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Fourier Transforms of Position-space Hybrids

To obtain the momentum distribution 7ty(p) due to a single hybrid orbital ip(f), it is necessary to perform a Dirac-Fourier transform. The square magnitude of the resulting momentum orbital ip(p) is the contribution of ip to the momentum density  [Pg.214]

that the hybrid ip(r) is real. It may be written as a combination of an inversion-symmetric part ips and an antisymmetric part ipa. The Fourier transform will map the former onto the real part of ip, and the latter on its imaginary part, both of which are symmetric themselves. As a result, the square-magnitude of ip is inversion symmetric with respect to p = 0 (as momentum densities should be) [7], [Pg.214]

Commonly (in position space), hybrid orbitals are written in terms of single-center linear combinations of basis functions that axe themselves products of radial parts and real spherical harmonics. Let us consider [Pg.214]

the following assumptions are made the radial function Ri r) is the same for all basis functions of the same ul quantum number , and its dependence on a shell quantum number n is of no consequence. The coefficients a/ describe the contribution of s, p, d,. .. character to the hybrid, and the bim govern the shape and orientation of that contribution. Sim are the real surface harmonics, defined in terms of the spherical harmonics (Y m). [Pg.214]

The radial behavior of the hydrogenic eigenfunctions in position and momentum space is exponential and Lorentzian , respectively, and their nodal structure depends on the associated Laguerre and Gegenbauer polynomials, respectively. [Pg.214]


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