Time Dependent HartreeEdit

Time-Dependent Hartree (TDH) is a mean-field approach to the quantum dynamics of many-body systems, most often electrons in atoms, molecules, or solids, as well as nuclei in some nuclear-physics contexts. It extends the static Hartree method into the time domain by letting single-particle orbitals evolve under a self-consistently generated mean field produced by the instantaneous density. In its simplest form, the many-body wavefunction is approximated by a product of time-dependent orbitals, which makes the method computationally tractable and interpretable, but at the cost of neglecting exchange effects and dynamic correlations that can be important in certain regimes. TDH thus sits at a practical baseline in the family of mean-field theories, with time-dependent Hartree–Fock (TDHF) and time-dependent density functional theory (TDDFT) offering ways to incorporate additional physics beyond the original Hartree picture. The method has found uses across quantum chemistry, materials science, and nuclear physics, especially for real-time simulations under external fields, photoexcitation, and rapid dynamical processes.

Theoretical foundation

Ansatz and equations

In the TDH approximation, the N-particle wavefunction is represented as a product of N single-particle orbitals that evolve in time: Ψ(r1, r2, ..., rN, t) ≈ ∏{i=1}^N φ_i(r_i, t). From this, the one-particle density is ρ(r, t) = ∑{i=1}^N |φ_i(r, t)|^2, and the interparticle Coulomb (or more general) interaction contributes a mean-field Hartree potential: v_H(r, t) = ∫ d^3r′ ρ(r′, t) w(|r − r′|), where w is the two-body interaction (for electrons, typically the Coulomb kernel 1/|r − r′|).

The effective single-particle Hamiltonian then reads hρ = −(1/2m) ∇^2 + v_ext(r, t) + v_H(r, t), with v_ext representing external fields or nuclei. The orbitals satisfy the time-dependent Schrödinger equation in this self-consistent field: i ∂_t φ_i(r, t) = hρ φ_i(r, t) for i = 1,...,N, together with appropriate spin structure if spin is included. The TDH framework enforces normalization of each orbital and, in practice, evolves the set {φ_i} under the evolving mean field.

Density and mean-field potential

Because all orbitals contribute to ρ(r, t), the Hartree potential is a nonlinear functional of the orbitals themselves, which makes the dynamics intrinsically nonlinear. The resulting equations are often solved on real-space grids or in a basis, using numerical time propagation schemes that preserve unitarity and stability. In many implementations, the Coulomb interaction is computed efficiently via Poisson solvers or fast Fourier transform techniques to handle the convolution implied by v_H.

Special cases and extensions

For bosonic systems with contact interactions, the TDH equations reduce to a form analogous to the Gross–Pitaevskii equation, a nonlinear Schrödinger equation widely used in Bose–Einstein condensate physics. For fermions, the Hartree approximation neglects antisymmetry effects; the time-dependent Hartree–Fock (TDHF) extension restores antisymmetry and includes exchange (Fock) terms, yielding a more accurate but more complex set of equations. The TDH framework thus functions as a baseline, with TDHF and post-HF methods building in additional physics.

Relationship to related theories

Hartree vs Hartree–Fock

The original Hartree method (static, self-consistent field) uses a product of spatial orbitals to approximate the N-body state. It ignores exchange due to particle indistinguishability, which can lead to qualitative and quantitative errors for many systems. Hartree–Fock, by contrast, uses a Slater determinant to enforce antisymmetry, introducing exchange terms that significantly affect orbital energies and dynamics. TDHF extends this antisymmetry to the time domain, improving accuracy for many dynamical processes where exchange plays a role.

TDHF and TDDFT

TDHF is a natural successor to TDH when one seeks a more faithful treatment of fermionic exchange in time evolution. Time-dependent density functional theory (TDDFT) replaces the explicit many-body wavefunction with a time-dependent electronic density, using exchange–correlation functionals to incorporate many-body effects beyond a mean-field description. TDDFT is often more accurate for a broad range of excitations and real-time responses, but is also dependent on the quality of the chosen functionals. TDH remains valuable as a transparent, computationally efficient baseline and as a tool for gaining intuition about dynamical processes without the ambiguity that can accompany functional choices.

Relation to other mean-field approaches

TDH is part of the broader mean-field and self-consistent-field family, which includes classical field theories for many-body systems and quantum-mechanical mean-field models. In nuclear physics, mean-field time evolution captures collective motion and reaction dynamics, providing a tractable framework where fully correlated methods would be intractable for the systems of interest.

Applications and domains

Quantum chemistry and molecular dynamics

TDH is used to study real-time electron dynamics under intense laser fields, ultrafast charge migration, and other dynamical phenomena where a fast, interpretable description of the electronic response is valuable. It serves as a computationally cheaper alternative or a stepping stone to more advanced methods, helping researchers screen systems and interpret results in light of a straightforward mean-field picture. In many cases, TDH results are complemented by more sophisticated techniques to assess the importance of exchange and correlation.

Nuclear dynamics

In nuclear physics, TDH-like mean-field treatments provide insight into collective motion, fusion, and fission dynamics in regimes where a full many-body treatment is impractical. The approach can capture general trends and spectroscopic features that arise from mean-field rearrangements of nucleons over time.

Solid-state and materials science

For extended systems, TDH can be used to study ultrafast charge transport, plasmonic excitations, and electron dynamics in response to external perturbations. Here, the efficiency of TDH makes it attractive for exploring large or complex materials where more exact methods would be computationally prohibitive.

Real-time dynamics and strong fields

Real-time TDH is particularly appealing for exploring dynamics under strong, time-dependent fields, such as laser-driven processes, where a clear, time-resolved view of the evolving density and mean-field potential yields valuable physical insight.

Practical considerations and limitations

Strengths and drawbacks

The main strength of TDH is its combination of physical transparency and computational efficiency. By solving a set of coupled, nonlinear single-particle equations, TDH scales more favorably with system size than methods that explicitly treat all many-body correlations. It provides an intuitive picture: each particle moves in a self-consistent field created by the instantaneous density.

The major limitation is the neglect of exchange and dynamic correlation beyond the mean field. For many systems, especially those with strong correlations, multireference character, or processes where exchange effects are central, TDH can give misleading results. In such cases, TDHF, TDDFT with carefully chosen functionals, or more advanced many-body techniques are preferred.

Numerical and methodological aspects

Numerically, TDH requires stable time propagation of a set of orbitals and the efficient evaluation of the Hartree potential. The choice of basis or grid, the treatment of boundary conditions, and the handling of long-range Coulomb interactions all influence accuracy and efficiency. Energy and norm preservation are important numerical checks, as well as sensitivity to the external field and to the size of the basis set. In practical workflows, TDH results are often used as a baseline to gauge the impact of exchange and correlation by comparison with TDHF or TDDFT calculations.

When to use or avoid TDH

TDH is well suited for large systems where a transparent baseline dynamical picture is desirable and where exchange and correlation effects are not anticipated to dominate the dynamics. It is less appropriate for systems with strong static or dynamic correlations, near-degeneracies, or where precise spectroscopic properties depend sensitively on exchange interactions. In such cases, practitioners typically turn to TDHF, TDDFT, or higher-level correlated methods, possibly in a multiscale or hybrid framework that leverages the strengths of each approach.