Self-consistent averaging procedure

The physical reasoning for why these densities were frequently employed in the earlier days of density functional theory was that in this way the degeneracy of the partially filled d-orbitals could be retained. A technical reason why these densities still have to be employed in some recent investigations is that calculations with integral orbital occupations simply do not converge in the self consistent field procedure (see, e. g., Blanchet, Duarte, and Salahub, 1997). Such densities correspond to a representation of a particular state 2S+1L with Mg = S and a spherical averaging over ML. 

Now we have written down a wave function appropriate for use in the case where H = h(i). In HF theory, we make some simplifications so many-electron atoms and molecules can be treated this way. By tacitly assuming that each electron moves in a percieved electric field generated by the stationary nuclei and the average spatial distribution of all the other electrons, it essentially becomes an independant-electron problem. The HF Self Consistent Field procedure (SCF) will be bent on constructing each x(x) to give the lowest energy. 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. 

As we have satisfied the plateau condition (24) and the condition (25) for reaching the optimal value of M, we have defined a self-consistent procedure leading to the optimal averaged density matrix D°. ... 

Howells (1974) restricted his attention to fixed particles, extending the method of Childress (1972) by considering a given number of particles chosen from an infinite set. This partly self-consistent scheme furnishes terms valid in the small-solids concentration limit. In a very readable paper, Hinch (1977) combined some of the above procedures in formulating an averaged-equation approach to particle interactions, providing expressions for the bulk stress, average sedimentation velocity, and effective permeability in suspensions and fixed beds. 

In the independent-partlcle-model (IPM) originally due to Bohr [1], each particle moves under the Influence of the outer potential and the average potential of all the other particles in the system. In modem quantum theory, this model was first Implemented by Hartree 12], who solved the corresponding one-electron SchrSdlnger equation by means of an iterative numerical procedure, which was continued until there was no change in the slgniflcant figures associated with the electric fields involved so that these could be considered as self-consistent this approach was hence labelled the Self-Conslstent-Fleld (SCF) method. In order to take the Pauli exclusion principle into account. Waller and Hartree [3] approximated the total wave function for a N-electron system as a product of two determinants associated with the electrons of... 

An efficient terminator technique is certainly desirable in the application of recursion methods to the study of disordered systems. It has been shown recently that a self-consistently determined terminator can be fruitfully applied to calculate the electronic states in the Anderson model and to evaluate the vibrational spectrum of lattices with isotopic disorder. The basic idea is to extend the procedures discussed in Section IV to ensemble averages. In this case a useful generalization of Eq. (4.5), satisfied by the terminator t(E), is... 

In order to derive the average chain dimensions, we need a suitable expression of the chain free energy wherefrom the relevant quantities will be obtained through a self-consistent procedure [10, 13, 23], Since it can be extended to the perturbed case, we shall drop the zero suffix in the remainder of the section. The reduced excess free energy turns out to be... 

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