Non-Adiabatic Quantum Molecular Dynamics

Quantum mechanical calculations of the properties of solids are nearly always based on the Born-Oppenheimer or adiabatic approximation, which assumes that the electrons remain in their ground state wavefunction as the nuclei move around. This is a good approximation because the electrons move much faster than the nuclei, but breaks down in some circumstances. If, for example, the nuclei pass through an arrangement in which two different electronic states are very close in energy, the slow nuclear motion can induce transitions from one electronic state to the other.

The forces felt by Born-Oppenheimer nuclei are always conservative: the total energies of the nuclei and electrons depend on the nuclear positions only and return to their starting values if the nuclei are moved around a closed loop, no matter how fast. In reality, a fast-moving nucleus in a solid feels a drag-like force, which irreversibly transfers nuclear kinetic energy into excitations of the electron gas. At very high nuclear velocities, these non-conservative drag forces are more important than collisions with other nuclei.

Electronic drag forces have often been incorporated into classical molecular dynamics simulations by adding -βv terms to Newton's second law. We have tested this approach using the Ehrenfest approximation, in which the nuclei obey Newton's laws but the forces are calculated from quantum mechanical electronic wavefunctions that feel the time-varying potential of the moving nuclei and evolve according to the time-dependent Schrodinger equation. Although the Ehrenfest approximation does not describe the transfer of energy from electrons to atoms accurately, it provides a good description of energy transfer from high-energy ions to lower-energy electrons. Our results so far suggest that the -βv approximation for the drag force is surprisingly good on average, but that the instantaneous forces depend quite strongly on the ionic position and velocity.

The direction and position dependence of the electronic drag coefficient in copper

The dependence of the electronic drag coefficient β on the position within the unit cell and direction of motion of a copper atom in metallic copper.

The Ehrenfest approximation can also be used to incorporate electronic drag into full-scale quantum molecular dynamics simulations. The figure below shows two snapshots of a simulation of the formation of a radiation damage cascade. Both snapshots were taken at the same time, 95fs into the simulation, but from different directions. At the beginning of the simulation, a single atom in a perfect crystal of metallic copper was set moving with an initial KE of 1keV in a random direction, just as if it had been struck a glancing blow by a fast neutron or ion. Only atoms that have moved significantly are shown.

Radiation damage cascade in copper

The radiation damage cascade created 95fs after a single atom in a perfect crystal of metallic copper has been set moving with kinetic energy 1keV. Only the atoms that have been moved significantly are shown (from two different directions), with the atomic charges denoted using colours. The spikes emerging from the cascade region are self-focussing replacement collision sequences, in which atoms displace each other as in a Newton's cradle. After the replacement collision sequence has petered out, a vacancy is left near the beginning of the spike and an interstitial near the end.

spICED code logo Our new Ehrenfest dynamics code, spICED, written by Daniel Mason, is based on a simple but qualitatively accurate tight-binding description of the electronic quantum mechanics. Capable of simulating large systems of tens of thousands of atoms, spICED can be used to study many interesting areas of science, including radiation damage, sputtering, and the behaviour of matter in intense laser fields.