| Summary: | 博士 === 國立臺灣大學 === 物理研究所 === 95 === The methods for including the many-body interaction are important in studying many-particle problems. A popular approach is to map the problem to a single particle picture and introduce a mean field potential. Alternatively, the quantum Monte Carlo (QMC) methods, which treat the correlation more direct and accurate, are a powerful computational tool for studying an interacting many-body system. The focus of this thesis is variational Monte Carlo and diffusion Monte Carlo methods.
In this thesis, two works were presented :
• We use the combination of the coupling-constant
integration procedure and the variational Quantum Monte
Carlo method to study the exchange-correlation (XC)
interaction in small molecules: Si2, C2H2, C2H4, and
C2H6. We report the calculated XC energy density, a
central quantity in density functional theory, as
deduced from the interaction between the electron and
its XC hole integrated over the interaction strength.
Comparing these“exact”XC energy densities with results
using the local-density approximation (LDA), one can
analyze the errors in this widely-used approximation.
Since the XC energy is an integrated quantity, error
cancellation among the XC energy density in different
regions is possible. Indeed we find a general error
cancellation between the high-density and low-density
regions. Moreover, the error distribution of the
exchange contribution is out of phase with the error
distribution of the correlation contribution. Similar
to what is found for bulk silicon and an isolated
silicon atom, the spatial variation of the errors of
the LDA XC energy density in these molecules largely
follows the sign and shape of the Laplacian of the
electron density. Some noticeable deviations are found
in Si2 in which the Laplacian peaks between the atoms,
while the LDA error peaks in the regions “behind”
atoms where a good portion of the charge density
originates from an occupied 1sigma_u antibonding
orbital. Our results indicate that, although the
functional form could be quite complex, an XC energy
functional containing the Laplacian of the energy is
a promising possibility for improving LDA.
• We use VMC method to study the excitation energies of
trans-polyacetylene. QMC have been used for the
calculation of excited-states of molecules and bulk
silicon, but little is known about applying it to
conjugated polymers. trans-polyacetylene is the simplest
one and has been studied by many theoretical and
experimental works. In theoretical calculation for
trans-polyacetylene, GW results are accurate in the
direct band gap and Bethe-Salpeter equation (BSE),
including the electron-hole interaction, is accurate for
the singlet optically active state 11Bu. However, the
excitation energy for optically inactive state 21Ag is
higher than the experimental value, resulting a
optically active state is lower than the optically
inactive state. In our VMC calculation, the direct band
gap for the isolated polymer is higher than the GW value
for 0.76 eV. For the previous calculation of
polythiophene by GW method, the direct band gap for a
single chain was also higher than the bulk calculation
for 1.1ev. Therefore, our VMC result in single chain
should be consistent with their study. The VMC
excitation energies are also higher than the
experimental or GW-BSE values, but the binding energy of
optically active state is comparable to their results.
This may be due to the error cancellation of our
calculation. In general, the quality of our VMC trial
wave functions dominate our results and the nodal
structure of the wave function is also important.
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