Centroid path integral approach

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. 

The treatment of NQM effects can be accomplished on a much more quantitative level by including the adiabatic limit and modifying the centroid path integral approach [54-56]. The centroid path integral represents the unifying approach, which is valid both in the adiabatic and diabatic limits. This is done in a way that... 

In Paper IV, several algorithms were developed and explored for the efficient computation of the centroid force in the CMD equations for many-body systems. Two of the algorithms are direct numerical path-integral approaches [17-19] in which the path averaging explicit in Eq. 

To enhance convergence of free-particle sampling in centroid path integral simulations, a bisection sampling technique was used for a ring of beads by extending the original approach of Pollock and Ceperley for free particle... 

The centroid path integral method described above enable us to conveniently determine KIEs by directly computing the ratio of the quantum partition functions for two different isotopes through free energy perturbation (FEP) theory. The use of mass perturbation in free-particle bisection sampling scheme results in a major improvement in computation accuracy for KIE calculations such that secondary kinetic isotope effects and heavy atom isotope effects can be reliably obtained. The PI-FEP/UM method is the only practical approach to yield computed secondary KIEs sufficiently accurate to be compared with experiments. ... 

A free energy perturbation approach in Feynman centroid path integral simulations has been developed to incorporate nuclear quantum effects. 

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

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. 

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. 

For another perspective we mention a second approach of which the reader should be aware. In this approach the dividing surface of transition state theory is defined not in terms of a classical mechanical reaction coordinate but rather in terms of the centroid coordinate of a path integral (path integral quantum TST, or PI-QTST) [96-99] or the average coordinate of a quanta wave packet. In model studies of a symmetric reaction, it was shown that the PI-QTST approach agrees well with the multidimensional transmission coefScient approach used here when the frequency of the bath is high, but both approaches are less accurate when the frequency is low, probably due to anharmonicity [98] and the path centroid constraint [97[. However, further analysis is needed to develop practical PI-QTST-type methods for asymmetric reactions [99]. 

A direct numerical path-integral computation of the centroid force as outlined above will undoubtedly provide an accurate value of the centroid potential surface. For low-dimensional systems, the centroid force might indeed be calculated for each point in space and stored on a grid in computer memory and later recalled in the CMD calculation. As the dimensionality of the system increases, however, this straightforward procedure is no longer feasible, due to the exponential growth of the computational effort. The real issue, therefore, is how to carry out such a computation efficiently within the context of the time integration of the CMD equations [Eq. (3.59)]. Consequently, more specialized approaches are required [6]. 

The probability PXqr q ) fo move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated [108] by PIMC or PIMD techniques [17-19] combined with umbrella sampling [77,108,123] of the reaction coordinate centroid variable. In the latter computational technique, a number of windows are set up which confine the path centroid variable of the reaction coordinate to different regions. These windows connect in a piecewise fashion the possible centroid positions in going from the reactant state to the transition state. A series of Monte Carlo calculations are then performed, one for each window, and the centroid probability distribution in each window is determined. These individual window distributions are then smoothly joined to calculate the overall probability function in Eq. (4.11). An equivalent approach is to calculate the centroid mean force and integrate it from the reactant well to barrier top (i.e., a reversible work approach for the calculation of the quantum activation free energy [109,124]). 

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