Self-consistent field approach

B3.6.2.3 SELF-CONSISTENT FIELD APPROACH AND GINZBURG-LANDAU MODELS... 

Bowman, J. M. (1986), The Self-Consistent-Field Approach to Polyatomic Vibrations, Acc. Chem. Res. 19, 202. 

P. Poulin, W. Essafl, and J. Bibette On the Colloidal Stability of Water-in-Oil Emulsions. A Self-Consistent Field Approach. J. Chem. Phys. B 103,5157 (1999). 

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. 

G. Stock. A semiclassical self-consistent-field approach to dissipative dynamics - the spin-boson problem. J. Chem. Phys., 103(4) 1561-1573, Jul 1995. 

Ehrenreich, H. and Cohen, M.H. (1959). Self-consistent field approach to the many-electron problem, Phys. Rev. 115, 786-790. 

Iron), and Eq. (3.12) is a one-electron differential equation. This has been indicated by writing F and T>, as functions of the coordinates of electron 1 of course, the coordinates of any electron could have been used. The operator F is peculiar in that it depends on its own eigenfunctions, which are not known initially. Hence the Hartree-Fock equations must be solved by an iterative process. One obtains approximate solutions for the ( ), and from these constructs the first approximation to F. Equation (3.13) is then solved to obtain a new set of 4>, (which are generally occupied according to their order of e,) and a new F is constructed. Such a process is called a self-consistent-field approach, and the process is terminated when the orbitals output from one step are virtually identical to those that are input from the preceding step (in practice, the energy E is usually monitored). 

We will have more to say later about the self-consistent field approach, but first we will see how the atomic orbitals for polyelectronic atoms can be used to account for the form of the periodic table of the elements. 

R.B. Gerber, V. Buch and M.A. Ratner, Time-dependent self-consistent field approximation for intramolecular energy transfer. I. Formulation and application to dissociation of van der Waals molecules, J. Chem. Phys., 77 (1982), 3022 M.A. Ratner and R.B. Gerber, Excited vibrational states of polyatomic molcecules the semiclassical self-consistent field approach, J. Phys. Chem., 90 (1986) 20 R.B. Gerber and M.A. Ratner, Mean-field models for molecular states and dynamics new developments, J. Phys. Chem., 92 (1988) 3252 ... 

The self-consistent field approach in relativistic quantum chemistry provides one of the most convenient and useful computational tools for the study of the electronic structure and properties of atoms, molecules and solids just as it does in nonrelativistic quantum chemistry. This chapter describes only methods in which the motion of electrons is described by the Dirac operator, namely... 

While 8 hence does not depend on the chain length N, the period D depends on both N and %. The theoretical analysis of this problem [41, 42, 56, 57, 59, 313-316] is rather complicated and requires somewhat restrictive assumptions, such as the self-consistent field - approach [56] for polymers. We shall not describe these theories here, but restrict ourselves to a scaling-type plausibility argument, using the fact that the interfacial tension between fully segregated phases is also depending on the x parameter only but not on chain length [305-313]... 

The variation is quite complicated since the various quantities appearing in Eq. (128) such as c, and w, are themselves functions of the coefficients a/. The averages are constrained (cf. Eq. (121)) to the centroid defined by the aj s. To circumvent this complication one may use a self-consistent-field approach. Given an initial choice of the coefficients, one varies the Lagrangian in Eq. (128), ignoring the dependence of the c/s, and the frequencies u) and ojq on the transformation coefficients, to find a new set of transformation coefficients. The new set is then used to generate a new set of c/s, etc., and the procedure is repeated until convergence is obtained. 

Fig. 15.2. The concentration profile between two flat plates according to the self-consistent field approach (after Joanny et al., 1979).

Menna, Moccia and Randaccio 44), using the molecular-orbital approach in the one-centre, self-consistent-field approach (OCE—SCF— MO) developed by Moccia 45—47), reports the calculated vibrational frequencies of HF and HCl (using the FG-method (7)) ... 

The problem of the searching for the optimal one-electron representation is one of the oldest in the theory of multielectron atoms. Three decades ago, Davidson had pointed the principal disadvantages of the traditional representation based on the self-consistent field approach and suggested the optimal natural orbitals representation. Nevertheless, there remain insurmountable calculational difficulties in the realization of the Davidson program (see, e.g. Ref. [12]). One of the simplified recipes represents, for example, the DPT method [18,19]. Unfortunately, this method does not provide a regular refinement procedure in the case of the complicated atom with few quasiparticles (electrons or vacancies above a core of the closed electronic shells). For simplicity, let us consider now the one-quasiparticle atomic system (i.e., atomic system with one electron or vacancy above a core of the closed electronic shells). The multi-quasiparticle case does not contain principally new moments. In the lowest second order of the QED PT for the A , there is the only one-quasiparticle Feynman diagram a (Fig. 12.1), contributing the ImAZ (the radiation decay width). 

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