Volume 25, number 3 CIiJZhIICAL PHYSICS LETTERS 1 April 1974

BENDING FORCE CONSTANT FORMULAS FROM A BOND-CHARGE MODEL OF TRIATOMIC MOLECULES

Gary SIMONS and Charles E. CHOC Deprrttnent of Chemistry. Wichita Stare University. Wichita, Kansas 67208. USA

Received 2 January 1974 Revised manuscript received 22 January 1974

.A bond-charge pointdipole model of motecular electronic structure is used to derive a genenl relation between the bending force constant, Kee, and stretching force constant. K,, , of linear and non-linear triatomic molecules.

l_ Introduction

Although models of chemical bonding which avoid wavefunctions cannot be completely correct, they may be useful if they (1) provide a new basis for discussing or defining theoretical constructs, (2) possess predictive capaci- ty, or (3) establish a theoretical basis for empirical relations or suggest new semi-empirical relations concerning measurable quantities_ A recently proposed [ 1,2] simple bond-charge (SBC) model of diatomics has been shown to be useful in ail three respects. The model employs bond charges which are reasonable measures of bond order [ I,2 1, allows the determination of core radii [2], and can be used to estimate effective nuclear charges [3]. Cubic and higher force constants are predicted semiquantitatively [4], as are vibrational frequencies of heteronuclear di- atomics [2] _ The model rationalizes the old experimental observation that through ah electronic states of a mole- cule, Rgo, = constant [4,5]; supports the Fox and Martin rule, RzKe = constant [I, 2,6] ; and, in a more complex form [7], provides a basis for the observations made by Calder and Ruedenberg regarding the values of the Dun- ham coefficients [S] .

One [9] of several recent polyatomic applications of the SBC approach [9-l 1] has shown that a bond-charge model which includes a point-dipole and invokes the polyatomic virial theorem [ 121 can describe both bends and stretches of triatomic molecules. Ray and Parr [ 131 have recently shown that one can derive from the SBC model a new, surprisingly accurate semi-empirical relation between the bond bending and bond stretching force constants of linear symmetric triatomics,

K 08 = 0.026Kll . (1)

In the following we demonstrate that a more general relation for bending force constants, which reduces to (1) for linear species, may be derived from the point-dipole formulation of the model, and that for most non-linear tri- atomics one obtains

K ee = O.lOK,, .

Numerical evidence of the validity of (2) is presented.

(2)

2. Theory

For a triatomic with bond lengths RI and.Rz-and bond angle 6; the poiyatomic @aI theorem [12) stat& that ,:

: .-’ . . .4li

.’ . . . I

: _. .-. :‘. : .: . . _- :. .. :- _ ,_.‘.~ .-.,:y 1 ‘. .-. ., ~., . .

Volume 25, number 3 CHEMICAL PHYSICS LETTERS 1 April 1974

V~R,,R2,~)=2W(R1,R2,~)+RllaW~R1,R2.~)/aR,I +R21a~u(R,,R2,e)laR21 , (3)

where Vis the potential energy component of the total Born-Oppenheimer energy W. A series of derivatives of (3) may be taken to obtain a set of relations between derivatives of Vand W [9] _ If we make the approximation that W is represented by a conventional valence force field,

I~=~~,,(R,-Rl,)2 +?$2(R,-R2e)2 +K12(Rl-Rl,) (R2-R2,)

fKle R,,(R,--R,,) (e-e,) +K2* R,,@,-R2,) @-@,I +%Q,, RleR2e(B--8e)2 , (4)

we obtain, for symmetric species, the following relations:

(K,,+K12)=(1/R,)(av/aR,),,. , (5)

(6)

~,e = (112~:) (a2viae2),quil _ (7)

By evaluating the appropriate derivatives of a model of V, the force tieid constants may be interrelated. We have chosen the model potential energy expression of ref. [9]; a point charge, -qe, is placed in the middle

of each bond, a charge p,e is piaced on each atom to obtain electroneutrahty, and a point-dipole is placed on the central atom to simuiate lone-pair effects and provide a mechanism for a non-linear equihbrium geometry. For linear molecules the dipole is not needed and is automatically set equal to zero. Thus we have

v= _ w2g 20q2e' ioqpe cost6 1 oqpe c0s+e ---- 9R, 9R, 3R: - 3R;

+ 22q2e2 2q2e2 2q2e2

Y(Rf+R;-2R1Rz ~0~0)“~ - 3(R;+$R;-R, Rz cod)” - 3(JRf+R;-RI R2 cos@“* - (8)

The two model parameters 4 and p may be evaluated by combining eqs. (S), (6), and (8). These values may then be employed in eq. (7) to obtain the result

-where

qie,) = 22w--C0se,)+q(1 --c02e,) 3(cose,-*j 3(1 --c0s2ee)1’2

P*(I -cose,)5f2 + (&c0se,)3/2 - (~-cose,y’* I

: and-

Fi(ee) = -2 [(I +&eej/(i -Case,)] TF~(~,) _ (11)

:E&. ($-o 1) thus represent-a generaI relation between K,*, K, 1, Kli, and KIe , valid’for all’sykmet& t&tomics,

subject to the as&med~model potential :V [eq. (8)] and the vaIence force field. [eq. (4) ] _ If a molecule is linear,

Volume 25, number 3 CHEMICAL PHYSICS LETTERS 1 April 1974

Tab12 1 Predicted bending force constants

Molecule f%

Hz0 104.3 H2S 92.2

H2Se 90.9 03 116.7 So2 119.3

Se02 113.6

NO2 134.2

Cl02 117.6

-2 104.9 NF2 104.2

Oh 103.0 SiF2 101.0 GeF2 97.2

Kge [eq. (1411 .Kee [es. (131 Kee (expt.) Ref.

-___

- 0.844 0.845 0.761 (141 0.431 0.428 0.429 1141

0.350 0.349 0.327 (141 0.75 057 1.27 I151 1.050 1.033 0.815 114;

0.697 0.692 0533 IlSl

1.338 1.104 I.109 1141 0.685 0.702 0.65 1 1141

0.76 0.60 1.40 0.617 0.48 1.08 I::)

0.504 0.405 0.707 1161 0538 OJO2 0.440 0.434 0.408 0.3 16

or, more approximately,

Kee =O.O26K,, . 03)

The accuracy of eqs. (12) and (13) has been demonstrated [ 131. Eqs. (9)-( 11) could, of course, be evaluated separately for each non-linear triatomic. However, in the region 90°-13S0 which includes the bond angles of most non-linear triatomics, Fi (0) and Fz(O) are relatively slowly varying functions with approximate average values of -0.05 and -1-0.08, respectively. Substituting these values into eq. (Y), we obtain

K ee “O.lO(KII+KIZ) +0.04K1, .

Since K1, and K,, are generally much smaller than K, I) eq. (14) can be further simplified to

K ee 2 0.10 K,, _

(14)

(15)

3. Results

We have tested eqs. (14) and (15) using 14 non-linear symmetric triatomics; the results are presented in table l_ In only three cases - 03, CF,, and NFz - are the predicted bending force constants in poor agreement with ex- perimental results, and the data on CF2 and NF, may contain sizable uncertainties. In all other cases the agree- ment between predicted and observed results is highly satisfactory. The essence of eq. (15) has, of couise, long been known to spectroscopists, but only as an empirical rule, without a theoretical or semi-empirical basis. We therefore suggest that the two relations, (1) for linear and (15) for-non-linear triatomics, when taken together, testify td the usefulness and predictive capacity of the bond-charge point-dipole model and estab!ish a semi- empirical scheme for the estimation of b.ending force constants.

A&iowledgement :

:

Volume 25, number 3 CHEMICALPHYSICS LETTERS- ~. .. .. : ‘. 1 A~prEl972. -_

-References :

.: ._ . . ,- _. :

111 R.~:~arrand R.~.‘Bcrlima~~,J. fzhem. phys.49 (1968) 1055; [2] R.F.Borkman, G:Simonsand R.G. Parr, J. Chem. Phys. 50 (1969) 58.. [3] P. PoIitzer.J. Chemi-P!tys. 50 (1969) 2780; Sl(l969) 459; 52 (1970) 2157.

.[4] R-F. Borknian and R.G. Parr, J. Chem. Phys. 48 (1968) 1116.: [S ] R-T. Birge, Phys. Rev. 25 (l-925) 240;

. :

R. Meike, 2. Physik 32 (1925) 823. r61J.J. Fox and A-E. Martin, J. Chem. Sot. (1939) 884. ‘. [7] G. Simons and R.G. Parr, J. Chem. Phys. 55 (1971) 4197. [S] G-V. Calder and K. Ruedenberg, J. Chem. Phys. 49 (1968) 5399. [9] G. Simons, J. Chem. Phys. 56 (1972) 4310.

[lo] R.F. Borkman, I. Chem: Phys. 51 (1969) 5596. [ll] R.F. Borkman and F-A. Settle Jr., J. Am. Chem. Sot. 93 (1971) 5640. (12 ] B. Nelander, J. Chem. Phys. Sl(l969) 469. [13] N.K. Ray and R.GI Parr, J. Chem. Phys.59 (1973) 3934. [14 J K. Kuchitsu and Y. Merino. BuB. Chem. Sot Japan 38 (1965) 814. [15] Y. Morino, Pare Appl. Chem. 18 (1969) 323.

(161 D-E. Mill&an and M.E. Jacox, J. Chem. Phys. 48 (1968) 2265. [17] V.M. Khanua, R. Hauge, R.F. Curl Jr. and J-L. Margrave, J. Chem. Phys. 47 (1967) 5031. 1181 H. Takeo. R-F. Curl Jr. and D.W. Win. J. Mol. Spectry. 38 (1971) 464.