Page 184 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CORE-HOLE STATES AND THE KOOPMANS THEOREM 167

point turn out to be eigenfunctions of a certain ‘effective’ Hamiltonian (which em-
bodies the constraints); and this leads to an iterative procedure parallel to that of
the usual closed-shell SCF theory. For molecules, these ‘canonical’ orbitals are nor-
mally the delocalized MOs which extend over the whole molecular framework; but
is invariant against unitary mixing of the orbitals within each shell (which leaves
the matrices unchanged) and this freedom may be exploited in the usual way
to obtain alternative orbitals with a high degree of localization in different regions
of the molecule (e.g. inner shells, bonds, lone pairs). Clearly, in discussing phe-
nomena related to physically well-defined regions, we shall be more concerned with
the localized orbitals than the canonical MOs. The question that then arises is that
of what localization criterion to adopt: the one to be used in working with atomic
inner shells is simply that the inner-shell orbital be constructed from basis functions
located on the atom in question. All other orbitals are easily orthogonalized against
inner shells (e.g. by the Schmidt method) and among themselves (e.g. by the Löwdin
transformation).
Instead of using repeated solution of a suitable eigenvalue equation to optimize the
orbitals, as in conventional forms of SCF theory, we have found it more convenient
to optimize by a gradient method based on direct evaluation of the energy functional
1
(4), orthonormalization being restored after every parameter variation . Although
many iterations are required, the energy evaluation is extremely rapid, the process is
very stable, and any constraints on the parameters (e.g. due to spatial symmetry or
choice of some type of localization) are very easily imposed. lt is also a simple matter
to optimize with respect to non-linear parameters such as orbital exponents.
3. Some results
We have considered K-shell ionizations from the atoms of carbon, oxygen, and nitro-
gen in a series of small molecules, typically using basis sets of ‘double-zeta’ quality
(as tabulated by, for example, Dunning [12]), with the addition of polarization func-
tions for the smallest systems. The total energies for the neutral molecules and some
of their core-hole positive ions are collected in Table 1. Energies for the molecular
ground states, as calculated by standard (RHF) SCF methods, are also shown for
comparison. It is evident that the localization constraint for the inner shells has a
negligible effect on the energies.
The energies in the last column of 1 show the effect of modifying the basis
set, after the ionization, to allow for the increased central field to which the valence
electrons are then exposed. Some of the early work on the interpretation of ESCA
and Auger spectroscopy employed an ‘equivalent-core’ approximation (Shirley [13])
in which, with a minimal basis set, valence orbitals were given exponents appropriate
to an effective atomic number Z + 1 instead of Z: the inner shell, with only one
Is electron, was thus ‘modelled’ by an 'equivalent core' with two 1s electrons but
one extra unit of positive charge on the nucleus. This simple model has been found
equally effective in the case of a DZ basis: in describing the valence electrons of
an atom with a core hole it is sufficient to use contracted gaussians with tabulated
exponents and contraction coefficients for the atom of atomic number Z + 1 instead
of Z. For the Is orbital, on the other hand, appeal to ‘screening constant’ rules
(Slater [14]; Clementi and Raimondi [15]) suggests that the Gaussian exponents for
1
The valence set is orthogonalized against the core set, so as not to ‘contaminate’ the core
orbitals, while symmetric orthonormalization is employed within each set.