2.6M - 1. Institut für Theoretische Physik - Universität Stuttgart

Summary
conventions and detailed transformations of several equations. Utmost importance is
attached to practical implementation. The complete printout of the source code of the
used program would not be helpful. However, detailed flowcharts (Nassi-Shneiderman
diagrams [22]) and listings are shown at different places to provide the reader with a
deeper understanding of the simulation method.
This thesis is a part of the project A15 “Numerical Methods for Many-Electron Atoms in
Neutron Star Magnetic Fields” in the context of the “Sonderforschungsbereich” 382 “Procedures
and Algorithms for the Simulation of Physical Processes on High-Performance
Computers” - a joint project of the Universities of Tübingen and Stuttgart.
Quantum Monte Carlo Method: The aim was to obtain as accurately as possible the
ground state energy E0 of the many body Schrödinger equation with the Hamiltonian:
ˆH = − 1
N�
�∇
2
i=1
2 i +
� �� �
kinetic energy
1
N�
N�
1
1
− Z
2 |�ri − �rj | |�ri|
i,j
i=1
j �=i
� �� �
� �� � Coulomb potential
Coulomb interaction
. (1)
A statistical approach is given by Quantum Monte Carlo simulations. The technique
is based on walkers moving in 3N -dimensional space. N stands for the number of
electrons of the atom considered and for the neutral atoms treated here is equal to the
atomic number Z . The chapter starts with basics of the Quantum Monte Carlo method
including
� the Variational Principle [23], providing an upper bound (see equation (2.5))
for the energy expectation value ET of a trial function ΨT,
� the Monte Carlo Integration [18], perfectly suitable for the high dimensional
integrals used here,
� the Importance Sampling, increasing efficiency of the Monte Carlo integration
and
� the Metropolis Algorithm [17], making a statement about the acceptance probability
(see equation (2.22)).
The above points lead directly to the Variational Quantum Monte Carlo method
(VQMC, see figure 2.1). This is one of the two simulation techniques used in this thesis.
It yields an upper bound of the ground state energy
126
ET = lim
N →∞
1
N
N�
EL( � R) (2)
i=1