# 1. Introduction to RPMDrate¶

## 1.1. What is RPMD?¶

RPMD is short for Ring Polymer Molecular Dynamics: an approximate quantum mechanical simulation technique [1] to compute approximate values of various dynamics properties, such as chemical reaction rate coefficients [2], [3], [4] and diffusion coefficients [5]. RPMD is based on the isomorphism between the quantum statistical mechanics of the physical system and the classical statistical mechanics of a fictious ring polymer consisting of n copies of the system connected by harmonic springs [6].

The resulting RPMD reaction rate theory is essentially a classical rate theory in an extended (discretized imaginary time path integral) phase space, and thus gives it some very desitable features:

• First, the RPMD rate becomes exact in the high temperature limit, where the ring polymer collapses to a single bead. It is also exact for a parabolic barrier bilinearly coupled to a bath of harmonic oscillators at all temperatures for which a rate coefficient can be defined.
• Second, the theory has a well-defined short-time limit that provides an upper bound on the RPMD rate, in the same way as classical transition state theory provides an upper bound on the classical rate. Indeed when the dividing surface is defined in terms of the centroid of the ring polymer the short-time limit of the RPMD rate coincides with a well-known (centroid-density) version of the quantum transition state theory (QTST).
• Finally, and perhaps most importantly, the RPMD rate coefficient is rigorously independent of the choice of the transition state dividing surface that is used to compute it. This is a highly desirable feature of the theory for applications to multidimensional reactions for which the optimum dividing surface can be very difficult to determine.

The application of RPMD to the study of thermally activated gas-phase bimolecular reactions is one of the most recent developments [2], [3], [4]. The rate coefficients obtained so far with RPMD

• are reliable (predictive) at high temperatures;
• correctly capture the zero-point energy effects;
• are within a factor of 2-3 of accurate QM results even at very low temperatures in the deep tunneling regime (when such comparison is available).

RPMDrate is the name of this software package, which provides functionality for conducting RPMD simulations in order to compute the bimolecular reaction rate coefficients for thermally activated processes in the gas phase. The RPMD rate methodology has been developed in Refs. [2], [3]. The RPMD codebase is split into a Fortran 90/95 core used to efficiently conduct the RPMD simulations, and a Python layer that provides a more user-friendly, scriptable interface.

 [1] Craig, I. R.; Manolopoulos, D. E. Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics, J. Chem. Phys. 2004, 121, 3368.
 [2] (1, 2, 3) Collepardo-Guevara, R.; Suleimanov, Yu. V.; Manolopoulos, D. E. Bimolecular reaction rates from ring polymer molecular dynamics, J. Chem. Phys. 2009, 130, 174713.
 [3] (1, 2, 3) Suleimanov, Yu. V.; Collepardo-Guevara, R.; Manolopoulos, D. E.; Bimolecular reaction rates from ring polymer molecular dynamics: Application to $$\mathrm{CH_4 + H} \rightarrow \mathrm{CH_3 + H_2}$$, J. Chem. Phys. 2011, 134, 044131.
 [4] (1, 2) Perez de Tudela, R.; Aoiz, F. J.; Suleimanov, Yu. V.; Manolopoulos, D. E. Chemical reaction rates from ring polymer molecular dynamics: Zero point energy conservation in $$\mathrm{Mu + H_2} \rightarrow \mathrm{MuH + H}$$, J. Phys. Chem. Lett. 2012, 3, 493.
 [5] Suleimanov, Yu. V. Surface Diffusion of Hydrogen on Ni(100) from Ring Polymer Molecular Dynamics, J. Phys. Chem. C 2012, 116, 11141.
 [6] Chandler, D. Exploiting the Isomorphism between Quantum Theory and Classical Statistical Mechanics of Polyatomic Fluids, J. Chem. Phys. 1981, 74, 4078.

RPMDrate is distributed under the terms of the MIT license. This is a permissive license, meaning that you are generally free to use, redistribute, and modify RPMDrate at your discretion. If you use all or part of RPMDrate in your work, we only ask that proper attribution be given (in addition to following the terms of the license as stated below).

The MIT license is reproduced in its entirety below:

Copyright (c) 2012 by Joshua W. Allen (jwallen@mit.edu)
William H. Green (whgreen@mit.edu)
Yury V. Suleimanov (ysuleyma@mit.edu, ysuleyma@princeton.edu)

Permission is hereby granted, free of charge, to any person obtaining a
copy of this software and associated documentation files (the "Software"),
to deal in the Software without restriction, including without limitation
the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
DEALINGS IN THE SOFTWARE.


## 1.3. References¶

### 1.3.1. RPMDrate references¶

The recommended reference for the current version of the RPMDrate code is to give Reference [1]. Reference [2] is also recommended in certain cases because it provides published explanations of the methodology used in the code.

[1] Suleimanov, Yu. V.; Allen, J. W.; Green, W. H. RPMDrate: bimolecular chemical reaction rates for ring polymer molecular dynamics, Comp. Phys. Comm. 2013, 184, 833.

[2] Suleimanov, Yu. V.; Collepardo-Guevara, R.; Manolopoulos, D. E.; Bimolecular reaction rates from ring polymer molecular dynamics: Application to $$\mathrm{CH_4 + H} \rightarrow \mathrm{CH_3 + H_2}$$, J. Chem. Phys. 2011, 134, 044131.

### 1.3.2. References for incorporated algorithms¶

1. Bennett-Chandler factorization of the rate coeffiicent.

[3] Bennett, C. H. In Algorithms for Chemical Computations, ACS Symposium Series No. 46; Christofferson, R. E., Ed.; American Chemical Society: Washington DC, 1977; p 63.

[4] Chandler, D. Statistical mechanics of isomerization dynamics in liquids and the transition state approximation, J. Chem. Phys. 1978, 68, 2959.

1. Umbrella integration along the reaction coordinate.

[5] Kästner, J.; Thiel, W. Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: “Umbrella integration” J. Chem. Phys. 2005, 123, 144104.

[6] Kästner, J.; Thiel, W. Analysis of the statistical error in umbrella sampling simulations by umbrella integration J. Chem. Phys. 2006, 124, 234106.

[7] Kästner, J. Umbrella integration in two or more reaction coordinates, J. Chem. Phys. 2009, 131, 034109.

1. RATTLE algorithm for constrained molecular dynamics simulations.

[8] Andersen, H. C. Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations, J. Comput. Phys. 1983, 52, 24.

1. Andersen thermostat.

[9] Andersen, H. C. Molecular dynamics simulations at constant pressure and/or temperature, J. Chem. Phys. 1980, 72, 2384.

1. Colored-Noise, generalized Langevin equation thermostats.

[10] Ceriotti, M.; Bussi, G.; Parrinello, M. Langevin equation with colored noise for constant-temperature molecular dynamics simulations, Phys. Rev. Lett. 2009, 102, 020601.

[11] Ceriotti, M.; Bussi, G.; Parrinello, M. Nuclear quantum effects in solids using a colored-noise thermostat, Phys. Rev. Lett. 2009, 103, 030603.

[12] Ceriotti, M.; Bussi, G.; Parrinello, M. Colored-noise thermostats a la carte, J. Chem. Theory Comput. 2010, 6, 1170.

[13] Ceriotti, M.; Manolopoulos, D. E.; Parrinello, M. Accelerating the convergence of path integral dynamics with a generalized Langevin equation, J. Chem. Phys. 2011, 134, 084104.