Self-consistent field theory convergence

Roothaan, C. C. J., Detrich, J. H. (1983). General quadratically convergent multiconfiguration self-consistent-field theory in terms of reduced matrix elements. Phys. Rev. A 27,9-56. 

The physical reasoning for why these densities were frequently employed in the earlier days of density functional theory was that in this way the degeneracy of the partially filled d-orbitals could be retained. A technical reason why these densities still have to be employed in some recent investigations is that calculations with integral orbital occupations simply do not converge in the self consistent field procedure (see, e. g., Blanchet, Duarte, and Salahub, 1997). Such densities correspond to a representation of a particular state 2S+1L with Mg = S and a spherical averaging over ML. 

Since the electric field, computed from the filled / is used to construct the Coulomb operator in T, the electric field that is used to constmct T from the converged orbitals is the same as the electric field that is computed from the orbitals that solve Eq. 3 for this Fock operator. Therefore, at convergence, both the orbitals and the electric field computed from them are self-consistent. Consequently, HF theory is also known as self-consistent field (SCF) theory. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

An important aspect of density-functional theory (like Hartree-Fock and other mean-field theories) is that the Schrodinger equation (9) has to be solved self-consistently, because the potentials Vn and V,c depend on the wavefunctions themselves. In its simplest form this means making an initial guess at the effective potential (7), then using the output wavefunctions and charge density to construct a new potential and iterating till convergence. 

The equilibrium in this theory entails self-consistent solutions of the potentials. The potential fields are themselves functionals of local monomer densities, which, at the same time, are determined by the potential field. Hence, the corresponding equations are solved self-consistently, with the potentials being updated at every iteration step until convergence. This technique amounts to the assumption that a single-field configuration dominates, and consequently all other configurations (or fluctuations ) can be neglected. 

---