ABSTRACT
We established the algebraic task of solving the Roothaan-Hall self-consistent field (SCF) equations [1] in a previous chapter, and illustrated the process of solving the equations for some very simple systems. Here we show how the program PCLOBE processes the SCF equations and evaluates derivatives of the SCF energy. These allow efficient search for relative minima in the potential energy and also allow estimation of the harmonic frequencies of a molecule. For the perfect pairing case the energy is
E ¼ 2 X imn
CimCinhmjhjni þ X ij
X mnrs
CimCjrCinCjs (2hmrjnsi hmrjsni)
The Fock matrix in the basis is
Fmn ¼ hmjhjni þ X jrs
CjrCjs(2hmrjnsi hmrjsni)
and the energy of orbitals-the diagonal elements of the Fock matrix-and the total energy are
Fii ¼ X mn
CimCinFmn
E ¼ X imn
CimCin(Fmn þ hmn)
The equations to be solved to self-consistency are
X FmnCvi ¼ SmnCnilii
The purpose of this section is to make our way through a full SCF calculation on our central example, formaldehyde. This molecule is small and symmetric enough to treat in detail. It displays a broad array of interesting experimental properties, which we will explore further in later chapters. The structure of the calculation is, of course, similar to that we sketched for SCF1s, but we will delve a bit deeper at each step.
