8. Born–Oppenheimer and Car–Parrinello dynamics
Position in the course: Lesson 8 of 11. Complete the preceding derivation and use the explained exercises to check understanding.
1. What changes when electrons supply the force?
In Born–Oppenheimer molecular dynamics (BOMD), solve an electronic ground-state problem at each nuclear configuration, then propagate classical nuclei on the resulting energy surface. This imports electronic-model errors, basis errors and SCF errors into every force evaluation. It does not automatically include nuclear tunneling, electronic transitions or excited-state chemistry. The Born–Oppenheimer separation requires an appropriate electronic state and separation of electronic and nuclear scales.
Define \(E_{BO}(R)=\min_\Psi\langle\Psi|\hat H_e(R)|\Psi\rangle+E_{NN}(R)\) for normalized electronic states. The nuclear Hamiltonian is \(H_N=\sum_I P_I^2/(2M_I)+E_{BO}(R)\). The electronic surface may be approximated by DFT, HF or a correlated method; the method determines the model rather than the integrator.
2. Derive stationary-state forces
Differentiate the electronic expectation value:
For an exact normalized eigenstate, the second term vanishes. In a stationary variational representation, orbital-response terms cancel only within the permitted variations. An atom-centered finite basis changes with nuclear positions and contributes Pulay terms; incomplete SCF convergence leaves additional residuals. Thus differentiating a nominal energy expression without its basis and solver dependence can produce inconsistent forces.
3. A controlled force check
For a one-dimensional harmonic surface \(E(q)=E_0+kq^2/2\), the centered energy difference gives exactly \(-kq\) in exact arithmetic. For a quartic term \(aq^4\), its derivative estimate gains a bias \(-4aq\delta^2\). Real electronic calculations add energy noise: if each energy uncertainty is at most \(\eta\), the difference can contain force error of order \(\eta/\delta\). Choose a displacement scan balancing curvature truncation and electronic convergence, rather than reducing \(\delta\) indefinitely. Keep the same basis, grid and electronic state on both sides.
4. Car–Parrinello is extended dynamics
Car–Parrinello dynamics introduces fictitious orbital inertia, schematically
The fictitious mass \(\mu\) is algorithmic, not the physical electron mass. The orbitals should remain close to an instantaneous ground-state manifold through adiabatic separation. Large fictitious mass, small gaps and energy transfer between orbital and nuclear modes can defeat this separation. Conserving an extended energy does not by itself prove motion on the BO surface. BOMD and CP dynamics have different control parameters and should not be compared using only the same nominal nuclear timestep.
5. A practical error budget
At fixed physical model, vary timestep and SCF tolerance independently in a short NVE diagnostic. Compare force differences to the thermal force scale, energy drift per atom per unit time and state continuity. Then test basis, cutoff, cell and electronic-method choices. A thermostat may hide SCF heating. Metals require occupation and electronic-temperature conventions; the force may derive from a free-energy functional rather than an uncorrected internal energy. A changing spin state needs deliberate tracking, not accidental SCF hopping.
6. Exercises
SCF energy noise is \(10^{-6}\) energy units; halve a finite-difference displacement. What happens to its noise contribution to force? It approximately doubles because the energy difference is divided by the displacement. Smaller displacement is not automatically a better test.
Does accurate DFT BOMD include zero-point vibrations? Classical nuclear motion does not. Nuclear quantum sampling requires additional methods such as path integrals, with their own convergence and dynamical interpretation.
7. References
Original analytic teaching illustration under the stated assumptions; no simulation data.
8. Related theory and practice
Quantum mechanics · Monte Carlo · Quantum Monte Carlo · CP2K · Quantum-Espresso