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vanvleck_1982 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1982 THE VAN VLECK TRANSFORMATION IN PERTURBATION THEORY 1 Although frequently it is desirable to carry a perturbation treatment to second or third order for a degenerate state, the required calculations often become very complicated. A simple procedure due to Van Vleck makes this task considerably easier. I. Perturbation Theory and the Problem of Degeneracy[?, ?, ?] In many quantum mechanical problems, the Hamiltonian may be written H = H + H (1) where the solution for the unperturbed Hamiltonian H is known, and H is a small perturbation. Pertur- bation theory may be used to find the small changes in the energy levels and wave functions introduced by H . If the matrix elements of H are evaluated in the H representation, i.e., calculated with the unperturbed wave functions, the H matrix is diagonal, H = E k δ k j . k j Several quantum numbers may be needed to label the states of H ; here k represents the entire set. Some of the unperturbed states may be degenerate and k must then include an index to distinguish the members of the degenerate set. The perturbation matrix H will have o ff -diagonal terms which couple the various unperturbed states as well as diagonal terms which directly shift the energy levels. In the usual perturbation theory, the shift of a particular unperturbed energy level E k and its wave function ψ k is evaluated by expanding the Hamiltonian in powers of a parameter λ , H = H + λ H + λ 2 H ′′ + ... (2) The solution of the Schr¨odinger equation, H λ = E λ (3) 1 These notes were written by Professor Dudley Herschbach.
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Handout: Van Vleck Transformation Spring, 1982 Page 2 is sought, where E k = E k + λ E k + λ 2 E k ′′ + ... λ k = λ k + λψ k + λ 2 ψ ′′ k + ... (4) which reduces to the unperturbed solution as λ 0. On substituting (4) into (3) and equating the coe - cients of like powers of λ , the perturbed energy levels are found to be summationdisplay H H E k = E k + λ H kk + λ 2 H ′′ k j jk + ... (5) kk + λ 2 E k E j j k to second order. The first contributions are merely the perturbation averaged with the unperturbed wave function of state k . In the second order approximation, there is a sum over the influence of the other states. The energy level is displaced upward by the states of lower energy and downward by those of higher energy; the displacements are proportional to the square of the coupling term as given by the matrix elements of H and inversely proportional to the corresponding energy di ff erences.
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