The Hartree Equation

The cornerstone of semiempirical and ab initio molecular orbital methods is the Harhee equation and its extensions and variants, the Harhee-Fock and Roothaan-Hall equations. We have seen that the Hamiltonian for the hydrogen atom. 

Computational Chemistry Using the PC, Third Edition, by Donald W. Rogers ISBN 0-471-42800-0 Copyright 2003 John Wiley Sons, Inc. 

For a larger molecule wdth V nuclei and n electrons, we unite 

Undei the Born Oppenheinier approximaii.on. the nuclei ai e assumed to be so much more massive and slow moving than the electrons that their motions are independent and can be treated separately. This permits the Hamiltonian in Hq, (9-4) to be separated into tw o parts, one that refers to nuclei only 

Even w hen we are working solely w ith the electronic Hamiltonian (9-6), w e must remember that the nuclei do move, albeit relatively slowly, with respect to 

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The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

Since the potential depends on the wave functions, and vice versa, the Hartree equation can only be solved by an iterative procedure. 

The condition that the energy becomes a minimum through a variational procedure, together with the conditions f fa(j>jdvidvj = Sij gives the Hartree equations... 

Note that Eq. (1.7) defines a set of equations, one for each electron.) Solving for the set of functions 4>j is nontrivial because itself depends on all of the functions. An iterative scheme is needed to solve the Hartree equations. First, a set of functions (d i, 02> > is assumed. These are used to produce the set of effective potential operators V , and the Hartree equations are solved to produce a set of improved functions 0,. These new functions produce an updated effective potential, which in turn yields a new set of functions d ,. This process is continued until the functions 4>i no longer change, resulting in a self-consistent field (SCF). 

This is closely analogous to the Hartree equations (Eq. (1.7)). The Kohn-Sham orbitals are separable by definition (the electrons they describe are noninteracting) analogous to the HF MOs. Eq. (1.50) can, therefore, be solved using a similar set of steps as was done in the Hartree-Fock-Roothaan method. 

Then the variational calculation leads to just the Hartree equation Eq, (A-7), with an exchange potential called free-electron exchange—or exchange, since in this field, n(r)... 

A partial justihcation for the interpretation of the KS eigenvalues as starting point for approximations to quasi-particle energies, common in band-structure calculations, can be given by comparing the KS equation with other self-consistent equations of many-body physics. Among the simplest such equations are the Hartree equation... 

It is directly analogous to the Hartree equation (8.1), except for an additional term 14c(t) to the average Hartree electrostatic potential Vn r) (denoted by VJf(r) in (8.1)). This means that each electron feels an extra attractive potential, the... 

Since the Hartree-Fock equation is a nonlinear equation, similarly to the Hartree equation, it is usually solved by the SCF method. The Hartree-Fock SCF method is carried out via the following process ... 

Later, J. C. Slater showed that the Hartree equations can be obtained if the variation principle is applied to a product of spin orbitals. The Russian theoretical physicist Vladimir A. Lock pointed out that certain symmetry conditions are not obeyed in the Hartree method, of which the most important one is the antisymmetric property of the total wave function. The variation principle was now applied to an antisymmetrized product of spin orbitals, that is, a Slater determinant. This is a fnndamen-tal method in electronic structure calculations and is referred to as the Hartree-Fock method or simply Hartree-Eock. ... 

The most common initial approach to molecular quantum mechanics is the Hartree-Fock (HF) calculation, introduced in Section 4.2. Applying the Born-Oppenheimer approximation, we permit ourselves to solve only the electronic wavefunction initially, for some fixed geometry of the nuclei. Therefore, the approach is no diflferent for molecules than for atoms, except that the potential energy includes contributions from more than one nucleus. For simplicity here, we will confine ourselves to the Hartree equation, rather than the HF equation that is applied to Slater determinants. Our Eq. 4.26 for the atom becomes... 

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