Hartree-Fock approximation operator

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Calculations show that cross-sections obtained in the Hartree-Fock approximation utilizing length and velocity forms of the appropriate operator, may essentially differ from each other for transitions between neighbouring outer shells, particularly with the same n. However, they are usually close to each other in the case of photoionization or excitation from an inner shell whose wave function is almost orthogonal with the relevant function of the outer open shell. In dipole approximation an electron from a shell lN may be excited to V = l + 1, but the channel /— / + prevails. For configurations ni/f1 n2l 2 an important role is... 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

From here it is obvious that for the Hartree-Fock approximation the parametrization of the energy not referring directly to the wave function is nevertheless possible (the Hartree-Fock density is a projection operator and it can be directly written using say eq. (1.107)), but the cost is fixing x = 0 with the consequences of this. (We can say that the works devoted to foundations of DFT basically reduce to developing a more or less widely applicable form of the two-electron density.)... 

In recent years, dramatic advances in computational power combined with the marketing of packaged computational chemistry codes have allowed quantum chemical calculations to become fairly routine in both prediction and verification of experimental observations. The 1998 Nobel Prize in Chemistry reflected this impact by awarding John A. Pople a shared prize for his development of computational methods in quantum chemistry. The Hartree-Fock approximation is a basic approach to the quantum chemical problem described by the Schrodinger equation, equation (3.10), where the Hamiltonian (//) operating on the wavefunction OP) yields the energy (E) multiplied by the wavefunction. 

The potentials V = V n) and V2 = V(r2) include all the nucleus and extra electron contributions to the Coulomb field acting on electrons 1 and 2. In other words, we operate in the framework of the Hartree-Fock approximation i.e., the action of extra electrons over electrons 1 and 2 is taken into account through a mean-field approximation. A similar Hamiltonian Hxb may be written for the fragment XB (with Hax = Uxb when A = B). In this article we shall exclusively consider the fragment AX of the entity AXB. Thus, all the physical quantities globally labelled Qxb and derived from the Hamiltonian Hxb will... 

Average values of the four-operator terms are factorized in the Hartree-Fock approximation and are expressed in terms of the polarization functions and the populations defined by... 

Generally speaking, it is probably impossible to give any general recommendations concerning the search for the observables, the operators of which commute with the Hamiltonian. The exceptions are the cases when the observable to be found characterizes the properties of the spatial symmetry of the ket-vector v /(t)> (in Schroedinger s spatial presentation this vector is called the wave function). It should be noted that a more or less precise pattern of the wave function /(t) is known for very few molecular system. At present the most widespread is the i /(t) presentation in the Hartree-Fock approximation as a symmetrized linear combination of atomic orbitals (LCAO) [16]. 

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 ... 

Analysis of Green s functions can be useful in seeking to establish model hamil-tonians with the property of giving approximately correct propagators, when put in the equations of motion. In this section, we explore a particularly simple model in order to familiarize the reader with various molecular orbital concepts using the terminology of Green s function theory. We employ the Hartree-Fock approximation and seek the molecular Fock operator matrix elements... 

There is a formal exact correspondence between the matrix elements fra and the Fock operator in the Hartree-Fock approximation, i.e., the expression... 

In the Hartree-Fock approximation, the wave function of an atom (or molecule) is a Slater determinant or a linear combination of a few Slater determinants [for example, Eq. (10.44)]. A configuration-interaction wave function such as (11.17) is a linear combination of many Slater determinants. To evaluate the energy and other properties of atoms and molecules using Hartree-Fock or configuration-interaction wave functions, we must be able to evaluate integrals of the form (H b D), where D and D are Slater determinants of orthonormal spin-orbiteils and B is an operator. 

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