Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
Department of Chemistry and Biochemistry, Kettering University, Flint, Michigan 48504, USA.
J Chem Phys. 2017 Jun 7;146(21):214116. doi: 10.1063/1.4984229.
A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature β limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.
基于费曼路径质心概念，针对广义密度算符的情况，利用 Blinov 和 Roy [J. Chem. Phys. 115, 7822-7831 (2001)]的投影密度算符形式，提出了量子统计力学的一种推导方法。在 Jang 和 Voth [J. Chem. Phys. 111, 2357-2370 (1999)]的正则平衡图像的背景下，对质心密度、质心符号和质心相关函数进行了公式化和分析。讨论了密度算符投影到系统的特定能量本征态的情况，并表明可以从双库珀变换相关函数中提取微观正则动力学信息。还表明，所提出的投影算符方法可用于在零倒数温度β极限下正式连接质心和维格纳相空间分布。提出了一种状态投影精确量子动力学的质心分子动力学（CMD）近似，并证明在简谐极限下是精确的。还对四次谐振子和双势阱进行了数值测试，并发现该方法比正则 CMD 更准确。在基态投影的情况下，该方法可以解析在正则 CMD 失效的较高势垒区域中双势阱问题的隧道分裂。最后，将状态投影的 CMD 框架表示为路径积分形式。
