The normal coordinates of nitromethane-like molecules

JOURNAL OF MOLECULaR SPECTROSCOPY31, 378-387

The

Normal

Coordinates J. W.

The Chemical

of Nitromethane-Like

FLEMING’

Laboratory,

(1969)

AND C.

N.

The University,

Molecules

BANWELL

Brighton,

England

Sussex,

The form of the vibrational kinet,ic and potential energy matrices for molecules with strllctures similar to nitromethane is discussed, and the kinetic energy matrix elements tabulated. Symmetry aspects of the vibrational problem are discllssed in terms of the group defined by Longuet-Higgins. The possible forms of the normal coordinates are derived, and their relevance to the associated internal rotation problem indicated. There

has been

ordinates

much

of molecules

(1, 2). In this paper, symmetric of all with

Molecular

Reference

inertial To

The

describe

nuclear

axes, which

the vibrational

motions

vectors;

the displacements

aligned

parallel

in which

a

framework

; to

shall deal

first

it proves

cow

motions

configurations

superimposed of the complete

One set comprises

also be chosen

the

to be the principal

this set is fixed in the frame

part

of the

fixed in the top part of the molecule.

of the molecule,

we consider

from the equilibrium

to the frame,

follows

top equilibrium

We

of these molecules,

that is, for atoms comprising

n-e expect the nuclear

rotation

may

molecule;

be a useful procedure

rotation),

motions

second set is a local system

of the molecule,

molecules.

co-

rotation

type of molecule.

of the nuclear

reference

displacement

axis systems

of molecules molecular

two sets of axes fixed in the molecule.

axes of the complete

molecule.

the normal

free, internal

Axes

to introduce

molecular

coordinates to a planar

and toluene

nitromethane

concerning

free, or nearly

the normal

the nitromethane

the simpler

literature

exhibit

of atoms is bonded

In any consideration main

in recent

may

we consider

top grouping

such a class belong

venient

interest

which

initially

the frame,

position

local

or top part

are measured

or top axes respectively.

That

in

this will

from the fact that, for any given vibrationalmode, to involve

small displacements

(arbitrary

on which

with

respect

from the frame

to the internal

will be the large amplitude

top configuration

about

the bond

motion

joining

and

angle

of

of internal

this group

to

the frame. The separation be considered

of the vibrational

from a viewpoint

and internal

rotational

similar to that adopted

1Present address: National Physical

Laboratory, 378

modes of motion

may

in the Born-Oppenheimer

Teddington,

Middlesex,

England.

NITItOMETH4NJ.CLIKE

MOLECULES

:379

separation of electronic and nuclear motions (3). Since the period of internal rotation will be much greater than that of a vibration, we can consider the internal angle to be effectively fixed during the course of one vibration. Thus the analog of the BorrlPOppenheimer approximation would be to determine the vibrational motion for a series of fixed internal orientations. One would therefore determine vibrational eigcnfunctions which depend parametrically on the internal angle, and rotorsional (or torsional) functions which depend on the vibrational state.

111 terms of local nuclear displacement vectors d, (for frame atom i) and d,’ (for toI, atom i), the vibrational kinetic energy is simply:

\\-herec’ and c’ indicate a sum over all frame and all top atoms respectively. The Wilson I’(; method of determining the normal modes of vibration uses internal coordinates rather than Cartesian displacement coordinates, and the kinetic energ!’ is expressed as folio\\-sin terms of the inverse kinetic energy matrix: L’Tvi,, = S’g-‘i

(2)

s represents a column vector of internal coordinates, and the sl, are the Wilson st vectors for the internal coordinates chosen. The q matrix defined in this manner u-ill contain elements which are functions of the internal angle LY,since sL vectors for internal coordinates in the top and frame parts of the molecule are defined in terms of the two axis systems which are, in general, rotated with respect to ~~~1~ other. In order to obtain a clear zeroth order separation of the vibrational and internal rotational motions, we note that, by a suitable choice of internal symmetry coordinates, it is possible to obtain the kinetic energy matrix in a form completely independent of 01,and furthermore in a form bvhich has been reduced to blockdiagonal form to the maximum possible extent. The method of obtaining these symmetq coordinates is essentially as described by Crawford and Wilson (4) ; such a set for nitromethane-type molecules is given in Table I, and the internal coordinates used are depicted in Fig. 1. The kinetic energy matrix elements for nitromethane appropriate to this set of symmetry coordinates are tabulated in Table II (assuming angles of 109” 24’ and 120” at the methyl and nitro groups, respectively ). It is immediatel\- clear from the form of this matrix that there can be no exact vibrat,ional degcneracies in such a yvstem, quite irrespective of the form assumed by hhc force constant matrix.

380

FLEMING

AND BANWELL TABLE I

THE

VIBRATIONAL

FORNITROMETHANE

SYMMETRYCOORDINATES

1

1

1

&

= L 46

813= Y ;6

(2Arl -

(2A~3 -

(ZAP1 -

Symmetq

Arz -

Ara) cos (Y -

ACYIZ- Ad

A& -

1 42

cos CI -

A&) cos 01-

(Arg -

p=

;a Y

Sa

Ad

sin O(

(ALU, - AC& sin CY

(A& -

Apa) sin (Y

Considerations

This conclusion is also reached by considering the symmetry species of these coordinates. To classify the vibrational motions of this type of molecule, we use the group of 12 elements and G classes defined by Longuet-Higgins (5), and we denote this group by Gn. Noting that the internal angle a!is defined by the relation: a = Xt -

Xf

(3)

where 0,& xt and 0,@, xf are the three Eulerian angles defining the orientation of the two sets of axes relative to a space-fixed set of axes, and also noting that the

NITROMETHANE-LIKE TABLE THE

G MATRIX

CT,q, s are, respectively

FOR

the C-H,

MOLECULES II NITKOMETHANE

C-N,

N-O bond lengths)

352

FLEMING

AN11 BANWELL

FIG. 1. The internal coordinates. AC measures defined by the -NO, group and the -C-N bond.

the change

in angle between

the plane

axes are fixed in the molecule as shown in Fig. 1, then the transformation properties of a representative set of molecular coordinates under bhe three operations (123), (23)“, and (45),’ which together are sufficient to generate all the group operations, are readily deduced, and given in Table III. Using these results, we find that symmetry coordinates S1 to SE are of species 141’, S6 -& are AIN, and &I -Sl, are Az”, where we have used the same irreducible representation labels as Longuet-Higgins.

The Force Constant Matrix

Although the G matrix may be written in a form completely independent of the internal angle, there is no reason to suppose that this is true of the force constant matrix; indeed, it is evident that force constants representing the interaction of internal coordinates in the two ends of the molecule must be functions of (Y.Thus, in principle, the force constants may belong to symmetry species other than the 2 The notation (pqr) indicates a cyclic interchange of particles labelled p, q, and T, in the sense q -+ p, r + q, p --t T, while an asterisk as in (pq)* implies t.he interchange of p and q with simultaneous inversion of all particles at the center of mass.

NITROMETHANE-LIKE TABLE TRANSFORMATION

PROPERTIES (123)

MOLECULEH III

OF THE

MOLECULAR (23)*

COORDINATES W

totally symmetric species (which is not possibleforrigid molecules, ofcourse), and hence the invariance of the potential energy under all group operations no\\ implies that the triple product of the species of the force constant and the two associated coordinates (assuming a quadratic representation is sufficient) must contain the totally symmetric representation of the molecular point group. P’urthermore, since the force constants must be periodic functions of a! n-ith a period of ?a, we conclude that there may be force constants belonging to any one of the symmetry species in Cl2 . Thus the most general form of the symmetry factored fi matrix for a nitromethane type molecule will be:

This, therefore, must also be the form of the matrix product GF, and to obtain the form of the normal coordinates, we must investigate the type of eigenvectors to which this matrix may give rise.

In order to deduce the possible form that the eigenvectors may assume, w folio\?- similar arguments to those proposed by Hougen (1) for the vibrations of dimethylacetylene. We consider initially the special case when CY= -r/L’, \vhich corresponds to an eclipsed configuration. The effect of the symmetry operation (33 j* is to change 01into r - cy.Thus at (Y = -a/2, the effect of applying (2X )* is to make 01 = 3~/2, which, modulo %T, is again --K/2, and the configuration is unchanged. Therefore the potential energy function must also remain unaltered, which implies that all AZ’ and AzNF matrix elements are zero at t’his eclipsed ron-

384

FLEMING

AND

BANWELL

figuration, and likewise at any eclipsed configuration. Similarly, at the configuration where ar = - %/3, which is a staggered configuration, the effect of applying the operation (123) (45) may be shown to be exactly the same, modulo 2~, as applying (33)*. Now, therefore, as at any staggered configuration all A,” and A,’ F matrix elements must be zero. Consider now the effect of applying the operation (23)* to the matrix product GF as set up for any arbitrary value of the angle cy.If an eigenvector of this matrix is represented initially by

r...1 Vl ... VZ

LJ v3

then after applying (23)*, since the Ap’ and AT” blocks of GF change sign, the corresponding eigenvector of the transformed matrix must be

Vl ...

f

!1 vz ...

-

v3

EO0 z!zoE’o 1

In other words, by applying this operation to a given eigenvector, this eigenvector must be multiplied by the matrix:

L0

0

-E

where E represents the unit matrix of the appropriate dimension for each symmetry block. Now, at CY= -r/2, the GF matrix product is of the form:

A;

1

A,” 0

and consequently,

Al” A;0

0 1

A, 0

I

the eigenvectors must be of either of the two forms:

at

oc=-7r/2

(4)

But, at this configuration, the operation (23)” we have shown to be equivalent to

NITROMETHANE-LIKE

MOLECULE8

:is.i

the identity operation, which must, of course, leave all eigenvectors unchanged. We conclude, therefore, that eigenvectors of the type T1 must be multiplied by the matrix

when (23 ) * is applied at cy = - r/2, and eigenvectors of the type

EOO -0EO 0 -E [ 0

1

Tzby the matrix

also at Q = --a/.). The arguments of Hougen indicate that the eigenvectors carresponding to a given eigenvalue must change continuously as o( is varied. Thus, having chosen the correct matrix transformation for each type of eigenvector, at the special orientation where a! = --9/L?, we know that the effect of this symmetr? operation on the eigenvectors must be represented by the chosen matrix for an! value of CY,or else some of its elements would not vary continuously as cxis allowed to change. In an exactly similar manner, we may consider the operation (13 ) (45 ), \vhich induces the transformation CY+ cy + a/3. When (Y = -%!3 (a staggered con figuration), the eigenvectors may be of either of the k-o forms:

:

and for any arbitrary matrix

00 f[-E 0EO 1

configuration,

the eigenvect,ors must be mult.iplied by t,he

0

OE

when (12.3) (45) is applied. Now, the effect of applying operation (123) (43) a!= -2=/S must be the same as that obtained after applying operation (23 Thus, suppose there is an eigenvector of type Tl at a = -a/2, which becomes the form of a type Tsvector at Q! = -&r/2; under (23 )* it will be multiplied the matrix

:

EO 0 OE 01 [00--E

at )*. of b!-

386

FLEMING

AND

BANWELL

and so, by the argument of continuous behavior as (Yvaries, it must be multiplied by the matrix:

under the influence of the operation (123) (45). Therefore, this eigenvector is composed of elements whose symmetry properties may be written symbolically as:

Al ... tl =

...1

Al"

(6)

A," In this way, Iv-ededuce that there are in general four possible types of column eigenvectors of the GF matrix product. Since the individual elements of each eigenvector are the coefficients in the linear combinations which form the normal coordinates, the symmetry species of the normal coordinates are determined by the direct product of the species of the coefficients and the symmetry coordinates involved. The four types of eigenvectors, and the species of the associated normal coordinates are :

Normal Coordinate Species:

Thus, we see that using three types of symmetry coordinates (Al’, Al",and AzR), it is possible that normal coordinates of four different symmetry species exist. However, if no part of the chosen force field depends on o(, then it is evident that the normal coordinates for a nitromethane type molecule must form the reduced representation 5A1'+ 5AlN+ 4AzN in Glz , and the vibrational eigenvalues would be independent of the angle (Y.On the other hand, if a-dependent force constants are included in the force field for the system, then the vibrational eigenvalues will be functions of a, and the Lu-dependence of the total vibrational energy for a given vibrational state may be taken to define a dynamic potential barrier hindering internal rotation for that state. Hougen has shown that the vibrational eigenvalues must obey a noncrossing rule; in order to conform with this require-

NITROMETH~4NE-LIKE

MOLIKULES

:zs7

ment when oc-dependent force constants are important, eigenvectors of type f4 may exist. Eigenvectors of type fl and tI correlate vibrations which involve .I polarized vibrational motions at the eclipsed configuration with y polarized motions at the staggered configurations, or vice versa. Thus, in the most general case, the symmetry of the normal coordinates may bc regarded as being det,rrmined by the molecular force field. Similar arguments ma?_ be applied to the normal vibrations of toluene type molecules, although hew, we must consider symmetry coordinates belonging to the four nondegenerate irreducible representations of (:I” . The same conclusions will apply; that is the syn1metr.v species of the normal coordinates depends on the molcculer force field. The form of the normal coordinates of these molecules is important when COIP sidering \vhxt effects free, or slightly hindered, internal rotation will have on the irrf’r:wed spectra. This is because Coriolis interactions, the extent of Lvhich depends on the normal coordinates, play an important role irr determining the efYect, of such motions OII t’he spectra (6 1.

RECEIVEI)

.Janu:try

27, 1969