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Born-Oppenheimer and Car-Parrinello Dynamics

📘 Deriving Classical Molecular Dynamics

The starting point is the time-dependent Schrödinger equation:

\[ i\hbar \frac{\partial}{\partial t} \Phi(\{ \mathbf{r}_i \}, \{ \mathbf{R}_I \}; t) = \mathcal{H}(\{ \mathbf{r}_i \}, \{ \mathbf{R}_I \}; t) \Phi(\{ \mathbf{r}_i \}, \{ \mathbf{R}_I \}; t) \tag{2.1} \]

With the Hamiltonian:

\[ \begin{aligned} \mathcal{H} &= - \sum_I \frac{\hbar^2}{2 M_I} \nabla^2_I - \sum_i \frac{\hbar^2}{2 m_e} \nabla^2_i \\ &\quad + \frac{1}{4 \pi \varepsilon_0} \left( \sum_{i < j} \frac{e^2}{|\mathbf{r}_i - \mathbf{r}_j|} - \sum_{I, i} \frac{e^2 Z_I}{|\mathbf{R}_I - \mathbf{r}_i|} + \sum_{I < J} \frac{e^2 Z_I Z_J}{|\mathbf{R}_I - \mathbf{R}_J|} \right) \end{aligned} \]

This can be rewritten as:

\[ \mathcal{H} = - \sum_I \frac{\hbar^2}{2 M_I} \nabla^2_I - \sum_i \frac{\hbar^2}{2 m_e} \nabla^2_i + V_{n-e}(\{ \mathbf{r}_i \}, \{ \mathbf{R}_I \}) \]

Or split into nuclear and electronic parts:

\[ \mathcal{H} = - \sum_I \frac{\hbar^2}{2 M_I} \nabla^2_I + \mathcal{H}_e(\{ \mathbf{r}_i \}, \{ \mathbf{R}_I \}) \tag{2.2} \]

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🔹 Born-Oppenheimer Molecular Dynamics (BOMD)

BOMD is based on the Born-Oppenheimer approximation. This separates the motion of electrons and nuclei, assuming electrons respond instantaneously to nuclear motion.

• Solves the electronic structure at every nuclear step
• Forces on nuclei are derived from ground-state energy surface
• Requires full SCF convergence at every time step

🔹 Car-Parrinello Molecular Dynamics (CPMD)

CPMD evolves both nuclear and electronic degrees of freedom together in time using fictitious mass for the electrons. This allows time integration of both on the same footing without solving SCF in each step.

• Couples nuclear and electronic motion via extended Lagrangian
• Faster than BOMD but requires tuning of fictitious mass and time step
• Often used for long simulations where SCF cost is too high