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Approximate energy relationships for molecules Ajit J. Thakkar Citation: J. Chem. Phys. 79, 523 (1983); doi: 10.1063/1.445506 View online: http://dx.doi.org/10.1063/1.445506 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v79/i1 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 26 Aug 2013 to 128.252.67.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

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Page 1: Approximate energy relationships for molecules

Approximate energy relationships for moleculesAjit J. Thakkar Citation: J. Chem. Phys. 79, 523 (1983); doi: 10.1063/1.445506 View online: http://dx.doi.org/10.1063/1.445506 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v79/i1 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

Downloaded 26 Aug 2013 to 128.252.67.66. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 2: Approximate energy relationships for molecules

NOTES

Approximate energy relationships for molecules Ajit J. Thakkar

Department o/Chemistry. University 0/ Waterloo. Waterloo. Ontario. Canada N2L 3G1 (Received 9 March 1983; accepted 18 March 1983)

According to Hartree-Fock theory, the total energy E of a molecule is related to its orbital energies to I by

E=LllltOl+ Vnn - Vee, (1) I

where Vnn and Vee are the total nuclear and electronic repulsion energies respectively, and the III are orbital occupation numbers. In semiempirical theories the approximation

(2)

is frequently used with A = 1. Convincing arguments have been presented to suggest that A = 3/2 provides a good approximation for atoms1 and molecules at their equilibrium geometries. 2 ,3 An empirical value of A = 1. 55 has also been proposed. 2

Applying approximation (2) to the separated atoms as well as the molecule at equilibrium leads to the following expression for the binding energy:

(3)

Using the atomic version (Vnn = 0) of the exact relation­ship (1) to eliminate Esa from Eq. (3), one obtains

The special case A = 1 of this approximation (4) has been known for many years, 4 and was shown recently5 to be numerically superior to approximation (2). However, binding energies obtained from Eq. (3) with A = 1 are much too large. 4 Thus it seems that approximation (4) with a smaller value of A would be an even better energy formula.

Approximations (2) and (4) with various values of A have been tested on SCF data for 55 molecules. 6 Table I shows that Eq. (4) with A'" 1/3 is almost three times as accurate as Eq. (4) with A = 1 which, in turn, is roughly four times as accurate as Eq. (2) with A = 1. 55.

It is interesting to consider a formula in which all three terms of Eq. (4) have empirical weights. Sur­prisingly a fit to such a formula indicates that the first

TABLE I. Root mean square percent errorsa

for various energy formulas.

Formula

Eq. (2), A = 3/2 Eq. (2), A = 1. 55b

Eq. (4), A=l Eq. (4), A = 1/3 Eq. (4), A=0.28b

Eq. (5), B=1.048, C=0.924b

rms % error

4.89 4.05 0.941 0.343 0.338 0.163

aBased on SCF data for 55 molecules. See Ref. 6.

IIoptimum parameters.

term has a coefficient that almost vanishes. This sug­gests a molecular energy formula in terms of purely atomic quantities

(5)

Table I shows that Eq. (5), with best fit parameters, is the most accurate energy formula considered in this paper being about twice as accurate as Eq. (4) with a best fit value of A. It is worth reiterating that Eq. (5) is applicable to molecules at their equilibrium geom­etries only.

This work was supported by the Natural Sciences and Engineering Research Council of Canada.

IN. H. March and J. S. Plaskett, Proc. R. Soc. London Ser. A 235, 419 (1956).

2K. Ruedenberg, J. Chem. Phys. 66, 375 (1977). 3N. H. March, J. Chem. Phys. 67, 4618 (1977). 4J. Goodisman, J. Am. Chem. Soc. 91, 6552 (1969). 5E. Donati, E. A. Castro, andF. M. Fernandez, Int. J.

Quantum Chem. 22, 429 (1982). 6The SCF data are of double zeta quality and were taken from

L. C. Snyder and H. Basch, Molecular Wave Functions and Properties (Wiley, New York, 1972). H2 was excluded followingP. Politzer, J. Chem. Phys. 64,4239 (1976), and Ref. 5. Note that the orbital energy sums needed were ob­tained from values of E, Ve8 and V", using Eq. (1). The atomic SCF data were also taken from Snyder and Basch.

J. Chern. Phys. 79(1), 1 July 1983 0021-9606/83/130523-01$02.10 © 1983 American Institute of Physics 523

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