Hartree-Fock approximation self energy

Invoking the Hartree-Fock approximation (HFA) (Salem 1966), means that we can replace Una-a in (4.33) by an averaged self-energy U na-a), whereby an effective adatom level of spin a is defined by2... 

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 one for its mean potential energy in the field of the nuclei, and a... 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

Practical calculations require approximations in the self-energy operator. Perturbative improvements to Hartree-Fock, canonical orbital energies can be generated efficiently by neglecting off-diagonal matrix elements of the selfenergy operator in this basis. Such diagonal, or quasiparticle, approximations simplify the Dyson equation to the form... 

This equation is the main result of the present considerations. In order to define the two-particle self energy (w) and for establishing the connection to the familiar form of Dyson s equation we adopt a perturbation theoretical view where a convenient single-particle description (e. g. the Hartree-Fock approximation) defines the zeroth order. We will see later that the coupling blocks and vanish in a single-particle approximation. Consequently the extended Green s function is the proper resolvent of the zeroth order primary block which can be understood as an operator in the physical two-particle space ... 

Since the effective potential is a function of the density, which is obtained from Eq. (2.79) and hence depends on all the single-particle states, we will need to solve these equations by iteration until we reach self-consistency. As mentioned earlier, this is not a significant problem. A more pressing issue is the exact form of [n(r)] which is unknown. We can consider the simplest situation, in which the true electronic system is endowed with only one aspect of electron interactions (beyond Coulomb repulsion), that is, the exchange property. As we saw in the case of the Hartree-Fock approximation, which takes into account exchange explicitly, in a uniform system the contribution of exchange to the total energy is... 

The variational principle Even when the wavefunctions are not accurate, as in the case of the Hartree-Fock approximation, the total energy is not all that bad. Energy differences between states of the system (corresponding to different atomic configurations, for which the single-particle equations are solved self-consistently each time), turn out to be remarkably good. This is because the optimal set of single-particle states contains most of the physics related to the motion of ions. The case of coherent many-body states... 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

In the latter expression, the derivative is evaluated at the converged energy. Diagonal self-energy approximations therefore subject a frozen Hartree-Fock orbital < F(x) to an energy-dependent correlation potential Epp(E). 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

---