A fourth-order real-space algorithm for solving local Schrodinger equations
A fourth-order real-space algorithm for solving local Schrodinger equations
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DOI：
10.1063/1.1404142
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发表时间：
2001-10-15
影响因子：
4.4
通讯作者：
Chin, SA
中科院分区：
文献类型：
--
作者：
Auer, J;Krotscheck, E;Chin, SA
给出了一种求解实际空间中具有局域势的薛定谔方程的快速收敛算法。该算法基于通过将演化算子e（-epsilonH）分解为纯正系数的四阶来求解虚时薛定谔方程。波函数\ psi (j)>和从归一化因子e(j)（-epsilonE）提取的相关能量收敛为O（epsilon(4)）。直接从期望值< psi (j)\H \ psi (j)>计算的能量收敛为O（epsilon(8)）。与现有的二阶分割算子方法相比，我们的算法至少要高效100倍。我们检查并比较了四种不同的四阶分解，用于解决一维中的sech(2)（ax）势，并得出结论，所有四种算法在大时间步长下都能很好地收敛，但其中一种更有效。我们还求解了具有相同势的球面模拟的最低四个本征态的三维薛定谔方程。我们得出结论，该算法在求解三维低洼束缚态谱时同样有效。在有20个电子的球形凝胶簇的情况下，我们的四阶算法允许使用非常大的时间步长，从而大大加快了收敛速度。这种快速收敛使得我们的方案对于求解密度泛函理论的Kohn-Sham方程和任意几何中稀玻色-爱因斯坦凝聚体的Gross-Pitaevskii方程特别有用。(C) 2001年美国物理研究所。
We describe a rapidly converging algorithm for solving the Schrodinger equation with local potentials in real space. The algorithm is based on solving the Schrodinger equation in imaginary time by factorizing the evolution operator e(-epsilonH) to fourth order with purely positive coefficients. The wave functions \ psi (j)> and the associated energies extracted from the normalization factor e(j)(-epsilonE) converge as O(epsilon (4)). The energies computed directly from the expectation value, < psi (j)\H \ psi (j)>, converge as O(epsilon (8)). When compared to the existing second-order split operator method, our algorithm is at least a factor of 100 more efficient. We examine and compare four distinct fourth-order factorizations for solving the sech(2)(ax) potential in one dimension and conclude that all four algorithms converge well at large time steps, but one is more efficient. We also solve the Schrodinger equation in three dimensions for the lowest four eigenstates of the spherical analog of the same potential. We conclude that the algorithm is equally efficient in solving for the low-lying bound-state spectrum in three dimensions. In the case of a spherical jellium cluster with 20 electrons, our fourth-order algorithm allows the use of very large time steps, thus greatly speeding up the rate of convergence. This rapid convergence makes our scheme particularly useful for solving the Kohn-Sham equation of density-functional theory and the Gross-Pitaevskii equation for dilute Bose-Einstein condensates in arbitrary geometries. (C) 2001 American Institute of Physics.