Formulas for integration and differentiation

Many classes of formulas can be represented most concisely in terms of the differences of the values of the function at a set of equally spaced grid points. Of the possible notations, the central difference notation is the most convenient and is the one used throughout.

Let f_j = f(x_j) where x is used to denote either r or$\rho$. By definition, the central difference operator is given by

$$\delta f(x) = f(x+h/2) + f(x-h/2).$$ (54)

Then

$$\delta f_{j+1/2} = f_{j+1} - f_j$$ (55)

and

$$\delta^2 f_j \delta f_{j+1/2} - \delta f_{j-1/2}$$ =

= f_j+1 - 2 f_j + f_j-1. (56)

Many atomic properties require the evaluation of an integral, say,

$$I = \int_0^\infty f(x)dx \approx \int_0^{x_M} f(x)dx$$ (57)

where x_M is a suitably chosen large value of x, such that $\int_{x_M}^\infty f(x)dx$ is negligible. When f(x) is tabulated at equally spaced points $x_j = jh, j =0,1,\ldots, M$, where M is even, the integration may be represented by the composite rule

$$\int_0^{x_M}f(x)dx = \sum_{ \mbox{$j$\space odd}} \int_{x_{j-1}}^{x_{j+1}} f(x)dx$$ (58)

where each individual integral is over two adjacent intervals. It can be shown that

$$\int_{x_{j-1}}^{x_{j+1}} f(x)dx = 2h\left(f_j + \frac{1}{6} \delta^2f_j -\frac{1}{180} \delta^4 f_j\right) + {\cal O}(h^7).$$ (59)

This formula uses values of the integrand outside the range of integration and cannot be used for the first and last integral. In these cases the $\delta^4$ term may be omitted leading to the well-known Simpson's rule

$$\int_{x_{j-1}}^{x_{j+1}} f(x)dx =\frac{h}{3}\left[f_{j-1} +4 f_j + f_{j+1}\right] + {\cal O}(h^5).$$ (60)

For certain integrals the accuracy of Simpson's rule is satisfactory.
 

2001-01-09