Path integral molecular dynamics (PIMD) is a prevailing computational tool to study nuclear quantum statistics for realistic molecular systems at thermal equilibrium. We have employed the Kolmogorov operator to develop a unified framework that covers both stochastic and deterministic thermostatting algorithms, including such as the Andersen thermostat, Langevin dynamics, NoséHoover chain (including NoséHoover thermostat), etc. [J. Chem. Phys. 145, 024103 (2016); 147, 034109 (2017)] Compared to most conventional PIMD algorithms that can be unified in the “side”/“end” schemes of the framework, the “middle” thermostat scheme introduced by us significantly improves the sampling efficiency as well as accuracy for general molecular systems. In addition to numerical examples for realistic systems, we have presented the analytical analysis to prove that the “middle” scheme leads to exact results for PIMD in the harmonic limit regardless of the thermostat parameter and of the finite time interval (as long as the propagation is stable).
In the unified framework we have revealed that, in addition to the usual density evolution, there exists another type of discrete evolution that may not correspond to a continuous, real dynamical counterpart of the Langevin equation [J. Chem. Phys. 147, 184104 (2017)]. This virtual dynamics case is also amenable to the desired Boltzmann distribution for various stochastic thermostatting methods [Chin. J. Chem. Phys. 30, 735 (2017)]. By analyzing the asymptotic distribution and characteristic correlation time, we have shown that, for good numerical performance in efficiency as well as accuracy, one may choose the thermostat parameter (of the “middle” scheme) in a wide range from around the optimal value to the high value limit.
More recently, we have further used the unified framework to develop a novel practical PIMD methodology in either of the diabatic and adiabatic representations for studying exact quantum statistics of large multielectronicstate systems in thermal equilibrium when the BornOppenheimer approximation, Condon approximation, and harmonic bath approximation are no longer valid [J. Chem. Phys. 148, 102319 (2018)].

"A simple and accurate algorithm for path integral molecular dynamics"
Jian Liu*, Dezhang Li, Xinzijian Liu
Journal of Chemical Physics, 145, 024103 (2016)
http://dx.doi.org/10.1063/1.4954990
Note: In the article we propose a simple algorithm for PIMD for studying quantum statistics, which significantly improves the performance of sampling accuracy as well as efficiency over conventional PIMD algorithms.

"A unified thermostat scheme for efficient configurational sampling for classical/quantum canonical ensembles via molecular dynamics"
Zhijun Zhang, Xinzijian Liu, Zifei Chen, Haifeng Zheng, Kangyu Yan, Jian Liu*
Journal of Chemical Physics, 147, 034109 (2017)
http://dx.doi.org/10.1063/1.4991621
Note: In this article we employ the Kolmogorov operator to develop a unified framework that covers both stochastic and deterministic thermostatting algorithms, which leads to the efficient “middle” thermostat scheme for performing either PIMD or MD.

"Path integral molecular dynamics for exact quantum statistics of multielectronicstate systems"
Xinzijian Liu, Jian Liu*
Journal of Chemical Physics, 148, 102319 (2018)
[Invited article for the JCP special topic issue on "Nuclear Quantum Effects"]
https://doi.org/10.1063/1.5005059
Note: In this article we propose a novel practical PIMD methodology in either of the diabatic and adiabatic representations for studying exact quantum statistics of large multielectronicstate systems in thermal equilibrium when the BornOppenheimer approximation, Condon approximation, and harmonic bath approximation are no longer valid.

"Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics"
Dezhang Li, Xu Han, Yichen Chai, Cong Wang, Zhijun Zhang, Zifei Chen, Jian Liu*, Jiushu Shao*
Journal of Chemical Physics, 147, 184104 (2017)
https://doi.org/10.1063/1.4996204
Note: By analyzing the asymptotic distribution and characteristic correlation time, we show that, for good numerical performance in efficiency as well as accuracy, one may choose the thermostat parameter (of the “middle” scheme) in a wide range from around the optimal value to the high value limit.

"Understanding molecular dynamics with stochastic processes via real or virtual dynamics"
Dezhang Li, Zifei Chen, Zhijun Zhang, Jian Liu*
Chinese Journal of Chemical Physics, 30, 735 (2017)
[Invited article for the CJCP special topic issue on "Chemical Dynamics"]
http://dx.doi.org/10.1063/16740068/30/cjcp1711223
Note: In the unified framework [J. Chem. Phys. 147, 034109 (2017)] we reveal that, in addition to the usual density evolution, there exists another type of discrete evolution (so called “virtual dynamics”) that may not correspond to a continuous, real dynamical counterpart (of such as the Langevin equation), which is also amenable to the desired Boltzmann distribution for various stochastic thermostatting methods.