Atomic Structure Theory

The matrices A and B are given by
Aij =
Bij =
R
0
R
0
dBi
dr
7.2 B-Spline Basis Sets 205
v =(p1,p2, ··· ,pn) . (7.52)
dBj
dr +2Bi(r)
V (r)+
l(l +1)
2r2
Bj(r) dr, (7.53)
Bi(r)Bj(r)dr. (7.54)
It should be mentioned that the matrices A and B are diagonally dominant
banded matrices. The solution to the eigenvalue problem for such matrices
is numerically stable. Routines from the lapack library [3] can be used to
obtain the eigenvalues and eigenvectors numerically.
Solving the generalized eigenvalue equation, one obtains n real eigenvalues
ɛ λ and n eigenvectors v λ . The eigenvectors satisfy the orthogonality relations,
which leads to the orthogonality relations
for the corresponding radial wave functions.
i,j
v λ i Bijv µ
j = δλµ , (7.55)
R
0
P λ
l (r)P µ
l (r)dr = δλµ, (7.56)
Table 7.1. Eigenvalues of the generalized eigenvalue problem for the B-spline approximation
of the radial Schrödinger equation with l = 0 in a Coulomb potential
with Z = 2. Cavity radius is R = 30 a.u. We use 40 splines with k =7.
n ɛn n ɛn n ɛn
1 -2.0000000 11 0.5470886 ···
2 -0.5000000 12 0.7951813 ···
3 -0.2222222 13 1.2210506 ···
4 -0.1249925 14 2.5121874 34 300779.9846480
5 -0.0783858 15 4.9347168 35 616576.9524036
6 -0.0379157 16 9.3411933 36 1414036.2030934
7 0.0161116 17 17.2134844 37 4074016.5630432
8 0.0843807 18 31.1163253 38 20369175.6484520
9 0.1754002 19 55.4833327
10 0.2673078 20 97.9745446
The first few eigenvalues and eigenvectors in the cavity agree precisely
with the first few bound-state eigenvalues and eigenvectors obtained by numerically
integrating the radial Schrödinger equations; but, as the principal