# The Path Integral molecular Dynamics (PIMD)

### This page gives hints on how to perform a PIMD calculation with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (GG)

Mentioned in help_features#4.

### Table of content:

- 1. Introduction.
- 2. Related input variables.
- 3. Selected input files.
- 4. References.

### 1. Introduction.

Path-Integral Molecular Dynamics (PIMD) is a technique allowing to simulate the quantum fluctuations of the nuclei at thermodynamic equilibrium [Marx1996]. It is implemented in ABINIT in the NVT ensemble since v7.8.2.

In the Path-Integral formalism of quantum statistical mechanics, the (quantum) nuclei are replaced by a set of images (beads) treated by means of classical mechanics, and interacting with each other through a specific effective potential. In the limit of an infinite number of beads, the quantum system and this many-beads classicle system have the same partition function, and thus the same static observables. In PIMD, the classical system of beads is simulated by standard Molecular Dynamics. The PIMD equations of motion are integrated by using the Verlet algorithm. At each time step, a ground state DFT calculation is performed for each image. PIMD can be used with any XC functional and works in the PAW framework as well as in the norm-conserving pseudopotential (NCPP) case.

PIMD in ABINIT follows the set of numerical schemes developed by several authors in the 90's [Marx1996],[Tuckerman1996]. PIMD in the canonical ensemble needs specific thermostats to ensure that the trajectories are ergodic: the Nose-Hoover chains are implemented, as well as the Langevin thermostat (controlled by the value of imgmov). Also, it is possible to use coordinate transformations (staging and normal mode), that are controlled by the keyword pitransform. In standard equilibrium PIMD, only static observables are relevant (quantum time-correlation functions are not accessible): the masses associated to the motion of the beads are controlled by the keyword pimass, whereas the true masses of the atoms are given by amu. The values given in pimass are used to fix the so-called fictitious masses [Marx1996]. In the case where a coordinate transformation is used, the fictitious masses are automatically fixed in the code to match the so-called staging masses or normal mode masses. The number of time steps of the trajectory is controlled by ntimimage, the initial and thermostat temperature by mdtemp. Except if specified, the images in the initial configuration are assumed to be at the same position, and a random distribution of velocities is applied (governed by mdtemp) to start the dynamics.

At each time step, ABINIT delivers in the output file:

- (i) information about the ground state DFT calculation of the ground state for each image
- (ii) the instantaneous temperature, the instantaneous energy as given by the primitive and virial estimators, and the pressure tensor as given by the primitive estimator.

The number of images (keyword nimage) is associated to a specific parallelization level (keyword npimage).

PIMD has been used with ABINIT to reproduce the large isotope effect on the phase transition between phase I and phase II of dense hydrogen [Geneste2012], and also some aspects of diffusion at low and room temperature in proton-conducting oxides for fuel cells [Geneste2015]. PIMD in the NPT ensemble is not available yet.
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### 2. Related input variables.

Compulsory input variables:

... dtion [Delta Time for IONs]
... imgmov [IMaGe MOVEs]
... mdtemp [Molecular Dynamics TEMPeratures]
... nimage [Number of IMAGEs]
... ntimimage [Number of TIME steps for IMAGE propagation]

Basic input variables:

... nnos [Number of NOSe masses]
... optcell [OPTimize the CELL shape and dimensions]
... qmass [Q thermostat MASS]
... vis [VIScosity]

Useful input variables:

... amu [Atomic Mass Units]
... npimage [Number of Processors at the IMAGE level]
... pimass [Path Integral fictitious MASSes]
... pimd_constraint [Path-Integral Molecular Dynamics: CONSTRAINT to be applied on a reaction coordinate]
... pitransform [Path Integral coordinate TRANSFORMation]
... restartxf [RESTART from (X,F) history]
... vel [VELocity]

Relevant internal variables:

... %ndynimage [Number of DYNamical IMAGEs]

Input variables for experts:

... adpimd [ADiabatic Path-Integral Molecular Dynamics]
... adpimd_gamma [ADiabatic Path-Integral Molecular Dynamics: GAMMA factor]
... dynimage [DYNamics of the IMAGE]
... irandom [Integer for the choice of the RANDOM number generator]
... istatimg [Integer governing the computation of STATic IMaGes]
... vel_cell [VELocity of the CELL parameters]
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### 3. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v7/Input: t08.in
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### 4. References.

[Geneste2012] G. Geneste, M. Torrent, F. Bottin and P. Loubeyre, "Strong Isotope Effect in Phase II of Dense Solid Hydrogen and Deuterium", Phys. Rev. Lett. 109, 155303 (2012).
[Geneste2015] G. Geneste, A. Ottochian, J. Hermet and G. Dezanneau, "Proton transport in barium stannate: classical, semi-classical and quantum regimes", Phys. Chem. Chem. Phys. 17, 19104 (2015).
[Marx1996] D. Marx and M. Parrinello, "Ab initio path integral molecular dynamics: Basic ideas", J. Chem. Phys. 104, 4077 (1996).
[Tuckerman1996] M. E. Tuckerman, D. Marx, M. L. Klein and M. Parrinello, "Efficient and general algorithms for path integral Car–Parrinello molecular dynamics", J. Chem. Phys. 104, 5579 (1996).

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