Computational Physics - Department of Physics

16.12 Exercises 533

φ 2 p(ri) =αrie−αri/^2. (16.53)

Observe thatri=

√

r^2 ix+r^2 iy+ri^2 z.

6. Set up the Hartree-Fock equations for the ground state of the beryllium and neon atoms with
   four and ten electrons, respectively, occupying the respective hydrogen-like orbitals. There
   is no spin-orbit part in the two-body Hamiltonian. Find alsothe experimental ground state
   energies using atomic units.


7. Solve the Hartree-Fock equations for the beryllium and neon atoms. Use again as input for
   the first iteration the hydrogen-like single-particle wavefunction. Compare the results with
   those obtained using the hydrogen-like wave functions only(first iteration).


8. Write a function which sets up the Slater determinant for beryllium and neon. Use the
   Hartree-Fock single-particle wave functions to set up the Slater determinant. You have only
   one variational parameter,β. Compute the ground state energies of neon and beryllium. The
   calculations should include parallelization, blocking and importance sampling. Compare the
   results with the Hartree-Fock results. How important is thecorrelation part? Is there a dif-
   ference compared with helium? Comment your results.
   16.2.The aim of this project is to use the Variational Monte Carlo (VMC) method to evaluate
   the ground state energy, onebody densities, expectation values of the kinetic and potential
   energies and single-particle energies of quantum dots withN= 2 ,N= 6 andN= 12 electrons,
   so-called closed shell systems.
   We consider a system of electrons confined in a pure two-dimensional isotropic harmonic
   oscillator potential, with an idealized total Hamiltoniangiven by

Ĥ=

N

∑

i= 1

(

−

1

2 ∇

2

i+

1

2 ω

(^2) r 2
i

)

+∑

i<j

1

ri j, (16.54)

where natural units (h ̄=c=e=me= 1 ) are used and all energies are in so-called atomic units

a.u. We will study systems of many electronsNas functions of the oscillator frequencyω

using the above Hamiltonian. The Hamiltonian includes a standard harmonic oscillator part

Ĥ 0 =

N

∑

i= 1

(

−

1

2 ∇

2

i+

1

2 ω

(^2) r 2
i

)

,

and the repulsive interaction between two electrons given by

Ĥ 1 =∑

i<j

1

ri j

,

with the distance between electrons given byri j=

√

r 1 −r 2. We define the modulus of the

positions of the electrons (for a given electroni) asri=

√

r^2 ix+r^2 iy.

1a) In exercises 1a-1e we will deal only with a system of two electrons in a quantum dot with a
frequency of ̄hω= 1. The reason for this is that we have exact closed form expressions for the
ground state energy from Taut’s work for selected values ofω, see M. Taut, Phys. Rev. A 48 ,
3561 (1993). The energy is given by 3 a.u. (atomic units) when the interaction between the
electrons is included. If only the harmonic oscillator partof the Hamiltonian, the so-called
unperturbed part,
Ĥ 0 =

N

∑

i= 1

(

−

1

2

∇^2 i+

1

2

ω^2 r^2 i

)

,

the energy is 2 a.u. The wave function for one electron in an oscillator potential in two dimen-

sions is