Density-Based Methods Potential Functions

FIGURE 20 Density-based methods individual and cumulative potentials in the univariate case when (A) triangular or (B) Gaussian functions are used. 

Extensions of this model in which the atomic nuclei and core electrons are included by representing them by a potential function, V, in Equation (4.1) (plane wave methods) can account for the density of states in Figure 4.3 and can be used for semiconductors and insulators as well. We shall however use a different model to describe these solids, one based on the molecular orbital theory of molecules. We describe this in the next section. We end this section by using our simple model to explain the electrical conductivity of metals. 

In the simulation, the density of the silicon structure is chosen as that of c-Si (2.33 g/cm ), since the density of a-Si without voids is close to that of c-Si [17]. The atoms are initially arranged as the diamond structure with periodic boundary conditions. They move according to the intermolecular forces based on the potential function, Eq. (3), and these movements can be described by the classical momentum equations. The momentum equations are integrated by the Gear algorithm with a time step of 0.002 ps and the average temperature of the structure is kept constant by the momentum scaling method. 

Our description thus far of DOS-based methods has centered on calculation of the density of states. A particularly fruitful extension of such methods involves the calculation of a potential of mean force ( ), or PMF, associated with a specified generalized reaction coordinate, (r) [28,29]. A PMF measures the free energy change as a function of (r) (where r represents a set of Cartesian coordinates). This potential is related to the probability density of finding the system at a specific value of the reaction coordinate (r) ... 

In the preceding sections we have studied diatomic interactions via U(R). However, the study of diatomic interactions can also be carried out in terms of the force F(R) instead of the energy U(R), where R denotes the internuclear separation. Though there are several methods for the calculation of the force, the electrostatic theorem of Hellmann (1937) and Feynman (1939) is of particular interest in this section, since the theorem provides a simple and pictorial method for the analysis and interpretation of interatomic interactions based on the three-dimensional distribution of the electron density p(r). An important property of the Hellmann-Feyn-man (HF) theorem is that underlying concepts are common to both the exact and approximate electron densities (Epstein et al., 1967, and references therein). The force analysis of diatomic interactions is a useful semiclassical and therefore intuitively clear approach. And this results in the analysis of diatomic interactions via force functions instead of potential ones (Clinton and Hamilton, 1960 Goodisman, 1963). At the same time, in the authors opinion, it serves as a powerful additional instrument to reexamine model diatomic potential functions. 

Efficient use of symmetry can greatly speed up localized-orbital density-functional-exchange-and-correlation calculations. The local potential of density functional theory makes this process simpler than it is in Hartree-Fock-based methods. The greatest efficiency can be achieved by using non-Abelian point-group symmetry. Such groups have multidimensional irreducible representations. Only one member of each such representation need be used in the calculation. However efficient localized-orbital evaluation of the chosen matrix element requires the sum of the magnitude squared of the components of all the members on one of the symmetry inequivalent atoms, based on Eq. 13. 

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