Four-Dimensional Spherical Harmonics

The construction of spherical harmonics can be extended to other dimensions. For example, V. Fock uses four-dimensional spherical harmonics in his article on the S O (4) symmetry of the hydrogen atom — see Chapter 9. Spherical harmonic functions of various dimensions are used in many spherically symmetric problems in physics. 

We would now like to show that equation (9.8) is nothing but the integral equation for the four-dimensional spherical harmonic functions. 

We choose the following representation for the four-dimensional spherical harmonics. We set... 

We proceed to estahUsh the addition theorem for four-dimensional spherical harmonics. Equation (9.19) is an identity with respect to r. Expanding the integrand in powers of r... 

In his work on the wave equation of the Kepler problem in momentum space (ZS. f. Phys. 74, 216, 1932), E. Hellras has derived a differential equation [Equations (9g) and (10b) in his article] which—after a simple transformation — can be understood as the differential equation of the four-dimensional spherical harmonics in stereographic projection. [With the gracious approval of E. Helleras, we correct the following misprints in his article the number E that appears in the last term of his equations (9f) and (9g) should be multiplied by 4.]... 

The actual implementation does not involve the explicit calculation of the polar angles, we calculate the spherical harmonics in term of the Cartesian coordinates X, y, z and zo- The first two four-dimensional spherical harmonics are... 

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. 

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. 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

In the case of the icosahedral point groups, Ih and I, Table 3.10, the analysis is more complicated and there is a need to identify the combinations of the spherical harmonics, which will generate higher dimensional irreducibile subspaces. For example, at level 3, there are 7 harmonics, but the irreducible subspaces in icosahedral symmetry are four-fold [Gu] and three-fold [T2u]. It is found that three of the original functions can be carried over to provide basis functions in icosahedral, symmetry but that four distinct linear combinations of... 

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