Liu Jian, Miller William H
Department of Chemistry and K. S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720-1460, USA.
J Chem Phys. 2007 Jun 21;126(23):234110. doi: 10.1063/1.2743023.
It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.
展示了量子力学时间关联函数[例如由式(1.1)定义]如何能在无近似的情况下，以与半经典初值表示(LSC-IVR)或经典维格纳模型的线性化近似相同的形式来表示关联函数[参见式(2.1)]，即作为对应于两个算符的维格纳函数在相空间上(对轨迹的初始条件)的平均。不同之处在于，LSC-IVR中涉及的轨迹按经典方式演化，即根据经典运动方程演化，而在精确理论中，它们根据在此处推导的广义运动方程演化。然后引入精确运动方程的近似，以得到适用于复杂(即大的)分子系统的实用方法。本文提出了四种这样的方法——全维格纳动力学(全WD)和基于“维格纳轨迹”的二阶WD[H. W. 李和M. D. 斯库利，《化学物理杂志》77, 4604 (1982)]以及全多诺索 - 马滕斯动力学(全DMD)和基于“多诺索 - 马滕斯轨迹”的二阶DMD[A. 多诺索和C. C. 马滕斯，《物理评论快报》8722, 223202 (2001)]——所有这些都可视为原始LSC-IVR方法的推广。针对两个非谐模型问题对这种新方法的四个版本进行了数值测试，并且针对每个问题计算了动量自关联函数(即坐标或动量算符的线性算符)和力自关联函数(非线性算符)。确实可以看到这四种新的近似处理方法是对原始LSC-IVR近似的显著改进。
