# time-dependent density functional theory

> quantum mechanical theory to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials

**Wikidata**: [Q4120992](https://www.wikidata.org/wiki/Q4120992)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory)  
**Source**: https://4ort.xyz/entity/time-dependent-density-functional-theory

## Summary
Time-dependent density functional theory (TDDFT) is a quantum-mechanical extension of ordinary density functional theory that lets researchers compute how the electron density of a many-electron system evolves when the external potential changes with time. By mapping the real, time-dependent many-electron problem onto a set of single-particle equations, TDDFT provides an efficient, first-principles route to excited-state energies, optical spectra, charge-transfer dynamics and other non-equilibrium properties of atoms, molecules and solids.

## Key Facts
- Sub-class of density functional theory (DFT); explicitly incorporates time-dependent external potentials
- Formal foundation is the Runge–Gross theorem (1984), which proves a one-to-one mapping between time-dependent external potentials and time-dependent electron densities
- Standard acronym: TDDFT; also written TDLDA (time-dependent local-density approximation) when the adiabatic LDA functional is used
- Solves for the time-dependent electron density rather for the full many-electron wavefunction
- Recognised IUPAC Gold-Book entry ID TT07484
- Covered in 5 Wikipedia language editions (en, es, ja, ko, tr) as of the last data snapshot
- Library of Congress Subject Heading sh2021005651 lists "TDDFT" as a variant term
- Microsoft Academic tracked the topic under ID 20853536 until the service was discontinued

## FAQs
### Q: How does TDDFT differ from regular DFT?
A: Ordinary DFT finds the ground-state electron density, whereas TDDFT propagates the density in real time or extracts excited-state energies via linear-response theory, allowing calculation of optical absorption, excitation energies and non-equilibrium dynamics.

### Q: What can TDDFT actually calculate?
A: Typical outputs are vertical excitation energies, UV–visible absorption spectra, photo-emission intensities, charge-transfer times and laser-driven electron motion, all from first principles and at a computational cost similar to a ground-state DFT calculation.

### Q: Is TDDFT limited to small systems?
A: No. Because it scales favourably (formally N³–N⁴), TDDFT is routinely applied to molecules with hundreds of atoms, nanoclusters, and periodic solids; plane-wave codes even treat surfaces and 2-D materials.

## Why It Matters
TDDFT bridges the gap between accurate but expensive many-electron wave-function methods and fast but ground-state-only DFT. By adding time dependence, chemists can predict how molecules absorb light, estimate fluorescence yields, and follow ultrafast electron transfer in photovoltaic materials. Physicists use the same framework to model plasmonics, strong-field laser interactions and attosecond spectroscopy. Because TDDFT retains the favourable scaling of DFT, it is the only first-principles excited-state method practicable for medium-to-large systems, making it the de-facto standard for theoretical spectroscopy in computational chemistry, materials science and condensed-matter physics.

## Notable For
- First rigorous extension of Hohenberg–Kohn DFT to time-dependent realms via the Runge–Gross theorem
- Delivers excited-state energies with an accuracy of ~0.1–0.3 eV for valence excitations at a cost only modestly higher than ground-state DFT
- Dominant method for computing optical spectra in quantum-chemistry packages (Gaussian, ORCA, NWChem) and materials codes (VASP, Quantum-ESPRESSO, GPAW)
- Adiabatic approximation allows use of existing ground-state exchange-correlation functionals, giving TDDFT an immediate "plug-and-play" advantage over competing excited-state methods

## Body
### Theoretical Foundation
The Runge–Gross theorem, published in 1984, extends the Hohenberg–Kohn existence theorem to potentials that depend on time. It guarantees that, for a given initial state, the time-dependent electron density uniquely determines the external potential, so an auxiliary system of non-interacting electrons (the Kohn–Sham system) can reproduce the exact density. This legitimises a set of time-dependent Kohn–Sham equations whose orbitals evolve under an effective potential containing the external field, Hartree term, and an unknown time-dependent exchange-correlation (xc) potential.

### Practical Implementation
Most calculations adopt the adiabatic approximation: the xc potential at time t is taken as the ground-state functional evaluated on the instantaneous density. Linear-response TDDFT then yields poles (excitation energies) and residues (oscillator strengths) by solving a non-Hermitian eigenvalue problem whose dimension equals the product of occupied and unoccupied Kohn–Sham orbitals. Real-time propagation, an alternative route, directly integrates the time-dependent Kohn–Sham equations and Fourier-transforms the resulting dipole signal to obtain spectra.

### Limitations and Extensions
Known weaknesses include underestimation of Rydberg and charge-transfer excitations, absence of double excitations within adiabatic functionals, and incorrect description of conical intersections. Long-range-corrected hybrid functionals, many-body corrections (TDDFT+MBPT), and spin-flip or Δ-SCF variants partially remedy these issues while keeping the favourable scaling intact.

## References

1. National Library of Israel Names and Subjects Authority File
2. [OpenAlex](https://docs.openalex.org/download-snapshot/snapshot-data-format)