eFF, a method to simulate large scale excited electron dynamics

New: Parallel eFF

eFF has now been officially implemented into the LAMMPS molecular dynamics package, by Dr. Jaramillo-Botero at Caltech. Using LAMMPS, computations on millions of particles over thousands of processors are now possible and practical.

Scope of application

eFF is a molecular dynamics model that includes electrons [1]. It can simulate highly excited systems where the Born-Oppenheimer approximation may break down over long times, and where excitations may be distributed in a spatially heterogeneous nonequilibrium fashion over tens of thousands of electrons. Possible applications include modeling shock compressed dense matter, semiconductor etching by plasmas, and radiation damage of materials (Figure 1).

Only three universal parameters are used, for all elements. Currently, elements with Z=1-6 are well described. Work is ongoing to extend the scope of the force field, with preliminary results for Z=1-10 detailed in a separate section (eFF2).

Figure 1: Schematic diagram of excited electron processes: (a) electron etching of semiconductor surface, (b) chemical vapor deposition of diamond layers, (c) electron transport across a semiconductor junction, (d) thermoelectric, (e) organic wire, (f) electride or solvated electron, (g) fuel cell half reaction, (h) electrochemical sensor for dopamine, (i) lithium battery half reaction, (j) electron transport in cellular respiration, (k) hydrogen plasma.

The eFF method

In eFF, electrons are represented by Gaussian wave packets, and nuclei are represented by point charges. Both the electrons and nuclei are mobile, and in addition the electrons can grow and shrink in size over time. The electrons and nuclei interact via an effective potential that includes a repulsion between pairs of electrons due to the Pauli exclusion principle; a kinetic energy for individual electrons arising from the Heisenberg uncertainty principle; and electrostatics. The energy function is so simple that forces acting between thousands of nuclei and electrons can be computed in less than a second on a modern processor.

Relation to conventional force fields

eFF stands for ``electron force field'', a term intended to suggest an analogy to conventional force fields, which are effective potentials between nuclei commonly used to iterate the dynamics of systems in their ground state.

In ground state systems -- and excited state systems between curve crossings -- the Born-Oppenheimer approximation applies, and the electronic wavefunction is set by the instantaneous position of the nuclei. In conventional force fields, the total energy is parameterized as a function of the nuclear coordinates alone. Since a wavefunction does not need to be explicitly computed, a significant time savings over quantum mechanics methods is possible, and the dynamics of large systems can be simulated in a practical way.

However this approach fails to describe highly excited systems, where there may be many curve crossings and a high density of states, causing the Born-Oppenheimer approximation to break down over long time intervals (Figure 2).

Figure 2: (a) Systems with only a few states follow the Born-Oppenheimer approximation between curve crossings, but (b) in highly excited systems with many states, the wavefunction needs to include nuclear and electron coordinates together.

In eFF, a very simple time-dependent wavefunction containing both nuclear and electron parameters of motion is used. Substituting the wavefunction into the time-dependent Schrodinger equation generates semiclassical equations of motion which specify how nuclear positions and electron positions and sizes change over time.

Since electrons and nuclei in eFF can move independently of each other, the method can simulate the chemistry of large scale excited systems undergoing nonadiabatic as well as adiabatic dynamics. The energy function is simple enough that the speediness of conventional force field/MD schemes is preserved, which allows practical computations on large systems (up to millions of electrons, Figure 3).

Figure 3: Linear scaling of calculation time is achieved using a 20 bohr pairwise force cutoff; shown is the time spent on a single energy/force iteration on bulk lithium solid. The memory required also scales linearly with the number of electrons and nuclei.

Context and relation to prior work

Heller pioneered the use of semiclassical wave packet dynamics of nuclei to solve spectroscopic problems in a time-dependent framework; eFF uses equations of motion similar to his to propagate nuclear and electron coordinates [2].

eFF falls into a category of methods called ``fermionic molecular dynamics'' [3], where the interaction energy of fermions is computed using effective potentials, with particular emphasis placed on accounting for the effects of Pauli exclusion and Heisenberg uncertainty.

These methods have been used to study the scattering and combination of nucleons in nuclear reactions (Dorso and Randrup [4,5], Boal and Glosli [6,7], Maruyama et al [8], Wilets et al [9]); the dynamics of hydrogen plasmas and liquid lithium (Hansen and McDonald [10], Klakow and Knaup [11,12]); atomic shell structures (Kirschbaum and Wilets [13], Cohen [14]); and proton passage through beryllium (Beck and Wilets [15]).

eFF represents an advance over these methods in that it is accurate enough to reproduce a wide variety of ground state structures containing covalent, ionic, multicenter, and/or metallic bonding; and further that its excited state dynamics for warm dense hydrogen [1] and core-ionized hydrocarbons show good agreement with experiment and higher level theory.

eFF is also related to the ab initio floating spherical Gaussian orbitals (FSGO) methods developed by Frost in 1964, who showed that it was possible using a minimal basis set of floating Gaussians to obtain reasonable geometries of hydrocarbons and atom hydrides [16]. Overall, we find that eFF produces geometries as good or better than FSGO, with much reduced computational cost ( $ \mathrm{N^{2}}$ versus $ \mathrm{N^{4}}$, if no cutoffs are used).