## 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
where the matrix of *H*_{0} 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

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 *H*_{0}.