Momentum-Dependent Potentials

Large-scale simulations are needed to model heterogeneous, non-equilibrium electronic dynamics in a wide variety of scenarios, including stopping power experiments in dense plasmas, multi-species mixing under extreme conditions, laser-matter experiments (e.g., core ionization in x-ray free electron laser experiments), and ultracold neutral plasmas (short-pulse laser photoionization of a cold atomic gas), just to name a few. The need for such modeling is made more important by the advent of recent high energy-density experimental facilities, such as the Z machine, National Ignition Facility, Linac Coherent Light Source and Deutsches Elektronen-Synchrotron (DESY). Furthermore, diagnostic capabilities such as imaging XRTS will provide unprecedented information about the evolution of electronic states in these experiments. Coupled with recent advances in computational power that allows  molecular dynamics simulations to span unprecedented length (multi-trillion particles) and time scales (pico to micro-seconds), a detailed knowledge of the non-equilibrium dynamics of such systems is, in principle, obtainable; however, it is currently not possible to perform such large scale simulations for electronic dynamics because of the computational overhead in modeling quantum systems.

High fidelity approaches such as time-dependent density functional theory (TD-DFT) can be employed for modeling non-equilibrium electronic dynamics, but typically have O(N^3) scaling for an N-particle system, limiting its applicability to small-scale systems. Alternatively, effective classical approaches have been formulated to model quantum systems, such wave packet molecular dynamics (WPMD) and quantum statistical potentials (QSP); however, WPMD has some undesirable properties (e.g., persistent wavepacket spreading) and the statistical nature of QSPs (i.e., temperature dependence) is inapplicable to non-equilibrium systems.

Before going on, let’s get some intuition for what the goal is. Using QSPs, we simulated an electron-ion plasma under conditions for which the electrons are barely free. That is, the electron-ion coupling is large but not enough to form bound states. Watch this movie which shows red electrons and blue ions, noting the quasi-bound states of some of the electron-ion pairs (orbiting red-blue pairs).

While QSPs are known to provide an accurate description of static properties (through, for example, comparisons with path-integral Monte Carlo results), our concern is that QSPs don’t describe correctly the dynamics seen in this movie: the quantum nature of the potentials used here was statistical and the same for each electron; however, as is evident, each electron is in a very different state (e.g., momentum) from the others at any given time.

An alternative framework is an effective classical Hamiltonian treatment with momentum-dependent potentials (MDP). The notion behind the MDP approach is the observation that quantum systems have more degrees of freedom than classical systems; thus, we should use all of the degrees of freedom available in the most general way. Retaining only the classical phase-space degrees of freedom, this suggests non-separable potentials that depend on position and momentum. Such a non-separable potential, in fact, reflects nicely the non-commutative properties of the position and momentum operators of quantum mechanics.

MDPs have been successfully applied to atomic, molecular and nuclear systems but little development has been carried out for plasma-like systems. Our objective is to identify a MDP that can adequately model both bound atomic-like states and continuum scattering states for modeling of partially ionized plasmas in which electrons persistently transition among all states. To connect with existing knowledge, we began our studies with the well known Kirschbaum-Wilets (KW) MDPs that are empirical in nature. By training their parameters, we found them to be good models for ground state energies and first ionization energies for the elements we considered. To investigate the electron-ion scattering properties, we tested KW MDP with momentum transfer cross section (MTCS) which is an important quantity connected to stopping power and transport properties. Despite having a strong influence on the electron-ion interaction, MTCS of KW MDP are closer to the purely classical MTCS than the quantum values, thereby suggesting that their efficacy in modeling bound states does not transfer to the scattering properties.

You can learn more about our recent work here:


Based on our recent findings, we have identified a new functional form for the MDP that appears to be promising, perhaps mitigating most of the outstanding issues. Stay tuned!

This work is lead by Gautham Dharuman.


Atomic bound state and scattering properties of effective momentum-dependent potentials
Physical Review E 94, 043205, (2016)
Gautham Dharuman, John Verboncoeur, Andrew Christlieb and Michael S. Murillo

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