Essentials of Computational Chemistry

8.4 EXCHANGE-CORRELATION FUNCTIONALS 261
Choose basis set(s)
Choose a molecular geometry q (0)
Compute and store all overlap
and one-electron integrals
Replace P (n–1) with P (n)
Choose new geometry
according to optimization
algorithm
no
no
yes
Optimize molecular geometry?
Does the current geometry
satisfy the optimization
criteria?
yes
Output data for optimized
geometry
yes no
Guess initial density matrix P (0)
Construct and solve Kohn–Sham
secular equation
Construct density matrix from
occupied KS MOs
Is new density matrix P (n)
sufficiently similar to old
density matrix P (n–1) ?
Output data for
unoptimized geometry
Figure 8.1 Flow chart of the KS SCF procedure
orbitals are formed. In addition, the density itself may be expanded in an ‘auxiliary’ basis
set. Such a concept may at first seem strange, since we know the density can be represented
as the product of AO basis functions and density matrix elements, as it is in the Coulomb
integrals of HF theory. However, in HF theory this is a natural choice because one needs to
evaluate both Coulomb and exchange integrals, and in the latter the interchange of electronic
coordinates requires that orbitals be used, not densities (which are the product of orbitals).
In Eq. (8.21), however, there are no exchange integrals, so it is computationally convenient
to represent ρ(r) with an auxiliary basis set, i.e.,
ρ(r) =
M
cii(r) (8.28)
i=1