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.MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1982 THE VAN VLECK TRANSFORMATION IN PERTURBATION THEORY1 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◦= Ek◦δkj.kj 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 off-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 Ek◦and its wave function ψ◦k is evaluated by expanding the Hamiltonian in powers of a parameter λ, H = H◦ + λH′ + λ2H′′ + . . . (2) The solution of the Schr¨odinger equation, Hλ = Eλ (3) 1These notes were written by Professor Dudley Herschbach.Handout: Van Vleck Transformation Spring, 1982 Page 2 is sought, where Ek = Ek◦+ λEk ′ + λ2Ek ′′ + . . . λk = λk◦+ λψk ′ + λ2ψ′′ k + . . . (4) which reduces to the unperturbed solution as λ 0. On substituting (4) into (3) and equating the coeffi-→cients of like powers of λ, the perturbed energy levels are found to be X H′ H′ Ek = Ek◦+ λHkk ′ + λ2H′′ kj jk + . . . (5) kk + λ2 Ek − Ejj�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 diff erences. However, if some states are degenerate with state k, the treatment must be modified because of the vanishing energy denominators which make (5) infinite (except, of course, in the special case that the offending terms have zero matrix elements, e.g., due to symmetry considerations). If the unperturbed Ek◦belongs to a group of gk degenerate levels, its wave function ψ◦k is not completely determined, but may be arbitrarily chosen to be any linear combination of a set of gk orthonormal functions associated with the degenerate level. The perturbation will in general cause energy splittings which at least remove the degeneracy to some extent. However, as λ 0, the perturbed wave function ψk will not, in → general, reduce to the arbitrarily chosen initial ψ◦k, as in (5), but will become a linear combination of the initial set of gk degenerate functions, the particular combination depending on the perturbation. To apply perturbation theory, one must then first determine this correct limiting linear combination and make the functions for the other unperturbed states that are degenerate with state k be orthogonal to state k. This requirement means that the transformation of matrix elements accompanying the transformation to these correct zero-order wave functions should uncouple the degenerate states in H′. Thus H′ becomes diagonal for the block of states that are degenerate with state k, and all the terms with vanishing denominators in (5) disappear. The determination of the zeroth-order wave functions therefore requires solving the secular equation, |H′(k) − ξ1| = 0 (6) which corresponds to diagonalizing H′(k), which is the gk × gk block of H′ that corresponds to the gk degenerate unperturbed states of energy Ek◦. The solution immediately provides the diagonal term ξk in the transformed H′ which is needed for the first approximation in (5). However, if it is desired to evaluate theHandout: Van Vleck Transformation Spring, 1982 Page 3 second or higher order terms, then the off-diagonal matrix elements H′ coupling state k to all other states kj j must be transformed to correspond to the transformation to the correct zero-order wave function for state k. Moreover, in case the first order perturbation did not remove completely the degeneracy present in the unperturbed problem, as would happen if part of the H′matrix were diagonal and itself degenerate, then for the second order treatment it would be necessary to initially diagonalize the submatrix of the entire Hamiltonian that includes not only all second order terms H′′ , coupling state k to each state k′ degenerate kk′ with it, but also terms of the form H′ and H′ which couple the degenerate states in second order through kj k′ j other states j (see reference [?]). The program of the usual degenerate perturbation theory outlined above is often difficult or impossible to apply. The difficulty is the more serious because the upper, continuous portion of the energy spectrum can be handled only by introduction of an awkward box normalization or some equivalent. We shall now consider the Van Vleck procedure which brings great simplification because it allows any block of interest to be uncoupled (to third order) from the rest of the H′ matrix and afterwards diagonalized. This minimizes the size of the secular equations that must be solved, and there is ordinarily no difficulty in diagonalizing those blocks of H′ associated with the lower unperturbed energy levels in which there is greatest physical interest. II. The Van Vleck Transformation Although the Van Vleck method has been applied to a variety of