Orbitals Sturmians

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

We choose 2 = 2 so that io( ) is the radial orbital uio(r) of the ground state (4.27). The radial orbitals UnAA are known as the Sturmians (Roten-berg, 1962). They are sometimes called pseudostates . 

The configuration-interaction representation of the lower-energy states of an atom is the IV-electron analogue of the Sturmians in the hydrogen-atom problem. We choose an orbital basis of dimension P, form from them a subset of all possible A/ -electron determinants pk),k = 0,Mp, and use these determinants as a basis for diagonalising the IV-electron Hamiltonian. It may be convenient first to form symmetry configurations kfe) from the pfe). 

V. Aquilanti, A. Caligiana, S. Cavalli, and C. Coletti, Hydrogenic orbitals in momentum space and hyperspherical harmonics elliptic sturmian basis sets. Int. J. Quant. Chem., 92 212-228, 2003. 

V. Aquilanti, S. Cavalli, C. Coletti, D. Di Domenico, and G. Grossi, Hyperspherical harmonics as sturmian orbitals in momentum space a systematic approach to the few-body coulomb problem. Int. Rev. in Phys. Chem., 20 673-709, 2001. 

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

The reader will recognize that this is just the wave equation obeyed by the familiar hydrogenlike orbitals, except that Z/n has been replaced by the constant k. Thus, if we start with a hydrogenlike orbital and replace Z/n everywhere by the constant k, we will have generated a set of Coulomb Sturmians. They have the form... 

The reader may verify that these become the familiar hydrogenlike orbitals if k is replaced by Z n, where Z is the nuclear charge and n is the principal quantum number. It can be shown [19] that the Coulomb Sturmians obey a set of potential-weighted orthonormality relations of the form ... 

Molecular Orbitals Based on Sturmians 5.1 The One-Electron Secular Equation... 

Fig. 2 This figure shows the Sturmian molecular orbital corresponding to the ground state of the Hj ion, withS = 6, k = 1.16885, and/J = 5.13325 Bohrs...

We have just seen that the treatment of a single electron moving in the field of several nuclei has been developed by a number of authors. Let us now turn to the question of whether molecular orbitals based on Coulomb Sturmians can be used to treat /V-electron molecules. To answer this question, let us consider a Slater determinant of the form... 

Aquilanti V, Cavalli S, Coletti C, Grossi G (1996) Alternative sturmian bases and momentum space orbitals an application to the hydrogen molecular ion. Chem Phys 209 405... 

Caligiana A (2003) Sturmian orbitals in quantum chemistry. Ph.D. thesis, University of Perugia, Italy... 

Together with Osvaldo Goscinski they examine in their second chapter whether it is possible to generate natural orbitals for a system from an initial calculation using generalized Sturmians. 

In the present paper, we shall discuss a method for generating many-electron states of a given symmetry using Kramers pair creation operators and other symmetry-preserving pair creation and annihilation operators. We will first develop the formalism for the case where orthonormality between the orbitals of different configurations can be assumed. Afterwards we will extend the method to cases where this orthonormality is lost, so that the method also can be used in generalized Sturmian calculations [11-13] and in valence bond calculations. 

In the preceding discussion, the creation and annihilation operators always referred to a set of orthonormal spin-orbitals. We can ask how much of the formalism holds in cases where complete orthonormality can no longer be assumed, for example, in valence-bond calculations or in the generalized Sturmian method. 

Natural Orbitals from Generalized Sturmian Calculations... 

Atomic Densities, Polarizabilities, and Natural Orbitals Derived from Generalized Sturmian Calculations... 

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