Self-consistent field method total energy

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

The complete active space valence bond (CASVB) method is an approach for interpreting complete active space self-consistent field (CASSCF) wave functions by means of valence bond resonance structures built on atom-like localized orbitals. The transformation from CASSCF to CASVB wave functions does not change the variational space, and thus it is done without loss of information on the total energy and wave function. In the present article, some applications of the CASVB method to chemical reactions are reviewed following a brief introduction to this method unimolecular dissociation reaction of formaldehyde, H2CO — H2+CO, and hydrogen exchange reactions, H2+X — H+HX (X=F, Cl, Br, and I). 

The Hartree-Fock or self-consistent field (SCF) method is a procedure for optimizing the orbital functions in the Slater determinant (9.1), so as to minimize the energy (9.4). SCF computations have been carried out for all the atoms of the periodic table, with predictions of total energies and ionization energies generally accurate in the 1-2% range. Fig. 9.2 shows the electronic radial distribution function in the argon atom, obtained from a Hartree-Fock computation. The shell structure of the electron cloud is readily apparent. 

In the quantitative development of the structure in the self-consistent field approximation (S.C.F.) using the Hartree-Fock method the energy Ei is made up of three terms, one for the mean kinetic energy of the electron in ipi, one for its mean potential energy in the field of the nuclei, and a correction term for the effect of all the other electrons. The total energy... 

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. 

---