Direct correlation function, self-consistent

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

Note that the R-MMSA and R-MPY/HTA approximations for the direct correlation functions are now deterministic, that is, uncoupled from the determination of the full pair correlation functions gv,M (r). However, for the most complex R-MPY closure the direct correlation functions are still self-consistently linked to the full radial distribution functions and tail potentials. 

The proof of this result is based on exact bounds on the asymptotic behavior of the interfacial correlation functions, obtainable from the Bogoliubov inequality, and uses a reductio ad absurdum a self-maintained interface is assumed to exist and it is then shown that this assumption leads to a contradiction. The key step in the demonstration consists in deriving and making use of the asymptotic behavior at large separations of the direct correlation function of Ornstein-Zernike, c(r, r ), defined in terms of the more familiar pair correlation function ft(r, r ) which measures the probability of having a molecule at point r given that there is one at point r ... 

The basic approximation in the Percus-Yevick closure is that the direct correlation function is short range. In Fig. 7 we can observe that indeed C(r) for both the bead-spring model and polyethylene are short range approaching zero on a scale of 5 A. However, C(r) from self-consistent PRISM theory is even shorter range and... 

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

The problem of the sign of AR/R for the divalent tin compounds was investigated by Lees and Flinn (16). In the relationship between the quadrupole splitting and chemical shift for the stannous compounds, two distinct correlations became apparent—compounds with a linear covalent bond, and compounds with a predominantly planar bond. Furthermore, there exists a linear relationship between the number of 5 p electrons and the chemical shift and hence the total 5 electron density. Using free tin ion wave functions in a self-consistent field calculation, they showed that the direct eflEect of adding 5 electrons is considerably... 

We recently presented a correlation method based on the Wigner intracule, in which correlation energies are calculated directly from a Hartree-Fock waveftmction. We now describe a self-consistent form of this approach which we term the Hartree-Fock-Wigner method. The efficacy of the new scheme is demonstrated using a simple weight function to reproduce the correlation energies of the first- and second-row atoms with a mean absolute deviation of 2.5 m h. 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

Such a distinction is made here for two reasons a) In the cases where the theoretical considerations lead to a given vxc directly, the parent functional is not known, b) In the process of developing approximations to the exchange-correlation functional, it is frequently the case that the functional is tested on electron densities obtained with a potential corresponding to another exchange-correlation potential i.e. not self-consistently. 

Direct ab initio methods, in which data are recomputed when required, rather than being stored and retrieved, provide an alternative that seems more useful for parallel development. The simplest level of ad initio treatment (self-consistent field methods) can be readily parallelized when direct approaches are being exploited. Experience demonstrates, however, that data replication methods will not lead to truly scalable implementations, and several distributed-data schemes (described later) have been tried. These general approaches have also been used to develop scalable parallel implementations of density functional theory (DFT) methods and the simplest conventional treatment of electron correlation (second-order perturbation theory, MP2) by several groups. 3-118... 

The HF method treats electron-electron interactions at a mean field level, with the Hartree and exchange interactions exactly written. The method can be implemented either in its spin restricted form (RHF), for closed shell systems, or in the unrestricted form (UHF) for open-shell or strongly correlated systems. In the first case, the one-electron orbitals are identical for electrons of both spin directions, while UHF can account for a non-uniform spin density. The one electron orbitals, which are determined in the course of the self-consistent resolution of the HF equations, are expanded on an over-complete basis set of optimized variational functions. 

Later developments of linear methods have been in the direction of self-consistent calculations of ground-state properties utilising local spin-density-functional formalism [1.51,52] for exchange and correlation. The basis of the self-consistency procedure was given in papers by Madsen et al. [1.53], Vouisen et al. [1.54] and Andersen and Jepsen [1.55], and was soon followed by results for the magnetic transition metals [1.56], the noble metals [1.57], some lanthanides [1.58], the actinides [1.59,60], and the 3d transition metal monoxides [1.61,62]. In this context one should also mention calculations of the electronic structure in transition metal compounds [1.63,64], A15 compounds [1.65,66], rare-earth borides [1.67], Chevrel... 

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