Hartree equations

In this case, the individual orbitals, (ti/r), can be detennmed by minimizing the total energy as per equation Al.3,3. with the constraint that the wavefiinction be nomialized. This minimization procedure results in tire following Hartree equation ... 

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... 

While Dirac [3] chose to solve Eq. (4) as a quadratic equation for in terms of the Hartree potential yHC "), it was Slater in 1951 ([6] see also [4]) who chose an alternative, and more fruitful, route by regarding Eq. (4) as demonstrating that it could be viewed as a modified Hartree equation, with the Hartree potential Unfr) now supplemented by the exchange n -potential (the so-called Dirac-Slater (DS) exchange potential), to yield a total one-body potential energy... 

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)... 

Extension of Hartree s method to an iV-electron atom is straightforward, in principle. Each electron now moves in the field of the nucleus plus the overlapping charge clouds of the iV— 1 other electrons. The N coupled Hartree equations ... 

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 ... 

A numerical solution of the self-consistent Hartree equations (Nosanow and Shaw, 1962) showed that the exact one-particle wavefunction can be approximated very well by a gaussian. Therefore we use harmonic oscillator wavefunctions as trial wavefunctions, with parameters to be determined variationally ... 

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. ... 

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