Functional expansions

Inherent in the development of approximations by the described interpolation models is to assign polynomial variations for function expansions over finite elements. Therefore the shape functions in a given finite element correspond to a... 

When the MFA is used in absence of the external field (J,- = 0) the Lagrange multipliers //, are assumed to give the actual density, p, known by construction. In presence of the field the MFA gives a correction Spi to the density p,. By using the linear response theory we can establish a hnear functional relation between J, and 8pi. The fields Pi r) can be expressed in term of a new field 8pi r) defined according to Pi r) = pi + 8pi + 8pi r). Now, we may perform a functional expansion of in terms of 8pi f). If this expansion is limited to a quadratic form in 8pj r) we get the following result [32]... 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

This approximation amounts to truncating the functional expansion of the excess free energy at second order in the density profile. This approach is accurate for Lennard-Jones fluids under some conditions, but has fallen out of favor because it is not capable of describing wetting transitions and coexisting liquid-vapor phases [105-107]. Incidentally, this approximation is identical to the hypemetted chain closure to the wall-OZ equation [103]. 

Three different approximations for / EX[p(r)] have been employed. The first approximation, due to McCoy, Curro and coworkers [122-125], is to truncate the functional expansion at second order, that is,... 

From this wave function, one sees how even in the early beginning of molecular quantum mechanics, atomic orbitals were used to construct molecular wave functions. This explains why one of the first AIM definitions relied on atomic orbitals. Nowadays, molecular ab initio calculations are usually carried out using basis sets consisting of basis functions that mimic atomic orbitals. Expanding the electron density in the set of natural orbitals and introducing the basis function expansion leads to [15]... 

Increasing evidence suggests that evolution has used (and is using) the E2 fold for new purposes. In one apparent example of functional expansion, E2 core domains have been observed to be embedded within much larger polypeptide chains [140, 141]. The functional properties of these massive E2s remain poorly characterized, and it is likely that more of them will be discovered. But the clearest case of functional diversification is provided by the UEV proteins. UEVs are related to E2s in their primary, secondary, and tertiary structures, but they lack an active-site cysteine residue and therefore cannot function as canonical E2s [142]. Nonetheless they play several different roles in ubiquitin-dependent pathways. 

With regard to the basis function expansions, several standard basis sets were employed (namely 6-31G, 6-31G, 6-31IG and TZVP). 

Alternative GGA exchange functionals have been developed based on rational function expansions of the reduced gradient. These functionals, which contain no empirically optimized parameters, include B86, LG, P, PBE, and znPBE. 

Such problems arise in steady-state diffusion, such as the equilibrium radius of a vapor bubble attached to a heated surface in contact with liquid. This was considered by Bick (B5), who used spherical Bessel function expansions rather than integral methods. 

In Fig. 9.1, we see the er-function expansion of six important solvents. Apparently the aprotic solvents are almost perfectly described, while there are some small deviations remaining for the protic solvents. 

In 1951 Roothaan and Hall independently pointed out [26] that these problems can be solved by representing MO s as linear combinations of basis functions (just as in the simple Hiickel method, in Chapter 4, the % MO s are constructed from atomic p orbitals). Roothaan s paper was more general and more detailed than Hall s, which was oriented to semiempirical calculations and alkanes, and the method is sometimes called the Roothaan method. For a basis-function expansion of MO s we write... 

The previous basic observations have suggested to adopt a closure that allows us to interpolate continuously between two existing theories. In this framework, the famous HNC-SMSA(HMSA) closure [22] has been proposed for the Lennard-Jones fluid. It interpolates between HNC and SMSA. The HMSA has strong theoretical basis since it can be derived from Percus functional expansion formalism and its bridge function expresses as a functional of the remormalized indirect correlation function y (r) = y(r) — Pmi,r(/") so that... 

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