Real-Time TDDFT: Ultrafast Electronic Dynamics

• RT-TDDFT is a quantum many-body method that propagates time-dependent Kohn–Sham states to capture ultrafast electronic dynamics.
• It employs advanced propagation algorithms, including the parallel-transport gauge, to achieve numerical stability and scalable simulation performance.
• RT-TDDFT accurately models photo-induced phenomena, nonlinear optical responses, and excitonic dynamics across molecular and solid-state systems.

Real-Time Time-Dependent Density Functional Theory (RT-TDDFT) is a quantum many-body methodology for simulating and analyzing the ultrafast electronic dynamics of molecular and extended systems. It generalizes Kohn–Sham density functional theory (DFT) into the time domain by propagating electronic states under a time-dependent Hamiltonian, fully capturing linear and nonlinear response, non-equilibrium excitation, and collective phenomena under arbitrary time-dependent perturbations. RT-TDDFT has evolved into a central framework for investigating photo-induced phenomena, nonequilibrium carrier dynamics, high-harmonic generation, dynamical excitonic effects, and quantum transport, with implementations that enable accurate, large-scale simulations on heterogeneous supercomputing architectures across molecular, plasmonic, and solid-state domains (Pela et al., 2021, Kononov et al., 2022, Jakowski et al., 11 Oct 2024, Ji et al., 21 Dec 2025).

1. Theoretical Framework and Electronic Structure Formulation

RT-TDDFT is founded on the time-dependent Kohn–Sham (TDKS) equations,

$$i\hbar\,\frac{\partial}{\partial t}\,\psi_{i}(\mathbf{r}, t) = \hat{H}_{\mathrm{KS}}[\rho](\mathbf{r}, t)\,\psi_{i}(\mathbf{r}, t),$$

where the TDKS Hamiltonian $\hat{H}_{\mathrm{KS}}$ incorporates kinetic, ionic, Hartree, and exchange–correlation potentials, and couples to external fields via either the length (dipole) or velocity (minimal coupling) gauge. For solids and periodic systems, the velocity gauge is preferred: $$\hat{H}(t) = \frac{1}{2} [-i\nabla + \mathbf{A}(t)/c]^2 + v_{\mathrm{KS}}[\rho](\mathbf{r}, t),$$ with the vector potential $\mathbf{A}(t)$ encoding the electromagnetic field. For all-electron accuracy with periodicity, the linearized augmented plane wave + local orbital (LAPW+lo) basis achieves precise representation of both core and valence regions (Pela et al., 2021). Alternatively, localized numerical atomic orbitals (NAOs) with Bloch sums and explicit k-point resolution provide efficient access to band- and $\mathbf{k}$-resolved dynamics in extended phases (Lian et al., 2017, Pemmaraju et al., 2017, Ji et al., 21 Dec 2025).

Matrix representations use the generalized eigenproblem for coefficient vectors $C$,

$i S_k\,\partial_t C(t) = H_k(t)\,C(t),$

where $S_k$ and $H_k$ are the overlap and Hamiltonian matrices at each k-point. The basis expansion and block propagation strategies enable RT-TDDFT to faithfully treat multi-band, anisotropic, and spatially inhomogeneous fields in molecules, 2D materials, and bulk crystals.

2. Propagation Algorithms, Numerical Stability, and Scaling

RT-TDDFT requires integration of stiff, nonlinear matrix differential equations for the electronic wavefunctions or density matrix. Explicit Runge–Kutta (S-RK4), implicit midpoint/Crank–Nicolson (CN), enforced time-reversal symmetry (ETRS), Magnus expansion, and self-consistent exponential midpoint schemes have all been implemented (Pela et al., 2021, Lian et al., 2017, Jakowski et al., 11 Oct 2024, Kononov et al., 2022, Jia et al., 2018).

A key advance is the adoption of the parallel-transport (PT) gauge, which rotates the occupied manifold to minimize the norm of the time derivative: $$P\,\dot{\Phi} = 0, \quad \dot{\Phi} = H\Phi - \Phi(\Phi^* H \Phi),$$ eliminating fast phase oscillations and permitting 10–100x larger time steps compared to Schrödinger gauge explicit solvers (Jia et al., 2018, Jia et al., 2018, Liu et al., 6 Jan 2025). Implicit PT-integrators (PT-CN, PT-IM) remain time-reversible and nearly symplectic, yielding stable propagation for time steps up to 50–100 attoseconds, while retaining all physical observables invariant (Liu et al., 6 Jan 2025, Jia et al., 2018).

For density-matrix propagation, the Magnus or commutator expansion

$$P(t+\Delta t) = P(t) + (-i/\hbar)\Delta t [\bar{H}, P] + \cdots$$

is used, preserving unitarity and energy conservation up to long times with rapid convergence for modest expansion order (Jakowski et al., 11 Oct 2024).

Parallelization is achieved via domain decomposition of real-space grids, orbital, or band indices, with communication patterns (ring-based, shared-memory) reducing the cost of Fock exchange in hybrid functional calculations (Liu et al., 6 Jan 2025, Jakowski et al., 11 Oct 2024). Scaling efficiency is demonstrated for multi-thousand atom systems and up to $\sim$24,000 electrons (Jakowski et al., 11 Oct 2024, Liu et al., 6 Jan 2025).

Scheme	Max stable $\Delta t$	Parallel transport	Fock cost (hybrid)
RK4	$\sim$1 as	no	$\gg$ hybrid methods
PT-CN / PT-IM	10–100 as	yes	$<$10% with ACE/MLWF
Commutator/Magnus	2–5 a.u. (grid)	N/A	N/A

3. Exchange–Correlation Functionals and Extensions

The majority of RT-TDDFT implementations use adiabatic exchange–correlation (XC) approximations (ALDA or GGA). For excitonic dynamics and accurate band gaps in solids, range-separated hybrid (RSH), long-range corrected (LRC), and meta-GGA kernels are essential (Sun et al., 2021, Ji et al., 21 Dec 2025, Kononov et al., 2022). In solids, the time-dependent vector potential representation of LRC kernels introduces a nonlocal macroscopic XC field,

$$A_{\mathrm{xc}}^{\mathrm{LRC}}(t) = \alpha \int_0^t dt' \int_0^{t'} dt''\, J_{\text{macro}}(t'')$$

restoring the $-1/|k|^2$ behavior required for capturing excitonic binding (Sun et al., 2021, Kononov et al., 2022). Range-separated hybrids in periodic NAO bases account for the singularity of the Fock term via auxiliary-function corrections, enabling rapid convergence with respect to k-mesh and correct excitonic peak positions (Ji et al., 21 Dec 2025).

Hybrid functional RT-TDDFT, historically limited by the cost of Fock operator application, is now feasible for large-scale, finite-temperature systems through adaptively compressed exchange (ACE), occupation-matrix diagonalization, and domain-wise communication minimization (Jia et al., 2018, Liu et al., 6 Jan 2025). For dynamics involving core-electron excitations or relativistic effects (e.g., attosecond transient absorption spectroscopy at L-edges), fully four-component Dirac–Kohn–Sham or efficient atomic mean-field X2C (amfX2C) reconstructions are available, with sub-eV accuracy and substantial speed-up over explicit four-component propagations (Moitra et al., 2022).

4. Observables, Analysis Tools, and Physical Applications

RT-TDDFT enables direct calculation of time-domain currents, band-population transfer, time-dependent polarization, nonlinear optical response, and transient spectral signatures under arbitrary drive. The mapping from induced current/polarization $J(t) \to \sigma(\omega) \to \epsilon(\omega)$ yields dielectric response, while time-resolved projections onto ground-state Kohn–Sham (KS) basis states or band-resolved populations permit physical decomposition of spectra and identification of excitation processes (Rossi et al., 2017, Lian et al., 2017).

For plasmonic and collective modes, the Kohn–Sham electron–hole decomposition recovers Casida eigenvectors at spectral peaks, enabling assignment of transitions even in large nanospheres and nanoparticles (Rossi et al., 2017). Maximally localized Wannier function (MLWF) propagation, feasible on-the-fly, provides spatially resolved, chemically meaningful orbitals for interpreting bond-specific excitations, time-resolved charge transfer, and quantized charge pumping (Yost et al., 2019, Kononov et al., 2022).

RT-TDDFT is applied to:

• Linear and nonlinear optical responses, including third-harmonic generation (Pela et al., 2021).
• Ultrafast laser-induced band excitation and energy transfer in 2D, layered, and correlated materials (Lian et al., 2017, Krishtal et al., 2015).
• Time-resolved excitation energy transfer and decoherence using open-system (stochastic) extensions (Hofmann et al., 2012).
• Core-level spectroscopy in the extreme UV and X-ray using velocity gauge and explicit core pseudopotentials (Pemmaraju et al., 2017, Moitra et al., 2022, Ye et al., 2022).
• High-energy-density (HED) physics for dynamic structure factors, conductivity, and stopping power (Kononov et al., 18 Nov 2025).
• Plasmonic nanoparticles, nonlinear field enhancement, transient absorption, and ultrafast demagnetization (Jakowski et al., 11 Oct 2024, Ji et al., 21 Dec 2025).

5. Scalability, Parallelism, and Software Implementations

Contemporary RT-TDDFT achieves high scalability via all-electron LAPW+lo bases (Pela et al., 2021), numerical atomic orbitals (Lian et al., 2017, Pemmaraju et al., 2017, Ji et al., 21 Dec 2025), real-space multigrids (Jakowski et al., 11 Oct 2024), and plane-wave pseudopotentials (Kononov et al., 2022, Jia et al., 2018, Jia et al., 2018). Algorithmic advances include:

• Matrix-commutator expansion for density-matrix propagation, with per-step cost scaling linearly with basis size for large systems (Jakowski et al., 11 Oct 2024).
• Distributed-memory parallelization on MPI ranks with ring-based asynchronous communication, shared-memory reduction for Fock operators, and GPU offloading (Liu et al., 6 Jan 2025, Jakowski et al., 11 Oct 2024).
• Performance demonstrated for nanorods up to 24,000 electrons, with per-step times $\lesssim$10 seconds on 64 nodes (Jakowski et al., 11 Oct 2024).
• Near-ideal scaling of pure MPI up to 256 ranks, and efficient OpenMP parallelism (Pela et al., 2021).

Code	Basis	Specialty	Max System Size
exciting	LAPW+lo	All-electron, full-potential	$\sim$1,000 bands
RMG	Real-space	Multigrid + commutators, GPU capable	24,000 e−
Qb@ll	Plane-wave	MLWFs, velocity/length gauge, hybrids	2,000+ atoms
SIESTA	NAO	LCAO velocity gauge, core edges	1,000s of atoms

6. Limitations, Validation, and Future Directions

RT-TDDFT with adiabatic local/semilocal functionals does not capture electron–electron scattering and thermalization timescales, nor memory-dependent (nonadiabatic) response in correlated/metallic systems. Advanced XC kernels, dynamical (nonlocal in $\omega$) response, and embedding approaches are active research areas (Kononov et al., 2022, Pela et al., 2021). Subsystem RT-TDDFT and explicit domain/fragment partitioning schemes enable distributed simulations and energy transfer analysis for large molecular assemblies, but hinge on the accuracy of nonadditive kinetic-energy functionals (Krishtal et al., 2015).

For relativistic and high-energy electrodynamics, direct four-component propagation or amfX2C reduction enables attosecond pump–probe predictions for heavy elements (Moitra et al., 2022). Systematic control over gauge choice, discretization errors, and convergence—especially in the X-ray regime—requires high-order commutator-free propagators and automated step-size selection (Ye et al., 2022).

Ongoing and prospective directions include:

• Real-time Bethe–Salpeter (RT-BSE) for excitonic and correlated matter dynamics (Pela et al., 2021).
• Coupled quantum-classical (Ehrenfest, surface hopping) and quantum–Maxwell propagation for light–matter interaction beyond the dipole approximation (Shapira et al., 2023, Pela et al., 2021).
• Integration of GW/BSE within real-time density-functional frameworks (Jakowski et al., 11 Oct 2024).
• Machine-learning ODE solvers and advanced preconditioners for further acceleration (Kononov et al., 2022).

RT-TDDFT provides a mathematically rigorous, numerically robust, and physically transparent platform for probing the spectrum of electronic time-dependent phenomena, bridging fundamental quantum theory and emerging quantum technologies across chemical, condensed matter, and materials science domains.