On Hybrid Orbitals in Momentum Space

James Clark, Hartmut L. Schmider, and Vedene H. Smith, Jr. a  

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. 

Among the countless concepts that Linus Pauling introduced from Quantum Mechanics into chemistry[l,2], and that became standard principles of the trade, there is the idea of hybridization . In the framework of the valence-bond description of a system, it is useful to mix atomic orbitals of the same n-quantum number , or of similar spatial extent, to construct directed, asymmetric atomic contributions. Although hybrids are not needed in an LCAO-MO description of the system, they have so much become part of the language of both organic and inorganic chemistry, that people will go out of their way to arrive at descriptions that are compatible with them. 

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. 

The two concepts have on occasion been brought together Coulson and Duncanson[4] gave an explicit formula for sp-orbitals based on Slater type orbitals (STO s). Rozendaal and Baerends used hybrids to describe chemical bonding in a momentum representation [5], and more recently, Cooper considered the shape of sp hybrids in momentum space, and their impact on momentum densities [6], We would like to have a closer look at them, in terms of their functional behavior, their nodal structure and their topology. We will do 

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