)~

= C3

o t h e r w i s e R 1 and/or R 2 only c o n t a i n s the identity or (4)(5)(6)

(i.e.

(1)(2)(3)

). On the other hand the e q u i v a l e n c e of the rotors

states the f r a m e w o r k ' s y m m e t r y contains p e r m u t a t i o n s

:if rotors are e q u i v a l e n t

,S

that interchange nuclei b e l o n g i n g to

different rotors

C2v : s =I ~ ~ (2~)~56~

(14)(26)(35)(78)'® (14)(25)(36)(78) I

If rotors are not e q u i v a l e n t Cs = S = ~ E ~

(8 )

(23)(56~ 1

(9 )

E is used in every group to denote the identity element.

Table 3

contains the five p o s s i b l e groups the d e u t e r a t e d d e r i v a t i v e s of dimethyl ether can be c l a s s i e d into. Isomorphisms with point groups in the case of the m o l e c u l e s with asymmetric rotors are easily d e r i v e d from Table 3. C h a r a c t e r tables for groups of m o l e c u l e s having symmetric rotors GI8

) can be found e l s e w h e r e

(for example ref.6-8

)

( G36 and

265

Table 3 Molecule

( R 1 ~ R 2 )~

S =

CH3OCH 3

C3

C3

C2v

G36

G36

G36

CH3OCD 3

C3

C3

Cs

GI8

G36

G36

CH3OCD2H

C3

E

Cs

G6

G6

G36

CH2DOCH2D

E

E

C2v

G4

G4

G36

CH2DOCHD 2

E

E

Cs

G2

G2

G36

S y m m e t r y groups for the five types of d e u t e r a t e d d e r i v a t i v e s in the family of dimethyl ether. p o t e n t i a l operators

The symmetry of kinetic and

is shown in the last column.

SUMMARY AND D I S C U S S I O N Kinetic energy o p e r a t o r terms for the most r e p r e s e n t a t i v e asymmetric two-rotor molecules

in the family of d e u t e r a t e d

d e r i v a t i v e s of dimethyl ether have been calculated. terms are periodic

functions of

~land

The B 13

~ 2 ' and a few terms in

the Fourier e x p a n s i o n seem to be enough to stand for it. Potential energy is s u m m a r i z e d in table 2 as well. The symmetry of a n o n r i g i d m o l e c u l e can be d e t e r m i n e d from its chemical

formula as shown above.

In the present case it is

found that d e p e n d i n g on the e q u i v a l e n c e and symmetry of the rotors, every d e u t e r a t e d d e r i v a t i v e of like dimethyl ether molecules

is c o n s i d e r e d to belong to one of the five groups shown

in table 3. However a more d e t a i l e d comment on this subject seems to be necessary. The s y m m e t r y of a q u a n t u m m e c h a n i c a l means of the set of t r a n s f o r m a t i o n s

system is d e f i n e d by

that leave invariant the

H a m i l t o n i a n d e s c r i b i n g it, therefore the symmetry p r o p e r t i e s

for

the internal rotation m o l e c u l e s under study can be d e r i v e d from the e x p r e s s i o n s of ~ and ~ (in tables 1 and 2). Whereas q u a n t u m chemical p o t e n t i a l derivatives

the

is the same for all of d e u t e r a t e d

and commutes with G36 elements,

the kinetic energy

o p e r a t o r reduces the H a m i l t o n i a n s y m m e t r y when a s y m m e t r i c rotors occur,

giving rise to the five different groups found for that

family of molecules. The symmetry derived from e x p r e s i o n s of ~ in table 1 coincides with that f o r e s e e n from the chemical

formula in all cases

~66

except in CD3OCH 3. The obtained kinetic terms B 13 (i,j =1,2

)

for that molecule are constant due to the symmetry of the rotors, thus it is easy to see that operations which switch angles describing the two different rotors as: (14)(26)(35)(78) (14)(25

(36)(78)

f(

~i'

~ 2 ) = f(-

f(

change the Hamiltonian

~2'-

) = f(

~I )

(10)

)

(4) in such a way that the following

expression for the matrix elements holds: (kl/H( ~i, ~ 2 )/rt) = (ik/H( ~2' ~ i )/tr)

(ii)

where kl and rt stand for the eigenfunctions of a free two-rotor, used as basis set to build up the Hamiltonian matrix.

It is easy

to realize from (ii) that the Hamiltonian matrix does not change under operations like those in (10),so that the spectrum of the operator is not affected. A higher symmetry than expected frDm feasible P.I. group commute with the rovibronic molecular Hamiltonian of an isolated molecule,while the group of T derived from table 1 stands for the symmetry of the Hamiltonian obtained under the semirigid model assumptions

(neither coordinates nor momenta

different to those corresponding to internal mouvements are present

).

As far as the semirigid model is satisfactory to describe internal rotation this effective symmetry can be worthly used.

REFERENCES

(1)G.A.Natanson: "Symmetry Classification of Normal Vibrations in Molecules with INternal Rotation" in Symmetry and Properties of Non Rigid Molecules: A Comprehensive Survey (J.Maruani and J.Serre Ed.) Elsevier,Amsterdam,1983 (2)G.S.Ezra: "Symmetry Properties of Molecules ~ in Lecture Notes in Chemistry. v.28,Springer Verlag,Berlin,1982 (3)G.D.Carney,L.L.Sprandel,C.W.Kern:Adv.Chem. Phys. 37(19787305 (4)P.Gomez: Doctoral thesis .U.Complutense.Madrld 1987 (5)P.Groner,J.R.Durig:J.Chem.Phys. 66{1977)1856 (6)H.C.Longuet-Higgins: Mol.Phys.66(1963)445 (7)P.R.Bunker: "Molecular Symmetry and Spectroscopy ".Academic Press,New York 1979 (8)Y.G.Smeyers,M.N.Bellido : Int.J.Quantum Chem. 19(1981)553 Y.G.Smeyers,A.Nino :J.Comp.Chem. 8(1987)380