Page 182 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Core-Hole States and the Koopmans Theorem

C. AMOVILLI and R. McWEENY
Dipartimento di Chimica e Chimica Industriale, Via Risorgimento 35, 56100 Pisa,
Italy

1. Introduction
The theorems of Brillouin [1,2] and Koopmans [3], in both their original and gener-
alized forms, have provided a recurring theme in the work of Gaston Berthier who
always showed a profound appreciation of their significance and importance (see, for
example, [4,5]). Both theorems have been of immense value in the calculation and
interpretation of a wide range of molecular properties. But both are ‘first-order’ the-
orems, based originally on the Hartree-Fock model, and refer to the first-order effect
of perturbations that are considered ‘small’. When the perturbations become large
the theorems lose their value, except as a basis for rough approximations, but the
violations themselves are also of considerable practical importance. In particular, as
every quantum chemist knows, the ionization energy for removal of an electron from
orbital is related to the Hartree-Fock orbital energy according to the Koopmans
theorem, by

where is calculated using the ‘zero-order’ for the unperturbed (neutral) system.
The perturbation of the orbitals in passing from the neutral to the ionized system is
irrelevant to the first-order result. To calculate second- and higher-order corrections
to equation (1), however, it is necessary to allow the orbitals of the ionized system to
‘relax’ in order to describe the perturbation of the Hartree-Fock field caused by the
change in occupation number of orbital Such relaxation effects are often rather
small and the Koopmans result (1) can give a fairly satisfactory interpretation of the
ionization processes observed in valence-electron photoelectron spectroscopy (PES);
but for ‘deep’ ionizations, as observed in ESCA experiments (see, for example, Sieg-
bahn et al [6]) where electrons are knocked out of atomic inner shells, the relaxation
effects can be very large. The electron distribution tends to ‘collapse’ towards the
‘core hole’ – roughly equivalent to an increased nuclear charge – and the use of (1)
commonly yields ionization energies in error by 20–30 eV.
This note is concerned with the alternative procedure in which (1) is replaced by

where E (the electronic energy of a neutral molecule) and (that for the molecule
in a ‘core-hole’ state) are both calculated independently. It must be remarked at
165
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 165–173.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.