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

Summary
Jastrow factor was chosen in accordance with the estimate corresponding to equation
(3.70) and figure 5.76 in section 5.2.2. A horizontal solid line is printed in the figure if
other methods provide a comparable value. The quoted energy value in the caption of
each figure is the RPDQMC ground state energy value; the error gives the variation of
the block energy EB according to equation (4.3). This chapter also states the dependence
of ∆τ (see figures 5.74 and 5.75) and b JF (see figure 5.76). It is observable that ground
state energy increases slightly for smaller ∆τ but the statistical error increases too.
The implementation shows that ∆τ may not be chosen arbitrarily small. Regarding
the acceptance probability it can be said that a great acceptance probability leads to
a large standard deviation, and a small one to instabilities. Therefore ∆τ is chosen in
such a way that the acceptance probability does not sink below 98 % . The figures for
the Jastrow parameter b JF show a very good agreement with equation (3.70); it is not
advisable to choose a smaller b JF . Therefore the DQMC method b JF = √ β is set. The
chapter closes with a view on computing time (see figures 5.77 up to 5.80) increasing on
the whole with the cube of the number of electrons N .
Conclusion and outlook: It is obvious that the Diffusion Quantum Monte Carlo
method can also treat problems with greater atomic number than Z = 2 in the presence
of an external magnetic field. In regions where values for comparison are available for
an upper bound [8, 14] these values can be improved. Of course this method can also
calculate ground state energies for other magnetic fields and for ions. This is done in an
ongoing diploma thesis [25]; as a foretaste, for Fe 13+ results are given in figure 6.2. Figure
6.1 shows the scheme of this program. Only the short-time approximation is applied
but the influence can be neglected because of the tiny steps in ∆τ. Table 6.1 gives an
overview of the released-phase DQMC values. These values are the most comprehensive
and accurate ground state energies of medium-Z atoms up to iron (Z = 26) in neutron
star magnetic fields presented in literature.
An extension of the Quantum Monte Carlo method to calculate excited states was proposed
by Jones et al. [10]. Note that it is always possible to extract one excited state
by choosing a guiding wavefunction perpendicular to the true ground state. This is the
ground state within a sub-space. But in this context they are not to be understood as the
excited states. In the past excited states for helium (Z = 2) could be calculated in low
magnetic fields by means of this approach. The idea of the method is to create a basis
set of functions and to formulate a generalised eigenvalue problem. The diagonalisation
of these matrices as a function of projection time leads to the ground state and some
excited states simultaneously. The numbers of excited states are given by the number
of basis functions minus one. These basis functions can be realised by several input files
for the HFFEM calculations by using different quantum numbers. This idea offers the
opportunity of obtaining highly accurately calculated excited states of heavy atoms or
ions in strong magnetic fields.
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