Sturmian basis functions

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. 

Since the hydrogenlike Sturmian basis functions form a complete set, the term %i,o,o(xy-R) can be represented as a single-center expansion in terms of functions localized at the origin ... 

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. 

If we begin with the position-space Schrodinger equation, (1), and expand the wave function in terms of a set of many-particle Sturmian basis functions, so that... 

Since the Sturmian basis functions, 4> x), are solutions of (4), equation (15) can be rewritten in the form ... 

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

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. 

This book explores the connections between the theory of hyperspherical harmonics, momentum-space quantum theory, and generalized Sturmian basis functions and introduces methods which may be used to solve many-electron problems directly, without the use of the self-consistent-field approximation. ... 

The orthonormality relations for generalized Sturmian basis functions in direct space and reciprocal space can be written in the form [22-24]... 

Because of the scaling factor pK, which is different for each state, the Sturmian basis functions adjust in scale automatically For tightly bound states they are contracted, while for highly excited states they are diffuse. 

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... 

In a remarkably brilliant early paper, the Russian physicist V. Fock showed that the Fourier transforms of Coulomb Sturmian basis functions can be related in a simple way to 4-dimensional hyperspherical harmonics [38, 39]. Fock discovered this relationship by projecting momentum space onto the surface of a 4-dimensional hypersphere using the relationship... 

This is, of course, also consistent with the potential-weighted orthonormality relation of the Coulomb Sturmian basis function, (7), as can be seen by making use of (132) for the special case where Xfl- = X = 0 and making the substitution k = ZJn. Looking at Table 6, we can see that for the special case where 5 = 0, the diagonal elements of T< T are equal to 1, while the off-diagonal elements vanish, as is required by the orthonormality relations (94). The momentum-space orthonormality relations for Coulomb Sturmians can be used to make a weakly... 

The integrals over dp in (182) are simple enough to be evaluated by Mathematica and they can conveniently be stored as functions kR in the form of interpolation functions. Notice that the integrals depend only on n and /, and there are therefore fewer of them than there would be if they also depended on m. The first 105 of these functions are shown in Fig. 3. Equations (173), (180), and (182) give us a very rapid and convenient way of evaluating integrals of the form shown in (173), where the densities are formed from products of Coulomb Sturmian basis functions located respectively on the two centers, a and a. They constitute the largest contribution to the effects of interelectron repulsion. 

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 JS, Herschbach DR (1992) Hyperspherical sturmian basis functions. J Quantum Chem... 

Thus, we represent the true wave function as a superposition of isoenergetic (generalized Sturmian) basis functions ... 

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