# 16. Ring Polymer Molecular Dynamics (RPMD) Plugin¶

Ring Polymer Molecular Dynamics (RPMD) provides an efficient approach to include nuclear quantum effects in molecular simulations. When used to calculate static equilibrium properties, RPMD reduces to path integral molecular dynamics and gives an exact description of the effect of quantum fluctuations for a given potential energy model. For dynamical properties RPMD is no longer exact but has shown to be a good approximation in many cases.

For a system with a classical potential energy E(q), the RPMD Hamiltonian is given by

$H=\sum _{k=1}^{n}\left(\frac{{p}_{{k}^{2}}}{2m}+E({q}_{k})+\frac{m({k}_{B}Tn)^{2}}{2\hbar^{2}}({q}_{k}-{q}_{k-1})^{2}\right)$

This Hamiltonian resembles that of a system of classical ring polymers where different copies of the system are connected by harmonic springs. Hence each copy of the classical system is commonly referred to as a “bead”. The spread of the ring polymer representing each particle is directly related to its De Broglie thermal wavelength (uncertainty in its position).

RPMD calculations must be converged with respect to the number n of beads used. Each bead is evolved at the effective temperature nT, where T is the temperature for which properties are required. The number of beads needed to converge a calculation can be estimated using

$n>\frac{\hbar\omega_{max}}{{k}_{B}T}$

where $$\omega_{max}$$ is the highest frequency in the problem. For example, for flexible liquid water the highest frequency is the OH stretch at around 3000 cm-1, so around 24 to 32 beads are needed depending on the accuracy required. For rigid water where the highest frequency is only around 1000 cm-1, only 6 beads are typically needed. Due to the replication needed of the classical system, the extra cost of the calculation compared to a classical simulation increases linearly with the number of beads used.

This cost can be reduced by “contracting” the ring polymer to a smaller number of beads. The rapidly changing forces are then computed for the full number of beads, while slower changing forces are computed on a smaller set. In the case of flexible water, for example, a common arrangement would be to compute the high frequency bonded forces on all 32 beads, the direct space nonbonded forces on only 6 beads, and the reciprocal space nonbonded forces on only a single bead.

Due to the stiff spring terms between the beads, NVE RPMD trajectories can suffer from ergodicity problems and hence thermostatting is highly recommended, especially when dynamical properties are not required. The thermostat implemented here is the path integral Langevin equation (PILE) approach. This method couples an optimal white noise Langevin thermostat to the normal modes of each polymer, leaving only one parameter to be chosen by the user which controls the friction applied to the center of mass of each ring polymer. A good choice for this is to use a value similar to that used in a classical calculation of the same system.