## 4.5. Van Vleck Transformations

It is sometimes desirable to take into account perturbations arising from rather distant states without actually including these states in the matrix to be diagonalized exactly. Under these circumstances, it is convenient to use a Van Vleck transformation [26] (pp. 394-396) to correct the matrix elements within the submatrix to be diagonalized for effects arising from interactions with the distant states. (A Van Vleck transformation is essentially a transformation from the original Hamiltonian to a new Hamiltonian in which first-order interactions between the states under consideration and the distant states are eliminated. For more information, the reader is referred to reference [26].) If the Hamiltonian is divided into a zeroth-order part, a first-order part, and a second-order part

 (4.7)

where the matrix of H0 is diagonal in the basis set under consideration, then the corrected matrix elements (to second order) within the block actually to be diagonalized are given by

 (4.8)

In (4.8), the indices i and j are chosen from within the block to be diagonalized; the summation index k runs over all values outside this block. The energies are those calculated for each state from H0.