Atoms Sturmians

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

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 Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

Figure 1 This figure shows the ground-state energies of the 6-electron iso-electronic series of atoms and ions, C, iV, 0 +, etc., as a function of the atomic number, Z. The energies in Hartrees, calculated in the crudest approximation, with only one 6-electron Sturmian basis function (as in Table 1), are represented by the smooth curve, while dementi s Hartree-Fock values [10] are indicated by dots. 

If we wish to achieve high accuracy in atomic calculations it is necessary to use basis sets consisting of many Sturmian basis functions and with... 

Many-electron Sturmians applied to atoms and ions in strong external fields... 

There have been several papers published on a(co) and y(co) for the hydrogen atom[85]-[90]. Shelton[89] used an expansion in Sturmian functions to obtain y values for Kerr, ESHG, THG and DFWM at a number of frequencies. A more straightforward and simpler method is to use the SOS approach and a pseudo spectral series based on the wavefunctions formed by the linear combinations ... 

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). 

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. 

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

We have dropped the index i because for the moment we are dealing with a single electron). The use of Coulomb Sturmian basis functions located on the different atoms of a molecule to solve (63) was pioneered by C.E. Wulfman, B. Judd, T. Koga, V. Aquilanti, and others [30-37]. These authors solved the Schrodinger equation in momentum space, but here we will use a direct-space treatment to reach the same results. Our basis functions will be labeled by the set of indices... 

Here, n, l, and m are the quantum numbers of the Coulomb Sturmians, while a is the index of the atom on which the basis function is localized. Thus we write... 

Here, x has the meaning defined by (65), where the index a is the index of the atom on which a Coulomb Sturmian basis function is located. In the case of a general 4-center integral, all the a values may be different from one another. Integrals of this type fall... 

Avery JE, Avery JS (2006) Generalized Sturmians and Atomic Spectra. World Scientific. 256 pages, ISBN 981-256-806-9... 

The examples discussed here show that even in the case where the generalized Sturmian method is applied to atoms, a case in which radial orthonormality between different configurations is sometimes lost, simplified formulas for the matrix elements can often be derived. However, even when this is not possible, the... 

In making this table, the basis set used consisted of 63 generalized Sturmians. Singlet and triplet states were calculated simultaneously, 0.5 s of 499 MHz Intel Pentium III time being required for the calculation of 154 states. Experimental values are taken from the NIST tables (http //physics.nist.gov/asd). Discrepancies between calculated and experimental energies for the ions may be due to experimental inaccuracies, since, for an isoelectronic series, the accuracy of the generalized Sturmian method increases with increasing atomic number. 

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