In this review, we give a comprehensive comparison of the most widely used coherent state (CS) based methods to solve the time-dependent Schrödinger equation (TDSE). Starting from the fully variational coherent states (VCS) method, after a first approximation, the coupled coherent states (CCS) method can be derived, whereas an additional approximation leads to the semiclassical Herman–Kluk (HK) method. We numerically compare the different methods with another one, based on a static rectangular grid of coherent states (SCS), by applying all of them to the revival dynamics in a 1D Morse oscillator, with a special focus on the number of basis states (for the CCS and HK methods the number of classical trajectories) needed for convergence and the related issue of tight frames, which in principle allow the usage of CSs as if they were orthogonal. Different discretisation strategies for the occurring phase space integrals for systems with more degrees of freedom are also discussed and the apoptosis procedure that allows to circumvent the linear dependency problem in the VCS method is reviewed. The Holstein molecular crystal model serves to further illustrate the latter point.

中文翻译：

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时变薛定谔方程的基于相干状态的解：变分原理的近似层次

在这篇综述中，我们对最广泛使用的基于相干状态 (CS) 的方法进行了全面比较，以解决瞬态薛定谔方程 (TDSE)。从完全变分相干态 (VCS) 方法开始，经过一次近似后，可以导出耦合相干态 (CCS) 方法，而额外的近似导致半经典 Herman-Kluk (HK) 方法。我们将不同的方法与另一种方法进行数值比较，基于相干态的静态矩形网格 (SCS)，将所有方法应用于一维莫尔斯振荡器中的复兴动力学，特别关注基态的数量（对于CCS 和 HK 方法（经典轨迹的数量）需要收敛和紧框架的相关问题，原则上允许使用 CS，就好像它们是正交的一样。还讨论了具有更多自由度的系统发生相空间积分的不同离散化策略，并回顾了允许规避 VCS 方法中线性相关问题的细胞凋亡过程。荷斯坦分子晶体模型用于进一步说明后一点。