Partitioning and perturbation methods

Variational techniques, either in their general form (Section 2.4) or in the form corresponding to use of a linear variation function (Section 2.3), are 

For present purposes it is more useful to concentrate on other approaches, which start from the finite-basis form of the linear variation method. In many forms of variation-perturbation theory, exact unperturbed eigenfunctions are not required and the partitioning of the Hamiltonian into two terms is secondary to a partitioning of the basis. At the same time, as we shall see, it is possible to retrieve the equations of conventional perturbation theory by making an appropriate choice of basis. 

To develop the variation-perturbation approach we may suppose that 1 is an approximation to the required eigenfunction and that we wish to improve this approximation by adding functions 02 and solving 

Equation (2.5.2) is exactly equivalent to (2.5.1), but we have reduced the secular problem from one of n dimensions to one of dimensions by defining an effective Hamiltonian to allow for the other functions. 

Since Heff depends on E, which is not yet known, the solution must generally be obtained by iteration. Let us consider first, for example, the case .4 = 1, which will be appropriate if i is non-degenerate. We may then choose Ci = 1 since it is not necessary at this stage to normalize the eigenfunction. The effective Hamiltonian (2.5.3) then reduces to a single element (1x1 matrix), and (2.5.2) gives 

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