Madsen Niels Kristian, Hansen Mads Bøttger, Worth Graham A, Christiansen Ove
Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark.
Department of Chemistry, University College London, 20, Gordon St., WC1H 0AJ London, United Kingdom.
J Chem Phys. 2020 Feb 28;152(8):084101. doi: 10.1063/1.5142459.
The multiconfiguration time-dependent Hartree (MCTDH) method is a powerful method for solving the time-dependent Schrödinger equation in quantum molecular dynamics. It is, however, hampered by the so-called curse of dimensionality which results in exponential scaling with respect to the number of degrees of freedom in the system and, thus, limits its applicability to small- and medium-sized molecules. To avoid this scaling, we derive equations of motion for a series of truncated MCTDH methods using a many-mode second-quantization formulation where the configuration space is restricted based on mode-combination levels as also done in the vibrational configuration interaction and vibrational coupled cluster methods for solving the time-independent Schrödinger equation. The full MCTDH wave function is invariant with respect to the choice of constraint (or gauge) operators, but restricting the configuration space removes this invariance. We, thus, analyze the remaining redundancies and derive equations for variationally optimizing the non-redundant matrix elements of the constraint operators. As an alternative, we also present a constraint that keeps the density matrices block diagonal during the propagation and the two choices are compared. Example calculations are performed on formyl fluoride and a series of high-dimensional Henon-Heiles potentials. The results show that the MCTDH[n] methods can be applied to large systems and that an optimal choice of constraint operators is key to obtaining the correct physical behavior of the wave function.
多组态含时哈特里（MCTDH）方法是求解量子分子动力学中含时薛定谔方程的一种有力方法。然而，它受到所谓的维度诅咒的阻碍，这导致其计算量随系统自由度数量呈指数增长，从而限制了它对中小尺寸分子的适用性。为避免这种增长，我们使用多模二次量子化公式推导了一系列截断MCTDH方法的运动方程，其中配置空间基于模式组合水平进行限制，这与求解不含时薛定谔方程的振动配置相互作用和振动耦合簇方法类似。完整的MCTDH波函数对于约束（或规范）算符的选择是不变的，但限制配置空间会消除这种不变性。因此，我们分析了剩余的冗余性，并推导了用于变分优化约束算符非冗余矩阵元的方程。作为一种替代方法，我们还提出了一种在传播过程中保持密度矩阵块对角的约束，并对这两种选择进行了比较。对氟甲酰和一系列高维亨农 - 海尔斯势进行了示例计算。结果表明，MCTDH[n]方法可应用于大型系统，并且约束算符的最优选择是获得波函数正确物理行为的关键。
