For simplicity, in the following discussion we omit the label that indicates the selected stationary state and simply write j and Ej for the perturbation corrections of order j to the eigenfunction and

energy, respectively. That is to say, we expand a particular solution of Hˆ = E as

(r) = ∞∑ j=0

j(r)λ j , E =

Ejλ j . (2.3)

The method of Dalgarno and Stewart [10] consists of writing the perturbation corrections to the eigenfunction as

j(r) = Fj (r) 0(r), j = 0, 1, . . . (2.4) and solving the resulting equations for the functions Fj (r):

−1 2 ∇2Fj − 1

0 ∇ 0 · ∇Fj + V1Fj−1 −

EiFj−i = 0 . (2.5)

In this equation ∇ is the gradient vector operator and the dot stands for the standard scalar product. These equations are easier to solve than the original differential equations for the perturbation corrections j . In many cases the correction factors Fj are simple polynomial functions of the coordinates. Notice that F0 = 1 is a suitable solution to the equation of order zero, and that E0 does not appear in the perturbation equations (2.5).