Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

nucleus. Even in open-shell atoms, this can be imposed as an approximation,
by spherically averaging ρ k . The simplification is considerable, since we know
a priori that the orbitals will be factorized as in Eq.(2.9). The angular part is
given by spherical harmonics, labelled with quantum numbers l and m, while
the radial part is characterized by quantum numbers n and l. Of course the
accidental degeneracy for different l is no longer present. It turns out that even
in open-shell atoms, this is an excellent approximation.
Let us consider the case of two-electron atoms. The Hartree equations,
Eq.(6.14), for orbital k = 1 reduces to
− ¯h2
2m e
∇ 2 1φ 1 (1) − Zq2 e
r 1
φ 1 (1) +
[ ∫
φ ∗ 2(2) q2 e
r 12
φ 2 (2) dv 2
]
φ 1 (1) = ɛ 1 φ 1 (1) (6.19)
For the ground state of He, we can assume that φ 1 and φ 2 have the same spherically
symmetric coordinate part, φ(r), and opposite spins: φ 1 = φ(r)v + (σ),
φ 2 = φ(r)v − (σ). Eq.(6.19) further simplifies to:
− ¯h2
2m e
∇ 2 1φ(r 1 ) − Zq2 e
r 1
φ(r 1 ) +
6.4 Code: helium hf radial
[ ∫ q
2
e
r 12
|φ(r 2 )| 2 d 3 r 2
]
φ(r 1 ) = ɛφ(r 1 ) (6.20)
Code helium hf radial.f90 1 (or helium hf radial.c 2 ) solves Hartree equations
for the ground state of He atom. helium hf radial is based on code
hydrogen radial and uses the same integration algorithm based on Numerov’s
method. The new part is the implementation of the method of self-consistent
field for finding the orbitals.
The calculation consists in solving the radial part of the Hartree equation
(6.20). The effective potential V scf is the sum of the Coulomb potential of the
nucleus, plus the (spherically symmetric) Hartree potential
V scf (r) = − Zq2 e
r
+ V H (r), V H (r 1 ) = q 2 e
∫ ρ(r2 )
r 12
d 3 r 2 . (6.21)
We start from an initial estimate for V H (r), calculated in routine init pot (
V (0)
H
(r) = 0, simply). With the ground state R(r) obtained from such potential,
we calculate in routine rho of r the charge density ρ(r) = |R(r)| 2 /4π (note that
ρ is here only the contribution of the other orbital, so half the total charge, and
remember the presence of the angular part!). Routine v of rho re-calculates
the new Hartree potential VH
out (r) by integration, using the Gauss theorem:
∫ r
ṼH
out (r) = V 0 + qe
2 Q(s)
r max s
∫r