Centroid dynamics [11, 12] is based on the use of the path integral centroid density as a semiclassical phase space function
Time correlation function calculated using centroid phase space function, can be related to the true quantum result, , in frequency space by [11, 12]
because centroid dynamics generates an approximation to the Kubo transformed time correlation function[11, 12]. The expression is exact for quadratic potentials and is accurate to order for more complex systems. For operators other than ``x'' the centroid dynamics approximation is of lower order accuracy but has been found to generate useful results within the semiclassical operator approximation [11, 12] (for example, ). The manybody generalization within Boltzman Statistics goes through straight forwardly.
In order to complete the picture which allows centroid PIMD to be employed, it is necessary to develop a procedure which will generate the motion of the particle centroids on the potential of mean force, . This can be accomplished through the application of the normal mode variable transformations and the constant temperature path integral molecular dynamics method described above . A set of slow variables, the particle centroids, can be made to move on the potential of mean force of a set of fast variables, the other modes generated by the normal mode transformation, by adiabatically separating the two sets of degrees of freedom . The necessary adiabatic separation can be achieved under the normal mode dynamics described above by simply adjusting the mode masses so that the particle centroids, or center of masses, move slowly compared to the other modes (making progressively larger and testing the results for convergence). Thus, the particle centroids can be made to move on the appropriate potential of mean force, .
The particle centroids can be permitted to evolve without any Nosé-Hoover thermostats and the velocities of these degrees of freedom periodically resampled from the Boltzman distribution. This procedure both permits the centroid variables to undergo Newtonian evolution on the centroid potential of mean force (provided the necessary adiabatic separation has been achieved) and yet also performs the required canonical average over the initial conditions, [28, 5]. Time correlation functions are thus obtained for times less than or equal to the velocity refresh period which is assumed to be long compared to the decay time of the correlation function of interest. A second more approximate method involves coupling the centroids to a Nosé-Hoover chain and assuming that the dynamics is approximately Newtonian for times on the order of the time scale associated with the action of the centroid Nosé-Hoover chain, . The time scale parameter, is assumed to be appropriately long compared to the decay time of the correlation function of interest. Here, the average over initial conditions occurs as a natural part of the algorithm. Convergence can be checked by increasing the time scale parameter, . In practice, it is difficult to discern differences between the first rigorous and the approximate but more convenient approach that is employed commonly in classical molecular simulations. Nonetheless, it is advisable to be aware of the approximation and perform appropriate checks (i.e. ensure that is taken large enough).
Centroid dynamics can also be calculated under constant pressure. Rigorously, one could perform a constant pressure path integral molecular dynamics simulation, generate configurations (cell matrices and MN particle positions), and use these configurations to commence constant volume simulations of the type described in the first part of the previous paragraph. This procedure would generate the additional average over the volume required by isothermal isobaric ensemble and the required dynamics. Alternatively, an approximate but more convenient approach can be employed. Simply, perform a constant pressure path integral molecular dynamics simulation in the adiabatic limit, the centroids slow compared to the other modes but fast compared to the extended system variables, the volume and the centroid Nosé-Hoover chain, and collect the desired time correlation functions. This second, practical, convenient approach has been found to work well in classical molecular dynamics calculations. Again, it is advisable to be aware of the approximations and perform appropriate checks (i.e. the time scales associated with the extended system variables must be taken sufficiently long compared to the decay time of the correlation functions).