Uniquely defined harmonic force constants in redundant coordinates

Journal of MolecularStructure, 160 (1987) 159-177 Elsevier Science Publishers B.V., Amsterdam -Printed

in The Netherlands

UNIQUELY DEFINED HARMONIC FORCE CONSTANTS IN REDUNDANT COORDINATES

KRZYSZTOF

KUCZERA

Institute of Physics, Polish Academy (Poland)

ofSciences,

Al. Lotnikbw 32946,

02-668 Warsaw

(Received 15 July 1986; in final form 23 December 1986)

ABSTRACT The problem of describing molecular harmonic vibrations in the full dependent set of rectilinear internal valence coordinates is considered. Force constants in dependent coordinates are not uniquely defined; an infinite, continuous multitude of force constant values exists for the description of a given vibrational motion. In this paper coordinate values identically fulfilling the redundancy relations and corresponding force constants are obtained through a projection operation. This allows unique, characteristic values of harmonic force constants to be assigned to all internal valence coordinates. Use of the new parameters, called pure vibrational force constants, is recommended for describing the potential energy of harmonic vibrations, especially in ring molecules. Additionally, properties of traditionally used molecular force fields are analyzed. INTRODUCTION

The GF method is widely used in the description of molecular vibrations of mechanical harmonicity is applied. The basic parameters of the model, the harmonic force constants, are defined in terms of rectilinear internal valence coordinates (IVCs). This may lead to certain interpretational problems in the case when the number of equivalent IVCs is larger than the number of the vibrational degrees of freedom of the system, i.e. when redundancies appear. Two basic methods of description may be used in such a situation: (i) elimination of redundancies, and (ii) retention of the full set of equivalent IVCs. The purpose of this paper is to develop approach (ii) and to compare it with (i). In method (i) the description of the system is restricted to an independent subset of the full set of equivalent IVCs. The force constant matrix lacks invariance, i.e. the element F, is fully defined by the specification of all coordinates, not only i and j [2]. For each choice of the independent subset different values of all force constants may be obtained in the general case. Thus the problem of whether force constants defined in such independent coordinates may be considered to be characteristic of individual IVCs is open to discussion. Additional difficulties arise when the independent coordinates are obtained as non-local linear combinations of IVCs. This [ 11. In this method the approximation

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0 1987 Elsevier Science Publishers B.V.

160

leads to the loss of the simple physicochemical interpretation of the force constants as connected with deformations of local elements of molecular chemical structure (chemical bonds, bond angles, etc.). This simple physicochemical interpretation of force constants is retained in method (ii). However, the treatment of harmonic molecular vibrations in dependent coordinates gives rise to new problems. They include the appearance of a linear term in the potential energy expansion, the introduction of additional force constants, the indeterminacy of the F matrix and difficulties in the derivation of the vibrational eigenvectors and eigenvalues. In previous papers [3, 41 most of these problems have been solved. It has been shown that in dependent coordinates the linear term in the potential energy expansion may be neglected [3] and that the vibrational problem may be solved by the standard method [4]. The analysis of the indeterminacy of the force constants has led to a proposal of a uniquely defined i? matrix in dependent coordinates, the 8’: force field [4]. In ref. 4 the l?$ matrix was introduced on the basis of heuristic arguments. In this paper it is shown that %g, called here the pure vibrational force field, is the only theoretically correct form of the l? matrix in dependent coordinates. The approach presented here is analogous to that employed to eliminate rotational and translational contributions to motions of molecular nuclei [ 51. The analysis of the structure of the extended configuration space X of a vibrating system described by dependent coordinates is undertaken. It is shown that the redundancy relations constrain the system to a subspace of this configuration space, called the pure vibrational subspace X0. In order to restrict the description to physical motion, coordinates must be projected on X0. The form of the projection operator on X0 in the IVC basis R is found and the properties of the projected coordinates R*, called the pure vibrational coordinates, are analyzed. A consequent application of this projection to harmonic force constants leads to the conclusion that the force field l?$ proposed in ref. 4 is the correct form of the I? matrix in dependent coordinates. The results obtained and the methods introduced are used in an analysis of the properties of force constants in independent coordinates. The problem of obtaining harmonic force constant values characteristic of individual IVCs is discussed. As an illustration, the force field for the planar vibrations of benzene expressed in several types of coordinates is given. BASIC DEFINITIONS

In the following discussion a molecule having N vibrational degrees of freedom will be considered. The configurations of the molecular nuclei will be described in two principal coordinate systems. The first system, further called the R coordinates, will consist of the full set of equivalent rectilinear IVCS [ 3, 4, 61 Ri (i = 1, . . . , M). In the general case M > N, and the Ri satisfy p = M - N linear relationships (redundancy conditions) [6] , which can be expressed as

161 M

1

CkiRi=O

k = l,...,p

i=l

or CR=0

(1)

where C = [Cki] (k = 1,. . . ,p; i = 1,. . . , M) is a matrix of constant coefficients and R is a single-column matrix of the elements Ri. Further subscript and matrix notation will be used alternatively or together in order to specify clearly the range of appearing subscripts on the one hand and to briefly present the results on the other. The matrix C can be chosen to have orthonormalized rows, without loss of generality; this will further be assumed. The second coordinate system Si (i = 1, . . . , M) denoted as S, is derived from R by a linear orthogonal transformation S=UR

(2)

chosen so that UN + k,j =

k = l,...,

ckj

S;

l,...,

M

i=

l,...,N k=

=SN+k

where the S’ coordinates (5) we obtain

(4) l,...,p

(5)

are null coordinates

[7], because from eqns. (l)-

k = l,...,p

Sk = 0

(3)

Si (i = 1, . . . , M) may be divided into two subsets

The coordinates Sp= Si

p;j=

@a)

or S’= 0

(6b)

which is equivalent to eqn. (1). From eqns. (2)~-(5) we may obtain S=CR

S”=WR

(7)

where W I[Wij] (i = l,..., N;j= l,..., M) is a matrix built from the first N rows of U. From the orthogonality of fr several useful formulae may be derived. R=U’rS=$rS”+$S’

(3) C CT = &

6+=&,,

(9)

++=i,=W‘+++CTC where i M,

iN,

6,

(10) are Unit matriCeS

of order M, N and p respectively.

162 DEFINITION

OF THE PURE VIBRATIONAL

COORDINATES

R*

The extended configuration space X of our system may now be treated as an M-dimensional real vector space with the standard scalar product. Any vector (configuration) v e X may be expressed in the two coordinate systems introduced in the previous section M

V=

RieR

C

= ;

Sj ef

(11)

i=l

i=l

whereer andef (i= l,... It is easy to find that (efl$)=

, M) are the appropriate basis vectors.

Uij

Also from eqns. (7) and (12) we obtain e%+k = E

Chi eF

(13)

i=l

From the properties of the S coordinates it is easy to see that the configuration space may be represented as a sum of two orthogonal subspaces XOandX’,spannedbythevectorsef fori=l,...,Nandi=N+l,...,M respectively. Thus any vector v e X may be unambiguously decomposed into a sum v = v* + v’

(14)

where v* E X0 and v’ E X’, i.e. v* = :

&‘pef =

i=l

F

RT ey

(15)

i=l

M

M

c

v’ = i=N

Si$= i-1

g Seem+, k

=1

=

1

Rie”

(16)

i=l

Equations (15) and (16) may be treated as definitions of coordinates R* and R’. From eqns. (6), (15) and (16) it follows that the configurations of the system are constrained to belong to the subspace X,, , which shall further be called the subspace of pure vibrations. X’ on the other hand may be called the subspace of redundancies. The redundancy relations (1) require that the system cannot attain positions in X’. The coordinates R* of vectors v E X0 in the basis er (i = 1, . . ., M) will be called pure vibrational coordinates. From a series of transformations we obtain

163

v* =

2

(vIef)eF

2

C

Rj(eFleF)ep

j = 1

E RjUij$ = ; ; E Rj UijiYi,

; j=l

=

i=l

i=1

=

M

N

N

j=l

i=l

j=l

ef

k=l

so that RL$+R

(17)

and similarly R’ = C’ CR

(18)

Eqns. (lo), (17) and (18) give R=

R*+R’

(19)

The pure vibrational corrdinates R* do not constitute a new coordinate system. They represent configurations belonging to the pure vibrational subspace X0 of the configuration space X in the valence coordinate basis ef (i = 1,. . . ,M). R* are the result of the projection of an arbitrary configuration R on the subspace X 0, where the projection operator matrix in the eR basis (eqns. (lo), (17)-(19)) is ~=~T~T~M-CTC

(20)

and CT 6 is the matrix of the projection operator on the subspace of redundancies X’. As may be seen from eqn. (19) the R coordinates generally describe configurations having non-vanishing components in the subspace X’. This is because (e: 1ef ) for i = 1, . . . ,M, j = N + 1,. . . ,A4 is generally different from zero (see eqn. (12)). To ensure that molecular configurations represented in the valence coordinates correspond to physically allowed positions of nuclei, the projection operation (17) must be used. The natural coordinates for the description of vibrations are So. They form a set of independent coordinates orthogonal to the null coordinates S’; their number is equal to the number of vibrational degrees of freedom of the system. Their practical application is limited however. Generally the So are non-local linear combinations of IVCs (especially in the case of cyclic compounds). Also, the So coordinates are not uniquely defined; all coordinate systems obtained from So by orthogonal transformations are equivalent. PROPERTIES

OF PURE VIBRATIONAL

COORDINATES

The most important properties of the R* coordinates are given below.

164

Rectilinearity R* are rectilinear IVCs. This assures the simple form of the molecular vibrational problem in the R* coordinates [6] , the elimination of rotations and translations of the system from the description and a clear physicochemical interpretation of the molecular deformations and force constants (as associated with changes of bond lengths, bond angles, etc.). Dependence The configurations described by R* and by R coordinates (eqn. (17)) satisfying (1) are identical. For the R* coordinates the redundancy conditions (1) are identically fulfilled, for from eqn. (17) and the orthogonality of U we get CR*=

C(WTWR)=

(CWT)WR=

0

Equivalence to independent coordinates So The R* coordinates are equivalent to the independent So coordinates from eqn. (7). This follows from the existence of direct transformation R” = WT So

(21)

which is obtained from eqns. (7) and (17), and an inverse one So = WR*

(22)

This last formula is the result of eqns. (7), (9) and (17), for S”=WR~&WR=WWTWR~WRR* Invariance The values of the R* coordinates do not depend on the choice of the bases in the subspace of pure vibrations X0 or the subspace of redundancies X’. Let us take a non-singular transformation in X0 S’ = V SO The basic relations (7) and (21) coordinates as

may now be expressed through the S’

S’=VS”=VWR

This_ shows the R* coordinates and the form of the projection operator WT W do not depend on the choice of the basis in X0. Similarly it may be shown that they do not depend on the basis in X’.

165 DEFINITION

OF THE PURE VIBRATIONAL

FORCE

FIELD i;”

In describing molecular vibrations in the extended configuration space X care must be taken to take into account only configurations from the subspace of pure vibrations X0, corresponding to real physical motion. It was found that in the basis of rectilinear dependent IVCs ep (i = 1,. . . ,M) the correct coordinate values are given by the pure vibrational coordinates R* (ean. (17)). This condition may now be used to find the correct form I?‘; of the force constant matrix P in dependent IVCs. The desired I?$ matrix is +=+I$

*~=fi~G@~fiiirTfi

(23)

where I$$ and @a are force fields of the system expressed in the independent So coordinates (eqns. (7), (21)) and in the R coordinates respectively. The proof oteqn. (23) is given in the Appendix. The Fg matrix of eqn. (23), which may be called the pure vibrational force field, is the same as the canonic force field introduced in ref. 4. In that work it was shown that the vibrational eigenvalues and eigenvectors do not depend on the values of force constants corresponding to null coordinates. A whole family of physically equivalent force fields was found to exist. The choice of F$ as the representation of the potential energy in dependent coordinates was based on the heuristic argument of simplicity, since @$ corresponds to the vanishing of the indeterminate force constants. Now it has been shown that this choice is also based on physical arguments: the real motion of the system must be constrained to the pure vibrational subspace X0 . PROPERTIES

OF @;

The force field @a is uniquely defined by eqn. (23) and corresponds to vanishing of the force constants connected with null coordinates. I?% describes the same vibrations of the system as the force field I?: in independent coordinates from _eqn. (23)_[4]. In the Appendix it is shown that each of the two force fields Fg and F$ is necessary and sufficient to obtain the other one (eqns. (A3), (A4)). Thus, the force constants (Fg )ij are not independent; they satisfy certain linear relationships [4]. The appearance of the projection operator matrix 6 = L$‘L$ in eqn. (23) assures that @;ir is independent of the choice of bases in the subspaces X0 and X’ of the extended configuration space X. Thanks to the fact that R* are IVCs, the force_constants (I$ )ij have a simple physicochemical interpretation. Since the F matrix does not have the invariance property (see Introduction), the force constants (l?f )ii depend on the specification of the full set of coordinates, not only i and j. However, the set R* contains all equivalent IVCs and is fixed once and for all. For this reason the element (Fg )ij may be treated as characteristic of

166

coordinates i and j. In the pure vibrational force field F$ unique values are assigned to force constants associated with all the coordinates from the full set of equivalent IVCs. The properties presented above indicate that representation of the potential energy in Fg form is highly advantageous. It has been argued [4] that the parameters (Fg )ij should be used for comparing force fields obtained in different coordinate systems or by different methods, building model force fields and transfering force constants to similar compounds. All the specified properties of pure vibrational force fields are also retained in the case when the initial coordinates R are not individual IVCs but their local linear combinations. This preserves the rectilinearity of the coordinates (eqn. (1) is the fulfilled) and the simple physicochemical interpretation due to locality. An example of such coordinates are the standard CH planar deformations for benzene [S] (see also Example: The Planar Pure Vibrational Force Field of Benzene). In the next Section the concepts introduced in the preceeding part of the paper will be used in order to study the properties of force fields defined in independent coordinates. HARMONIC

FORCE

CONSTANTS

IN INDEPENDENT

COORDINATES

The properties of molecular force fields expressed in independent coordinates will now be analyzed. Two cases will be discussed, the general force field l?: and the special representation I@. The $

matrix

This is the force field in independent coordinates So (eqn. (7)). From the theoretical point of view Fg is the most proper form for the representation of the potential energy of harmonic vibrations of a molecule for which the full set of IVCs is dependent. The So coordinates are independent, they span the subspace of the pure vibrations and are orthogonal to the null coordinates. The F”, matrix may be represented in the full set of dependent coordinates, giving as the corresponding field I?, such that [4] tFS )ij=

(E

tFS)ij =

O

)ij

i,j=

l,...,N

i>Norj>N

i.e., the use of I?“, corresponds to the assumption that the null coordinate force constants vanish. The I?: form of the F matrix is not convenient for practical applications because the So coordinates depend on the choice of the basis in the subspace of the pure vibrations X0 ; also the So are generally non-local linear combinations of IVCs. Thus the force constants (Fg), are not uniquely defined and do not have a simple physical interpretation.

167

The general kg matrix A general method of constructing independent vibrational coordinates is to form linear combinations of IVCs R [7]. Let us consider an orthogonal transformation given by the M X M matrix II1 P= U,R

(24)

The P coordinates may be divided into two subsets PO = W, R

P’ = 6, R

(25)

where WV,and 6, are built from the first N and the last p = M --N rows of U1 respectively. e (i = 1 , . . . ,N) may be used as independent coordinates for the description of vibrations;Pi (i = 1,. . . ,p) are neglected. The coordinates P” and the corresponding force field @$ are widely used [9]. This representation of molecular vibrations is fully adequate. The force constants (P$)ij are uniquely defined in the sense that a change in their values will cause a modification of the vibrational eigenvalues and eigenvectors. However, the choice of N independent coordinates from M equivalent dependent coordinates is always to some extent arbitrary. Some interesting properties of fo_rce fields_of the Fg type may be obtained by considering the connections of F$ with F matrices defined in coordinates spanning the whole extended configuration space of the studied system. Generally, the F matrix does not possess the invariance property [2], i.e. the elements Fij do no_t depend only on the definitions of the coordinates i and_j. However, the F$ matrix has this property, as will now be shown. Let FR be a force field of the studied system in the dependent IVCs R. From eqns. (24) and (A2) the equivalent force field in the P coordinates may be obtained l?p = U, Fa $

(26)

and hence, from the definition (25) of P” l$? = WI,lY,WT

(27)

From eqn. (27) it follows that only the-elements of the ith and jth rows of the matrix W1 are used in obtaining (F$)ij. On the other hand, these are exactly the elements of W1 which define the coordinates e and Pi” (eqn. (25)). Thus the force constant (Fi), may be considered to be characteristic for the coordinates fl and fl. It must be stressed that this property holds only for independent coordinates obtained as orthogonal linear combinations of IVCs R. For non-orthogonal linear combinations W1 in eqn, (27) must be substituted by a matrix built from the first N rows _of (UT’)~ ; each of the elements of this matrix depends on the whole matrix U1 . A new insight into the interpretation of force constants in inGependent coordinates (@g)ij may be obtained by generating the force field F, corresponding to @ . From eqns. (25) and (Al) it follows that

This $a matrix may also be interpreted as the result of transformation of a matrix F, in the full P coordinates; the inversion of eqn. (26) gives I& = fif I&l tii

(29)

Comp_arisons of eqns. (28) and (29) leads to the conclusion that the force field FP corresponding to Pi is (FP)ij

=

(G)ij

fFP)ij

=

O

i,j=

1 ,***, N (39)

i>Norj>N

Thus it may be seen that the choice of P” as the independent coordinates implies the assumption of vanishing of the force constants associated with the neglected P coordinates. The P’ are not null coordinates. After transformation of FP to the S coordinates (eqn. (2)) the force constanks of the null coordinates S’ (eqns. (5)-(7)) will have non-zero values and F, from eqns. (29) and (30) will not be a pure vibrational force field. As has been shown above, the force constants (Fg), are certain invariant characteristics of the coordinates @ and e. In many cases some of the coordinates PO may be chosen to be equal to individual IVCs. The corresponding elements (J$)ij should be treated as characteristic for the IVCs in question. From the presented discussion it also follows that the force cons_tant va&es associated with the same IVCs obtained in the representations Fg and Fz will generally be different (see Example: The Planar Pure Vibrational Force Field of Benzene). The problem of determining the correct values of force constants of IVCs is considered under Conclusions. CONSTRUCTION

OF PURE VIBRATIONAL

FORCE FIELDS

The transformation of a force field into the Fg form is simple (eqn. (23)) if it is expressed in the R or So coordinates (see Introduction). If the F matrix is given in independent coordinates of the P” type (see Harmonic Force Constar+ in Independent Coordinates: the I$ matrix), then the corresponding F, force field must first be obtained (eqn. (28)). In the most general case, when the coordinates used cannot be related to R by an orthogonal linear transformation, the corresponding Fg force field may be calculated from the equivalent F matrix in Cartesian coordinates. Such transformations are performed by the program IMP0 [lo]. The basic_ problem in obtaining Fa* through eqn. (23) lies in finding the W matrix. W depends on the form of the redundancy relations (l)_which are generally not trivial to derive. A simple method for generating W cons$ts of using the normalized eigenvectors of the kinetic energy matrix GR corresponding to the non-zero eigenvalues [4, 111. The symmetry of the Ga matrix guarantees the orthogonality of these eigenvectors (which are put into the rows of W). This algorithm is used in the program KANON

169 [lo] which transforms force fields from I$ and @a form to pure vibrational form Pi. EXAMPLE: THE PLANAR PURE VIBRATIONAL

FORCE FIELD OF BENZENE

The influence of redundancies existing among the vibrational coordinates for ring molecules on the force constant values is presented here using the benzene molecule as an example. The transformation of the force field of benzene to the pure vibrational form has already been shown [4] . This calculation is now repeated using a more accurate force field [8] , new problems are also discussed. The internal coordinates for the description of the planar vibrations of benzene may be divided into four groups: ring bond stretching (b), CH bond stretching (r), CH planar deformation (4,~’ or p = Z-l’* (# - @‘), p’ = 2-“*(@ + $I’)) and ring angle deformation (a) (see Fig. 1 for definitions) The full set of equivalent IVCs R to be considered consists of M = 30 coordinates. The number of vibrational degrees of freedom is N = 21; p = 9 redundancies occur. The redundancy conditions are of two types. The first type, the ring redundancies, arise from the conditions of the closure of the ring and are linear combinations of the b and Q! coordinates [4, 121; their number is p1 = 3. The second type, the tree redundancies, are of the form $I’= (Yi+ f#Ji+ fp;S 0

i=

1 ,*-*,

6

(31)

Fig. 1. Internal valence coordinates for the planar vibrations of benzene. Coordinate xi may be obtained from x, through a counterclockwise rotation by (i - 1) 60”, (x = b, r, @,G’).

170

and their number is pz = 6 (see Fig. 1). The independent coordinates e (i = 1, . . . ,N) used to obtain the force field I$ for the planar vibrations of benzene [8] consist of all the b, r and /I coordinates and three orthogonal linear combinations of the (Y coordinates (Table 1). The simplest method of obtaining a pure vibrational fprce field corresponding to I$ is the one outlined in ref. 4. The matrix W1 in the relation PO = WIRl

(32)

is easily found from Table 1, where dinates consisting of all the 13, r, p obtain a special form of the force ponding to FE. By analogy with eqn.

R1 is a set of M1 = 24 (= N + pl) coorand c. Equation (32) may be used to field Fal in the R1 coordinates corres(28)

@a1 = *I$%,

(33)

From +a1 the pure vibrational force field $x1 may easily be derived (see Construction of Pure Vibcational Force Fields). Several force constants from the matrices I?:!, Fal and Fg, are presented in Table 2. In I?gl the influence of ring redundancies on the values of force constants has been eliminated. This does not hold for the tree redundancies because the CYand $J” coordinates are not orthogonal (eqn. (31)). The correct procedure would be to obtain the matrix Fg in the R* coordinates. This is inconvenient because M = 30 coordinates appear in the description of vibrations. A better approach is to introduce new planar ring deformation coordinates TABLE 1 Coordinate sets used in the description of the planar vibrations of benzene Symbol

Coordinate numbers

Definitiona

PO

l-6 7-12 13-18 19 20 21 l-18 19-24 25-30 l-24 l-18 19-24 l-24 25-30

b,,...,b,

R

RI R, R,

;:‘=2-;~q@1 -@;), . . . ,pg

6-“2(~, -LYE +,x3 -01~ +(u, -a,) 12-“Z(2ti, -oL2 --cxl + 201, -oL5 -cx,) 2-1((Y*-cY(y++oLg-(Y6) As in PO ;;

L;’ 2’q@,

+ @I), . . . , p;

As in R As in R alr...,a6

AsinR @:‘....,@I

=Coordinates of b, r and OLare defined in Fig. 1, 0~’by eqn. (34) and $I’ by eqn. (31).

171 TABLE 2 Values of the diagonal (b*) and offdiagonal (b - b) force constants of ring bond stretching obtained in different representations of the vibrational potential energy of benzenea

b= b-b

ortho meta para

6.578 0.710 -0.407 0.425

6.578 0.710 -0.407 0.425

5.212 0.027 0.276 1.791

4.927 -0.115 0.418 2.076

aThe coordinate sets PO, R, and R, are defined in Table 1. The force constants of I!: are taken from ref. 8; other values have been calculated in this paper. Units are aJ Be* .

&f = 6-“2

(2% - $i - 4;)

i=

1,...,6

(34)

The (Y’ are localized linear combinations of IVCs and thus retain the simple physicochemical interpretation of (Y. Moreover, the (Y’ are orthogonal to the null coordinates $“. The application of coordinates of this type is standard for ring substituents, e.g. the -NH2 group [9]. The use of CY’makes it possible to reduce the number of pure vibrational coordinates to M1 = 24 in the treatment of vibrations in the full extended configuration space. This may be done in the following way. Two new coordinate sets will now be introduced. The set Rz consists of all the b, r, fl and 0~’ coordinates, while R3 contains additionally the $J” coordinates (Table 1). A special form F,, of the F matrix in the R3 coordinates may be obtained from the following transformations (see Appendix)

with

where the coefficients in the relations R = V R3 and P” = Wir,R may easily be found from Table 1 (W, is equal to W, (eqn. (32)) with six columns of zeroes added). From I?,, the pure vibrational force field I?& is easily found by the methods given in Construction of Pure Vibrational Force Fields. The @” are null coordinates; pure vibrational force constants associated with them must vanish (see Properties of $2 ), Thus the non-zero part of l?& will correspond to the R2 coordinates. This matrix is the pure vibrational force field I?,, in the R2 coordinates. l?i$, is the simplest form of the force field of benzene in which the elimination of the influence of all redundancies on the force constant values has been accomplished with conservation of the local interpretation of vibrational coordinates. The (FE,), are fully correct values of force constants associated with IVCs i and j. For brevity, the complete force fields for planar vibrations of benzene

172

derived in this section have not been supplied. These results are available in the form of Supplementary Material from B.L.D.D. (S.U.P. 26337 (7 pages)). In Table 2 numerical results are presented for force constants of ring bond stretching found in different force fields: I?$, l?a1, I?gl and I?&. To check the correctness of transformations it was verified that all these force fields lead to the same vibrational eigenvalues in the Wilson GF method (see ref. 4). The following main conclusions may be drawn from the comparison of force fields Fz, FR1, @, and I?&. The force constants associated with the r coordinates (which do not appear in the redundancy relations) and /3 coordinates (which are orthogonal to null coordinates) are the same in all derived force_ fields. Conversely, the force ‘constants of b-and (Y coordinates in &! and Fal are markedly different from those of F$, and F& . The diagonal bond stretching force constant is much smaller, and the interaction of C-C bond +-etches in the pm-u position is much larger in F& and F& as compared to F$ (Table 2). The large differences indicate that the existence of redundancy relations in the full set of equivalent IVCs of a molecule has a significant influence on force constant_values assigned to these coordinates. The differences between I$i and F& show that the tree redundancies eqn. (31) also influence force constant v_alues. It may be noted that the existence of two different force fields, Fal and Fz, , describing the same vibrations of the system is an illustration of the indeterminacy relations for force constants in dependent coordinates (Appendix, [4lb CONCLUSIONS

Molecular vibrations in the extended configuration space X spanned by the full set of equivalent rectilinear IVCs Rj (i= 1,. . . , M) have been considered. It was shown that redundancy relations existing among the IVCs constrain the system to a subspace X0 of X, called the pure vibrational subspace. In order to obtain coordinate values corresponding to physically allowed configurations, projection on X0 must be performed. The form of the necessary projection operator has been found. The projected coordinate values, called the pure vibrational coordinates R*, identically fulfill the redundancy relations. R* represent configurations of the system belonging to the pure vibrational subspace X0 in the basis defined by IVCs R. R* are thus rectlinear internal valence coordinates. They give a description of molecular configurations equivalent to the one provided by the independent coordinates Sp (i = 1, . . . , N) spanning the subspace X0 and orthogonal to the null coordinates. The condition that the motion of the system must be restricted to the subspace X0 was next applied to derive the correct form of the harmonic force constant matrix I? in dependent coordinates. The desired matrix F$ , called the pure vibrational force field, was found to be the same as

173

the canonic force field introduced in ref. 4 on the basis of heuristic arguments. Thus the basic problems of formulating the vibrational harmonic eigenvalue problem in dependent rectilinear coordinates have been solved. In ref. 4 the method of obtaining vibrational eigenvalues and eigenvectors by diagonalization of the matrix GF in the R coordinates has be_en outlined. An unequivocal definition of the kinetic energy matrix G exists [l] , which may be used in both independent and dependent coordinates. In this work the proper form P$ of the F matrix in dependent coordinates has been determined. I$$ gives a description of the potential energy of the vibrating system equivalent to that provided by the matrix @j in independent coordinates So. @$ corresponds to vanishing of the null coordinate force constants. Since force constants (I?$)ij correspond to molecular deformations described by the pure vibrational coordinates R*, they have a simple physicochemical interpretation (bond stretching, valence angle bending, etc.). In the force field @g unique values may be assigned to force constants associated with all coordinates from the full set of equivalent IVCs. These force constants may be treated as characteristic of the IVCs. The ideas and methods used in the process of deriving I?$ were also used to study properties of force fields in independent coordinates. It was found that the most proper form for the representation of the potential energy is through the matrix I?: in the coordinates So. This approach has however a limited practical application. The most widely used independent vibrational coordinates PO are obtained as orthogonal linear combinations of the IVCs R (P” are not orthogonal null coordinates). The disadvantage of using P” for describing molecular vibrations lies in the ambiguity of choosing N independent coordinates from M equivalent dependent coordinates. Additionally, the simple physicochemical interpretation of force constants (Fg)ij may be lost if the PO coordinates are taken to be non-local linear combinations of the IVCs R. The advantage of using PO is that the number of coordinates is equal to the number of vibrational degrees of freedom of the system. Also, it has been proved that the values of the elements (Fg), depend only on the definition of coordinates i and j. Since some of the coordinates PO may be chosen equal to individual IVCs, the corresponding elements (@), may be treated as invariant characteristics of these IVCs. Further analysis has shown that values of such force constants are generally different from those appearing in the pure vibrational force field F$ . In order to illustrate the theoretical conclusions obtained in the paper, numerical calculations of several types of force fields for the planar vibrations of benzene were performed. Large differences were found between corresponding force constants in the @ and pure vibrational F matrices (in the case of IVCs which appear in redundancy relations). For instance, the diagonal ring bond stretching force constant of benzene was found to be 4.927 aJ a -2 in the $g2 matrix as compared to 6.578 aJ A -2 in Fg

174

(see previous section). The influence on force constant values of redundancies existing in the full set of equivalent IVCs was thus demonstrated. To eliminate a part of this influence a type of coordinate different from the one commonly used was proposed for the description of planar ring deformations. The advantage of using IVCs (or their local combinations) to describe vibrations lies in the simple physicochemical interpretation of force constants. For non-cyclic molecules local linear combinations of IVCs orthogonal to the null coordinates may easily be found [9] (the force fields correspond to I$ ; see Harmonic Force Constants in Independent Coordinates: the General %E Matrix). For ring molecules such as benzene however, it is necessary to use the @ representation in order to retain the local physical interpretation of all force constants. The traditional way of describing the potential energy of harmonic vibrations of polyatomic molecules is through the matrix Fg. Force fields of this type have been obtained for a large number of molecules by different methods. The transferability of F$ force constants has been checked on many examples [13-151. This is an empirical verification of the fact that (Fg), values are characteristic of the integral coordinates i and j. In this paper new characteristic values of force consta_nts connected with the IVCs have been proposed, the elements (Fg ),. In Fi$ the influence of redundancies existing in the full set of equivalent IVCs on force constant values has been eliminated. From the presented discussion of the properties of pure vibrational force fields it follows that the use of this-representation of the potential energy should be highly advantageous. The Fg form could be used for compa_ring force fields, transfer of force constants and construction of model F matrices [4]. This is especially the case for cyclic compounds. Recently the method of obtaining molecular force fields by empirical scaling of force constants calculated by quantum mechanical methods at the SCF level has been widely used, e.g. refs. ‘7 and 16. Better transferability of the scaling factors appearing in this approach might be obtained if these parameters were defined for pure vibrational force fields. The disadvantages of using the I?; representation are that it is more complex to derive than Fg, and that the number of coordinates under consideration exceeds the number of the vibrational degrees of freedom of the system. Also, the transferability of the (Fg )ij force constants must be empirically verified. As the first step towards such a verification, the pure vibrational for_ce field for planar vibrations of pyridine was determined on the basis of Fg from ref. 16. The results were similar to those obtained for benzene. In the pure vibrational force field (corresponding to @& for benzene from the previous section) very large positive interaction force constants were found between ring stretching coordinates in the paru position. The diagonal ring bond stretching force constants in I!& were 5.345, 4.832 and 5.062 aJ a-’ for the bonds NI-C2, CZ-C3, C3-C4 respectively (compared to 7.110, 6.529,6.610 aJ AW2in I?$! [16] ).

175 ACKNOWLEDGMENTS

Helpful discussions with Professor Krystyna Szczepaniak-Person are gratefully acknowledged. The investigation was financed by the Research Project CPBP-10.12. APPENDIX

The form @ of the harmonic force constant matrix corresponding to the description of vibrations in pure vibrational coordinates will be-derived here. The most general formula giving the transformation of the F matrix upon a linear transformation of coordinates P = A Q is [5]

ie =;i’i,ii

t-41)

The force fields @o and @r are physically equivalent, i.e, they describe the same harmonic potential in different coordinates. If A is a _rectangu&r matrix there is no unique one-to-one correspondence between F, and Fr. In this case eqn. (Al_) gives a special form of the F, matrix describing the same vibrations as FP ; two such matrices shall be_ called corresponding. The procedure for obtaining the general form of F, is presented below [4]. If A is a square non-singular matrix, a unique inverse transformation to eqn. (Al) exists. This is especially simple for orthogonal transformations, then I& = A&-$

(A2)

The existence of unique direct and inverse transformations connecting $r and F, ensures a one-to-one correspondence between them. Two such $’ matrices will be called equivalent. In order to study the effect of the presence of redundancies on force constant values, the potential _energy of the_ vib;ating system will be describe-d by three force fields: Fa , 6’: and Fg . Fa is the general form of the F matrix in the IVCs Ri (i = 1, . . . ,M) (see Basis Definitions) on which the conditions (1) have not been imposed. Thus F$ contains, generally, M(M + 1)/2 independent parameters. The force field Fg in the independent coordinates Sp (i = 1,. . . ,iV) (see Basic Definitions) consists of N(N + 1)/2 independent parameters in the general case. The same is true of the force field Fg in the dependent R* coordinates, because from eqns. (21), (22) and (Al) it follows that $2 = ~j$$fr

(A3)

$1 = *T $; ti

(A4)

If transformations of the type given by eqn. (Al) are to be used to obtain an unambiguous definition of I?*a care must be taken that in all steps the result contains a number of independent parameters not greater than in the initial matrix. For example eqn. (17) may not be used to obtain the relation-

176

ship between fig and I?,. Similarly, eqn. (7) may not be used to express I?, through @“,.-This is because the application of eqn. (Al) would lead to a definition of Fa (M(M + 1)/2_parameters) through & or %‘g (N(N + 1)/2 parameters). Such a force field Fa would not be a general one, but rather a special case, as discussed above. Thanks to the equivalence of the I$* and So coordinates (see Properties of Pure Vibrational Coordinates) formulae (A3) and (A4) have been obtained. The existence of unique direct (A3) and inverse (A4) transformations connecting $2 and FQ shows that these force fields are equivalent. The knowledge of one of these matrices is sufficient to obtain the other. Equation (A3) is the basic definition of the pure vibrational force field. The relation between Fg a_nd Fa may now be obtained. Thanks to the orthogonality of the matrix U (eqn. (2)) the force field Fs equivalent to Fa may be obtained from (A2) as Fs = U FR UT From the definition of the So coordinates 6’: is a sub-matrix of Fs (Ghj

=

i,j=

fFS)ij

(eqns. (4) and (7)) it follows that

1 >--*, N

and thus j?; = WFaWr Substitution

(A5) of eqn. (A5) into (A3) gives the desired formula

F~=?@sir&WG Finally, from eqn. (A5) it follows that all matrices @a = i+$ + I?

(A6) Fa of the form (A6a)

where Wj$Wr=o

(A7b)

will give the same matrix I?: after transformation to the So coordinates. Equation (A7) gives the general form of_the Fp. mat+_ des_cribing the same vibrations of the system as I?$ (or $8 = W F, W = W Fg WT ) [4]. REFERENCES 1 E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. 2 M. V. Volkenstein, L. A. Gribov, M. A. Eliasbevicb and B. I. Stepanov, Molecular Vibrations, Nauka, Moscow, 1972. 3 K. Kuczera, J. Mol. Struct, 117 (1984) 11. 4 K. Kuczera and R. Czermihski, J. Mol. Struct., 105 (1983) 269. 5 N. Neto, Chem. Phys., 87 (1984) 43. 6 M. Gussoni and G. Zerbi, Chem. Phys. I&t., 2 (1968) 145. 7 B. Crawford and J. Overend, J. Mol. Spectrosc., 12 (1964) 307. 8 P. Pulay, G. Fogarasi and J. E. Boggs, J. Chem. Phys., 74 (1981) 3999.

177 9 P. Pulay, G. Fogarasi, F. Pang and J. E. Boggs, J. Am. Chem. Sot., 101 (1979) 2550. 10 IMP0 and KANON are FORTRAN programs written by the author; listings are available upon request. 11 D. M. Adams and R. G. Churchill, J. Chem. Sot. A, (1970) 697. 12 J. C. Duinker and I. M. Mills, Spectrochim. Acta, Part A, 24 (1968) 417. 13 J. H. Schachtschneider and R. G. Snyder, Spectrochim. Acta, 19 (1963) 85. 14 V. J. Eaton and D. Steele, J. Mol. Struct., 48 (1973) 446. 15 H. Hollenstein and Hs. H. Giinthard, J. Mol. Spectrosc., 84 (1980) 457. 16 G. Pongor, P. Pulay, G. Fogarasi and J. E. Boggs, J. Am. Chem. Sot., 106 (1984) 2765.