Total electronic energy, self-consistent field

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 problem is now solved again by an iterative process, which starts with a choice of the x set and the expansion (6.58). The Hartree-Fock operator F and the matrix representation Fx are calculated, (6.64) is solved for the orbital energies, and these are used to compute a new set of coefficients in (6.63). If these are different from the starting set, the cycle is repeated until the self-consistent-field limit is reached. The total electronic energy is obtained by adding the SCF energy to the core energy for the lowest occupied n/2 levels ... 

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. 

A very important conceptual step within the MO framework was achieved by the introduction of the independent particle model (IPM), which reduces the AT-electron problem effectively to a one-electron problem, though a highly nonlinear one. The variation principle based IPM leads to Hartree-Fock (HF) equations [4, 5] (cf. also [6, 7]) that are solved iteratively by generating a suitable self-consistent field (SCF). The numerical solution of these equations for the one-center atomic problems became a reality in the fifties, primarily owing to the earlier efforts by Hartree and Hartree [8]. The fact that this approximation yields well over 99% of the total energy led to the general belief that SCF wave functions are sufficiently accurate for the computation of interesting properties of most chemical systems. However, once the SCF solutions became available for molecular systems, this hope was shattered. 

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. 

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