以量子蒙地卡羅方法探討電子的相關聯作用

Studying Electron Correlation with Quantum Monte Carlo Methods

邢正蓉(Cheng-Rong Hsing)

指導教授 ： 李定國

國立臺灣大學/理學院/物理研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.01577

摘要

在對真實材料的計算中，如何正確的探討電子間的交互相關聯作用是很重要的。對於不同的第一原理電子結構計算方法，量子蒙地卡羅方法是公認可以給出相當正確的電子相關聯效應。我們所用的量子蒙地卡羅方法是以平均場近似的結果 (如 Hartree-Fock 或是 密度泛涵理論)當作初始值，加入電子相關聯效應的部份來直接解多體粒子的薛丁格方程式。
本文中我們以量子蒙地卡羅方法來研究兩個問題：
第一個主題是結合耦合常數積分計算和量子蒙地卡羅方法來探討小分子 － Si2、C2H2、C2H4及C2H6 的交互相關聯能並且與局域密度泛涵 (LDA) 的結果做比較。我們發現當比較局域密度泛涵與量子蒙地卡羅方法的結果時，局域密度泛涵所得到的交互相關聯能的誤差值會在電子密度高的區域與密度低的區域間相互抵銷掉，這說明了為何局域密度泛涵方法在不同的系統中總是可以給出不錯的結果，因為交互相關聯能是一個積分量，所以在做積分運算時在不同區域的誤差是有機會相互抵銷的。在我們的計算中我們也發現局域密度泛涵的交互相關聯能的誤差值與電子密度的Laplacian量是非常相似的，這說明了在這些系統中，交互相關聯能的近似函數應該要將電子密度的Laplacian項考慮進去，這個結果與之前矽原子和塊矽 (bulk silicon) 的計算一致，稍有不同的是，這兩個系統其局域密度泛涵的誤差值與Laplacian量更為相似，而在小分子中，某些區域的誤差與Laplacian量並不是很一致的。
本文中另一個題目主要是探討聚乙炔的激發態問題。我們以量子蒙地卡羅方法來計算直接能隙及激子的結合能。聚乙炔是發光共軛高分子中結構最簡單的一個材料，早期的電子結構計算通常無法給出正確的能隙和結合能。最近，GW 方法及包含了電子與電洞交互作用能的Bethe-Salpeter equation 可以解出相當正確的直接能隙及激子的結合能，但是在聚乙炔這個系統中得到一個與實驗相反的結果，就是光學活性激子態的激發能比光學不活性激子態的激發能還要低。這樣的結果引發了我們研究的動機希望用量子蒙地卡羅方法來探討這個系統。在計算時，我們首先要確認不同激子態的波函數均有包含正確的對稱性，這使得量子蒙地卡羅方法可以找到正確的激子態。我們的計算結果顯示以量子蒙地卡羅方法得到的直接能隙及激子的激發能均大於實驗的結果，對於直接能隙的計算這樣的結果是可以預見的，因為我們只有計算單一個聚合物而忽略了塊材中聚合物間的作用力，對於GW方法在其他發光共軛高分子的計算中，加入聚合物間的作用能會縮小計算的直接能隙。對於激子的激發能計算，因為計算結果並不是收斂的很好，我們認為在未來也許使用不同於本文中所用的電子相關聯效應在需要被最佳化的波函數中是一個比較好的作法。

關鍵字

量子蒙地卡羅 ； 交互相關作用 ； 分子 ； 激發態 ； 聚乙炔

並列摘要

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.

並列關鍵字

Quantum Monte Carlo ； exchange-correlation ； molecules ； excitation ； polyacetylene

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