On regularizing the MCTDH equations of motion.
On regularizing the MCTDH equations of motion.
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DOI：
10.1063/1.5024859
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
H. Meyer;Haobin Wang
中科院分区：
文献类型：
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作者：
H. Meyer;Haobin Wang
当存在未被占据的所谓的单粒子函数（SPFs）时，多构型时变哈特里（MCTDH）方法导致运动方程（EOM）变得奇异。从Hartree产品开始，除了第一个spf之外，所有spf最初都未被占用。因此，为了数值求解MCTDH-EOMs，必须通过正则化过程去除奇点。通常密度矩阵的逆是正则化的。在这里，我们论证并展示了正则化系数张量，这反过来又正则化密度矩阵，导致EOMs的性能得到改善。最初未占用的spf在Hilbert空间中更快地旋转到“正确方向”，并且最终结果对正则化参数值的选择不太敏感。对于一个特殊的例子（一个用变换哈密顿量研究的自旋玻色子系统），我们甚至可以证明只有使用新的正则化方案才能得到正确的结果。最后，在附录A中讨论了Lubich及其同事开发的一种新的MCTDH-EOMs集成方案。本文认为该格式不能解决自然轨道的未占据问题，因为该格式忽略了自然轨道，并且根本不传播它们。
The Multiconfiguration Time-Dependent Hartree (MCTDH) approach leads to equations of motion (EOM) which become singular when there are unoccupied so-called single-particle functions (SPFs). Starting from a Hartree product, all SPFs, except the first one, are unoccupied initially. To solve the MCTDH-EOMs numerically, one therefore has to remove the singularity by a regularization procedure. Usually the inverse of a density matrix is regularized. Here we argue and show that regularizing the coefficient tensor, which in turn regularizes the density matrix as well, leads to an improved performance of the EOMs. The initially unoccupied SPFs are rotated faster into their "correct direction" in Hilbert space and the final results are less sensitive to the choice of the value of the regularization parameter. For a particular example (a spin-boson system studied with a transformed Hamiltonian), we could even show that only with the new regularization scheme could one obtain correct results. Finally, in Appendix A, a new integration scheme for the MCTDH-EOMs developed by Lubich and co-workers is discussed. It is argued that this scheme does not solve the problem of the unoccupied natural orbitals because this scheme ignores the latter and does not propagate them at all.