Coulomb Sturmians

Apart from these one-particle effects , additional complications arise in the case of two electrons, which may dissolve from a bound state to the positronic and electronic continua. A mathematically rigorous approach to avoid this so-called continuum dissolution is to use as basis functions the relativistic Coulomb Sturmians... 

Abstract The theory of Sturmians and generalized Sturmians is reviewed. It is shown that when generalized Sturmians are used as basis functions, calculations on the spectra and physical properties of few-electron atoms can be performed with great ease and good accuracy. The use of many-center Coulomb Sturmians as basis functions in calculations on N-electron molecules is also discussed. Basis sets of this type are shown to have many advantages over other types of ETO s, especially the property of automatic scaling. 

The reader will recognize that this is just the wave equation obeyed by the familiar hydrogenlike orbitals, except that Z/n has been replaced by the constant k. Thus, if we start with a hydrogenlike orbital and replace Z/n everywhere by the constant k, we will have generated a set of Coulomb Sturmians. They have the form... 

The reader may verify that these become the familiar hydrogenlike orbitals if k is replaced by Z n, where Z is the nuclear charge and n is the principal quantum number. It can be shown [19] that the Coulomb Sturmians obey a set of potential-weighted orthonormality relations of the form ... 

A Coulomb Sturmian basis set is isoenergetic. All the members of the set correspond to the energy... 

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... 

Here, n, l, and m are the quantum numbers of the Coulomb Sturmians, while a is the index of the atom on which the basis function is localized. Thus we write... 

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... 

Here, the functions R / are the Coulomb Sturmian radial functions (Table 5). If we let then... 

Table 7 The first few overlap integrals m., ., = J d jcy (x) /t(x) between displaced Coulomb Sturmians. The definitions of S, 6 and are the same as in Table 6. The integrals were evaluated by means of equation (124)...

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... 

As Vincenzo Aquilanti has shown, the Shibuya-Wulfman integrals can be related to the effect of a translation on Coulomb Sturmians. Combining (75) and (137), we obtain... 

In other words, if a Coulomb Sturmian located on one center is expanded in terms of Coulomb Sturmians located on another center, the expansion coefficients are Shibuya-Wulfman integrals. It should be noted, however, that this expansion is... 

We have just seen that the treatment of a single electron moving in the field of several nuclei has been developed by a number of authors. Let us now turn to the question of whether molecular orbitals based on Coulomb Sturmians can be used to treat /V-electron molecules. To answer this question, let us consider a Slater determinant of the form... 

We will now show that when the densities are produced by products of Coulomb Sturmians, interelectron repulsion integrals of the type shown in (165) and (167) can be readily evaluated using Fock s relationship and the properties of hyper-spherical harmonics. Suppose that... 

The series in (171) terminates and the expansion is exact. The coefficients -ifl form a large but very sparse matrix that can be precalculated and stored. What we have done here is to expand a product of two Coulomb Sturmians in terms of a single Coulomb Sturmian with double the k value. When this is done, the exponential part is automatically correct, and only the polynomial parts need to be taken care of. Hence, the sparseness of Cpt p p. Then... 

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... 

The alternative on which we focus here is the study of Coulomb Sturmian orbitals and their generalizations, functions of a type which have so far received limited attention in quantum theory. Coulomb Stiumians are formally similar to the well-known hydrogenic functions, but they are isoenergetic and their use has important and peculiar implications that are worthwhile investigating. 

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... 

R. Szmytkowski. The Dirac-Coulomb Sturmians and the Series Expansion of the Dirac-Coulomb Green Functions Application to the Relativistic Polarizability of the Hydrogen Like Atoms. /. Phys. B At. Mol Opt. Phys., 30 (1997) 825-861. 

Abstract A new compact two-range addition theorem for Coulomb Sturmians is presented. This theorem has been derived by breaking up the exponential-type orbitals into convenient elementary functions the Yukawa potential (e /r) and evenly-loaded solid harmonics, for which translation formulas are... 

Coulomb Sturmians (CSs) are an exponential-type complete set of basis functions which satisfy a Sturm-Liouville equation [2]. The main objective of the present work is to derive an ADT for the Slater-type orbitals (STOs), which are the fundamental ETO, and thereby for the CSs, which are a linear combination of STOs. The expression for the two-center overlap integral is then worked out for the CSs as an illustration and numerical results and conclusions are presented. 

Certain physical properties, such as NMR shielding tensor calculations directly involve the nuclear cusp and correct treatment of radial nodes, which indicates that basis sets such as Coulomb Sturmians are better suited to their evaluation than gaussians [4,16,33]. 

Coulomb Sturmians have the advantage of constituting a complete set without continuum states because they are eigenfunctions of a Sturm-Liouville equation involving the nuclear attraction potential i.e., the differential equation below. 

Here, N is the normalisation constant previously obtained for Coulomb Sturmians, L is the associated Laguerre polynomial of order 21 + 2 —a with suffix n-l-l-a (recall that a = 1 defines the Coulomb Sturmians. 

After a suitably accurate electron density has been obtained for the optimized geometry over a Coulomb Sturmian basis set, the second order perturbation defining the nuclear shielding tensor should be evaluated in a Coupled perturbed Hartree Fock scheme. 

The integrals involved may conveniently be evaluated using B-functions with linear combinations giving the Coulomb Sturmians. 

Some tests show that Slater type orbitals (STO) or B-functions (BTO) are less adequate basis functions that Coulomb Sturmians, because only the Sturmians possess the correct nuclear cusp and radial behavior. 

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