Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel.
J Chem Phys. 2011 Apr 7;134(13):134104. doi: 10.1063/1.3573566.
The frozen Gaussian approximation to the quantum propagator may be a viable method for obtaining "on the fly" quantum dynamical information on systems with many degrees of freedom. However, it has two severe limitations, it rapidly loses normalization and one needs to know the Gaussian averaged potential, hence it is not a purely local theory in the force field. These limitations are in principle remedied by using the Herman-Kluk (HK) form for the semiclassical propagator. The HK propagator approximately conserves unitarity for relatively long times and depends only locally on the bare potential and its second derivatives. However, the HK propagator involves a much more expensive computation due to the need for evaluating the monodromy matrix elements. In this paper, we (a) derive a new formula for the normalization integral based on a prefactor free HK propagator which is amenable to "on the fly" computations; (b) show that a frozen Gaussian version of the normalization integral is not readily computable "on the fly"; (c) provide a new insight into how the HK prefactor leads to approximate unitarity; and (d) how one may construct a prefactor free approximation which combines the advantages of the frozen Gaussian and the HK propagators. The theoretical developments are backed by numerical examples on a Morse oscillator and a quartic double well potential.
冻结的高斯近似量子传播子可能是一种可行的方法，可以获得具有多个自由度的系统的“实时”量子动力学信息。然而，它有两个严重的限制，它迅速失去归一化，人们需要知道高斯平均势，因此它不是力场中的纯局部理论。这些限制原则上可以通过使用半经典传播子的赫尔曼-克鲁克（Herman-Kluk，HK）形式来补救。HK 传播子在相对较长的时间内近似地保持幺正性，并且仅依赖于裸势及其二阶导数。然而，由于需要评估单径矩阵元，HK 传播子涉及更昂贵的计算。在本文中，我们：（a）基于一个适合“实时”计算的无预因子 HK 传播子，推导出一个新的归一化积分公式；（b）表明，冻结的高斯归一化积分版本不易“实时”计算；（c）提供一个新的见解，说明 HK 预因子如何导致近似的幺正性；以及（d）如何构建一个无预因子的近似，它结合了冻结的高斯和 HK 传播子的优点。理论发展得到了莫尔斯振荡器和四次双阱势的数值示例的支持。
