The SCF Method

Roothaan's basis set expansion method simplified the Hartree-Fock equations into a set of matrix equations comprised of matrix elements between basis functions and operators. The matrix elements are able to be evaluated using either analytical or numerical techniques, but Roothaan's method still has not solved one major problem: examination of the elements of the Fock matrix, equation (1.59), reveals that the Fock operator depends on the very LCAO-MO coefficients one is trying to find, creating a difficult non-linear problem. The solution to this problem is to guess an initial form of the Fock matrix, typically the core Hamiltonian matrix h_uv, and generate an initial set of LCAO-MO coefficients using the process discussed at the tail end of section 1.8. From this initial set of coefficients one generates a better Fock matrix that can be used to get new coefficients and so on. This process is iterated until the LCAO-MO coefficients change by an amount less than some tolerance, i. e. until the system reaches self-consistency. The name given to this method is the self-consistent-field (SCF) method and it is one of the most important techniques in modern quantum chemistry. The full SCF procedure is provided in Table 1.3 for convenience.

 

Table 3: The SCF procedure

1	Specify molecular geometry and basis set
2	Compute required one and two electron integrals
3	Construct initial Fock matrix by setting Duv in Eq. (1.45) to zero
4	Transform Fock matrix according to Eq. (1.66).
5	Diagonalize transformed Fock matrix to find C
6	Reverse the transformation in Eq. (1.68) to get C from C
7	Construct new density matrix and compute energy
8	Check convergence using energy differences or RMS differences of
	Duv elements
9	Quit if converged, otherwise recompute Fock matrix from Eq. (1.45)
	using the new density matrix, and then repeat steps 5-9