Introduction

Path Integral Molecular Dynamics

Molecular dynamics (MD) is a technique used to study the properties of a system of interacting particles by applying Newton’s equations of motion to produce trajectories which can be used to efficiently explore the phase space. This can be used to calculate many equilibrium and dynamical properties and to study systems from isolated gas molecules to condensed phase bulk materials.

However, while this technique has been very successful, in most MD implementations the assumption is made that the nuclei behave as classical particles, which for light nuclei such as hydrogen is often a very poor approximation as the effect of zero-point energy (ZPE) and quantum tunnelling can be large. For example, even at room temperature the vibrational frequency of an OH stretch in water is over 15 times larger than the available thermal energy, and so this motion will be highly quantized. The current state-of-the-art method to include nuclear quantum effects (NQE) in the calculation of static properties of condensed phase systems is path integral molecular dynamics (PIMD).

PIMD generates the quantum-mechanical ensemble of a system of interacting particles by using MD in an extended phase space. This is derived from the path integral formalism [FH64], which relates the statistics of a collection of quantum particles to those of a set of classical ring polymers, a ring polymer being a number of replicas of a particle coupled by harmonic springs. This so-called classical isomorphism is exact in the limit as the number of replicas goes to infinity, but in practice is converged numerically with only a finite number.

This then allows quantum phase space averages to be calculated from classical trajectories, with only about an order of magnitude more computing time than would be required for standard MD. Also, since PIMD is simply classical MD in an extended phase space, many of the techniques developed to improve the scope and efficiency of MD simulations can be applied straightforwardly to the equivalent PIMD calculations [CPMM10, MHT99]. Finally, several techniques designed specifically for PIMD simulations are now available to increase the rate of convergence with respect to the number of replicas used [CBRM11, CMP11, Chi97, MM08b, PerezT11, Suz95], further reducing the computational overhead of the method. All of these facts mean that it is now feasible to do PIMD simulations with thousands of molecules, or even to use ab initio electronic structure calculations to propagate the dynamics for small systems.

Furthermore, the framework used to run PIMD simulations can be adapted to generate approximate quantum dynamical information [BM06, CV93, CV94, CM04], and so can also be used to calculate correlation functions. While real-time quantum coherences cannot be captured, the inclusion of quantum statistical information and the rapid decoherence observed in condensed phase systems mean that in many cases very accurate results can be obtained from such approximate treatments of quantum dynamics [HMMM13].

Implementation

Automated evaluation (depend objects)

i-PI uses a caching mechanism with automatic value updating to make the code used to propagate the dynamics as simple and clear as possible. Every physical quantity that is referenced in the code is created using a “depend” object class, which is given the parameters on which it depends and a function used to calculate its value.

_images/ipi-depend.svg

Schematic overview of the functioning of the depend class used as the base for properties and physical quantities in i-PI. A few “primitive” quantities – such as atomic positions or momenta – can be modified directly. For most properties, one defines a function that can compute that property based on the value of other properties. Whenever one property is modified, all the quantities that depend on it are marked as tainted, so that when the value of one of the properties is used, the function can be invoked and the updated value obtained. If a quantity is not marked as tainted, the cached value is returned instead.

“Depend” objects can be called to get the physical quantity they represent. However, they have further functionality. Firstly, once the value of a “depend” object has been calculated, its value is cached, so further references to that quantity will not need to evaluate the function that calculates it. Furthermore, the code keeps track of when any of the dependencies of the variable are updated, and makes sure that the quantity is automatically recomputed when it is needed (i.e., when the quantity is assessed again).

This choice makes implementation slightly more complex when the physical observables are first introduced as variables, as one has to take care of stating their dependencies as well as the function that computes them. However, the advantage is that when the physical quantities are used, in the integrator of the dynamics or in the evaluation of physical properties, one does not need to take care of book-keeping and the code can be clean, transparent and readable.

_images/ipi-forces.svg

Schematic representation of the different objects that are involved in the evaluation of the forces. The multiple layers and complex structure are necessary to give the possibility of decomposing the evaluation of the forces between multiple different clients and using different imaginary time partitioning (e.g. one can compute the bonded interactions using one client, and use a different client to compute the long-range electrostatic interactions, contracted on a single bead [MM08b]).

Force evaluation

Within i-PI, the evaluation of the forces plays a crucial role, as it is the step requiring communication with the client code. In order to have a flexible infrastructure that makes it possible to perform simulations with advanced techniques such as ring-polymer contraction [MM08b], the force evaluation machinery in i-PI might appear complicated at first, and deserves a brief discussion.

A scheme of the objects involved in the calculation of the forces is presented in Figure 1.3. The infrastracture comprises a force provider class that deals with the actual subdivision of work among the clients, and a sequence of objects that translate the request of the overall force of the system into atomic evaluations of one component of the force for an individual bead: i-PI is built to hide the path integral infrastructure from the client, and so beads must be transferred individually.

Let us discuss for clarity a practical example – a calculation of an empirical water model where the bonded interactions are computed on 32 beads by the program A, and the non-bonded interactions are computed by client B, ring-polymer contracted on 8 beads. Each client “type” is associated with a object in the input. In the case of a interface, the forcefield object specifies the address to which a client should connect, and so multiple clients of type A or B can connect to i-PI at the same time. Each forcefield object deals with queueing force evaluation requests and computing them in a first-in-first-out fashion, possibly executing multiple requests in parallel.

On the force evaluation side, the task of splitting the request of a force evaluation into individual components and individual beads is accomplished by a chain of three objects, Forces, ForceComponent and ForceBead. is the main force Forces evaluator, that is built from the prototypes listed within the field of . Each item within the tag describe one component of the force – in our example one ForceComponent bound to a forcefield of type A, evaluated on 32 beads, and one ForceComponent bound to type B, evaluated on 8 beads. Forces contains the machinery that automatically contracts the actual ring polymer to the number of beads required by each component, and combines the various components with the given weights to provide the overall force, energy and virial where required. Note that in order to support ring polymer contraction (RPC), the RPC procedure is executed even if no contraction was required (i.e. even if all clients contain the full amount of beads). ForceComponent is a very simple helper class that associates with each bead a ForceBead object, that is the entity in charge of filing force requests to the appropriate ForceField object and waiting for completion of the evaluation.

Communication protocol

Since i-PI is designed to be used with a wide range of codes and platforms, it has to rely on a simple and robust method for communicating between the server and client. Even though other choices are possible, and it should be relatively simple to implement other means of communication, the preferred approach relies on sockets as the underlying infrastructure. Both Internet and Unix domain sockets can be used: the latter allow for fast communication on a single node, whereas the former make it possible to realise a distributed computing paradigm, with clients running on different nodes or even on different HPC facilities. In order to facilitate implementation of the socket communication in client codes, a simple set of C wrappers to the standard libraries socket implementation is provided as part of the i-PI distribution, that can be used in any programming language that can be linked with C code.

As far as the communication protocol is concerned, the guiding principle has been keeping it to the lowest common denominator, and avoiding any feature that may be code-specific. Only a minimal amount of information is transferred between the client and the server; the position of the atoms and cell parameters in one direction, and the forces, virial and potential in the other.

For more details about sockets and communication, see 3.3.

Internal units

All the units used internally by i-PI are atomic units, as given below. By default, both input and output data are given in atomic units, but in most cases the default units can be overridden if one wishes so. For details on how to do this, see 3.1.1.1 and 3.2.1.

Unit

Name

S.I. Value

Length

Bohr radius

5.2917721e-11 m

Time

N.A.

2.4188843e-17 s

Mass

Electron mass

9.1093819e-31 kg

Temperature

Hartree

315774.66 K

Energy

Hartree

4.3597438e-18 J

Pressure

N.A.

2.9421912e13 Pa

Regarding the specification of these units in the i-PI input files, the user is able to specify units both in the i-PI input file or in the structure file. If the structure file is of the .xyz format, the units specifications should be present in the comment line. Examples of such inputs can be found in examples/pes-regtest/io-units/ . The code then behaves in the following way, depending on the user’s choice:

  • If no units are specified in the input, i-PI tries to guess from the comment line of the structure file and it nothing is present, assumes atomic units, or angstrom for PDB files.

  • If units are specified in both input and structure file and they match, conversion happens just once. If they do not match, an error is raised and i-PI stops.

Core features

i-PI includes a large number of advanced molecular dynamics features, with an obvious focus on path integral molecular dynamics, but also several methods for sampling classical trajectories.

Features in version 1.0

  • molecular dynamics and PIMD in the NVE, NVT and NPT ensembles, with the high-frequency internal vibrations of the path propagated in the normal-mode representation [CPMM10] to allow for longer time steps;

  • ring polymer contraction [MM08a, MM08b], implemented by exposing multiple socket interfaces to deal with short and long-range components of the potential energy separately; treating different components that have different computational cost and characteristic time scale separately can reduce substantially the overall effort associated with a simulation

  • efficient stochastic velocity rescaling [BDP07] and path integral Langevin equation thermostats [CPMM10] to sample efficiently the canonical ensemble;

  • various generalized Langevin equation (GLE) thermostats, including the optimal sampling [CBP09a, CBP10], quantum  [CBP09b], and \(\delta\) [CP10] thermostats; the parameters for different GLE flavors and the conditions in which they should be applied can be obtained from a separate website [Cer10];

  • mixed path integral–generalized Langevin equation techniques for accelerated convergence, including both PI+GLE [CMP11] and the more recent and effective version PIGLET [CM12]; these techniques reduce the number of path integral replicas needed, while allowing for systematic convergence;

  • all the standard estimators for structural properties, the quantum kinetic energy, pressure, etc.;

  • more sophisticated estimators such as the scaled-coordinate heat capacity estimator [Yam05], estimators to obtain isotope fractionation free energies by re-weighting a simulation of the most abundant isotope [CM13], and a displaced-path estimator for the particle momentum distribution [LMCP10];

  • the infrastructure that is needed to perform approximate quantum dynamics calculations such as ring polymer molecular dynamics (RPMD) [CM04, HMMM13] and centroid molecular dynamics (CMD) [CV93, CV94].

Features added in version 2.0

Further details can be found in Ref. [KRM+19].

  • reweighted fourth-order path integral MD (M. Ceriotti, G.A.R. Brain) [CBRM11, JV01]; this method makes it possible to obtain fourth-order statistics by re-weighting second-order trajectories; attention should be paid to avoid statistical inefficiencies;

  • finite-differences implementation of fourth-order path integrals (V. Kapil, M. Ceriotti) [KBC16]; this schemes enables explicit fourth-order path integral simulations, that converge faster than conventional Trotter methods;

  • perturbed path integrals (I. Poltavsky) [PT16]; essentially, a truncated cumulant expansion of fourth-order reweighting, that often enables fast convergence avoiding statistical instability;

  • open path integrals and momentum distribution estimators (V. Kapil, A. Cuzzocrea, M. Ceriotti) [KCC18]; makes it possible to compute the particle momentum distribution including quantum fluctuations of nuclei;

  • quantum alchemical transformations (B. Cheng) [CBC16, LAM+13]; Monte Carlo exchanges between isotopes of different mass, useful to sample isotope propensity for different molecules or environments;

  • direct isotope fractionation estimators (B. Cheng, M. Ceriotti)  [CC14]; avoid thermodynamic integration to obtain isotope fractionation ratios;

  • spatially localized ring polymer contraction (M. Rossi, M. Ceriotti) [LDCR18]; simple contraction scheme for weakly bound molecules, e.g. on surfaces;

  • ring polymer instantons (Y. Litman, J.O. Richardson, M. Rossi); evaluation of reaction rates and tunnelling splittings for molecular rearrangements and chemical reactions;

  • thermodynamic integration (M. Rossi, M. Ceriotti) [RGC16]; classical scheme to compute free energy differences;

  • geometry optimizers for minimization and saddle point search (B. Helfrecht, R. Petraglia, Y. Litman, M. Rossi) [RGC16]

  • harmonic vibrations through finite differences (V. Kapil, S. Bienvenue) [RGC16]; simple evaluation of the harmonic Hessian;

  • multiple time stepping (V. Kapil, M. Ceriotti) [KVC16]; accelerated simulations by separating slow and fast degrees of freedom into different components of the potential energy;

  • metadynamics through a PLUMED interface (G. Tribello, M. Ceriotti); simulation of rare events and free energy calculations;

  • replica exchange MD (R. Petraglia, R. Meissner, M. Ceriotti) [PNicolaiW+16]; accelerated convergence of averages by performing Monte Carlo exchanges of configurations between parallel calculations

  • thermostatted RPMD [RCM14], including optimized-GLE TRPMD [RKC18]; reduces well-known artifacts in the simulation of dynamical properties by path integral methods;

  • dynamical corrections to Langevin trajectories (M. Rossi, V. Kapil, M. Ceriotti) [RKC18]; eliminates the artifacts introduced into dynamical properties by the presence of thermostats;

  • fast forward Langevin thermostat (M. Hijazi, D. M. Wilkins, M. Ceriotti); a simple scheme to reduce the impact of strongly-damped Langevin thermostats on sampling efficiency; [HWC18]

  • Langevin sampling for noisy and/or dissipative forces (J. Kessler, T. D. Kühne); suitable to stabilize and correct the artifacts that are introduced in MD trajectories by different extrapolation schemes;

Licence and credits

Most of this code is distributed under the GPL licence. For more details see www.gnu.org/licences/gpl.html. So that they can easily be incorporated in other codes, the files in the directory “drivers” are all held under the MIT licence. For more details see https://fedoraproject.org/wiki/Licensing:MIT.

If you use this code in any future publications, please cite this using [CMM14] for v1 and [KRM+19] for v2.

Contributors

i-PI was originally written by M. Ceriotti and J. More at Oxford University, together with D. Manolopoulos. Several people contributed to its further development. Developers who implemented a specific feature are acknowledged above.

On-line resources

Python resources

For help with Python programming, see www.python.org. For information about the NumPy mathematical library, see www.numpy.org, and for worked examples of its capabilities see www.scipy.org/Tentative_NumPy_Tutorial. Finally, see http://hgomersall.github.io/pyFFTW/ for documentation on the Python FFTW library that is currently implemented with i-PI.

Client code resources

Several codes provide out-of-the-box an i-PI interface, including CP2K, DFTB+, Lammps, Quantum ESPRESSO, Siesta, FHI-aims, Yaff, deMonNano, TBE. If you are interested in interfacing your code to i-PI please get in touch, we are always glad to help!

There are several Fortran and C libraries that most client codes will probably need to run, such as FFTW, BLAS and LAPACK. These can be found at www.fftw.org, www.netlib.org/blas and www.netlib.org/lapack respectively.

These codes do not come as part of the i-PI package, and must be downloaded separately. See chapter 2.2 for more details of how to do this.

i-PI resources

For more information about i-PI and to download the source code go to http://ipi-code.org/.

In http://gle4md.org/ one can also obtain colored-noise parameters to run Path Integral with Generalized Langevin Equation thermostat (PI+GLE/PIGLET) calculations.