Atomic Structure Theory

7.2 B-Spline Basis Sets 207
If we let VHF represent the HF potential, then its contribution to the action
integral S for an orbital a will be
R
0
Pa(r)VHFPa(r)dr =
b
2[lb]
R0(abab) −
k
ΛlaklbRk(abba)
, (7.57)
where the sum is over all occupied shells. This contribution to S leads to the
following modification of the potential contribution in the matrix element Aij,
R
drBi(r)VHFBj(r)
0
R
= drBi(r)
2[lb] v0(b, b, r)Bj(r)dr
0
b
−
k
Λlaklbvk(b,
Bj,r)Pb(r) , (7.58)
where vl(a, b, r) is the usual Hartree screening potential.
To solve the generalized eigenvalue problem in the HF case, we do a preliminary
numerical solution of the nonlinear HF equations to determine the
occupied orbitals Pb(r). With the aid of these orbitals, we construct the matrix
A using the above formula. The linear eigenvalue problem can then be
solved to give the complete spectrum (occupied and unoccupied) of HF states.
With this procedure, the states Pb(r) are both input to and output from the
routine to solve the eigenvalue equation. By comparing the eigenfunctions of
occupied levels obtained as output with the corresponding input levels, one
can monitor the accuracy of the solutions to the eigenvalue problem. It should
be noted that this consistency check will work only if the cavity radius is large
enough so that boundary effects do not influence the occupied levels in the
spline spectrum at the desired level of accuracy.
In Table 7.2, we compare low-lying levels for the sodium atom (Z = 11)
obtained by solving the generalized eigenvalue problem with values obtained
by solving the HF equations numerically. The potential used in this calculation
is the HF potential of the closed Na + core. It is seen that the B-spline
eigenvalues of the occupied 1s, 2s and 2p levels agree precisely with the corresponding
numerical eigenvalues. The B-spline eigenvalues of higher levels
depart from the numerical eigenvalues because of cavity boundary effects.
7.2.2 B-spline Basis for the Dirac Equation
Application of B-splines to obtain a finite basis set for the radial Dirac Equation
is described by [24] and follows very closely the pattern described above
for the radial Schrödinger equation. Several differences between the relativistic
and nonrelativistic expansions should be noted. Firstly, in the relativistic
case, we expand both P (r) andQ(r) in terms of B-splines: