General methods of analysing molecular vibrations

409 Journal of Molecalar @ Elsevier Scientific

GENERAL

Structure, 24 (1975)

Publishing

METHODS

L- S. MAYANTS AND

G.

1 February

OF

409-431

Amsterdam

ANALYSING

- Printed in The Netherlands

MOLECULAR

VIBRATIONS

B. SHALTUPER

The Institute of Eiemento-Organic (USSR)

(Received

Company,

Compounds

of the Acaderu_v of Sciences

of the USSR,

Moscow

1974)

ABSTRACT

An acute need may arise to develop for the complete analysis of molecular vibrations practically convenient general methods based on coordinates other than “chemica1 coordinates”. One reason is the proven proposition: Among independent internal coordinates corresponding to a molecule, there cannot be one which describes a small displacement of a chemical group as a whole relative to a certain molecular pIane, provided this group contains more than two linearly or three non-linearly arranged atoms. Two methods are presented in some detail. The first is based on the use of Xi coordinates which are components of “bond vectors” in the “own” (for each “bond”) Cartesian coordinate system. The second method utilizes X0 coordinates, i.e. the components of atomic displacements in the “own” (for each atom) Cartesian coordinate system. Computation of the torsional vibration of transdichioroethane is given as an example illustrating the first method. The Mayants treatment of the symmetry of a molecule, proceeding from elementary considerations which do not use the group theory explicitly and are valid for any coordinates, is expounded in a somewhat improved version. The peculiarities arising when considering the mean-square amplitude matrix, Z’, in Xi and X0 coordinates are also discussed.

INTRODUCTION

norma

The vibrational equation used to find the frequencies and modes vibrations of molecules can be written in the form 11-33 wq(i)

=

&q(i),

where W is the vibrational

of the

(1) matrix,

li = of is the square of the circular frequency

410 of the ith normal vibration of the molecule, and q”’ is the column matrix of coordinates that determines the mode of this vibration, i.e. relations between coordinates characteristic of the ith normal vibration_ The W matrix is always of the form W=GF,

(2)

where G is the matrix of kinematic coefficients and F is the matrix of force coeEicients (the potential energy matrix)*. The particular form of the matrices G and F, and therefore of the matrix W, depends on the choice of coordinates in which the calculations are made. In order that the computation of vibrations of a molecule may be complete, it is necessary that the modes of all n = 3N--6 (3iV-5 for a linear molecule) intramolecular normal vibrations can be expressed by means of the coordinates chosen. (Here iV is the number of atoms in the molecule.) This analysis is often carried out on the basis of internal coordinates q. “Chemical” coordinates are usually chosen. These are: changes in the chemical bond lengths, changes in the valency angles, and the so-called nonplanar coordinates which characterize the changes in the mutual arrangement of various parts of a molecule. The chemical coordinates have the virtue that the potential energy matrix F, based on them is especially simple in form. Namely, its diagonal elements corresponding to changes in the chemical bond lengths have the largest magnitudes**, while most of the nondiagonal elements, particularly those corresponding to coordinates describing remotely Iocated atoms, can as a general rule be equated to zero, thus considerably reducing the number of parameters in the computation. However, it is not always possible to compIeteIy analyse the moIecular vibrations in chemical coordinates, since a sufficient number of nonplanar coordinates cannot always be successfuIIy introduced. For instance, no one has so far succeeded, as far as we know, in introducing a coordinate describing the small rotation of the methyl group as a whole around the C-C bond, so that a complete computation in chemical coordinates (see the Appendix) is impossibIe even for such a simple moiecule as ethane. Furthermore, there are “noncIassica1 molecules”, i.e. those having an electronic structure which cannot even be approximated by a set of chemical bonds between pairs of atoms. For such molecules the very notion of chemica1 coordinates is devoid of any sense. AnaIysis of various torsional vibrations is becoming even more desirable, as such vibrations can be responsible for certain intramolecular rearrangements in particular, for the transition of a moiecule from one conformation to another. The computation of vibrations for the “nonclassical” moIecuIes, whose number is continuousIy increasing, is also becoming necessary. Moreover, the incom* The notation G and F proposed by Wilson [4-61 is used in this paper instead of T-’ utilized in our Russian publications. ** It is assumed that the elements of this matrix are written in the same units.

and U

411 pleteness of the vibrational analysis precludes, for instance, a complete computation of vibrational intensities or mean-square amplitudes. If the analysis is restricted to internal coordinates, of which n are independent (where n is the number of intramolecular normal vibrations), then the changes in distances between pairs of atoms might be proposed as suitable coordinates for a complete analysis of molecular vibrations. In practice, however, application of these coordinates is also difficult. In particular, it is not always easy even to determine what sets of such coordinates are sufficient in number for the complete computation of vibrations. Thus, an acute need arises to develop practically convenient methods for the complete analysis of the vibrations of any molecule on the basis of some coordinates other than the internal coordinates. We recently proposed one such method [I-/J.In this paper we present it in more detail with an example of the analysis of the torsional vibration of transdichloroethane, and another possible computational method.

1. xd” COORDINATES

The first method is based on the use of coordinates XI similar to those originally introduced to account for the symmetry in computing intensities in the vibration spectra of molecuies [8]. These coordinates are components of the changes of “bond vectors” in the “own” (for each “bond”) Cartesian coordinate systems. Now, let us consider the construction of the coordinates X,” . A “bond” is understood (as distinct from Ref. [S]) to be a segment of a line that joins any two atoms of a molecule. Therefore, it may or may not coincide geometrically with the chemical bond in the case of classical moIecules (in Ref. [S] only chemical bonds were treated as “bonds”). A vector directed from one atom of a molecule to another will be called a “bond vector”. The initial a and ultimate b atoms of the bond vector will also be considered as the initial and the uItimate atoms of the bond itself, respectively. If R, and R, are radius vectors drawn from the origin of some inertial coordinate system to the respective atoms, then the bond vector R is of the form:

R = R,--R, The bond vectors, as is cIear from eqn. (3), do not depend on the choice of the inertial coordinate system and therefore remain unchanged for the translatory motion of a molecule as a whole. For any other type of motion the change Y of each bond vector R is given by r

=

R-R,,,

where R, is the bond vector in the initial equilibrium cule.

(4)

configuration

of the mole-

412 It is always possibIe to choose N- 1 “bonds” for an N-atomic molecule in such a way that they will connect a11 atoms of the moIecuIe. The bond vectors corresponding to them are independent in the sense that none of them can be represented as an algebraic sum of the others. One can somehow number these bonds to obtain a set of bond vectors R(‘) (i = 1,2, . . . , IV- I). The set of changes r”’ (i = 1, 2, . . . , N- 1) of these vectors completely determines the deformation of the molecule and the change of its spatial orientation as a whole. By choosing the “own” coordinate system for each bond, let us represent rci) in the ith Cartesian coordinate system by the coIumn matrix

xgi = (xi,

y; , Zb),

(5)

where XL, & and zb are the respective Cartesian components of Y(~)_ Now, the set of coordinates X$? which, by definition, are the components of rfi) in their “own” coordinate systems can be represented by the column matrix A’,” which in the block form looks like:

x,” = (x,“‘, xi2, . . _,xy-1).

(6)

Computation of vibrations in the Xz coordinates yields, in addition to intramolecular normal vibration frequencies and modes, three (two for Iinear molecules) more zero frequencies and corresponding modes of external vibrations, i.e. smaJ1 rotations of the molecule as a whole. One of the possible choices for the bond vectors and “own” coordinate systems for transdichloroethane is shown in Fig. 1. Since this molecule can be regarded as cJassicaJ, it is expedient to direct the bond vectors along the chemical bonds. For the sake of uniformity the Z-axis of each “own” coordinate system wiI1 always be regarded as directed along the chemical bond (and, therefore, in the case under consideration as well). The X, Y and 2 axes of each “own” coordinate system will aIways form a right-handed system. Due to the choice of the Z-axes of the “own” coordinate systems aIong the corresponding bond vectors, the Z-coordinates, i.e. the Z-components of the vectors rci) (the last elements of the X$” column submatrices), coincide with the corresponding internal coordinates, namely with the changes in the Iengths of the bonds chosen. These coordinates do not change when the molecule as a whole rotates, and therefore do not participate in the description of the change of its spatial orientation. This means that only the x and y coordinates, i.e. the -‘c and y components of the vectors r”), can be non-zero for the modes of small rotations of the molecule. The relation between the internal coordinates Xi and 4 in the matrix form is q =

where Bt

B;Xf, is a matrix

(7) whose elements

are easy to find for those

qi

coordinates

413

Fig. 1. Transdichloroethane. The Cr , Cz, Cl, and Cl2 atoms are in the plane of the drawing; Hz and H3 are below while HI and H4 are above this plane. All the I axes are directed along the chemical bonds from the C atoms. The XI, X, and X5, as well as the Zr , Z., and Z5 axes are in the plane of the drawing. The X2 axis is directed into the HICICZ angle, the X6 axis into the H3C2C1 angle, the X3 axis into the HzCICz angle, and the X7 axis into the H4C2C1 angle. The Y axes are directed in accordance with the right-hand screw rule. The Y, , Y4 and Ys axes denoted short arrows are normal to the plane of the drawing. All angles between bonds are assumed to tetrahedral.

which can be introduced regard for (6) that 4i

=

in the analysis.

Indeed,

it follows from

c B&i) xgi,

where B,q,i, is the ith block of the jth row of the matrix three elements corresponding to the ith bond. On the other hand, qi can be expressed in the form qj =

(7) with due

c
Y(‘)),

&‘, which consists of

(9

where Zy’ are known three-dimensional vectors (for example, see Refs. 3 and 6). From a comparison of (8) and (9) it follows that the elements B,qii> are equal to the components of the vector Zy) in the ith “own” Cartesian coordinate system.

414 For example, if qi is the change of the kth bond length, then in eqn. (9) ll_‘) = 0 for i # k and Zy’ is the unit vector directed along the kth bond vector. Thus, alljth row elements of the Bf matrix are zero, except for the third element of the B0a(jk) bIock, which is equal to 1. To estabhsh a connection between the coordinates Xz and X (the latter are the Cartesian coordinates of atomic displacements, i.e. the components of atomic dispIacements along the axes of a Cartesian coordinate system common to the entire moIecuIe), we shall make use of the X6 coordinates, the components of the rti) vectors in the Cartesian coordinate system common to the entire moIecuIe [S]. The Xi and X, coordinates are related [8] as foliows: X, = AX;,

X,” = AX,

(10)

where A in the block form is A = M&J

(11)

(i,j= 1,2 ,__., N-l): Ai is a third order orthogonal matrix which transforms the vectors from the “own” Cartesian coordinate system for the ith bond into the Cartesian coordinate system common to the entire molecule; 6, is the Kronecker delta. The X, and X coordinates are related as follows: X, = EdX, where E, is a 3N-3 Ea =

(12)

by 3N matrix having the block form (i

ilEijll

=

. ., N-l;

1,2,.

j=

1,2,...,N)

(13)

Ei~ = ES, i.e. a third order unit matrix if the jth atom is the ultimate atom of the ith bond; Eii = -Es if the jth atom is the initial atom of the ith bond, and Eii = 0 in a11 other cases. From (10) and (12) it follows that X,” = AEaX

= IlAIiEiilIX

(i =

1,2,

. . ., N-1;

j = 1,2,

. . ., N)

(14)

LeL us note that the arbitrariness of the choice of an “own” coordinate system for each bond reduces, due to the quite definite choice of the 2 axis, to the arbitrariness in the choice of the angles, say, between the X2 plane and some other pIane passing through the 2 axis. Therefore, for different choices of the “own” coordinate system of the ith bond vector, the z component of any vector is the same, whereas its x and y components undergo corresponding two-dimensional orthogonal transformations.

2. VIBRATIONAL

MATRIX IN x,0

COORDINATES

It follows from (2) that the vibrational W” d = G°Fo I5 6,

matrix

W,” in Xt

coordinates

is (15)

415 where Gz is the matrix of kinematic coefficients and e is the potential energy matrix in these coordinates. Since the kinematic coefficient matrix in X coordinates is a diagonai matrix E of the inverse atomic masses, which in the block form is e = Il%W&I we

1,2, . _ ., N)

(k, j =

(16)

have, taking account of (14) Gdo= Ii~iEijllEllAiEjilI

= IIT~~Ai~ll

(i, j =

1,

2, . - -, N--l)

(17)

Here, rii = aa(i)+Ebci) (ati> and b(i) are the initial and ultimate atoms of the ith bond, respectively); rii = +ectii, (c(g) is the common atom of the ith and jth bonds; the plus sign will be taken if the common atom is either the initial or ultimate for both bonds, and the minus sign will be taken if it is the initial for one bond and the ultimate for the other); rii = 0 if the ith andjth bonds do not have common atoms; A, = XiAj is the orthogonal third order matrix that transforms the components of any vector from the jth to the i&h “own” coordinate system. ft is obvious that for any values of indices, Aii = E,,

A, A,

= A, )

Aji = ~ii

(W

For A, we have: t ejJ

(eix A,

=

Cei2c 3 e.jy)

(eix

9 e.jJ

(eiy,

ejx)

(eiy,

eiy)

(eiy,

ej=>

(ei=,

e_J

(%=

ei,.l

(ei,,

q,)

1 (1% I

where e, and eil. (a = x, y, z) are the unit vectors of the ith and jth “own” coordinate systems, respectively. To construct Gf, it is necessary, from (17), to find the matrices Ai; for all pairs of bonds chosen containing a common atom. This is not difficult to do on the basis of eqn. (19). This problem can be soIved with computers. The matrices for transdichloroethane, if the choice of the “own” coordinate systems is as shown in Fig. 1, are as follows:

A 43

3J3

$J2 0

5-J2

0 -1 0

-3

I’

_-T.3

-&/3

-rfJ2 -&J’S

,

4 A 41

0

=

=

1 $2

90

A42

A,,

=

=

=

-23

-&i2

+J34

0

-h/6 *J2

,

+

+J2

0

-3

0 -1 0

-W2 0

-&/3 % 0

342

-2234 ’

A 46

3

0

-3

A47

=

-sJ2

2;:

&

A,/3 -_gJ2

i

U6

-3

4

!

Y

The rest of the necessary A, matrices are obtained from the above ones by means of eqn. (18). It follows from (17) that Gz is a non-singular matrix. However, Wi

416 has three (or two) zero eigenvalues. matrix and satisfy the conditions: F;X;(X)

Therefore,

Ft,

from (15), must be a singular

= 0

(20)

where X,0’“’ are arbitrary linear combinations of modes of small rotations of a molecule as a whole in the X,” coordinates. In X, coordinates these linear combinations can be taken on the basis of considerations given in Ref. 8 as follows: Xp’

= (0, R’d=‘, -Rb;?,

_ _ .,O,Rb'=', -R',',!, ___,O,Rb;+, --Rb;-I')

A'?'= (-Rb’=!O,R’d,‘, . . . . -Rb'=',O,R$, .... --Rb;-",O,R:N,-I))(21) Xi='= (RL':, -R$,O,...,Rb':,-R$,O,. __,Rb;-I), -R$;+,O) where RE (a = x, y, Z; i = 1, 2, .._,N-I) are the respective components of the ith bond vector for the initial equihbrium configuration of the molecule in the Cartesian coordinate system common to the entire molecule. To go over from X, to Xi, we shall use some “own” coordinate system, say, the 1st “own*’ coordinate system as common for the entire molecule, and the second transform of eqn. (10). After some simple calculations, from eqn. (21) we obtain (taking account of the orthogonality of the A 1i matrices): X,0’“’ = (0, S1,0,

-f-)--a"'s0,._.,-a~~-l'SN__l, a'N-l'S,y_l) 0) xy L,Atari,

X,0"'= (-S*,O,O,.-.)--a~~~~i, n~~~i,O,. ..f-LI~~--L'~~__L,~~~-l'sN_* ,O) (22) X,0'=' =(O,O,O,-_-,-UgSi,UizSi,O, --.') -a~~-"S~-~,a~~-"'~,-,,O) where Si is the equiIibrium length eIements of Ali (i = 2, . . ., N-I). Eqn. (22) distinctly shows linear independent modes of small (i) = Xi”’ = 0 in this case, since a=,

of the ith bond, and a$ (a,p = x, y, z) are that for linear moIecuIes rotations of the molecule a(‘) =)’ = 0.

there are only two as a whoIe. Indeed,

To obtain the conditions that must be satisfied by the Fi matrix explicitly, eqn. (22) must be substituted into eqn. (20). One of the criteria for the Fz matrix to be regular is that these very conditions should be satisfied. Incidently, it is seen from eqn. (22) that these conditions concern only those columns of the F’: matrix which correspond to the x and y coordinates. In the case of transdichloroethane, with the choice of the “own” coordinate system made and for s1 = s5 = 1.624, s2 = s3 = S, = S, = 1; S, = 1.413, eqn. (22) yields: X,0’“’ = (0; 1.624; 0; -0.289; -00.289; -0.833; 0; 0; 0.471; X,o’Y’

=

(-

0.5; 0.289;

0.833; 0; 0.289; 0.833; 0; 0.289; -0.833; 0)

1.624; 0; 0; 0.5; -0.289; 0; 0.5; 0.289; 0; 0.5; -0.289; 0; 1.413; 0; 0)

x0(=) = (0; 0; 0; -0.816; --60.816; 0.471; 0; 0.816;

-0.471; 0; 0.816; -0.471; 0.471; 0; 0; 1.332; 0)

0; 0; -1.624;

0; - 1.624;

0;

0; 0; (23)

0; 0; 0; 0;

417 The form of the Ft matrix depends, first of all, on the elastic properties of the molecule (of course, it is assumed that there are no external elastic forces acting on the molecule), i.e. on the F4 matrix in a certain complete set of internal coordinates. Indeed, using (7), we obtain in view of the invariance of the potential energy: F: =Bpq~;

(24)

foIIows from eqn. (24), by virtue of the structure of the Bf matrix, that the block of the Fz matrix, which corresponds to z coordinates, coincides with that of the F4 matrix, which corresponds to changes in the lengths of the bonds’chosen. It is known ]9] that, in general, the true 8” matrix cannot be obtained from the experimental vibrational frequencies of one molecule, or even of a set of its isotopic species. The Fq matrix can be considered suitable, in practice, for computing the molecular vibrational frequencies, provided it satisfactorily describes the spectra of all isotopic species of the molecule and conforms to the experimental data obtained by other methods. For cIassica1 molecules the elements of such a matrix Fq must also be applicable to a certain degree to some homological series having the initial mofecuie as its term. Eqn. (24) shows that the same considerations also hold for the F: matrix. As a result of numerous computations of vibrations of classical molecules in chemical coordinates, a great amount of data on “force coefficients” has been accumulated that allows one to construct the approximate Fq matrices for computing the vibrations of other molecules. To match the computed frequencies for the new molecules with the experimental frequencies, as a rule, one has to change some old elements in the Fq matrix, and also introduce some new elements in it if necessary. The elements in the Fq matrices can be corrected purposefuIIy by utilizing the partial derivatives of vibrational frequencies with respect to the elements of F4. Computational methods for these derivatives (in any coordinates) have been developed, in particular, by Mayants [IO-13, 31. It is necessary to note that, if from the very beginning the Xt coordinates were used for computing the molecular vibrations, then the data on “force coefficients” could have been accumulated in these coordinates with the same success. Moreover, these data would be more complete as the computation is complete in the Xi coordinates. However, since moIecuIar vibrations were originally more often computed in the chemical coordinates, it is expedient to use eqn. (24) together with the known approximate matrix Fq to construct the matrix Ff, if oniy approximately_ If the molecular vibrations are computed using a number of internaf coordinates less than what is needed for the complete computation of the molecular vibrations, then the Fq matrix obtained as a result of such a computation turns out to be incomplete. Substituting such a matrix in eqn. (24), we obtain an incomplete Pz matrix which does not permit one to carry out the complete computation of molecular vibrations. However, it is sufficient to change only slightly It

418 some elements of this F: matrix which correspond to x and y coordinates (ensuring that the conditions (22) are not violated), and then the complete computation of vibrations is feasible. Then, it only remains to correct the Fi eIements making use of the above considerations. The torsional vibration of transdi~h~oroethane was computed by us in such a way. We used the results of incompIete computation of vibrations for this molecule [14], carried out without the coordinate describing the rotation of two chIoromethyI groups with respect to one another, for constructing the approximate Ff matrix with the help of eqn_ (24). Then only two eIements in the F$)(AU)block obtained from Fz had to be changed to match the computed and the experimentaI frequencies to a sufficient accuracy *. The nonnormalized mode of the torsional vibration in the Xf coordinates chosen, and its frequency ol, were obtained as follows X0(‘) = (0; 1; 0; -0.01; 0;O;O;O;

1;0;0.01;0.58;0.00;

0.58;

-00.00; 0.01; -0.01;0.58;

0.58; 0.00; -O.OO),

mt = Li8cm-”

whilst the experimental value of this frequency [lSJ is w1 = 122 cm-l. If the _Fz matrix is known, then the F4 matrix can be obtained independent coordinates by the formula:

Fq = yF,oY

from it in

(25)

which should satisfy the equality: 5tY = En. To find Y, first, one shoutd construct a non-singufar tzth order matrix P such that the rows of the P&’ matrix are orthonormaf. Then P&‘B~~ = E;, and therefore B’i?‘pP = EtZt - hence d d where Y is the matrix

Y = isgopP

(26)

The construction of the P matrix is a well-known probiem in linear algebra. It can be solved with a computer. The problem of finding this matrix is simpfified by the fact that the rows of the Bz matrix corresponding to bond length changes do not need any transformation.

3. X0

COf3RIXNATES

Another general method for a complete analysis of molecular vibrations is based on the use of the X0 coordinates. These coordinates are the components of atomic displacements in “own” Cartesian coordinate systems for each atom. Let * The torsional vibration of transdichloroethane beIongs to the AU-type symmetry as regards accounting for the symmetry properties, see Section 5.

419 us somehow number all N atoms in an N-atomic of the ath atom is

ra = Ra-RR,,,,

molecule.

The displacement

Y,

(44

where R,,,,is the radius vector directed from the origin of the chosen inertial coordinate system to the ath 7’ 7 in the initial equilibrium configuration of the molecule. By taking an “own coordinate system for each atom, we shall express r. by a column matrix

(5a) where xz, JJ,“, zz are the respective components of r, (to avoid confusion, the atomic displacement components are indicated by subscripts, whereas the components of bond vector changes are indicated by superscripts). Now, the set of the X0 coordinates can be expressed by the column matrix having the block form

In the X0 coordinates, as well as in the X coordinates, the molecular vibration computation gives six (or five) zero frequencies and the corresponding modes of external vibrations, i.e. translations and small rotations of the molecule as a whole, in addition to the frequencies and modes of the intramoIecuIar normal vibrations. For transdichloroethane (Fig. 1) one may take as the “own” coordinate systems for the H and Cl atoms the “own” coordinate systems for the C-H and C-Cl bonds respectively; for the Ctl) atom one may take the “own” coordinate system for the C-C bond, and for the Cctj atom the coordinate system obtained by rotating the latter through 180” around the Cc2) axis normal to the plane of the drawing _ The relation between the internal and X0 coordinates in the matrix form is q = Box0

(74

where B” is a matrix whose elements can be easily found from considerations similar to those described in Section 1. From (7a) and (6a) it follows that 4j = c @a x:,

@a)

where B$ is the ath block of thejth row in the B” matrix, which consists of three eiements and corresponds to the ath atom. Since qi can also be written in the form

whereZia(a= 1,2..., N) are the corresponding three-dimensional vectors, the elements of the Bio, matrix are equal to the components of the Zi, vector in the “own” coordinate system for the ath atom. The X0 and X coordinates are connected by the equalities

420 X = AOX’,

x0 =

jpx

(W

where A0 in the block form is

A0 = IlA:~a,ll

Wa)

Here, AZ is a third order orthogonal matrix which transforms the Cartesian components of any vector from the “own” coordinate system of the ath atom to the coordinate system common to the entire molecule_ The relation between Xz and X0 coordinates can easily be derived from expressions (10a) and (14), which give X-,0 = AE,A"Xo

4.

VIBRATIONAL

MATRIX

U4a)

IN x0

COORDINATES

It follows from (1) and (lOa)-(1 X0 coordinates has the form:

la) that the vibrational

matrix

IV0 in the

W0 = &FO

(Isa)

where E is given by eqn. (16). From the same considerations as in Section 2 it follows that the F” matrix must be singular and satisfy six (or five) conditions: ~OXO’” = 0

(2Oa)

where X0(,) are arbitrary linear rotations of the molecule as a follows from the considerations in Xcoordinates can be chosen

combinations of modes of translations and small whole in X0 coordinates. For small rotations it of Section 2 that the required linear combinations as follows:

X’“’ = (09 &(O>,> ---R,(O)).,

- - -3 O,K(,,=,

-R,(o),,

- - -,O,&(O)=,

-&v(OJ

x0)

* - *, -%7(O):,

0, &(0)X,

- * -9 --&v(O):,

0, Kv(O,,)

=

(-R*(o)=,

0, &(0,x,

(214 x'='

=

@lye

-.h(O)x,

0, - * .> &O)y,

-R7(0),,

0, * - -7 &qO)y,

-&(0)X,

0)

&(o)~ and &col= (a = 1, 2, . - -, IV) are the respective components in the Cartesian coordinate system common to the entire molecule. Now of &7(0, one can again take, for example, as such a coordinate system, the “own” coordinate system for the 1st atom, and then derive the required linear combinations where

JL(o),,

of modes of small rotations in the X0 coordinates from expressions (21a) by using the second equation of (lOa) together with eqn. (lla). However, the genera1 expressions obtained in this case are much more cumbersome than the expression (22); therefore we do not present them here.

421 The linear combinations as follows:

of translations

in the X coordinates

can be chosen

x$=) = (1, 0, 0, 1, 0, 0, . - ., 1, 0, 0, . - -, 1, 0,O) l,O,O,

l,o ,‘_.,

O,l,O ,.-.)

xy,

=(O,

x$)

= (O,O, l,O, 0, 1, . . -, o,o,

O,l,O)

1, . - -, o,o,

(21b) 1)

If the expressions (21b) are regarded as written in the “own” system for the 1st atom, then in X0 coordinates they take the form:

coordinate

where a$@ (CX,j3 = X, y, z ) are the elements of the A:, matrix (a = 1, 2, . . . , N) transforming the components of any vector from the uth atom coordinate system to the 1st atom coordinate system. One of the criteria for the F” matrix to be true is that the conditions (20a) must be fulfilled. Therefore, despite the fact that to construct the Go = E matrix one does not need to find the AZ matrices, nevertheless they are needed since they are contained in explicit expressions for the conditions (20a). If the F, matrix is known in the complete system of internal coordinates, then the I;’ matrix can be obtained by the equation: F” = B°F, B”

(244

For the incomplete matrix F4 (a result of incomplete computation due to the fact that some internal coordinates are neglected) the F” matrix obtained from it by the formula (24a) is also incomplete. Its elements are corrected in the same way as for the Fi matrix (taking account of the conditions (20a)). If the F” matrix is known, then the F4 matrix in independent coordinates can be obtained from it by the formula Fq = P°FoYo,

(254

yo --

(26a)

where BOjTJpO

and the P” matrix has similar significance

5.

SYMMETRY

as P in eqn. (26).

CONSIDERATIONS

The most rational

way to simplify the computation

of the normal molecular

422 vibration is based on elementary considerations which do not explicitly use the group theory proposed and developed by Mayants [2, 16,3]. The essence of these considerations valid for any coordinates is as follows_ When account is taken of symmetry, the elements of column matrices describing the normal vibrations for each symmetry type are reIated in a definite way due to the fact that each symmetry operation causes a quite definite transformation of these column matrices. Therefore, for vibrations of each symmetry type, the number of independent unknowns in the system of equations (1) is less than the number of these equations. AI1 coordinates are partitioned with respect to each symmetry element into sets such that the symmetry operations corresponding to this symmetry element transform the coordinates of each set into linear combinations of the same coordinates. By comparing the coordinate transformations for the vibrations of a given symmetry type performed by various symmetry operations we can easily find the mutually transforming sets of coordinates as well as the relations between these coordinates for vibrations of this symmetry type. By choosing some coordinates (in the required number) as the independent unknowns, and then expressing all other coordinates in terms of the chosen coordinates and substituting all of them into the set of equations (I), we obtain a redundant compatible set of equations with the required number of unknowns. To solve this set of equations, it is sufficient to choose as independent equations those having the same indices as the coordinates chosen as the independent unknowns. The square matrix IV’“’ of this new set of independent equations for cc-type symmetry vibrations is connected with the W matrix of eqn. (1) by the relation: WC’“’ = C’“‘I,+”

(27)

where C’=) is a rectangular matrix defined for the E-type symmetry vibrations. Rows of C’“’ correspond to coordinates in which the W matrix is written and its columns correspond to the coordinates chosen as the independent unknowns, which represent the sets of mutually transforming coordinates; the relations between elements of each column of the C’“’ matrix are the same as the relations between the respective coordinates in the a-type symmetry vibrations. The columns of the C’“’ matrix can be considered to be orthonormal (if they are not, then they can be made orthonormal by multiplying C@) on the right by an appropriate nonsingular matrix D’“‘). Therefore, multiplying (27) on the left by c@), we obtain: II/@’ = C’(a)I,j/C@)

(28)

Eqn. (28) shows how one can obtain the IV’“’ matrix having the eigenvaIues and eigencolumns which determine the frequencies and modes of the cc-type symmetry vibrations from the initial W matrix. However, it is simpler to obtain it using the equation w@“’ = G’“‘F’“’

(29)

423 where C”’ = C(@GC@) F”’

(30)

= e(‘“)FC@)

(31)

The validity of eqns. (29)-(31) follows from eqn. (2) and from the fact that the G and F matrices have the same symmetry properties as the W matrix. From the eigencolumns of the W’“’ matrix, which describe modes of the cr-symmetry normal vibrations in the “representatives” of the respective sets of coordinates (if then these representatives coincide with the the columns in C@) are orthonormal, so-called symmetry coordinates), one may obtain the corresponding eigencolumns of the W matrix, which determine the modes of the same’vibrations in the initial coordinates, by simply muhiplying the eigencolumns of the IV@) matrix on the left by the Cc3’ matrix. Thus, to construct the I@“’ matrix, one should find the C(@ matrices. To this end, some basic symmetry elements are selected from all the symmetry elements of a molecule, i.e. those by means of which we can obtain all the symmetry operations allowed by the molecule. For each of these symmetry elements the sets of equivalent coordinates corresponding to it are constructed, i.e. the sets of coordinates that transform into each other when performing the symmetry operations induced by this symmetry element. The equivalent coordinates, as mentioned above, are linear combinations of the initial coordinates in the general case. For thep order symmetry element aP, each set of the equivalent coordinates contains p coordinates (there may be repeated ones among them) and is so constructed that the kth equivalent coordinate is obtained by the ai symmetry operation from some coordinate chosen as the first. The relations between the coordinates equivalent relative to the symmetry element ap in vibrations of different symmetry types with respect to aP can be written as nonnormalized column matrices CcI,, the elements Ccrji of which are of the form: C (i)i = exp t0W(j

- 111

(32)

The vaIue 2 = 0 corresponds to symmetric vibrations, and I = &p (for even p) corresponds to antisymmetric vibrations with respect to the symmetry element up. AI1 other values of I correspond to vibrations degenerate with respect to this symmetry element. Since for vibrations degenerate with respect to ap,the elements of Ccl, are complex numbers, it is more convenient to use instead of the column matrices C,,,. for these vibrations the nonorthonormalized two-column matrices c trt0 wrth real elements CtrtlIki: c

(k

r&tr)kJ

Jointly

considering

the relations

=

1,2,

f . ‘, p;

between the coordinates

j

=

b2)

(33)

of different sets

424 equivaIent reIative to the chosen symmetry elements for vibrations of different symmetry types, we can construct the columns of C(=) matrices (to an accuracy of the normalization factor). In order to use them in eqns. (28), (30) and (311, they must, of course, be made orthonormai beforehand. Lf none of the columns of the C,,,, matrix separateIy satisfies the symmetry conditions with respect to other symmetry eIements following from the vibration symmetry type, then one shoufd attempt to satisfy them by an appropriate linear combination of these columns. If for vibrations of a certain symmetry type the symmetry conditions with respect to different symmetry eIements turn out to be incompatible, then the vibrations of this symmetry type are not possibfe for the molecule under consideration*. Symmetry imposes certain Iimitations on the form of the matrices G, Fand W. Indeed, each symmetry operation s causes the transformation of the column of coordinates by the appropriate matrix S corresponding to a given choice of coordinates. This matrix must commute with the matrices G, Fand IV. Therefore, for exampIe, the following equalities must hold for the matrix F FS = SF

(34)

and consequently F =

S-"FS

(35)

for any symmetry operation s and for any choice of coordinates. The set of equations (35) for all symmetry operations determines the relations between the elements of the F matrix. Due to the symmetry, the same refations also exist for the elements of the G and W matrices. One more criterion for the correctness of composition of the F (and G) matrix is the fulfilment of the relations (35). Of course, it is sufficient to check whether these equations are fulfilled for symmetry operations effected by the symmetry elements chosen as basic. In general, we can choose any set of coordinates for computing the normal vibrations of molecuIes. However, it is advisable to select a set which accounts for the symmetry of the moIecuIe in the best possible way, i.e. the elements of the transformation matrices S acting on the column of coordinates for various symmetry operations, and, consequently, also the relations between eIements of the F matrix that correspond to these symmetry operations, should be as simple as possible. Redundant (dependent) coordinates are usually introduced additionally to account for the symmetry of the molecule in the best possibIe way. (For an account of redundant coordinates see, for instance, refs. 17, 3 and 18.) Transdichloroethane has Czir symmetry. As the basic symmetry elements, * For a more detailed account of the symmetry, incIuding the derivation of eqns. (28)-(33), see refs. 16 and 3, which also give a table containing C,, and C,,,, for symmetry ekments from 2nd to 6th orders inclusive.

matrices in a numerical form

425 let us take the rotationaf symmetry axis C, passing through the centre of the C-C bond normai. to the piane of the drawing and the a,, symmetry plane coinciding with the plane of the drawing (Fig. 1). Table I shows the sets of Xz coordinates equivalent with respect to these symmetry elements. The numbering of coordinates corresponds to that of the bonds. TABLE

1

Symmetry element

C2

Gh

Set of equivafent coordinafes

A few words need to be said about the transformation of the x,, y4 and z1 coordinates in carrying out the C: symmetry operation, since difficuities may be encountered due to the fact that the C-C bond vector does not transform into itself. One shouId remember that the coordinates undergo transformations in the symmetry operations provided that the positions of the atoms and “own” coordinate systems chosen remain unchanged, i.e. the same as for the equilibrium configuration. In carrying out the Cl symmetry operation the following transformations take place: x& rS -x& , y& SJ$, , z& S -z& , where 01:~(a = x, y, z; i = 1, 2) are the respective ‘“own” dispraeement components of the C, and Cz atoms. Hence: x&-x:, -xI:-fx&yf:--+ --yz and zg -S 2:. On comparing the relations between the coordinates as given in Table 1 with the conditions imposed on the vibrations of different symmetry types, we obtain the respective C’“’ matrices. In particuiar, the C(AU)matrix after orthonormalization takes the form of eqn. (35a)

426 0

0

l/J2

0 0

0 0 0 0 0 0

0

0 0 0

0 0

0 0 ii

1- 0 0 -t -f*

0”

0

0

*

0

0

-3

0 0 0

0 0 0 0

0

0

VJz

0 0 -3 0

0

0

0

0

0

0

0

0 0

0 0 0 0 0 _t

0 0 0 0

0 --$

0 0

0

2% 0

0

0

0

+

0

0

0

0

ik

0

The number of columns in the Cfa) matrices is evidently equal to the number of cc-type symmetry normal vibrations. Out of 21 normal vibrations of transdichloroethane, obtained in computing in Xg coordinates, there are 18 intramofecuhrr vibrations, white 3 vibrations correspond to small rotations of the molecufe as a whole. From the expression (23) for the modes of smatl rotations it follows that Xp”’ and X,0’=’ belong to Bg symmetry type, while X,0”’ belongs to A,. Therefore, three conditions (20) are reduced by symmetry to two conditions for FiCBa) and one condition for F, o(Ad . Since these conditions bear no relation to matrix (which determine the frequency and the mode the eIements of the F,OCAU) of the torsional vibration of transdi~hIoroethane), these elements can be varred without any reference to the conditions (20). The initial matrix .F$$J) obtained by the formuta (30) from the initial matrix Fz which, in its turn, has been obtained by the formula (24) from the incomplete matrix E4 (being the resutt of incomplete vibration computation), is as foilows:

JqGy1

0.580 0.332 Ezz -0.666 0.453

As a result of variations

0.332 1.025 -0.381 -0.050

- 0.666 -0.381 0.765 -0.520

of two etements

0.453 - 0.050 - 0.520 8.250

I I

(W

I

on which the frequency

of torsionali

427 vibration

depends to a large extent, this matrix assumes its final form* 0.645 0.332 = / _ 0.615 fl 0.453

0.332 1.025 -0.381 - 0.050

1

egnu:

-0.615 -0.381 0.765 - 0.520

0.453 ‘1 -0.050 ’ -0.520 8.250

(359

Of course, one should not necessarily regard this matrix as physically significant, or even suitable for practical purposes. In order to draw the last conclusion, one has to use this matrix for computing the frequencies of all isotopic species of transdichloroethane and to verify whether the calculated and experimental values are in sufhcientiy good agreement with each other. However, the mode of the torsional vibration which is calculated with this matrix is highly characteristic for the y coordinates of the C-C1 and C-H bonds, and cannot change under reasonable variations of the elements of the FzcAU’ matrix. Thus the situation cannot change, for the frequency of this vibration depends mainly on those eIements of the Fg”‘“’ matrix which are varied. Table 1 can also be used to obtain easily the matrices which transform the A’: column matrix in carrying out the C$ and o,* symmetry operations. These matrices in the block form are as follows: /j 0 !,O

S=

0 0

0 0

10 0 0 = j,0 0 0 E,O 0 0 OE,O OE,O

0 0

E3 0

0

0 11

OE,j 0 0E30 / E; 0 0 0 / 0 0 0 0 1 0 0 0 0 0 0 0 il

E,O 0 0 0 0 OE,O 0 OE,O 0 0 0 0 OE,O 0 0 0 OE,O 0 0 0 0 0 OOOOOE,Ol

0 0 0 0

0' 01,/I 01 0 0

(354

(353

OE,

where

E;=

1 0 o! O-10 0 01

(36f 1

* The frequency and mode of the torsional vibration of the molecule given in Section 2 were obtained from this matrix. The values of the force coefficients are expressed in lo6 cms2.

428 Obviously, S(G)-

the equalities

= S(G)

7

SWfI)-’ = S(Wa)

hold true for the inverse matrices. S’cZ’F,oS’c”

= Fz,

Hence, the relations

‘+z,~;S(~‘Z,

(35) take the form

= FO d

(36)

Substituting the S”” and SCah) matrices in their explicit forms into eqn. (36), we obtain the reIations which must be satisfied by the elements of the Fi matrix owing to the symmetry of the transdichloroethane molecule. The use of the X0 coordinates instead of Xi coordinates is less convenient for transdichloroethane, since the symmetry can be taken into account with the same simplicity for both Xz and X0 coordinates, whilst a11 other stages of computation, including the treatment of reiations of the form (20), are simpIer in Xj coordinates.

6. THE MEAN-SQUARE AMPLITUDE MATRIX z IN x,” AND x0

COORDINATES

The mean-square amplitude matrix [19] Z is computed in the X,” and X0 coordinates by the same formulae as in any other coordinates. However, to avoid confusion, one should remember that, as required by the very meaning of the mean-square amplitudes, the summation is to be carried out in these formulae only for those indices which correspond to intramolecular normal vibrations (of course, it is assumed that all these vibrations have nonzero frequencies). Therefore, the Z matrices in Xi and X0 coordinates are singular. Since the W and F matrices in these coordinates are also singular, not a11 the expressions given in refs. 20 and 21 are valid”. Only the foIlowing expressions are valid (for the sake of simplicity we shall omit the indices denoting the coordinates in which they are written): ZF = *rZW* cth (ri W+/2kT) ZFXG-1

= $A2 cth’ (ri W+/2kT)

= CG-‘ZF

ZoFZoG-’

= CoG?ZoF

= $h2

E+sjl(-lY-l

az

-

dT -

L

-CFCkT2

(37)

- Ii2 4kT2

BsA2S

(2s)!(kT)2S G

(38) (39)

Ws)

(40) (41)

l TTheZG-’ matrix has the same eigencolumns as the W matrix, and therefore it commutes with any function of the latter. However, it is evidently not a function of W, since negative degrees of

W do not exist.

429 The relations internal coordinates

7. CONCLUDING

between the 2 matrices in X,” and X0 coordinates q, by virtue of eqns. (7) and (7a), are as follows:

and in the

REMARKS

Of the two proposed methods for the complete computation of molecular vibrations in Xz and X0 coordinates, in most cases preference must be given to the first method, primarily because of the greater number and greater complication of the conditions (20a) which must be satisfied by the elements of the F” matrix. However, the second method can compete with the first in certain cases, e.g. for very simple symmetric molecules (such as cyclopropane), owing to the simpler treatment of the symmetry. To compute the intensities in the molecular vibration spectra, the first method is also more convenient, since this computation can be carried out directly in Xf coordinates. The question whether it is possible to transfer the elements of Fi and F” matrices for identical chemical groups from one molecule into another can be clarified only after a large amount of computed data has been accumulated. At present, we can only assert that for the classical molecules, Ft elements which correspond to z coordinates can be transferred to the same degree as the force coeffcients corresponding to chemical bond length changes (provided, of course, the bond vectors are chosen along the chemical bonds).

APPENDIX

Some authors have reported complete molecular vibration computations where internal coordinates were used that were believed to describe small torsional displacements of methyl or other chemical groups as a whole around certain molecular axes. However, these authors were mistaken. Some linear combinations of changes in angles between certain molecular planes were taken by them as “torsional” coordinates. But a definite value of such a “torsional” coordinate does not determine uniquely every angle change in the combination. Therefore, “torsional” coordinates of such a kind have, in fact, neither chemical nor geometrical meaning. We have proved, here, the proposition that: Among independent internal coordinates corresponding to a molecule there cannot be one which describes a small dispiacement of a chemical group as a whole relative to a certain molecular plane, provided this group contains more than two linearly or three nonlinearly arranged atoms. This proposition, we think, is of great importance not only because it con-

430 firms the incorrectness of “torsional” coordinates mentioned above, but also because it should prevent any attempts at designing independent internal coordinates of such a kind. The proof of this proposition is as fohows. One internal coordinate corresponds to a chemical group of two atoms. When one more atom is added to this chemical group then two possibilities can occur: (a) the chemical group obtained consists of three atoms linearly arranged; (b) it consists of three nonhnearly arranged atoms. Four and three independent internal coordinates correspond to these chemicaI groups in cases (a) and (b) respectively. Every additional atom yieIds three more independent internal coordinates in both cases (in case (a), provided the linearity remains). Therefore, the number of independent internai coordinates corresponding to an iv-atomic chemical group can be written as 3(N-2)+ 1, if this group is linear, and as 3(N-3)+3, if it is nonIinear. Hence, there are at least N- 2 atoms in a linear and N- 3 atoms in a nonlinear chemical group such that displacement of each of them is invoIved in expressions (9) at least for three independent internal coordinates corresponding to this chemical group. The subsequent reasoning is the same for both cases. Let us assume that among the independent internal coordinates corresponding to the molecule under consideration there is a coordinate qd which describes a small displacement of the chemical group CG as a whole relative to the molecular pIane G, the CG group containing the number of atoms required by the proposition. The expression (9) for qd must involve displacements of ail atoms of the CG group and displacements of at least two more atoms in the moIecuIar plane CJ as well. The mutual independence of the coordinate qd and a11 internal coordinates corresponding to the CG group means that all these coordinates can take any values independent of one another; in particular, qd can take any nonzero value, while all other coordinates vanish. Now, let us choose one of the atoms in the CG group, say the atom g, whose displacement is involved in expressions (9) for at least three independent internal coordinates qgci, (i = I, 2, . ._, k; k & 3) corresponding to this group. Further, let the displacements of all atoms of the molecule, except for that of atom g, be equal to zero. Then for any displacement of the atom g resulting in nonzero value of qd, the vaIue of at least one of the coordinates qg(i) should aIso be nonzero, thus contradicting the initial assumption, and proving the proposition.

REFERENCES 1 L. S. Mayants, Doki. Akad. Nauk SSSR, 32 (1941) 120_ 2 L. S. Mayants, Theory of Characteristic Frequencies and its Applications, Doctorate thesis (in Russian), FIAN, 1947 [Trudy FIAN, 5( 1950)]. 3 L. S. Mayants, Theory and Calculation of Molecular Vibrations (in Russian), Jzd. Akad. Nauk

SSSR, 1960. 4 E. B. Wislon Jr., J. Chem. Phys.,

7 (1939)

1047.

431 5 E. B. Wilson Jr., J. Gem. Phys., 9 (1941) 76. E. B. Wilson Jr., J. C. Decius and P. C. Cross, Molec&r oibrations, McGraw-Hill, New York, 1955. 7 L. S. Mayants and G. B. Shaltuper, Dokl. Akad. Nauk. SSSR, 206 (1972) 657. 8 L. S. Mayants and B. S. Averbukh, Theory and Calchtion of Intensities in Vibration Spectra of Molecules (in Russian), Izd. Nauka, 1971. 9 B. S. Averbukh, L. S. Mayants and G. B. Shaltuper, f. Mol. Spectrosc., 30 (1969) 310. 10 L. S. Mayants, Dokl. Akad. Nauk SSSR, 50 (1945) 121. 11 L. S. Mayants, Opt. i. Spektr., 5 (1958) 378. 12 L. S. Mayants, Opt. i Spektr., 8 (1960) 199. 13 L. S. Mayants, Dokl. Akad. Narrk. SSSR, 131 (1960) 51. 14 M. V. Volkenstein, M. A. Eljashevitch and B. I. Stepanov, Kolebaniya Moiekul, Vol. 1, Moscow and Leningrad, 1949. 15 I. Ichishima, H. Kamiyama, T. Shimanouchi and S. Mizuchima,‘J. C/rem. Phys., 29 (1958) li90. 16 L. S. Mayants, Zlr. Eksp. Teor. Fix, 25 (1953) 393. 17 L. S. Mayants, Dokl. Akad. Natrk SSSR, 89 (1953) 423. 18 L. S. Mayants, Opt. i Spektr., 16 (1964) 753. 19 S. J. Cyvin, Molecular Vibrations and Mean-Square Amplitudes, Universitetsforlaget, Oslo, 1968. 20 L. S. Mayants, Dokl. Aknd- Nauk SSSR, 202 (1972) 124. 21 L. S. Mayants and S. J. Cyvin, f. hfof. Strrrct., 17 (1973) 1. 6