Hyperspherical harmonics 4-dimensional

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 ... 

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. 

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. 

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]. 

The Fourier transforms of the hydrogenlike orbitals were shown by Fock [18] to be expressible in terms of 4-dimensional hyperspherical harmonics when momentum space is mapped onto the surface of a 4-dimensional unit hypersphere by the transformation ... 

The first few 4-dimensional hyperspherical harmonics K i, ,m(u) are shown in Table 5. Shibuya and Wulfman [19] extended Fock s momentum-space method to the many-center one-particle Schrodinger equation, and from their work it follows that the solutions can be found by solving the secular equation (63). If Fock s relationship, equation (67), is substituted into (65), we obtain ... 

Then (using (76) and the orthonormality of the 4-dimensional hyperspherical harmonics) we obtain ... 

The full three-body problem in the physical three-dimensional space required development of hyperspherical harmonic expansions [39]. Crucial for further progress was the introduction of discrete analogues for the latter [40-43], based on hyperangular momentum theory [44,45] and leading to the efficient hyperquantization algorithm [46 19]. For other hyperspherical approaches to reaction dynamics, see [50-63],... 

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... 

To further illustrate the tree-method, consider the six-dimensional hypersphere which parameterizes the components of Jacobi vectors for the three-body problem the symmetric tree, see fig. 5, corresponds to the hyperspherical harmonics... 

The Schrodinger equation for the D-dimensional analogue of hydrogen (equation (88)) can be solved exactly, both in direct space and in reciprocal space and in both cases the solutions involve hyperspheri-cal harmonics. In this section we shall discuss the close relationship between hyperspherical harmonics, harmonic polynomials, and exact D-dimensional hydrogenlike wave functions. We shall also discuss the importance of these functions in dimensional scaling and in the hyperspherical method. 

S.P. Alliluev [29,30] was able to obtain exact D-dimensional hydrogenlike wave functions in momentum space by a generalization of Fock s method. In Alliluev s treatment, Fock s transformation was generalized in such a way as to project D-dimensional momentum space onto the surface of a (DH-l)-dimensional hypersphere [24]. The momentum-space hydrogenlike wave functions could then be shown to be proportional to (D-f-l)-dimensional hyperspherical harmonics. 

From the above discussion, we can see that, both in direct space and in reciprocal space, the D-dimensional hydrogenlike wave functions involve hyperspherical harmonics and we shall therefore devote a little space to discussing these functions. Hyperspherical harmonics are closely related to harmonic polynomials [24]. In fact, hyperspherical harmonics are nothing but harmonic polynomials, orthonormal-ized in an appropriate way, and divided by appropriate powers of the hyperradius. Let us therefore begin by looking briefly at the theory of harmonic polynomials. 

The angular functions iVA/i( ) called hyperspherical harmonics. When D = 3, they reduce to the familiar 3-dimensional spherical harmonics, yim 0,). There are many different ways of choosing a set of hyperspherical harmonics (i.e., a set of orthonormal eigenfunctions of A ). To illustrate this, we can consider the case where D = 4 and A = 1. The set of functions... 

Equation (97) can be solved in direct space by almost the same method which is used to solve the 3-dimensional Schrodinger equation for hydrogen If iV/ (f ) is a hyperspherical harmonic satisfying... 

In the theory of hyperspherical harmonics in a P-dimensional space, the Gegenbauer polynomials, C , with a = D/2 — 1, play a role analogous to that of the Legendre polynomials in the theory of three-dimensional spherical harmonics [4]. Since the Gegenbauer polynomials are the P-dimensional generalizations of Legendre polynomials, a possible choice of a P-dimensional multipole perturbation analogous to of equation (10), is... 

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