Centroid path integral dynamics

Variations on this surface hopping method that utilize Pechukas [106] formulation of mixed quantum-classical dynamics have been proposed [107,108]. Surface hopping algorithms [109-111] for non-adiabatic dynamics based on the quantum-classical Liouville equation [109,111-113] have been formulated. In these schemes the dynamics is fully prescribed by the quantum-classical Liouville operator and no additional assumptions about the nature of the classical evolution or the quantum transition probabilities are made. Quantum dynamics of condensed phase systems has also been carried out using techniques that are not based on surface hopping algorithms, in particular, centroid path integral dynamics [114] and influence functional methods [115]. 

Diffusion constants are enhanced with the approximate inclusion of quantum effects. Changes in the ratio of diffusion constants for water and D2O with decreasing temperature are accurately reproduced with the QFF1 model. This ratio computed with the QFF1 model agrees well with the centroid molecular dynamics result at room temperature. Fully quantum path integral dynamical simulations of diffusion in liquid water are not presently possible. 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

It seems that surface hopping (also called Molecular Dynamics with Quantum Transitions, MDQT) is a rather heavy tool to simulate proton dynamics. A recent and promising development is path integral centroid dynamics [123] that provides approximate dynamics of the centroid of the wavefunctions. Several improvements and applications have been published [123, 124, 125, 126, 127, 128). 

Jang S, Voth GA (2001) A relationship between centroid dynamics and path integral quantum transition state theory. J Chem Phys 112(8747-8757) Erratum 114, 1944... 

The Feynman-Hibbs and QFH potentials have been used extensively in simulations examining quantum effects in atomic and molecular fluids [12,15,25]. We note here that the centroid molecular dynamics method [54, 55] is related and is intermediate between a full path integral simulation and the Feynman-Hibbs approximation ... 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

Voth, G.A. (1996). Path-integral centroid methods in quantum statistical mechanics and dynamics. Adv. Chem. Phys. 93, 135-218... 

A disadvantage of using Car-Parrinello path integral methods is that the molecular dynamics is used only to compute averaged properties, the simulation dynamics having no direct physical meaning. A recently developed, albeit approximate method for generating fully quantum mechanical dynamics is the ab initio centroid molecular dynamics method (Marx et al., 1999 Pavese et al., 1999). Tlie application of Car-Parrinello methods to... 

In a path-integral Monte Carlo (PIMC) calculation, the centroid force can readily be calculated from Eq. (3.64) by using the importance sampling function exp[-5p(q,. . . , qp) h] and pairwise MC moves to enforce the centroid constraint. For a path-integral molecular dynamics (PIMD) calculation [71], one defines fictitious momenta p, for each of the quasiparticles q and then runs an MD simulation with Hamilton s... 

In this article a perspective on quantum statistical mechanics and dynamics has been reviewed that is based on the path centroid variable in Feynman path integration [1,3-8,21-23]. Although significant progress has been achieved in this research effort to date, much remains to be done. For example, in terms of the calculation of equilibrium properties it... 

The reactive flux method is also useful in calculating rate constants in quantum systems. The path integral formulation of the reactive flux together with the use of the centroid distribution function has proved very useful for the calculation of quantum transition-state rate constants [7]. In addition new methods, such as the Meyer-Miller method [8] for semiclassical dynamics, have been used to calculate the flux-flux correlation function and the reactive flux. 

Perez, A. Tuckerman, M. E. Muser, M. H., A Comparative Study of the Centroid and Ring-Polymer Molecular Dynamics Methods for Approximating Quantum Time Correlation Eunctions from Path Integrals. J. Chem. Phys. 2009,130,184105. 

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