Extended Hartree-Fock approximate

It should also be observed that there exists an approximation which is "intermediate between the unrestricted and the extended Hartree-Fock scheme. In starting from the former, the energy is increased by the mixing in of unappropriate spin states, and it can hence be essentially improved by selecting the component of the pure spin desired. It is clear that the energy obtained... 

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. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

The UHF ansatz is necessary because in case of neutral solitons one has to deal with a doublet state. Thus a DODS (different orbitals for different spins) ansatz, as the UHF one, is necessary to describe the system. However, in the UHF method described so far, one Slater determinant with different spatial orbitals for electrons of different spins is applied, which is not an eigenfunction of S2, i.e. S(S+l)h2. The best way to overcome this difficulty would be to use the PHF (Projected Hartree Fock) method, also called EHF method (Extended Hartree Fock) where before the variation the correct spin eigenfunction is projected out of the DODS ansatz Slater determinant [66,67a]. Unfortunately numerical solution of the rather complicated EHF equations in each time step seems to be too tedious at present. Moreover for large systems the EHF wavefunction approaches the UHF one [68], however, this might be due to the approximations used in [67a]. Another possibility is to apply the projection after the variation using again Lowdin s projection operator [66]. Projection and annihilation techniques were... 

Because of the very weak binding field and this complexity of detail, they pose a formidable challenge to theorists accurate predictions of their properties require an excellent understanding of electron-electron correlations, and computational methods extending beyond the customary basis normally, negative ions do not exist within the Hartree-Fock approximation, precisely because polarisation of the core is not allowed for. 

Accurate predictions of electronic structure require going beyond the independent electron picture given by the Hartree-Fock approximation and it is obvious that correlation and relativistic corrections should be included simultaneously in a coherent scheme. Not unexpectedly, methods that had proven their efficiency in non-relativistic calculations started to be extended to the relativistic domain. To give some examples ... 

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 ... 

This method should lead to results which are just as accurate as the results of the methods described in the previous sections, and can be used as a check on the computed potential-energy minimum E(R ) at R = Re if fl is determined from curve-fitting of the Morse potential with the computed R and De and this leads to a wrong we and/or w, then it can be assumed that De and/or Rg are/is wrong. It is to be emphasized (12) that the Morse curve can mostly not be used with essentially ionic compounds like NaF because the attraction given by the Coulomb term extends out in space to greater distances than the Morse exponential part for these compounds many other types of potential have been postulated (e.g. the Hellmann-potential or the Bom-Landd potential (77)). The reader can try to calculate cog, etc. of NaF from the SCF— LCAO—MO calculation of Matcha (72) in the Roothaan-Hartree-Fock approximation, using the Morse curve (E = —261.38 au, R =3.628 au experimental values Rg = 3.639 au, a)g=536 cm i, >g g=3.83 cm-i). 

Two types of theoretical and computational methods will be considered. The first one is the quantum-mechanical ab initio approach, in the particular version based on the Hartree-Fock approximation extended to periodic system[9]. This, as all first-principles theoretical schemes, is very attractive because does not require any kind of empirical knowledge. In the last years, it has proved to be able to account for the structural and electronic behaviour of a number of crystalline solids and minerals quite successfully[10, 11, 12]. The Hartree-Fock approach is particularly reliable and efficient, in this respect. The hypersurface of the ground-state total en-... 

In the preceding sections we have discussed how quantum effects of the electrons are treated first of all within density-functional theory, but we emphasize that, from a conceptual point of view, treatments based on the Hartree-Fock approximation are fairly similar. There exist some few methods where the quantum treatment of the electrons is extended by a quantum treatment of (some of) the nuclei (see, e.g., refs. 60-62) but whereas the electrons still are treated within the standard electronic-strueture approaehes, the path-integral method of Feynman is used for the nuclei. The basie ideas behind these will be briefly outlined here followed by some few examples of their implieations. 

For the sake of clarity we shall first discuss in Section 6.2.1 ground-state properties in the framework of the unrestricted Hartree-Fock approximation. In Section 6.2.2 the theory is extended to finite temperatures by using a functional integral formalism including spin fluctuations. Finally, in Section 6.2.3 we analyze the problem of electron correlations by exact diagonalization of the simpler single-band Hubbard model. 

In order to extend these methods to make them feasible for the study dynamical chemical processes in biopolymers, simplifying assumptions are necessary. The most obvious choice is the use of semi-empirical techniques within the Hartree Fock, linear combination of atomic orbitals framework. These methods can achieve speedups on the order of 1000 over typical ab initio calculations using split valence basis sets within the Hartree Fock approximation. Often greater accuracy can be achieved as well because of the parameterization inherent in the semi-empirical approaches. One semi-empirical approach which has proven successful in representing many chemically interesting processes is the AMI and MNDO Hartree Fock Self-Consistent Field methods developed and paramerterized by Dewar and coworkers [46]. These methods have recently been implemented in a mixed quantum/ classical methodology for the study of chemical and biochemical processes by Field et al. [47]. 

LOwdin, P.-O. (1955a). Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 97, 1474-1489 ibid. (1955b). Quantum theory of many-particle systems. II. Study of the ordinary hartree-fock approximation. Phys. Rev. 97, 1490-1508 ibid. (1955). Quantum theory of many-particle systems, in. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects. Phys. Rev. 97,1509-1520 ibid. (1960). Expansion theorems for the total wave function and extended Hartiee-Fock schemes. Rev. Mod. Phys. 32, 328-334. 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

A pitch is made for a renewed, rigorous and systematic implementation of the GW method of Hedin and Lundquist for extended, periodic systems. Building on previous accurate Hartree-Fock calculations with Slater orbital basis set expansions, in which extensive use was made of Fourier transform methods, it is advocated to use a mixed Slater-orbital/plane-wave basis. Earher studies showed the amehoration of approximate linear dependence problems, while such a basis set also holds various physical and anal3ftical advantages. The basic formahsm and its realization with Fourier transform expressions is explained. Modem needs of materials by precise design, assisted by the enormous advances in computational capabilities, should make such a program viable, attractive and necessary. 

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