ISSN:
0020-7608
Keywords:
Computational Chemistry and Molecular Modeling
;
Atomic, Molecular and Optical Physics
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Chemistry and Pharmacology
Notes:
A general formula for angular integrations in many-dimensional spaces (derived in a previous paper) is applied to several problems connected with solution of the Schrödinger equation for many-particle systems. Matrix elements of the Hamiltonian are derived for cases where the potential can be expressed in terms of functions of the generalized radius multiplied by polynomials in the m coordinates. The theory of hyperspherical harmonics is reviewed, and a sum rule is derived relating the sum over all the harmonics belonging to a particular eigenvalue of angular momentum to the Gegenbauer polynomial corresponding to that eigenvalue. A formula is derived for projecting out the component of an arbitrary function corresponding to a particular eigenvalue of generalized angular momentum, and the formula is applied to polynomials in the m coordinates. An expansion is derived for expressing a many-dimensional plane wave in terms of hyperspherical harmonics and functions which might be called “hyperspherical Bessel functions.” It is shown how this expansion may be used to calculate many-dimensional Fourier transforms. A formula is derived expressing the effect of a group-theoretical projection operator acting on a many-dimensional plane wave. Finally, the techniques mentioned above are used to expand the Coulomb potential of a many-particle system in terms of Gegenbauer polynomials.
Additional Material:
1 Tab.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/qua.560220406
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