15.6 Exercises 515
show that you can write the Hartree-Fock equations as
∑
γ
hHFα γCkγ=εkCkα.
Explain the meaning of the different terms.
Set up the Hartree-Fock equations for the ground state beryllium with the electrons occupy-
ing the respective ’hydrogen-like’ orbitals 1 sand 2 s. There is no spin-orbit part in the two-body
Hamiltonian.
- As basis functions for our calculations we will use hydrogen-like single-particle functions. In
the computations you will need to program the Coulomb interaction with matrix elements
involving single-particle wave functions withl= 0 only, so-calleds-waves. We need only the
radial part since the spherical harmonics for thes-waves are rather simple. Our radial wave
functions are
Rn 0 (r) =
(
2 Z
n
) 3 / 2 √
(n− 1 )!
2 n×n!
L^1 n− 1 (
2 Zr
n
)exp(−
Zr
n
),
with energies −Z^2 / 2 n^2. A function for computing the generalized Laguerre polynomials
L^1 n− 1 (^2 Zrn)is provided at the webpage of the course under the link of project 2. We will use
these functions to solve the Hartree-Fock problem for beryllium.
Show that you can simplify the direct term developed during the lectures
∫
r^21 dr 1
∫
r^22 dr 2 R∗nα 0 (r 1 )R∗nβ 0 (r 2 )
1
(r>)
Rnγ 0 (r 1 )Rnδ 0 (r 2 )
∫∞
0
r^21 dr 1 R∗nα 0 (r 1 )Rnγ 0 (r 1 )
[
1
(r 1 )
∫r 1
0
r^22 dr 2 R∗nβ 0 (r 2 )Rnδ 0 (r 2 )+
∫∞
r 1
r 2 dr 2 R∗nβ 0 (r 2 )Rnδ 0 (r 2 )
]
.
Find the corresponding expression for the exchange term.
- With the above ingredients we are now ready to solve the Hartree-Fock equations for the
beryllium atom. Write a program which solves the Hartree-Fock equations for beryllium.
You will need methods to find eigenvalues (see chapter 7) and gaussian quadrature (chap-
ter 5) to compute the integrals of the Coulomb interaction. Use as input for the first iteration
the hydrogen-like single-particle wave function. Comparethe results (make a plot of the 1 s
and the 2 sfunctions) when self-consistency has been achieved with those obtained using the
hydrogen-like wave functions only (first iteration). Parameterize thereafter your results in
terms of the following Slater-type orbitals (STO)
RSTO 10 (r) =N 10 exp(−α 10 r)
and
RSTO 20 (r) =N 20 rexp(−α 20 r/ 2 )
Find the coefficientsα 10 andα 20 which reproduce best the Hartree-Fock solutions. These
functions can then be used in a variational Monte Carlo calculation of the beryllium atom.
15.2.In this problem we will attempt to perform so-called densityfunctional calculations.
- The first step is to perform a Hartree-Fock calculation using the code developed in the previ-
ous exercise but omitting the exchange (Fock) term. Solve the Hartree equation for beryllium
and find the total density determined in terms of the single-particle wave functionsψias
ρH(r) =
N
∑
i= 1
|ψi(r)|^2 ,
