Page 183 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

166 C. AMOVILLI AND R. McWEENY
the outset that since the energy of a ‘core-hole’ state normally lies high in the con-
tinuum, relative to the lowest energy state in which the hole has been filled by an
Auger transition from a valence orbital, there are severe problems in calculating the
energy by conventional bound-state methods: indeed the corresponding ‘state’ is
not a true bound state at all, being at best metastable and subject to spontaneous
decay, with filling of the hole and ejection of a second (Auger) electron. A completely
satisfactory calculation would thus require the inclusion in the basis of continuum
functions, to admit the possibile presence of a scattered electron, and would employ
propagator methods which are well adapted to the description of such processes (see,
for example, Agren [7]). Nevertheless, bound-state methods have been widely and
successfully used in the interpretation of PES, ESCA and Auger spectra. In partic-
ular, the formulation of SCF methods for systems containing incompletely occupied
shells (McWeeny [8]) was applied by Firsht and McWeeny [9] to free atoms and ions,
with inner-shell holes, yielding results of much higher accuracy than those based on
the Koopmans theorem. The present paper reports applications of similar methods
to some small molecules.
2. Formulation
For inner-shell ionizations, where the energy change may be several hundred eV, it
is sufficient to use ensemble averaging (Slater [10]; McWeeny [8]) over the various
states of a configuration - which differ relatively little in energy. The corresponding
formulation of many-shell SCF theory is fully described elsewhere (McWeeny [11])
and will be summarized only briefly. We use to denote the orbitals of Shell
K, containing electrons, and express the orbitals of all shells in terms of a common
set of TO basis functions { }: thus, collecting the functions in row matrices,

where is an matrix, n being the total number of orbitals employed. It is
also convenient to partition the row matrix into subsets and the rectangular
matrix into corresponding blocks . The set of occupation numbers
then defines the electron configuration, while the average energy
(for all states with the same partitioning of electrons among shells) is given by

Here is the fractional occupation number of the spin-orbitals
of Shell K and is a suitably averaged electron interaction matrix (cf. the usual
Roothaan matrix') and depends on the density matrices of all
shells: in fact

where the modified occupation number (removing the self-interaction when
is The matrix G(2R) coincides with the usual G matrix
for a closed-shell system, while h in (4) is the usual 1-electron Hamiltonian matrix.
All matrices are defined with respect to the basis functions in
The energy expression (4) applies when the orbitals are orthonormal and in seeking
a stationary value it is thus necessary to introduce constraints to maintain orthonor-
mality during a variation. When this is done, the orbitals that give a stationary