Car–Parrinello molecular dynamics

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\sum _{I}^{\mathrm {nuclei} }\ M_{I}{\dot {\mathbf {R} }}_{I}^{2}+\mu \sum _{i}^{\mathrm {orbitals} }\int d\mathbf {r} \ |{\dot {\psi }}_{i}(\mathbf {r} ,t)|^{2}\right)-E\left[\{\psi _{i}\},\{\mathbf {R} _{I}\}\right]+\sum _{ij}\Lambda _{ij}\left(\int d\mathbf {r} \ \psi _{i}\psi _{j}-\delta _{ij}\right),}$[8]

where ${\displaystyle \mu }$ is the fictitious mass parameter; E[{ψ_i},{R_I}] is the Kohn–Sham energy density functional, which outputs energy values when given Kohn–Sham orbitals and nuclear positions.

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

where δ_ij is the Kronecker delta.

The equations of motion are obtained by finding the stationary point of the Lagrangian under variations of ψ_i and R_I, with the orthogonality constraint.[9]

${\displaystyle M_{I}{\ddot {\mathbf {R} }}_{I}=-\nabla _{I}\,E\left[\{\psi _{i}\},\{\mathbf {R} _{I}\}\right]}$

${\displaystyle \mu {\ddot {\psi }}_{i}(\mathbf {r} ,t)=-{\frac {\delta E}{\delta \psi _{i}^{*}(\mathbf {r} ,t)}}+\sum _{j}\Lambda _{ij}\psi _{j}(\mathbf {r} ,t),}$

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

In the formal limit where μ → 0, the equations of motion approach Born–Oppenheimer molecular dynamics.[10][11]