Sturmians generalized

Let X be a 3A-dimensional vector, representing the Cartesian coordinates of the electrons in an A-particle system, i.e., let 

We now suppose that we are able to solve the approximate N-particle Schrddinger equation 

In equation (3), the constants j8 are especially chosen in such a way as to make all of the eigenfunctions correspond to the same energy, E, regardless of their quantum numbers v [1,5,11-35]. The isoenergetic configurations can be used to build up solutions to the actual Schrddinger equation 

we represent the true wave function as a superposition of isoenergetic (generalized Sturmian) basis functions  

The advantage of choosing our basis set to be isoenergetic at (the energy of the true wave function) is that the turning points of all the basis functions will be at the same approximate hyperradius as the turning point of and the asymptotic behavior of all the basis functions will be correct in the regions of hyperspace where both Vq and V become small. In this asymptotic 

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A. Hernandez-Laguna, J. Maruani, R. McWeeny and S. Wilson (eds.) Quantum Systems in Chemistry and Physics. Volume 2 Advanced Problems and Complex Systems, Granada, Spain, 1998. 2000 ISBN 0-7923-5970-4 Set 0-7923-5971-2 J.S. Avery Hyperspherical Harmonics and Generalized Sturmians. 1999... 

The results obtained by a number of authors, though specially by John Avery and collaborators in the field of generalized Sturmians, illustrate the power... 

This book explores the connections between the theory of hyperspherical harmonics, momentum-space quantum theory, and generalized Sturmian basis functions and introduces methods which may be used to solve many-electron problems directly, without the use of the self-consistent-field approximation. ... 

J. Avery, Hyperspherical Harmonics and Generalized Sturmians Kluwer Academic Publishers Dordrecht (2000)... 

Abstract The theory of Sturmians and generalized Sturmians is reviewed. It is shown that when generalized Sturmians are used as basis functions, calculations on the spectra and physical properties of few-electron atoms can be performed with great ease and good accuracy. The use of many-center Coulomb Sturmians as basis functions in calculations on N-electron molecules is also discussed. Basis sets of this type are shown to have many advantages over other types of ETO s, especially the property of automatic scaling. 

Keywords Exponential-type orbitals Generalized Sturmians Hyperspherical harmonics Interelectron repulsion integrals Isoenergetic configurations Momentum space Quantum theory Sturmians... 

Use of Generalized Sturmian Basis Sets to Solve the Many-Particle... 

Ek is the same for all the members of a generalized Sturmian basis set, while the weighting factor jSv is chosen in such a way as to make all the members of the basis set isoenergetic. Generalized Sturmian basis sets can be shown to obey the following orthonormality relation [22, 23] ... 

The orthonormality relations for generalized Sturmian basis functions in direct space and reciprocal space can be written in the form [22-24]... 

The generalized Sturmian secular equation (23) has several remarkable features ... 

Thus, when Goscinskian configurations are used, the generalized Sturmian secular equation for atoms (23) takes on the form ... 

Since only Coulomb potentials are involved, the matrix T v, v turns out to be energy independent. Its elements are pure numbers that depend only on N, the number of electrons, and are independent of the nuclear charge Z. The roots lK of the energy-independent interelectron repulsion matrix T v, v are also pure numbers (Table 1). In the large-Z approximation, the generalized Sturmian secular equation (41) reduces to the requirement ... 

Can the Generalized Sturmian Method be applied to /V-electron molecules This is another question that we are starting to explore. Let us consider a single electron... 

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