# Car–Parrinello method

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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.

## Fictitious dynamics

### Lagrangian

$\mathcal{L} = \frac{1}{2}\left(\sum_I^{\mathrm{nuclei}}\ M_I\dot{\mathbf{R}}_I^2 + \mu\sum_i^{\mathrm{orbitals}}\int d\mathbf r\ |\dot{\psi}_i(\mathbf r,t)|^2 \right) - E\left[\{\psi_i\},\{\mathbf R_I\}\right],$

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

$\int d\mathbf r\ \psi_i^*(\mathbf r,t) \psi_j(\mathbf r,t) = \delta_{ij},$

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.

$M_I \ddot{\mathbf R}_I = - \nabla_I \, E\left[\{\psi_i\},\{\mathbf R_J\}\right]$
$\mu \ddot{\psi}_i(\mathbf r,t) = - \frac{\delta E}{\delta \psi_i^*(\mathbf r,t)} + \sum_j \Lambda_{ij} \psi_j(\mathbf r,t),$

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.