Ring polymer dynamics in curved spaces

Abstract

We formulate an extension of the ring polymer dynamics approach to curved spaces using stereographic projection coordinates. We test the theory by simulating the particle in a ring, T(1), mapped by a stereographic projection using three potentials. Two of these are quadratic, and one is a nonconfining sinusoidal model. We propose a new class of algorithms for the integration of the ring polymer Hamilton equations in curved spaces. These are designed to improve the energy conservation of symplectic integrators based on the split operator approach. For manifolds, the position-position autocorrelation function can be formulated in numerous ways. We find that the position-position autocorrelation function computed from configurations in the Euclidean space R(2) that contains T(1) as a submanifold has the best statistical properties. The agreement with exact results obtained with vector space methods is excellent for all three potentials, for all values of time in the interval simulated, and for a relatively broad range of temperatures.

Quantum simulations of the hydrogen molecule on ammonia clusters

“…This conclusion, however, needs to be carefully tested against simulations describing the ammonia skeleton as fully flexible, which ought to involve additional development for DMC or the curved manifolds version of the ring polymer dynamics. 88 Work along these directions is in progress in our laboratories.…”

“…The latter remain stable upon decreasing the pressure to 100 KPa from the 200 MPa needed for the synthesis provided T remains below 248 K. Clearly, a more definitive conclusion should be sought by simulating the quantum ground and low temperature states of our clathratelike species employing either DMC or a curved manifolds version of the ring polymer dynamics. 88…”

Infinite swapping in curved spaces

“…The Matzubara frequencies grow linearly with the Trotter number. 84 The main contribution we make in this article is the development of an efficient integrator for spaces generally constructed from the parameters of continuous Lie groups and implement these into RE, INS, or PINS strategies to simulate molecular aggregates with curved spaces.…”

“…In this paper, we are interested in more general curved spaces beyond those generated by holonomic constraints, 81, 82 as these may arise in path integral simulations, for example. 83,84 In a recent application of Ring Polymer Dynamics to curved spaces, 84 we develop a method to integrate directly from generalized coordinates, and we observe that direct application of traditional symplectic integrators gives rise to numerical methods with significant amount of energy drift. The method we have developed in Ref.…”

“…The method we have developed in Ref. 84 to improve the long term energy drift of the symplectic integrator is to compute the state vector by using the Liouville operator. The latter is not split as customary.…”