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Application of Quantum Mechanical Perturbation Theory to Molecular Vibrational-Rotational Analysis
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The quantum mechanical Schrödinger equation for the vibrational motion of a diatomic molecule is solved to arbitrary order of Rayleigh-Schrödinger perturbation theory by means of symbolic formula generation using Mathematica. Current state-of-the-art calculations of this type allow treatment only up to second-order for polyatomic molecules (i.e., those composed of three or more atoms). It is demonstrated that, by using Mathematica, the lengthy algebraic equations resulting from high-order perturbation theory can be accurately and efficiently treated to the appropriate level of approximation as dictated by the molecular Born-Oppenheimer potential energy surface. Requisite integrated forms are generated to arbitrary order using a compact procedural program. A calculation through 15th-order results in over 3,000 lines of formulas representing a total of 32 terms. FortranForms of the resulting expressions are incorporated into a large, general-purpose program for execution on a mainframe computer. Such a procedure can be applied to other problems governed by equations of the form treated here.
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http://www.mathematica-journal.com/issue/v2i2/
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