J. Am. Chem. Soc. 1990,112, 8533-8542
8533
Additivity Methods in Molecular Polarizability Kenneth J. Miller Contribution from the Department of Chemistry, Rensselaer Polytechnic Institute, Troy, New York 12180-3590. Received January 8. 1990
Abstract: Empirical methods to calculate average molecular polarizabilities based on atomic hybrid parameters are reexamined.
New sets of optimized atomic hybrid components, sA(ahc),and atomic hybrid polarizabilities, aA(ahp),are presented. They yield average molecular polarizabilities with complementary empirical formulas, a(ahc) = (4/N)[E:,T~(ahc)]*( N = number of electrons in the molecule) and a(ahp) = x:,cYA(ahp), which reproduce the experimental polarizabilities to an average error of 2.2%and 2.8% respectively for approximately 400 compounds containing H, C, N, 0,S, F, CI, B, I, and P. Only one parameter, 7 A or aA, in each formula is required for each atom, A, in a particular atomic hybridization state. Conjugate parameters, cyA*(ahp)= ( 4 / N ~ ) T ~ ( a hand c ) ~~ ~ * ( a h=p )[NAuA(ahp)/4]'/*(NA= number of electrons in atom A) used as alternate sets in these two empirical formulas, yield average molecular polarizabilities to a root-mean-square standard deviation of 5.6% and 6.4%, respectively. These two partitioning schemes are discussed. The atomic hybrid polarizabilities can be combined to generate bond polarizabilities, bp and two possible sets are discussed. In addition, the atomic polarizabilities are related to group polarizabilities,gp. The atomic hybrid methods require fewer parameters than the bond or group polarizability methods, but the ahc, ahp, bp, and gp methods yield comparable results.
Introduction The most successful empirical approaches to the calculation of average molecular polarizabilities include the environment through atomic hybridization about the central atom or with the atoms defining a bond or groups. Miller and Savchik' were the first to propose a method that features atom hybridization (ah) in which each atom is characterized by its state of atomic hybridization. They used a functional form a(ahc) = ( 4 / N ) [ z T ~ ( a h C ) ] ~ A
(1)
where 7 A is an atomic hybrid component (ahc) for each atom A in a given state of hybridization, and N is the total number of electrons. This formula was based on a theoretical interpretation of variational perturbation results'4 and interpreted with molecular orbital theory.' Their optimized atomic hybrid components, ~ ~ ( a h creproduce ), a(ahc) to I-3% and within experimental error for most of the approximately 240 molecules studied. They wrote the average atomic polarizability as using the set of f A from the ahc optimization, where N A is the number of electrons in atom A. aA*(ahc) will be referred to as the conjugate (*) ahc polarizability. In this paper it will be useful to compute the conjugate ahc molecular polarizability a*(ahc) = xCVA*(ahC) A
(3)
to compare the two schemes of partitioning, eqs 1 and 3, and the conjugate relationship through eq 2. Subsequently, Yoffe and MaggioraS used eqs 2 and 3 in a study of London dispersion interaction energies and Kang and Jhon6 tabulated values of a*(ahc) for approximately 100 molecules. a*(ahc) are consistently larger than a(ahc) by only I-5% in most cases studied. Kang and Jhon6 refined the atomic polarizabilities to obtain an optimum set of atomic hybrid polarizabilities, aA(ahp), which reproduce the molecular polarizabilities 4 a h p ) = xaA(ahp) A
to approximately 1-3%.
(4)
In this paper it will be convenient to
( 1 ) Miller, K. J.; Savchik, J. A. J . Am. Chem. SOC.1979, 101, 7206. ( 2 ) Hylleraas, E. Z . Phys. 1930,65, 209. ( 3 ) Hasse, H. R . Proc. Cambridge Philos. SOC.1930, 26, 542. ( 4 ) Hirshfelder, J. 0.;Curtiss, C. F.; Bird, R . B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954; pp 942-948. ( 5 ) Yoffe, J. A.; Maggiora, G. M. Theor. Chim. Acta 1980, 56, 191. ( 6 ) Kang. Y . K.; Jhon, M. S.Theor. Chim. Acia 1982, 61, 4 1 .
define a conjugate (*) ahc parameter TA*(
a hp) =
a hp) /4] ' I 2
[NACYA(
(5)
calculated with the optimum parameters aA(ahp) from which the conjugate (*) ahp molecular polarizability becomes a*(ahp) = ( 4 / N ) [ x T ~ * ( a h P ) l ~
(6)
to relate eqs 4 and 6 through eq 5 . Comparisons between the ahc and ahp methods of eqs 1 and 4 calculated with newly optimized sA(ahc)and aA(ahp) and the conjugate methods of eqs 3 and 6 will be made to examine the interchangability of parameters, the equivalence of the two methods of partitioning the molecular polarizability, and the atomic size. The simplicity of the ahc and ahp atomic hybrid methods is understood by noting that the molecular polarizability cannot be written as a sum of atomic polarizabilities7~*when atoms are defined only by the atomic number. Ei~enloh?.~ and more recently Vogel'O set up an extensive system of atomic refractions" that have been supplemented by Batsanov12in which each atom must be assigned a polarizability depending on the atoms to which it is bonded. Silberstein7 pointed out that atomic polarizabilities are not additive unless the atomic environment is considered in great detail. The methods that consider atoms in atomic hybrid states eliminate the need to develop an extensive set of rules that consider the specific atoms involved in bonding. A third method of calculating molecular polarizabilities originated with the early work by von Steiger,I3 Smyth,I4 and Denbigh,Is who demonstrated additivity of bond polarizabilities (bp)
(7) where the sum is over all bonds B in the molecule. Denbighis and Vickery and Denbigh16 obtained a set of aB(bp) that reproduce a(bp) to I-2% for 52 molecules, and Vogel et al." have obtained ~
(7) Silberstein. L. Philos. Mag. 1917, 33, 92. (8) Eisenlohr, F. Z . Phys. Chem. (Leipzig) 1911, 75, 5 8 5 . ( 9 ) Eisenlohr, F. 2.Phys. Chem. (Leipzig) 1912, 79, 129. (IO) Vogel, A . J . Chem. SOC.1948, 1833. ( 1 I ) Molar refraction R (cm') is related to polarizability through Avo-
gadro's number, L, by a@') = 3R/(4nL) = 0.3964R. ( I 2 ) Batsanov, S.S.Refracfometry and Chemical Structure; Consultants Bureau: N e w York, 1961. ( 1 3 ) von Steiger, A. L. Berichte 1921, 54, 1 3 8 1 . ( 1 4 ) Smyth, C. Phil. Mag. 1925, 50. 361. ( 1 5 ) Denbigh, K. G.Trans. Faraday SOC.1940. 36. 936. ( I 6 ) Vickery. B.;Denbigh, K. Trans. Faradqv SOC.1949, 45. 61,
0002-7863/90/15 12-8533$02.50/0 0 1990 American Chemical Societv
Miller
8534 J . Am. Chem. SOC.,Vol. 112, No. 23, 1990 Table I. Parameters for Atoms in Hybrid Configurations TA(ahc),d aA(ahp),d symbol’ hybridb groupC A312 A)
aA*(ahc):
A’
TA*(ahp):
pA(ahc)/
,4312
A
pA(ahp)/
A
pA(exp)/
A
aA(exp),I A3
0.3 13 1.089 3.165 5.566 8.593 1.294
0.387 0.296 2.315 3.013 5.415 1.061
0.392 0.527 2.357 3.541 5.573 1.116
0.31 1 0.816 3.137 5.135 8.470 1.262
17227 I .322 1.922 2.128 2.383 1.594
1.223 1.144 1.913 2.044 2.366 1.574
1.200 1.470 1.750 1.850 1.980 1.700
0.358 0.807 1.620 2.023 2.655 1.443
\c-
1.433
1.352
1.369
1.424
1.678
1.673
1.700
1.443
