Semiempirical Models - Molecular Modelling - page 324

18.18 Effective Core Potentials

The ground state of the silver atom has electron configuration
Ag : (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 (3d) 10 (4s) 2 (4p) 6 (4d) 10 (5s) 1

and much of its chemistry can be explained in terms of the outer 5s electron. The remaining
46 electrons seem to form an inert core and this suggests that we might try to model the
silver atom as a nucleus of charge Z

47 and an inner shell of 46 electrons. This idea was
tried in the very early days using potentials such as

=

n V
4πε 0 r +

A exp (

− 2 kr )

U core =− (18.17)

r

where n V is the number of valence electrons and A and k are constants that have to be
determined by fitting an atomic property. Use of such a potential means that we have
eliminated most of the two-electron integrals.
The idea of dividing electrons into groups is quite common in chemistry; in CNDO
theory we treat the valence electrons only, without explicitly treating the core electrons. In
ZDO π -electron models we treat the π -electrons explicitly and assume that the effect of the
σ -electrons can be taken into account in the parameter choice. There is a serious problem
that I can explain by reference to silver. The 5s orbital has to be orthogonal to all the inner
orbitals, even though we do not use them (and do not necessarily want to calculate them).
Many ab initio packages allow the use of effective core potentials (ECPs), which replace
the atomic cores in valence-only ab initio calculations; traditionally they were represented
as linear combinations of functions of the type
r − n exp − α r 2

with coefficients and exponents determined by fitting the potential generated from accurate
HF-LCAO wavefunctions. In recent years it has become fashionable to represent the core
potentials by one- and two-term Gaussians obtained directly from the appropriate atomic
eigenvalue equation.

18.19 Problem Set

A set of problems, together with solutions, is available on the associated website.

References

Bingham, R.C., Dewar, M.J.S. and Lo, D.H. (1975) J. Am. Chem. Soc. , 97 , 1285.
Del Bene, J. and Jaffe, H.H. (1968) J. Chem. Phys. , 48 , 1807.
Dewar, M.J.S. and Thiel, W. (1977) J. Am. Chem. Soc. , 99 , 4907.
Dewar, M.J.S., Jie, C. and Yu, J. (1993) Tetrahedron , 49 , 5003.
Dewar, M.J.S., Zoebisch, E.G., Healey, E.F. and Stewart, J.J.P. (1985) J. Am. Chem. Soc. , 107 , 3902.
Hoffmann, R. (1963) J. Chem. Phys. , 39 , 1397.
Hückel, E.P. (1931) Z. Physik , 70 , 204.