Generalized finite basis representation

In general, theoretical studies of triatomic and tetra-atomic molecules employ analytical PESs carefully fitted to large grids of ab initio data points, and curvilinear vibrational coordinates, to take into account large-amplitude motions. On the other hand, larger polyatomic molecules are investigated with simple polynomial PES, whose parameters are obtained from ah initio data, and with normal coordinates, possibly considering only the active ones. Finite basis representations (FBR),... 

Early non-relativistic many-body perturbation theory studies of correlation energies in molecules established that the error associated with truncation of the finite basis set is most often much more significant than that resulting from truncation of the perturbation expansion.15 The chosen basis set is required to support not only an accurate description of the occupied Hartree-Fock orbitals but also a representation of the virtual spectrum. Over the past twenty years significant progress has been reported on the systematic design of basis sets for electron correlation studies in general and many-body perturbation theory calculations in particular.18... 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

Let G be a group functor, X a set functor. An action of G on X is a natural map G x X - X such that the individual maps G(R) x X(R)- X(R) are group actions. These will come up later for general X, but the only case of interest now is X(R)= V R, where V is a fixed k-module. If the action of G(R) here is also R-linear, we say we have a linear representation of G on V. The functor GLV(R) = Aut (F R) is a group functor a linear representation of G on V clearly assigns an automorphism to each g and is thus the same thing as a homomorphism G - GLK. If V is a finitely generated free module, then in any fixed basis automorphisms correspond to invertible matrices, and linear representations are maps to GL . 

Independently of the approximations used for the representation of the spinors (numerical or basis expansion), matrix equations are obtained for Equations (2.4) that must be solved iteratively, as the potential v(r) depends on the solution spinors. The quality of the resulting solutions can be assessed as in the nonrelativistic case by the use of the relativistic virial theorem (Kim 1967 Rutkowski et al. 1993), which has been generalized to allow for finite nuclear models (Matsuoka and Koga 2001). The extensive contributions by I. P. Grant to the development of the relativistic theory of many-electron systems has been paid tribute to recently (Karwowski 2001). The higher-order QED corrections, which need to be considered for heavy atoms in addition to the four-component Dirac description, have been reviewed in great detail (Mohr et al. 1998) and in Chapter 1 of this book. 

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