Schrodinger equation self-consistent field theory

This is the basic equation of the self-consistent field theory the reader may notice that it has the same form as Schrodinger s equation. This is the basis of the analogy between polymer chain conformations in interacting systems and the properties of interacting electrons. 

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. 

Lifshitz (1968) has developed a now formalism of polymer theory, b/used on the mathematical analogy between the state equation of a molecular coil with Schrodinger s equation of a quantum mechanical particle placed in an external potential field. This analogy was mentioned in section 3.1 when speaking of Edwards solution of the conformational problem of a molecular coil in a good solvent with the help of the solutions of the diffusion equations for Green s functions in the self-consistent field approximation. 

The use oE self-consistent field molecular orbital theory (SCF-MO) implies a number of approximations which are necessary, since the Schrodinger equation is generally unsolvable for three or more bodies. Thus SCF MO s represent an optimum approximation to the many-electron wave function. 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

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. 

A self-consistent reaction field method (SCRF) has been developed at the level of Hartree-Fock theory to solve the respective Schrodinger equation ... 

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