Page 257 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

240 R. CARBÓ AND E. BESALÚ
where the primed index vectors mean that the n-th element has been erased from the
initial unprimed vector.

Thus, there is the possible relationship between both n-th and (n-l)-th order
density functions:

It is straightforward to deduce, in general, how to obtain the (n-m)-th term of
the sequence:

The zero-th order term being, finally, the norm of the Slater determinant,
which by means of equation (23) becomes n!, a well known result.
Generalization of this one determinant function to linear combinations of
Slater determinants, defined for example as these discussed in the previous section 5.2,
is also straightforward. The interesting final result concerning m-th order density
functions, constructed using Slater determinants as basis sets, appears when obtaining the
general structure, which can be attached to these functions, once spinorbitals are described
by means of the LCAO approach.

5.3.2. LCAO expression of density functions
Taking into account equation (23), and supposing the Slater determinants
normalized, one can write, calling the initial constant factor v(n,m)=1/(n-m)!:

and using the LCAO approach for the spinorbitals, written as:

where each spinorbital has been expressed as a linear combination of atomic spinorbitals
from a given M-dimensional basis set Then, a product of spinorbitals like
(12) can be structured by means of the linear combination (25) as: