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J. Phys. Chem. 1983, 87, 1694-1696
found that the free energy of moving q, from infinity to charge contribution A VQq,(r,,lreLr,O')which is about 7 rF, &(rs,lm+r.sOx), is approximately 2 kcal/mol. The value kcal/mol. ~ be much smaller than 2 kcal/mol since ACY, of A C Ymust Neglecting Aaw and using the arguments presented in involves displacement of r,,l from rs,lred to r?,lox and not the first part of txis Appendix we can approximate the from infinity to r,,lox.This point can be rationalized in reorganization energy due to the water and the surface a different way by noting that the value of Ag(r.s,lm-+rp) charges, Aawqap,by the reorganization energy, a,p, evaluis almost two orders of magnitude smaller than the ated without the surface charges. charge-charge interaction AV9&,,lm-+rp) (- 150 kcal/ It should be mentioned that the experimentally obmol). This means that AVQ,~is almost entirely compenservedIgchanges in exchange rates with pH are consistent sated by the change in solvation energy, AAG,,, (this with the above arguments. That is, on going from an pH compensation is the origin of the high effective dielectric 7 to the isoelectric point at pH 10, where approximately constant for charge-charge interactions in solutions and eight positively charge residues are deprotonated (at low proteins." If Agaol(r,,lm--vp,ox) is about two orders of ionic strength), the rate increase by a factor of -3. This magnitude smaller than AVQ&,,lm+rp) than Aawq, is corresponds to only -1 kcal/mol change in &*at room about two orders of magnitude smaller than its chargetemperature. And probably only part of this can be due to the reorganization energy. (29)Rees, D.C. J.Mol. Biol. 1980,141,323-6. (30)Adman, E.Biochim. Eiophys. Acta 1979,549,107-44.
Registry No. Cytochrome c, 9007-43-6.
Correlations of Bending Mode Force Constants among Polyatomic Molecules Chln-An Chang IBM T. J. Watson Research Center, Yorktown Heights, New York 10598 (Received: September 21, 1982; In Final Form: December 17, 1982)
A Coulombic interaction potential is used to describe the bending mode force constant, fd, of triatomic and larger molecules. It gives the relation f+/r1r2= c z , ~ ~ R - ~ [ 3sin' r , r ~4 - R2 cos 41, z1 and z2 being the charges of the two bonds involved, 4 the bond angle, rl and r2 the bond lengths, R the distance between the two end atoms, and c a proportionality constant. For many different types of molecules, the product czlzz is shown to be nearly constant among molecules of each type. The above relation thus allows a ready estimation off@ for a molecule from data available for molecules of the same type.
Introduction Vibrational frequencies of polyatomic molecules are important for an understanding of the structures and thermodynamic properties of these molecules. When spectroscopic data are lacking, an estimation of the various force constants is needed. Taking triatomic molecules for illustration, one finds it is usually a good approximation to consider only the stretching and bending force constants and to neglect the interaction constants. This is generally known as the valence field method.' The stretching force constants of these molecules can be either estimated from similar molecules2* or approximated by their counterparts in diatomic molecules. The latter, in turn, can be estimated by various empirical methods if necessary.24 The bending force constant, on the other hand, has no counterpart in diatomic molecules. This quantity has been calculated with semiempirical methods? but no empirical approach similar to those for the stretching force constants is known. It is, therefore, helpful to establish some em(1)G. Herzberg, "Infrared and Raman Spectra of Polyatomic Molecules", Van Nostrand-Reinhold, New York, 1945. (2) R. M.Badger, J. Chem. Phys., 2, 128 (1934). (3) K. M. Guggenheimer, Proc. Phys. SOC. London, 58,4561 (1946). (4)G.R. Somayajula, J. Chem. Phys., 28,814 (1958). (5)D.R. Hershbach and V. W. Laurie, J. Chem. Phys., 35,458(1961). (6) Chin-An Chang, High Temp. Sci., 6,276 (1974). (7) R.J. Bartlett and R. G. Parr, J. Chem. Phys., 67,5828(1977);K. Ohwada, ibid., 72, 1 (1980),and references therein.
pirical correlations among polyatomic molecules for ready estimation of the bending force constants. In this paper a Coulombic interaction potential is used to describe the bending mode force constant. A simple equation is derived from such an approach which relates the bending force constant to the bond lengths and bond angles involved. This equation is shown to correlate well molecules of the same type for many different types of polyatomic molecules, both linear and nonlinear.
Method The potential energy describing the vibrational motions of a molecule can be expressed by the quadratic relation8
c fijqi qj 3N
2v =
1J=1
with
Here, f i j is the force constant, and qi's are the massweighted Cartesian displacement coordinates. A complete potential function should be able to describe all the stretching, bending, and interaction modes of a triatomic (8)E.B. Wilson, J. C. Deciw, and P. C. Cross, "Molecular Vibrations". McGraw-Hill, New York, 1955.
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The Journal of Physical Chemlstry, Vol. 87, No. 10, 1983
Bending Mode Force Constants of Polyatomic Molecules
molecule, for example. Here, we make an approximation by considering only that part of V which determines the bending mode force constant, fm.In other words, we choose a certain form of the potential function to describe the bending mode and treat it separately from the other vibrational motions. We then proceed to find an analytical form off+ from the chosen potential function through a correlation among molecules of the same type. As described later, the parameters involved in the derived is shown to be equation can be evaluated empirically. f,+, a simple function of the bond lengths and bond angle of the molecules concerned. The bending mode force constant f,+,has, according to eq 2, the form (3)
It is a measure of the change in potential energy with respect to the bond angle 4. In this work such a potential energy change is described by the change in interaction between the two bonds involved with respect to the angle 4, We choose a potential V in the Coulombic form (4)
Here z1and z2 are the charges of the two bonds, R' the distance between these charges, and c' the proportional constant. For a triatomic molecule XYZ, R' is approximated to be proportional to the distance R between the two end atoms X and Z; the proportionality constant is absorbed into c', to make the latter c. The approximation introduced by this approach is shown later to be justified when molecules of the same type are compared. Using the relation R = (r12 r 2 - 2r1r2cos 4)lI2,rl and r2 being the bond lengths, we obtain
+
f,+,/rlr2= c ~ , z , R - ~ [ 3 rsin2 ~ r ~4 - R2 cos 41
(5)
Equation 5 can be used to evaluate f,+, provided z1and z2 are known, and c is determined empirically for different types of molecules. The bond charges, z1and z2 are determined by the type of chemical bonds and number of bonding electrons involved. Here, we make a further assumption that both c and z's are closely related to the type of chemical bonds involved. The product czlz2would then remain constant among molecules of the same type, such as C02,CS2,and OCS or CC14,CBr4,and C14,etc. We will show later that the calculated czlz2indeed remains nearly constant among similar molecules of the same type, and allows an estimation of f,+, for a molecule of the same type. For larger polyatomic molecules, the bending motions involve all parts of the molecule. In order to apply eq 5 we make the following simplification using methyl halides for illustration. For those molecules, we compare the bending force constant of the X-C-H bending motion, X being the halogen. The X-C-H part of the molecule is treated as a triatomic entity and its interaction with the remaining part of the molecule is neglected. The observed f,+, for the X-C-H bending motion of these molecules is used to calculate czlzz by using the bond lengths and bond angle of X-C-H. For symmetric molecules, e.g., NH, or CHI, only the H-N-H or H-C-H part is used for the correlations. Results and Discussion The calculated czlzz for the various linear and nonlinear triatomic molecules are shown in Table I. The molecules ~~~
~
(9)D.F.Smith and J. Overend, Spectrochim. Acta, Part A , 28,471 (1972). (10)K. Kuchitsu and Y. Morino, Bull. Chem. SOC.Jpn., 38,814(1965).
1695
TABLE I: Calculated c z , z , Values and Parameters Used for Triatomic Molecules molecule H2O H,S H,Se 0 3
SO, NSF NSCl ONF ONCl ON Br
CO, CS, CSe,
ocs
OCSe SCSe SCTe ClCN BrCN ICN HgF, HgC1, HgBr, HgI, BeF, BeC1, BeBr, HCN HCP
q f5 ir lrz, mdyn
2.1 2.0 2.1 10.3 10.0 14.1 14.4 8.5 8.9 8.7
0.810 0.429 0.338 1.311 0.822 0.98 0.83 1.065 0.585 0.460
r1, a Nonlinear 0.9572 1.336 1.47 1.272 1.431 1.448 1.45 1.136 1.139 1.146
7.2 7.1 6.3 7.3 6.7 6.8 7.3 4.9 5.0 4.2 9.5 8.6 7.8 9.2 2.6 2.9 3.0 2.2 2.6
0.581 0.237 0.158 0.361 0.284 0.195 0.175 0.20 0.17 0.12 0.138 0.089 0.070 0.066 0.118 0.067 0.054 0.202 0.148
Linear 1.160 1.553 1.711 1.158 1.160 1.558 1.557 1.760 1.930 2.120 2.05 2.29 2.41 2.59 1.40 1.75 1.91 1.064 1.067
cz,z,
.&-I
deg
ref
0.9572 1.336 1.47 1.272 1.431 1.643 2.161 1.512 1.975 2.140
105 92 91 117 120 116.6 117.7 110 113.3 114.5
9 10 11 12 13 14 14 15 16 17
1.160 1.553 1.711 1.560 1.711 1.711 1.904 1.150 1.150 1.150 2.05 2.29 2.41 2.59 1.40 1.75 1.91 1.156 1.542
180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180
18 13 19 20 19 19 19
r,, A
$,
1 1 1 21 21 21 21 22 22 22 23 23
are grouped according to the type of bonding involved as described above. The values of czlzz are seen to remain nearly constant within each group of molecules. For the C02group, it is interesting to note that the Se-containing molecules all show consistently lower values of czlzz than the other molecules, giving rise to two sets of czlz2values. The H 2 0 group and O3group respectively have different center atoms; excellent correlations are obtained for these groups. The experimental data listed in Table I show another interesting relation among similar molecules: An approximate inverse dependence off on the bond lengths involved. Such a trend has often teen noticed in spectroscopic studies, but no simple quantitative equations have been proposed to account for it. In our work, an explicit inverse dependence off,+, on bond lengths is derived in eq 5. This becomes more clear for linear triatomic (11)K.Venkateswarlu and P. T h i i a n s a m b a n d a m , 2. Phys. Chem., B212, 138 (1959). (12)T.Tanaka and Y. Morino, J. Mol. Spectrosc., 33, 538 (1970). (13)D.F. Smith and J. Overend, J. Chem. Phvs., 54,3632 (1971). (14)A. Muller. N. Mehan. S. J. Swin. N. Weinstock. and Glemser. J. Mol. Spectrosc., 59, 161 (1976). (15)L. H.Jones, L. B. Asprey, and R. R. Ryan, J. Chem. Phys., 47, 3371 (1967). (16)L. H.Jones, R. R. Ryan, and L. B. Asprey, . J. Chem. Phys., 49, 581 (1968). (17)J. Laane, L. H. Jones, R. R. Ryan, and 1. B. Asprey, J. Mol. Spectrosc., 30,485 (1969). (18)I. Suzuki, J.Mol. Spectrosc., 25,479 (1968). (19)T.Wentink, J. Chem. Phys., 30, 105 (1959). (20)K. Machida and J. Overend, J. Chem. Phys., 50,4429 (1969). (21)A. Loewenschuss, A. Ron, and 0. Schnepp, J. Chem. Phys., 49, 272 (1968). (22)A. Snelson, J.Phys. Chem., 70,3208 (1966);72,250 (1968). (23)G. Herzberg, "Electronic Spectra of Polyatomic Molecules", Van Nostrand, Princeton, NJ,1966. "
1
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The Journal of Physical Chemistry, Vol. 87, No. 10, 1983
Chang
T A B L E 11: Calculated c z , z , Values and Parameters Used for Larger Polsatomic Molecule
CH,F CH,CI CH,Br CH,I CF, CCl, CBr,
H-C-F H-C-C1 H-C-Br H-C-I F-C-F c1-c-CI Br-C-Br
XH,
PH,
ASH, SbH,
H-N-H H-P-H H-AS-H H-Sb-H
BC1, BBr, NF,
3.7 4 .O 5.3 6.3 5.9
0.638 0.354 0.293 0.232 0.713 0.331 0.237
1.113 1.113 1.113 1.106 1.332 1.781 1.939
C1-B-C1 Br-B-Br
2.2 2.2 2.2 2.3 3.4 3.6
0.660 0.37 0.310 0.235 0.16 0.1 3
F-N-F F-P-F F-AS-F
6.8 6.2 5.7
1.04 0.62 0.40
1.0173 1.421 1.523 1.712 1.7157 1.887 1.37 1.54 1.71
1.37 1.54 1.71
(100) (102) (104)
CH,
H-C-H H-Si-H
1.9 1.9
0.461 0.189
1.09 1.46
1.09 1.46
109 109
1 1
SiF, SiCI,
3.0 4.2 4.1
0.252 0.157 0.1 27
1.54 2.02 2.15
1.54 2.02 2.1 5
109 109 109
1
SiBr,
F-Si-F Cl-Si-C1 Br-Si-Br
GeF, GeC1, GeBr, GeI,
F-Ge-F C1-Ge-C1 Br-Ge-Br I-Ge-I C1-Sn-C1 Br-Sn-Br
0.15 0.125 0.095 0.07 0.075 0.064
1.68 2.09 2.30 2.49 2.31 2.44
1.68 2.09 2.30 2.49 2.31 2.44
109 109 109 109 109 109
26
SnC1, SnBr,
2.3 3.7 3.7 3.5 3.0 3 .O
PF, AsF, SiH,
4 .O 4.O
molecules. When two linear triatomic molecules A and B of the same type are compared, eq 5 gives
We have thus found an empirical expression for such a dependence of f 4 on the bond lengths among similar molecules. The successful correlations shown in Table I support the usefulness of the relation obtained. We have also calculated the czlz2 values for some larger polyatomic molecules. The results are shown in Table 11. The molecules are grouped similarly to the triatomic molecules. The hydrides are compared as one group. (24) M. Pariseau, E. Wu, and J. Overend, J . Chem. Phys., 39, 217 (1963). (25) A. M. Mirri, J . Chem. Phys., 47, 2823 (1967). (26) F. Koniger, A. Mtiller, and W. J. Orville-Thomas, J. Mol. Stmct., 37, 199 (1977).
1.332 1.781 1.939 2.1396 1.332 1.781 1.939 1.0173 1.421 1.523 1.712 1.7157 1.887
108 110.5 111.2 111.2 108 110.5 111.2
1 1 1 1 1 1 1
107.8 93.3 92 91.5 120 120
24 24 24 24 1 1
25 25 25
1 1 1 1
26 1 1
Except for some flourine-containing species, the correlations obtained are quite satisfactory. It is interesting to note the excellent correlation among all the hydrides, including those of Table I. The values of czlzz for these molecules are all close to 2.0.
Conclusions We have developed an empirical equation to correlate the bending mode force constants of polyatomic molecules. The correlations obtained among similar molecules for many different types of triatomic and larger molecules are both encouraging and useful. The equation derived thus allows a ready estimation off+for a molecule by using data available for similar molecules. The combined use of the present work on the bending force constant and those on the stretching force constants should therefore assist considerably structural and thermodynamic studies of polyatomic molecules.
