Self-consistent variational potentials

Two additional Gaussian families of quantum-effective pair potentials v fR) arise from the variational principle for the free energy. These are the isotropic (ISVP) and the anisotropic (ASVP) self-consistent variational potentials. They were proposed by Giachetti and Tognetti [124] and, independently, by Feynman and Kleinert [125]. These potentials can be derived via an extensive use of the Fourier decomposition of the particle paths in modes characterized by the Matsubara frequencies = 2nn PHn 1), the zero frequency mode being the intercentroid... 

The Dirac-Hartree-Fock iterative process can be interpreted as a method of seeking cancellations of certain one- and two-body diagrams.33,124 The self-consistent field procedure can be regarded as a sequence of rotations of the trial orbital basis into the final Dirac-Hartree-Fock orbital set, each set in this sequence forming a basis for the Furry bound-state interaction picture of quantum electrodynamics. The self-consistent field potential involves contributions from the negative energy states of the unscreened spectrum so that the Dirac-Hartree-Fock method defines a stationary point in the space of possible configurations, rather that a variational minimum, as is the case in non-relativistic theory. 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

Figure 4 Overview of several theoretical predictions for the SE Brueckner-Hartree-Fock (continuous choice) with Reid93 potential (circles), self-consistent Green function theory with Reid93 potential (full line), variational calculation from [9] with Argonne Avl4 potential (dashed line), DBHF calculation from [16] (triangles), relativistic mean-field model from [22] (squares), effective field theory from [23] (dash-dotted fine).

Here Zg is the number of tt electrons provided by atom is essentially an ionization potential for an electron extracted from in the presence of the part of the framework associated with atom r alone (a somewhat hypothetical quantity), is a framework resonance integral, and is the coulomb interaction between electrons in orbitals < >, and <(>,. The essential parameters, in the semi-empirical form of the theory, are cug, and and from their definition these quantities are expected to be characteristic of atom r or bond r—s, not of the particular molecule in which they occur (for a discussion see McWeeny, 1964). In the SCF calculation, solution of (95) leads to MO s from which charges and bond orders are calculated using (97) these are used in setting up a revised Hamiltonian according to (98) and (99) and this is put back into (95) which is solved again to get new MO s, the process being continued until self-consistency is achieved. It is now clear that prediction of the variation of the self-consistent E with respect to the parameters is a matter of considerable difficulty. 

One can be easily convinced that this prescription is correct, if one compares the variation of the functionals minimized in Eqs. (87) and (64) the last minimization being equivalent to that in Eq. (28), solved via Eq. (33). Because the effective external potential (89) depends functionally on n r), an iterative method, leading to self-consistency, must be employed. 

The simplest version of the self-consistent field approach is the Hartree method, in which the variational principle is applied to a non-symmetrized product of wave functions, and the orthogonality conditions for functions with different n are neglected. This leads to neglecting the exchange part of the potential, which causes errors in the results. 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

Flexible polyelectrolytes exhibit conformational variations when the interaction strength (ZB) or the density of a system is changed. Hence, the structure factor is no longer an a priori known quantity but has to be determined in a self-consistent manner. This is achieved by casting the underlying multichain interactions into a medium-induced interaction potential among the... 

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