Sturmians construction

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 construct 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... 

We must remember that the subscript v represents a set of indices, and the constants jS may be independent of some of them. Orthogonality with respect to these minor indices must be established or constructed in some other way. Assuming that this has been done, we next need to normalize the generalized Sturmian basis set. It turns out that the most natural and convenient choice of normalization is that which yields the potential-weighted orthonormality relations in the form... 

To illustrate this method, we have calculated the natural orbitals of the ground state of lithium (Fig. 1). The basis of one-electron orthonormal spin-orbitals first-order density matrix consisted of 25 spin-up and 25 spin-down orthogonalized Coulomb Sturmians. The first-order density matrix flius constructed was block-diagonal. The eigenvalues (occupation numbers) corresponding to the spin-up block were... 

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