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Edited by: Francesco Aquilante, Uppsala University, Sweden

Reviewed by: Francesco Aquilante, Uppsala University, Sweden; K. R. S. Chandrakumar, Bhabha Atomic Research Centre, India

*Correspondence: Ryan Babbush and Alán Aspuru-Guzik, Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Room M113, Cambridge, MA 02138, USA e-mail:

This article was submitted to Theoretical and Computational Chemistry, a section of the journal Frontiers in Chemistry.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the

The energy and mass scales of chemical motion lie in a regime between quantum and classical mechanics but for reasons of computational complexity, molecular modeling (MM) is largely performed according to Newton's laws. When classical Hamiltonians are chosen to reproduce properties of real material, classical MM is an efficient compromise. An increasing amount of MM uses highly accurate Born-Oppenheimer (BO) potential energy surfaces, which allow one to study complex bond rearrangements where experiment cannot motivate a potential (Car and Parrinello,

Many approaches already exist to bridge this gap and study quantum equilibrium properties and dynamics: path integral Monte Carlo (PIMC), ring polymer molecular dynamics (RPMD), centroid molecular dynamics (CMD), variational path-integral approximations, discretized path-integral approximations, semi-classical approximations, thermal Gaussian molecular dynamics and colored-noise thermostats (Whitlock et al.,

An alternative philosophy is suggested by density functional theory (DFT) (Hohenberg and Kohn,

The bargain of our proposed effective classical potential is similar to that posed by DFT. One sacrifices access to rigorous momentum based-observables and abandons the route to systematic improvement. In exchange, the two properties which are physically guaranteed, the equilibrium particle density and the partition function, are obtained at a cost equivalent to classical sampling but with improved accuracy. As a practical tool, the map is an easy way to transform BO-based force fields into a form which is well-suited for classical sampling. Perhaps the most promising aspect of this mapping would be its scalability which could potentially extend the ability to treat quantum propagation effects to all systems that can be sampled classically. It is even possible that the fictitious trajectories of particles moving on such a potential would, like Kohn-Sham orbitals, have somewhat improved physicality over their classical counterparts, although we will not examine that possibility here.

First, we show the uniqueness of an equilibrium effective potential that gives the exact equilibrium quantum density via classical sampling. Next, we demonstrate that the equilibrium effective potential may be approximated by a linear operator acting on the true potential. Finally, we numerically approximate the map in a rudimentary way, and obtain surprisingly good results and transferability for both one dimensional potentials and a model of liquid

In their seminal work on path integral quantum mechanics, Feynman and Hibbs introduced the concept of an effective classical potential that allows for the calculation of quantum partition functions in a seemingly classical fashion (Feynman and Hibbs,

where ^{1}

where the Wick-rotated (

By integrating over only closed paths at each coordinate we obtain the scalar equilibrium density,

Finally, we define the partition function as a normalization factor which is obtained by integrating over

We are now in a position to define an equilibrium effective potential, which encapsulates knowledge of the physical quantum density into a form amenable to classical sampling. We choose the equilibrium effective potential,

Note that this definition associates the Boltzmann factor, ^{−βW(q)}, with the _{0}(

Using Equation 7, one can exactly calculate the equilibrium effective potential whenever one can evaluate the path integral. Unfortunately that is usually numerically intractable. Thus, it is useful to wonder if a ^{2}

_{Q}(_{0}(

Our first step toward developing a theory of force-field functors is to show that the proposed mapping, _{0}(

Both steps in this proof take the form of

where

which is minimum when ρ is equal to the quantum equilibrium density matrix ρ_{0} associated with the Hamiltonian, _{0}(^{3}

Using the definition of the quantum equilibrium particle density,^{4}

we see that,

But we see that this relation is still true if we interchange over-scored variables,

This leads to the contradiction,

and therefore only one _{0}(_{0}(

Equation 7 shows the existence the equilibrium effective potential,

where

which is minimum when η_{0}(

So we see that,

If we interchanged all over-scored quantities, we would also find the following,

Adding these equations together leads to the result,

Thus, we see that no two _{0}(

Because the physical potential _{0}(

The results of the above section establish the possibility of reversing

We begin by rewriting Equations 4 and 6,

and introduce several definitions which break apart the action term into a kinetic part and a potential part,

We now employ a notation due to Feynman and Hibbs, for the equilibrium average of a path functional weighted by

Jensen's inequality tells us that that, 〈^{f}〉 ≥ ^{〈f〉} with an error on the order of the variance of

Because any potential is unique only up to a constant, we can use properties of logarithms to remove

with corrections on the order of ^{2}. We also see from this that the equilibrium effective potential is a temperature dependent correction to the true potential.

In the multi-dimensional case, the path integral couples all 3

Approximate separability of this mapping is one of the key differences between our method and approaches such as Feynman-Kleinert, which introduces higher ordered many-body terms into the effective potential, or RPMD, which avoids the issue at the cost of introducing ancilla particles. Our

It is far from obvious that a transferable map between

Effective potentials were calculated using Equation 7 with densities obtained from the efficient real-space discrete variable representation (DVR) of the path integral (Thirumalai et al.,

In order to obtain the simplest possible

Consider the short time Trotterization of the path-integral, which we use to generate exact quantum densities for our test sets (Thirumalai et al.,

In order to obtain a linear functor capable of transforming a one-dimensional potential at fixed

The linear approximation to

Figure

The resulting radial distribution functions,

We have shown that for each physical potential, there is a unique effective potential which reproduces the quantum density and free energy when sampled with classical statistics. Other properties of a classical simulation of the effective Hamiltonian are not designed to approximate reality by the mapping, but the effective potential may be advantageous to the status quo: classical simulation on a Born-Oppenheimer surface. In this paper we have shown that the implied mapping between the physical and effective potential,

All authors conceived and designed the research project. With guidance from John Parkhill and Alán Aspuru-Guzik, Ryan Babbush wrote the proofs, first draft of the manuscript, and code for obtaining and characterizing the numerical functor. All authors interpreted the results and co-wrote the article.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank David Manolopoulos, Eugene Shakhnovich, and Eric Heller for helpful conceptual discussions regarding direction of the project. We also thank Jarrod McClean, Joseph Goodknight and Nicolas Sawaya for discussions regarding revision of the manuscript. Research sponsored by the United States Department of Defense. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressly or implied, of the U.S. Government.

In practice, path integral expressions are analytically intractable except in a few cases. Feynman proposed to simplify Equation 5 by changing from an integral over all closed paths that start and end at point

So that we only integrate over each closed path once, we must change our expression for the partition function to only calculate paths that match the centroid,

While the partition functions given by Equation 5 and Equation A2 are exactly equal, the two expressions are associated with subtly different scalar density functions. Equation 5 is associated with the true equilibrium density in Equation 4 and A2 is associated with the path centroid density,

The Dirac delta function in this equation enforces the requirement that integrating the Boltzmann factor associated with this density over the path centroid,

The quantum Bogoliubov inequality is proved for the grand canonical ensemble in the Appendix of (Mermin,

if

which is minimum only when ρ is equal to the quantum equilibrium density matrix ρ_{0} associated with the Hamiltonian,

where,

We see that ρ_{λ} = ρ_{0} if λ = 0 and ρ_{λ} = ρ if λ = 1. Accordingly,

by the fundamental theorem of calculus. To evaluate the derivative we use,

The first trace is stationary for variations of ρ_{λ} about the corresponding density matrix. Thus, we only need to differentiate the second trace,

We evaluate

where

Therefore,

By cyclically permuting operators within the trace, one can verify that

With these identities, we can rewrite Equation A15,

This integral is non-negative and can be zero only if Δ is a multiple of the unit operator, i.e., if ρ_{0} = ρ. This proves that the minimum of the free energy must occur when ρ_{λ} = ρ_{0}.

If η_{0}(

where

To see that this is the case we start by writing,

The difference between the right and left sides of this equation is,

Because

We can simplify this further to,

We know that,

where

We may safely assume that

The matrix which was ultimately used to transform the Silvera-Goldman potential was obtained by fitting 1000 random potentials with

where

Parameters for the Silvera-Goldman potential are provided in Table

α | 1.713 |

δ | 1.5671 |

γ | 0.00993 |

_{6} |
12.14 |

_{8} |
215.2 |

_{9} |
143.1 |

_{10} |
4813.9 |

_{c} |
8.321 |

Exponential functions cannot be easily represented in a polynomial basis and the Silvera-Goldman potential diverges exponential as

^{1}Throughout this paper the variable “^{3N} where

^{2}A functor differs from a functional in that a functor maps one vector space to another whereas a functional maps a vector space to a scalar. In this context, “operator” is a more common term than “functor” but we prefer to call this “force-field functor theory” to evoke the connection with DFT.

^{3}That the corresponding equilibrium density matrices are not equal is obvious in Equation 1

^{4}Recall that ^{3N} so, |^{3N}_{i = 1} |_{i}〉.