Sturmian function

Thus Pv must be real, and whenever Pu-, the two Sturmian functions axe orthogonal with respect to potential-weighted integration over the coordinates. It is convenient to normaJize our Sturmian basis sets in such a way that... 

The author is grateful to Professsor John Avery for his interest in Sturmian functions and in the unpublished report from 1968. He is further indebted to him for a Latex translation of the original manuscript where the conjugate eigenvalue problem as well Sturmians are introduced. 

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 ... 

Edmonds, A.R. (1973). Studies of the quadratic Zeeman effect I. Application of the sturmian functions, J. Phys. B6, 1603-1615. 

Rotenberg M (1970) Theory and application of sturmian functions. Adv At Mol Phys 6 233-268... 

L-spinors, [87], are related to Dirac hydrogenic functions in much the same way as Sturmian functions [93,94] are related to Schrodinger hydrogenic functions. The radial parts are given by [87, III]... 

E. A. Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenherg... 

The results above with a two term polynomial imply that only two Sturmian functions are necessary when expressing the Coulomb propagator in the first procedure described above on how to solve the inhomogeneous equation. 

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. 

Sturmian basis set obeys a potential-weighted orthogonality relationship analogous to equation (10). This still does not tell us how to normalize the functions, and in fact the choice is arbitrary. However, it will be convenient to choose the normalization in such a way that in momentum space the orthonormality relations become ... 

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... 

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. 

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 ... 

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. 

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 ... 

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.

---