Electrons Sturmians

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

In order to illustrate the discussion given above, let us consider the simplest possible example - the H2 molecule with a basis consisting of a single two-electron Sturmian ... 

The results presented in this paper seem to indicate that it will be possible to apply successfully the method of many-electron Sturmians to molecules. Momentum-space methods, pioneered by Shibuya and Wulfman [7], seem very well suited to solving the one-electron part of the problem. When the basis potential used in constructing the many-electron Sturmian basis set is taken to be the nuclear attraction potential experienced by the electrons in the molecule, the method of many-electron Sturmians has the following advantages ... 

Basis sets of the type discussed in this paper can only be applied to bound-state problems. It is interesting to ask whether it might be possible to constmct many-electron Sturmian basis sets appropriate for problems in reactive scattering in an analogous way, using hydrogenlike continuum functions as building-blocks. We hope to explore this question in future publications. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

This gives us a prescription for constructing many-electron Sturmians provided we are able to solve the single-electron Schrodinger equation, (22), and provided that the parameters k and b satisfy the subsidiary relations, (25) and... 

The determinential wave functions shown in equations (42)-(44) have the correct normalization for many-electron Sturmians (i.e. the normalization required by equation (6)). To see this, we can make use of the Slater-Condon rules, which hold for the diagonal matrix elements of... 

Table 1 shows analogous equations for po for the ground states of higher isoelec-tronic series, derived in the crude approximation where only one many-electron Sturmian basis function is used. Figure 1 shows the dementi s values [10] for the Hartree-Fock ground state energies of the 6-electron isoelectronic series... 

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. 

Tables 2, 3 and 4 show the first few excitation energies for the ions and again calculated in the crudes approximation Only one many-electron Sturmian basis function is used for the ground state, and only one for the excited state. As can be seen from the tables, where the experimental values [13] are also listed, even this very crude approximation gives reasonable results.

If we wish to malce accurate molecular calculations with many-electron Sturmians, we must pay careful attention to the question of normalization Suppose that we wish to find the normalization constant M for the basis function ... 

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

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