Hamiltonian Hartree-Fock approximation

The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. 

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. 

Recent application of the TB method to transition metal clusters often made use of a convenient formulation in the language of second quantization.14 In this formalism, the TB Hamiltonian in the unrestricted Hartree-Fock approximation can be written as a sum of diagonal and nondiagonal terms15... 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

Computationally, the simplification introduced by the Hartree-Fock approximation concerns the expression for the component of the potential energy resulting from the repulsion between electrons. These terms in the Hamiltonian depend on the instantaneous positions of all the electrons, because the energy... 

A measure of the extent to which any particular ab initio calculation does not deal perfectly with electron correlation is the correlation energy. In a canonical exposition [79] Lowdin defined correlation energy thus The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. This is usually taken to be the energy from a nonrelativistic but otherwise perfect quantum mechanical procedure, minus the energy calculated by the Hartree-Fock method with the same nonrelativistic Hamiltonian and a huge ( infinite ) basis set ... 

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 ... 

Quite a few years have now gone since the Tarfala Workshop in 1981, and many new results have appeared in the method of complex scaling. The question of the occurrence of complex eigenvalues corresponding to so-called resonance states in the Hartree-Fock approximation for various types of many-electron Hamiltonians is, however, still not completely solved, and the authors feel that time is now mature to bring up these problems for more intense discussions. 

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 ... 

The total Hamiltonian can be solved by Green s function techniques using the Hartree-Fock approximation for the Coulomb repulsion terms... 

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. 

Finally we must examine the property of additivity of the exchange Hamiltonian. Let us consider, for instance, four interacting electrons and let us specifically examine the exchange coupling between electrons 1 and 2. The effect of electrons 3 and 4 is treated through the effective (mean) potentials SV n) and SV r2) seen by electrons 1 and 2 (Hartree-Fock approximation). Under these conditions the general Hamiltonian for electrons 1 and 2 may be written ... 

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... 

A rigorous mathematical model for the relativistic electron-positron field in the Hartree-Fock approximation has been recently proposed (Bach et al. 1999). It describes electrons and positrons with the Coulomb interaction in second quantization in an external field using generalized Hartree-Fock states. It is based on the standard QED Hamiltonian neglecting the magnetic interaction A = 0 and is motivated by a physical treatment of this model (Chaix and Iracane 1989 Chaix et al. 1989). 

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]. 

The sum on the right hand side of this expression is precisely the expectation value of the Hamiltonian in the independent-particle approximation. Since the sum on the right-hand side of Eq. (83) is the starting point for a variational treatment of the (Dirac) Hartree-Fock approximation, it follows that choosing U = Vhf leads to - - = Ehf-... 

Several approximations are used to compute (cxc/i, r) j(F). In the Hartree--Fock approximation it follows directly from substitution of Eq.(2.9) in the exact n-electron Hamiltonian and depends on the molecular orbital considered. For molecular orbital j the result becomes ... 

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