卡氏座標下三維時變薛丁格方程式的平行化程式發展及其於雷射與分子間交互作用之應用研究

Parallel Solver for Three-dimensional Cartesian-grid Based Time-Dependent Schrödinger Equation and Its Applications in Laser-Molecule Interaction Study with Single-Action-electron Assumption

李允民(Yum-Min Lee)

指導教授 ： 吳宗信 ； 江進福

國立交通大學/工學院/機械工程系所/博士(2009年)

https://doi.org/10.6842/NCTU.2009.00527

摘要

我們以有限體積法來離散化一個在卡氏座標中的三維時變薛丁格方程式(TDSE)，假設分子的原子核不移動且只有一電子有作用(SAE)，並用一個交錯時間法(stagger-time)來處理波方程式的實部與虛部在時間上演進，以平行計算的程式來分析此三維時變薛丁格方程式，此程式利用多層圖型切割法來作區域切割(domain decomposition)，以此分配不同處理器(processor)的計算區域。此程式以一個H2+系統來作驗證，在沒有雷射(laser)的交互作用下，總電子機率和總能量的守恆。與其它二維薛丁格方程式的計算比較，在雷射的作用下離子化(ionization rates)的機率比較。此程式的平行化效率在使用128顆處理器下可以達到75%。最後以H2+系統，原子核距離9 au，電子在不同雷射入射角度(= 0 和 90)下的機率隨時間的分佈，及原子核距離2 au，和諧光譜(HHG)在不同雷射入射角度(= 0, 30, 60 and 90)，以及不同雷射入射角度對N2, O2及CO2分子離子化機率的影響並和實驗資料比較來說明此三維平行化的程式的能力以及應用。

關鍵字

時變薛丁格方程式 ； 雷射 ； 行平化 ； 有限體積法 ； 離子化

並列摘要

A parallelized three-dimensional Cartesian-grid based time-dependent Schrödinger equation (TDSE) solver for molecules with single active electron assumption, assuming freezing the motion of nucleus is presented in this thesis. An explicit stagger-time algorithm is employed for time integration of the TDSE, in which the real and imaginary parts of the wave function are defined at alternative times, while a cell-centered finite-volume method is utilized for spatial discretization of the TDSE on Cartesian grids. The TDSE solver is then parallelized using domain decomposition method on distributed memory machines by applying a multi-level graph-partitioning technique. The solver is validated, using a H2+ molecule system, both by observing total electron probability and total energy conservation without laser interaction, and by comparing the ionization rates with previous 2D-axisymmetric simulation results with an aligned incident laser pulse. Parallel efficiency of this TDSE solver is presented and discussed, in which the parallel efficiency can be as high as 75% using 128 processors. Finally, examples of temporal evolution of probability distribution of laser incidence onto a H2+ molecule at inter-nuclei distance of 9 a.u. (= 0 and 90) and spectral intensities of harmonic generation at inter-nuclei distance of 2 a.u. (= 0, 30, 60 and 90) and the angle effect of laser incidence on ionization rate of N2, O2 and CO2 molecules are presented to demonstrate the capability of the current TDSE solver.

並列關鍵字

TDSE ； Laser ； Parallelized ； FVM ； Ionization ； HHG

參考文獻

2. S. X. Hu and L. A. Collins, Phys. Rev. Lett. 94, 123902 (2005).

9. J. Muth-Böhm, A. Becker , and F. H. M. Faisal, Phys. Rev. Lett. 85, 2280 (2002)

12. L. B. Madsen, Phys. Rev. A 65, 053417 (2002)

13. J. H. Posthumus, Rep. Prog. Phys. 67, 623 (2004).

15. D. Dundas, Phys. Rev. A 65, 023408 (2002).