Sturmians secular equation

The definition of the matrix in equation (60) requires some explanation The minus sign is motivated by the fact that H(x) is assumed to be an attractive potential. Division by Po is motivated by the fact that for Coulomb systems, when is so defined, it turns out to be independent of po, as we shall see below. The Sturmian secular equation (61) has several remarkable features In the first place, the kinetic energy has vanished Secondly, the roots are not energy values but values of the parameter po, which is related to the electronic energy of the system by equation (52). Finally, as we shall see below, the basis functions depend on pq, and therefore they are not known until solution... 

For systems interacting through Coulomb forces, as defined by equation (19), is independent of po. Notice that the eigenvalues of the Sturmian secular equation (20) are not values of the energy but values of the parameter po, which is related to the binding energy of bound states by equation (2). 

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 ... 

To obtain the generalized Sturmian secular equations, we substitute the superposition (6) into the many-particle Schrodinger equation (5) ... 

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. 

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. 

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

In the present paper, generalized Sturmians are introduced in Section 2. Their potential-weighted orthonormality relations (Section 3) permit us to write down a peculiar secular equation (Section 4). Examples of its solution for atomic problems are given in Section 5. In these calculations, generalized... 

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