Harmonics hyperspherical

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

Equation (24) can be derived from the theory of hyperspherical harmonics and Gegenbauer polynomials but for readers unfamiliar with this theory, the expansion can be made plausible by substitution into the right-hand side of equation (23). With the help of the momentum-space orthonormality relations, (17), it can then be seen that right-hand side of (23) reduces to the left-hand side, which must be the case if the integral equation is to be satisfied. Let us now consider an electron moving in the attractive Coulomb potential of a collection of nuclei ... 

In (38), hi(uj) is an harmonic polynomial of order / in ui, U2 and M3, while hi sj) is the same harmonic polynomial with uj replaced by Sj. The Shibuya-Wulfman integrals can then be calculated by resolving the product of hyperspherical harmonics in (37) into terms of the form u hi(uj). To illustrate this second method for the evaluation of... 

Avery, J., Hyperspherical Harmonics Applications in Quantum Theory, Kluwer Academic Publishers, Dordrecht, Netherlands, (1989). 

Hyperspherical Harmonics as Atomic and Molecular Orbitals in Momentum Space... 

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. 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

As already noted in some preceding papers [10,13] from the theory of the 0(4) group, [8], the 3-sphere admits different systems of hyperspherical coordinates to which correspond alternative harmonics. In Ref. [14] we have classified the hyperspherical harmonics of the group 0(4), and shown that there are 15 distinct bases. In the following, we consider in more detail the spherical, the Stark and the Zeeman basis sets. 

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

In the following, we pay special attention to the connections among the spherical, Stark and Zeeman basis. Since in momentum space the orbitals are simply related to hyperspherical harmonics, these connections are given by orthogonal matrix elements similar (when not identical) to the elements of angular momentum algebra. 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

As is well known, conventional hydrogenoid spherical orbitals are strictly linked to tetradimensional harmonics when the atomic orbitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree structures of hyperspherical harmonics for R (Fig. 1). 

J. Avery, Hyperspherical Harmonics, Applications in Quantum Theory Kluwer Academic Publishers Dordrecht, The Netherlands, (1989). 

Hyperspherical harmonics as atomic and molecular orbitals in momentum 291 space... 

We can pass from tree a to b using the suitable Clebsch-Gordan coeficient (eq. 12). The tree (c) illustrates the hyperspherical parametrization that leads to the hyperspherical harmonics Yn- Xm(, W 9) They are related to the harmonics of tree a through the Z coeficient defined in eg. (15). The connection between (b) and (c) requires a Clebsch Gordan coefficient and a phase change related to a (see eq. (14)). 

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