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Advances in Ultrafast Condensed Phase Physics

Electron thermalization in TDDFT and Ehrenfest molecular dynamics

Tobias Zier, Uday Panta, David A. Strubbe

Published May 26, 2026• 0 citations
Publisher

Abstract

A non-equilibrium electronic state will, in general, thermalize toward an equilibrium state due to electron-electron interactions as well as interactions with ions. The description of such processes has remained unclear and controversial in time-dependent density functional theory (TDDFT), given that the occupation numbers remain fixed over time and that adiabatic functionals do not include explicit dissipation. Here, we study the explicit time propagation of graphene after ultrafast laser pulses, using the Octopus real-space code, with and without ionic motion in Ehrenfest molecular dynamics. This work has implications for the treatment of statistical mechanics in a TDDFT framework.

DOI
10.1117/12.3106298

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A non-equilibrium electronic state will, in general, thermalize toward an equilibrium state due to electron-electron interactions as well as interactions with ions. The description of such processes has remained unclear and controversial in time-dependent density functional theory (TDDFT), given that the occupation numbers remain fixed over time and that adiabatic functionals do not include explicit dissipation. Here, we study the explicit time propagation of graphene after ultrafast laser pulses, using the Octopus real-space code, with and without ionic motion in Ehrenfest molecular dynamics. This work has implications for the treatment of statistical mechanics in a TDDFT framework.
Classical molecular dynamics (MD) provides a powerful and widely used approach to determining thermodynamic properties by integrating the classical equations of motion of a system of atoms. Time-Dependent Density Functional Theory (TDDFT) provides a powerful and increasingly useful approach to integrating the quantum equations of motion for a system of electrons. TDDFT efficiently captures the unitary evolution of a many-electron state by mapping the system into a fictitious non-interacting system. In analogy to MD, one could imagine obtaining the thermodynamic properties of an electronic system from a TDDFT simulation in which the electrons are excited from their ground state by a time-dependent potential and then allowed to evolve freely in time while statistical data are captured from periodic snapshots of the system. For a variety of systems (e.g., many metals), the electrons reach an effective state of internal equilibrium due to electron-electron interactions on a time scale that is short compared to electron-phonon equilibration. During the initial time-evolution of such systems following electronic excitation, electron-phonon interactions should be negligible, and therefore, TDDFT should successfully capture the internal thermalization of the electrons. However, it is unclear how TDDFT represents the resulting thermal state. In particular, the thermal state is usually represented in quantum statistical mechanics as a mixed state, while the occupations of the TDDFT wavefunctions are fixed by the initial state in TDDFT. We work to address this puzzle by (A) reformulating quantum statistical mechanics so that thermodynamic expectations can be obtained as an unweighted average over a set of many-body pure states and (B) constructing a family of non-interacting (single determinant) TDDFT states that approximate the required many-body states for the canonical ensemble.
Quasiparticles are fundamental in describing the complex many-body interactions underlying single-particle excitations, playing a key role in understanding the macroscopic properties of quantum materials. Recent advancements in ultrafast laser technology, combined with various detection methods, have enabled precise monitoring and manipulation of quantum states and their dynamic processes. These developments not only enhance our comprehension of quasiparticle excitation mechanisms but also facilitate driving materials into non-equilibrium states, uncovering novel physical phenomena and advancing the design and application of innovative functional materials. The computational approach based on real-time time-dependent density functional theory (rt-TDDFT) transcends the limitations of linear response theory, enabling first-principles simulations of the dynamic response of strongly coupled quasiparticles under non-equilibrium conditions. The proposed method provides accurate predictions of highly nonlinear phenomena induced by strong laser fields, offering deeper insights into the underlying physical mechanisms. These theoretical insights are crucial for optimizing the performance of optoelectronic devices. This paper introduces the theoretical foundations of Ehrenfest dynamics within the framework of rt-TDDFT and presents case studies on the evolution of various quasiparticles under laser excitation. The results demonstrate the significant potential of ultrafast lasers for probing and controlling the properties of quantum materials, while also underscoring the vital role of dynamic simulations for studying non-equilibrium states and their time-domain characteristics.
Real-time time-dependent density functional theory provides a first-principles description of coherent electron dynamics in laser-driven solids, but its unitary formulation cannot capture the irreversible scattering, relaxation, and decoherence processes that drive excited carriers toward equilibrium. Here, we develop a dissipative rt-TDDFT framework in which first-principles electron-phonon interactions enter the evolution of the reduced one-body density matrix through self-energy-derived collision integrals within the Born-Markov approximation. The approach retains the quantum-coherent real-time propagation of the electronic system while introducing phonon-mediated transitions that redistribute carriers in energy and crystal momentum, thereby incorporating the microscopic momentum-transfer processes responsible for relaxation in real materials. The resulting framework provides a practical first-principles route to simulate relaxation, decoherence, and time-resolved spectroscopic signatures in realistic crystalline materials.
Ultrafast electronic dynamics in solids lies at the core of modern condensed matter and materials physics. To build up a practical \textit{ab initio} method for studying solids under photoexcitation, we develop momentum-resolved real-time time dependent density functional theory (rt-TDDFT) algorithm using numerical atomic basis, together with the implementation of both length and vector gauge of electromagnetic field. When applied to simulate elementary excitations in graphene, different excitation modes, only distinguishable in momentum space, are observed. The momentum-resolved rt-TDDFT is important and computationally efficient for the study of ultrafast dynamics in extended systems.
Time-dependent density-functional theory (TDDFT) treats dynamical exchange and correlation (xc) via a single-particle potential, Vxc(r,t){V}_{\mathrm{xc}}(\mathbf{r},t)Vxc​(r,t), defined as a nonlocal functional of the density $n({\mathbf{r}}^{\ensuremath{'}},{t}^{\ensuremath{'}})$. The popular adiabatic local-density approximation (ALDA) for Vxc(r,t){V}_{\mathrm{xc}}(\mathbf{r},t)Vxc​(r,t) uses only densities at the same space-time point (r,t)(\mathbf{r},t)(r,t). To go beyond the ALDA, two local approximations have been proposed based on quantum hydrodynamics and elasticity theory: (a) using the current as the basic variable (C-TDDFT) [G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4847 (1997)], (b) working in a comoving Lagrangian reference frame (L-TDDFT) [I. V. Tokatly, Phys. Rev. B 71, 165105 (2005)]. In this paper we illustrate, compare, and analyze both nonadiabatic theories for simple time-dependent model densities in the linear and nonlinear regime, for a broad range of time and frequency scales. C- and L-TDDFT are identical in certain limits, but, in general, exhibit qualitative and quantitative differences in their respective treatment of elastic and dissipative electron dynamics. In situations where the electronic density rapidly undergoes large deformations, it is found that nonadiabatic effects can become significant, causing the ALDA to break down.
Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.