Hydrogen momentum-space representation

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

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). 

A much lesser known contribution of Pauling to the chemical knowledge, is his explicit expression for the momentum representation of the hydrogenic wave function [3]. Momentum space concepts are common among scattering physicists, some experimental chemists and a few theoreticians however, they have not won over the bulk of chemists nearly as efficiently as the hybrid concept. The reason is that they are somewhat counter intuitive and molecular structure is expressed in a rather indirect and (in the truest sense of the word) convoluted manner. 

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