Hartree-Fock theory. Slater determinant

In the second place, a quite useful characteristic of LS-DFT is that it renders possible to transform an arbitrary wavefunction, say, the Hartree-Fock single Slater determinant into a locally-scaled one associated with a given one-particle density such as the exact one. Thus, one can easily generate a locally-scaled Hartree-Fock wavefunction that yields the exact p. In this sense, one finds much common ground between LS-DFT and those constructive realizations of the constrained-search approach which reformulate the Hartree-Fock method as well as with those developments which pose the optimized potential method as a particular instance of density functional theory [42,43,57-61]. 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

In Hartree-Fock theory, the many-electron wave function is expressed as a single Slater determinant, conveniently abbreviated as... 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

Ab initio Hartree-Fock theory is based on one single approximation, namely, the N-dectron wavefunction, F is restricted to an antisymmetrized product, a Slater determinant, of one-electron wavefunction, so called spin orbitals,... 

The approximation schemes that are discussed in this paragraph are all based on the Hartree-Fock theory, i.e., the many-electron wavefunction is described in terms of a single Slater determinant. 

In its simplest formulation at the level of the Hartree-Fock theory, the wave function has the form of a Slater determinant... 

As an approach analogous of nonrelativistic Hartree-Fock theory, the four-component Dirac-Hartree-Fock wave function is described with a Slater determinant of one-electron molecular functions ( aX l= U Nelec, ... 

To this end, Kohn and Sham assumed that the electrons in these so-called Kohn-Sham orbitals are non-interacting, such that the total electronic wave function can be written as a Slater Determinant. This allows the kinetic energy functional to be split into two parts, one of which, Ts, can be evaluated exactly, in a fashion very similar to the way it is done in Hartree-Fock theory, and a small correction term, which is formally absorbed in the exchange-correlation energy term. Thus, a general DFT energy expression can be written as... 

Table 3. Total ground-state energies of noble gas and closed s-subshell atoms as determined within Slater theory, the Work-interpretation Pauli-correlated approximation, and Hartree-Fock theory. The negative values of the energies in atomic units are quoted...

In Hartree-Fock theory, each electron is assigned to a molecular orbital, and the wave function is expressed as a single Slater determinant in terms of the molecular orbitals. For a system with nei electrons the wave function is then given as... 

Hartree-Fock theory employs a single Slater determinant. In the restricted Hartree-Fock (RHF) method, one spatial function 4>i is multiplied by an a (representing spin up, spin quantum number ms = +j) or P (representing spin down, nis = — ) spin function with the properties... 

Section 3.2 constitutes a derivation of the results of the previous section. The order of presentation of these two sections is such that the derivations of Section 3.2 can be skipped if necessary. For a fuller appreciation of Hartree-Fock theory, however, it is recommended that the derivations be followed. We first present the elements of functional variation and then use this technique to minimize the energy of a single Slater determinant. A unitary transformation of the spin orbitals then leads to the canonical Hartree-Fock equations. 

Until now the Hartree-Fock approximation has been viewed as an approximation in which the Hamiltonian is exact but the wave function is approximated as a single Slater determinant. For later use in the perturbation theory of Chapter 6, we now preview a different but equivalent view of Hartree-Fock theory that focuses on the Hamiltonian. 

Chapter 2 introduces the basic techniques, ideas, and notations of quantum chemistry. A preview of Hartree-Fock theory and configuration interaction is used to motivate the study of Slater determinants and the evaluation of matrix elements between such determinants. A simple model system (minimal basis H2) is introduced to illustrate the development. This model and its many-body generalization N independent H2 molecules) reappear in all subsequent chapters to illuminate the formalism. Although not essential for the comprehension of the rest of the book, we also present here a self-contained discussion of second quantization. 

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