## Introduction

The time scale problem is a familiar one to quantum chemists. Rooted into our common basic training, the Born Oppenheimer approximation is the essential tool needed to unravel fundamental problems that would otherwise be intractable. In its most familiar form, it allows one to separate the electronic degrees of freedom, associated with a set of much lighter bodies, from the heavier nuclear degrees of freedom. With some notable exceptions of marked nonadiabatic behavior, the unraveling of which remains an very active field of research, the Born Oppenheimer approximation yields very accurate results, and it is often used to build accurate models for the phenomenological potential energy surface felt by the nuclei. The latter set of bodies is traditionally treated by classical mechanics. Generally, at sufficiently elevated temperatures, for sufficiently heavy nuclei, and for sufficiently harmonic interactions, classical mechanics provides reasonably accurate answers. In this article, however, we depart from the traditional theme and explore the time scale problem one encounters when by necessity the quantum laws of motion are applied to the nuclei themselves. This implies the nuclei are light, the temperature is too low, and the interactions are highly anharmonic. There is a vast number of fundamental problems in chemical physics, where all these conditions take place at the same time, and insight into these is fundamental to a myriad of disciplines ranging from astrophysics to computational biology. More specifically, the set of problems that have preoccupied our two groups intensely for a number of years are the theoretical estimation of physical properties of molecular clusters. Clusters in general and molecular clusters in particular are models of condensed matter that can be studied both theoretically and experimentally. Insight gained from these investigations has already created a vast improvement in our understanding of the complicated phenomena that take place in the assembly process, thermodynamic stability as function of size of condensed matter, microsolvation, adsorption versus absorption, the effects of surface tension just to name a few. At the temperature and pressure conditions that produce stable molecular aggregates, nuclei of elements in the first three periods are sufficiently light to produce significant quantum effects, whereas at the same time, the weak interaction between the molecules in the aggregate are far from harmonic. More importantly for the discussion in this article, most intramolecular degrees of freedom are associated with relatively deeper dissociation energies, as in the case of stretching modes, and relatively stiff force constants compared to the intermolecular degrees of freedom. Atomistic quantum simulation of molecular aggregates is rendered either particularly challenging or practically intractable by the large difference in the time scales.

The classical equations of motion satisfied by a typical molecular aggregate are stiff. A small time step is required to sample properly the high frequency dynamics, whereas a long time scale is required to capture the effects on the system from the set of lower frequency events. Additionally, this problem exacerbates the lack of ergodicity and the occurrence of rare events when exploring thermodynamic properties in the classical limit with random walks. Therefore, judicious use of holonomic constraints is routine in classical simulations of molecular clusters. The typical outcome for thermodynamic properties is a significant reduction in the statistical error, making simulations more efficient. In quantum simulations, there is an added benefit gained by constraining stiff modes, at least for the two stochastic approaches we discuss in this article. The convergence properties of the diffusion Monte Carlo[1-12] (DMC) and the imaginary time path integral[13-31] (MCPI) are greatly enhanced by constraining high frequency modes.

In a recent article,[25] we investigate a simple one-dimensional (1D) harmonic chain with 1000 particles. All particles have the same mass, and every particle is connected to two neighboring particles on a line with one stiff harmonic spring on one side and a one soft one on the other side. The simple harmonic chain model constructed this way mimics a set of condensed molecules. Using analytical solutions of the imaginary time path integral at finite Trotter number, we compare the convergence of the analytical solutions of the path integral expression for the heat capacity as a function of temperature and Trotter number. Our results show that the adiabatic approximation is accurate for temperatures below

where is the smallest of the high frequency set. Below this temperature, a fully flexible simulation is highly inefficient as it requires many hundreds of time slices to converge, where the constrained simulations require less than 20 slices to converge for most of the temperature range. Consequently, the efficiency gains produced by performing the MCPI simulation with constraints are massive. Similar gains are quantifiable for DMC.[1] In Figure 1, we show the estimate of the ground state energy of a particle of unit mass in a harmonic potential with a.u. as a function of simulation time for various values of the step size . This is the parameter that has to be systematically reduced until convergence is achieved in DMC simulations.

To properly interpret the result in Figure 1, it is important to keep in mind that a similar computation with a.u. is fully converged with a a.u. (the largest value of in the set of simulations represented). It is evident that an increase in frequency by a mere factor of 3.2 requires a DMC step smaller than a factor of 10. As is also evidenced in Figure 1, DMC simulations must also converge with respect to simulation pseudotime (i.e., the number of DMC steps, on the *x-*axis in Fig. 1) and a smaller increases proportionally the total number of DMC steps required to reach the asymptotic distribution, that is, the ground state wavefunction in this case. When DMC is used to simulate systems with multiple time scales, the stiffer mode demands a small , while the softer one demands a relative longer time scale to reach convergence, and consequently a much greater number of steps. Therefore, when in a system, the subdivision of stiff and soft modes is clear, and the adiabatic approximation is expected to work well, it is highly advantageous to use the proper holonomic constraints, and the efficiency gained can be on the same order of magnitude as what we measure in MCPI simulations.

However, constraining intermolecular degrees of freedom, like bond stretches, bond angles, and the like, create curved spaces and special techniques have to be developed to handle the geometric complexity of these. The purpose of this article is to review some of the recent advances in the form of algorithms specifically designed to carry out quantum Monte Carlo simulations in curved spaces, and to demonstrate their powers. In the methodology section, we review briefly the basic objects needed for the geometry and the dynamics in general non-Euclidean spaces. Then, we develop the general theory for DMC and MCPI in non-Euclidean spaces. Our results section contains selected numerical examples and in the conclusions section, we discuss a set of directions that our group is currently undertaking to further develop our tools and continue to explore the rich and important field of condensed molecular matter.