# Car–Parrinello molecular dynamics

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Car–Parrinello molecular dynamics or CPMD refers to either a method used in molecular dynamics (also known as the Car–Parrinello method) or the computational chemistry software package used to implement this method.[1]

The CPMD method is related to the more common Born–Oppenheimer molecular dynamics (BOMD) method in that the quantum mechanical effect of the electrons is included in the calculation of energy and forces for the classical motion of the nuclei. However, whereas BOMD treats the electronic structure problem within the time-independent Schrödinger equation, CPMD explicitly includes the electrons as active degrees of freedom, via (fictitious) dynamical variables.

The software is a parallelized plane wave / pseudopotential implementation of density functional theory, particularly designed for ab initio molecular dynamics.[2]

## Car–Parrinello method

The Car–Parrinello method is a type of molecular dynamics, usually employing periodic boundary conditions, planewave basis sets, and density functional theory, proposed by Roberto Car and Michele Parrinello in 1985, who were subsequently awarded the Dirac Medal by ICTP in 2009.

In contrast to Born–Oppenheimer molecular dynamics wherein the nuclear (ions) degree of freedom are propagated using ionic forces which are calculated at each iteration by approximately solving the electronic problem with conventional matrix diagonalization methods, the Car–Parrinello method explicitly introduces the electronic degrees of freedom as (fictitious) dynamical variables, writing an extended Lagrangian for the system which leads to a system of coupled equations of motion for both ions and electrons. In this way an explicit electronic minimization at each time step, as done in Born-Oppenheimer MD, is not needed: after an initial standard electronic minimization, the fictitious dynamics of the electrons keeps them on the electronic ground state corresponding to each new ionic configuration visited along the dynamics, thus yielding accurate ionic forces. In order to maintain this adiabaticity condition, it is necessary that the fictitious mass of the electrons is chosen small enough to avoid a significant energy transfer from the ionic to the electronic degrees of freedom. This small fictitious mass in turn requires that the equations of motion are integrated using a smaller time step than the one (1–10 fs) commonly used in Born–Oppenheimer molecular dynamics.

## General approach

In CPMD the core electrons are usually described by a pseudopotential and the wavefunction of the valence electrons are approximated by a plane wave basis set.

The ground state electronic density (for fixed nuclei) is calculated self-consistently, usually using the density functional theory method. Then, using that density, forces on the nuclei can be computed, to update the trajectories (using, e.g. the Verlet integration algorithm). In addition, however, the coefficients used to obtain the electronic orbital functions can be treated as a set of extra spatial dimensions, and trajectories for the orbitals can be calculated in this context.

## Fictitious dynamics

CPMD is an approximation of the Born–Oppenheimer MD (BOMD) method. In BOMD, the electrons' wave function must be minimized via matrix diagonalization at every step in the trajectory. CPMD uses fictitious dynamics[3] to keep the electrons close to the ground state, preventing the need for a costly self-consistent iterative minimization at each time step. The fictitious dynamics relies on the use of a fictitious electron mass (usually in the range of 400 – 800 a.u.) to ensure that there is very little energy transfer from nuclei to electrons, i.e. to ensure adiabaticity. Any increase in the fictitious electron mass resulting in energy transfer would cause the system to leave the ground-state BOMD surface.[4]

### Lagrangian

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\sum _{I}^{\mathrm {nuclei} }\ M_{I}{\dot {\mathbf {R} }}_{I}^{2}+\mu \sum _{i}^{\mathrm {orbitals} }\int d\mathbf {r} \ |{\dot {\psi }}_{i}(\mathbf {r} ,t)|^{2}\right)-E\left[\{\psi _{i}\},\{\mathbf {R} _{I}\}\right],}$

where E[{ψi},{RI}] is the Kohn–Sham energy density functional, which outputs energy values when given Kohn–Sham orbitals and nuclear positions.

### Orthogonality constraint

${\displaystyle \int d\mathbf {r} \ \psi _{i}^{*}(\mathbf {r} ,t)\psi _{j}(\mathbf {r} ,t)=\delta _{ij},}$

where δij is the Kronecker delta.

### Equations of motion

The equations of motion are obtained by finding the stationary point of the Lagrangian under variations of ψi and RI, with the orthogonality constraint.[5]

${\displaystyle M_{I}{\ddot {\mathbf {R} }}_{I}=-\nabla _{I}\,E\left[\{\psi _{i}\},\{\mathbf {R} _{J}\}\right]}$
${\displaystyle \mu {\ddot {\psi }}_{i}(\mathbf {r} ,t)=-{\frac {\delta E}{\delta \psi _{i}^{*}(\mathbf {r} ,t)}}+\sum _{j}\Lambda _{ij}\psi _{j}(\mathbf {r} ,t),}$

where Λij is a Lagrangian multiplier matrix to comply with the orthonormality constraint.

### Born–Oppenheimer limit

In the formal limit where μ → 0, the equations of motion approach Born–Oppenheimer molecular dynamics.[6]

## Application

(1) Studying the behavior of water near a hydrophobic graphene sheet.[7]

(2) Investigating the structure of liquid water at ambient temperature.[8]

(3) Solving the heat transfer problems (heat conduction and thermal radiation) between Si/Ge superlattices.[9][10]

(4) Probing the proton transfer along 1D water chains inside carbon nanotubes.[11]

(5) Evaluating the critical point of aluminum.[12]

## References

1. ^ Car, R.; Parrinello, M (1985). "Unified Approach for Molecular Dynamics and Density-Functional Theory". Physical Review Letters. 55 (22): 2471–2474. Bibcode:1985PhRvL..55.2471C. PMID 10032153. doi:10.1103/PhysRevLett.55.2471.
2. ^ "CPMD.org". IBM, MPI Stuttgart, and CPMD Consortium. Retrieved 15 March 2012.
3. ^ David J. E. Callaway; Aneesur Rahman (30 August 1982). "Microcanonical Ensemble Formulation of Lattice Gauge Theory". Phys. Rev. Lett. 49: 613. Bibcode:1982PhRvL..49..613C. doi:10.1103/PhysRevLett.49.613.
4. ^ The CPMD Consortium. "Car-Parrinello Molecular Dynamics: An ab initio Electronic Structure and Molecular Dynamics Program" (PDF). Manual for CPMD version 3.15.1.
5. ^ Callaway, David; Rahman, Aneesur (1982). "Microcanonical Ensemble Formulation of Lattice Gauge Theory". Physical Review Letters. AIP. 49 (9): 613. Bibcode:1982PhRvL..49..613C. doi:10.1103/PhysRevLett.49.613.
6. ^ Kühne, T. D.; Krack, M.; Mohamed, F. R.; Parrinello, M (2007). "Efficient and Accurate Car-Parrinello-like Approach to Born-Oppenheimer Molecular Dynamics". Physical Review Letters. 98: 066401. doi:10.1103/PhysRevLett.98.066401.
7. ^ Rana, Malay Kumar; Chandra, Amalendu (2013-05-28). "Ab initio and classical molecular dynamics studies of the structural and dynamical behavior of water near a hydrophobic graphene sheet". The Journal of Chemical Physics. 138 (20): 204702. ISSN 0021-9606. doi:10.1063/1.4804300.
8. ^ Lee, Hee-Seung; Tuckerman, Mark E. (2006-10-21). "Structure of liquid water at ambient temperature from ab initio molecular dynamics performed in the complete basis set limit". The Journal of Chemical Physics. 125 (15): 154507. ISSN 0021-9606. doi:10.1063/1.2354158.
9. ^ Ji, Pengfei; Zhang, Yuwen (2013-05-01). "First-principles molecular dynamics investigation of the atomic-scale energy transport: From heat conduction to thermal radiation". International Journal of Heat and Mass Transfer. 60: 69–80. doi:10.1016/j.ijheatmasstransfer.2012.12.051.
10. ^ Ji, Pengfei; Zhang, Yuwen; Yang, Mo (2013-12-21). "Structural, dynamic, and vibrational properties during heat transfer in Si/Ge superlattices: A Car-Parrinello molecular dynamics study". Journal of Applied Physics. 114 (23): 234905. ISSN 0021-8979. doi:10.1063/1.4850935.
11. ^ Dellago, Christoph (2003-01-01). "Proton Transport through Water-Filled Carbon Nanotubes". Physical Review Letters. 90 (10). doi:10.1103/PhysRevLett.90.105902.
12. ^ Faussurier, Gérald; Blancard, Christophe; Silvestrelli, Pier Luigi (2009-04-03). "Evaluation of aluminum critical point using an \textit{ab initio} variational approach". Physical Review B. 79 (13): 134202. doi:10.1103/PhysRevB.79.134202.