Multi-Configuration Time-Dependent Hartree¶

A variational quantum-dynamics method that expands a multidimensional wavefunction in configurations built from time-dependent single-particle functions and evolves both coefficients and basis functions.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2322
Origin domain: quantum dynamics
Subdomain: wavepacket propagation
Aliases: MCTDH, Multiconfigurational time-dependent Hartree

Core Idea¶

Multi-Configuration Time-Dependent Hartree (MCTDH) approximates a multidimensional time-dependent wavefunction with a sum of Hartree products whose one-coordinate factors move with time:

\[ \Psi(q_1,\ldots,q_f,t)= \sum_{j_1=1}^{n_1}\cdots\sum_{j_f=1}^{n_f} A_{j_1\cdots j_f}(t) \prod_{\kappa=1}^{f} \varphi^{(\kappa)}_{j_\kappa}(q_\kappa,t). \]

Both the configuration coefficients \(A_J(t)\) and the single-particle functions are determined variationally. A fixed primitive product basis can require exponentially many coefficients; adapting the one-dimensional bases to the evolving wavepacket can represent correlated dynamics with fewer configurations. Meyer, Manthe, and Cederbaum introduced the method in 1990.

Scope of Application¶

MCTDH is used for molecular wavepacket dynamics, photodissociation, reactive scattering, vibrational energy redistribution, nonadiabatic dynamics, and model open-system Hamiltonians. It is especially effective when a moderate number of modes have structured coupling and the wavefunction occupies a low-rank adaptive subspace.

The method requires choices: coordinate system, mode combination, primitive grids, SPF counts, integrator, and Hamiltonian representation. A poor partition can leave the coefficient tensor exponentially large. Mode combination groups coordinates into multidimensional SPFs to balance correlation and basis cost.

Clarity¶

The coefficient tensor captures correlation among modes relative to the current SPFs. The SPFs themselves rotate to follow occupied directions. These adaptations must be separated: more configurations expand correlation rank, while better SPFs improve the moving modal subspaces.

For orthonormal SPFs, a common gauge is

\[ \left\langle\varphi_j^{(\kappa)} \middle| \dot\varphi_l^{(\kappa)} \right\rangle=0. \]

Manages Complexity¶

A direct product basis with \(N\) functions for each of \(f\) modes contains \(N^f\) coefficients. MCTDH replaces full primitive occupancy with \(n_\kappa\) adaptive SPFs, yielding \(\prod_\kappa n_\kappa\) configurations plus SPF coefficients. When \(n_\kappa\ll N_\kappa\), this is a major reduction.

Abstract Reasoning¶

Let the trial manifold consist of normalized ansatz states \(\Psi(z)\) parameterized by \(A\) and the SPFs. Dirac–Frenkel variation requires

\[ \left\langle\delta\Psi\middle| i\frac{\partial}{\partial t}-H \middle|\Psi\right\rangle=0 \]

for all tangent variations. Variations in \(A^\ast\) yield a Schrödinger-like coefficient equation in the SPF product basis. SPF variations yield projected mean-field equations involving reduced density matrices.

Knowledge Transfer¶

The adaptive low-rank idea transfers to tensor trains, time-dependent variational matrix-product states, and multilayer tensor networks: evolve a state on a structured low-rank manifold rather than in the full tensor product. The precise MCTDH identity remains the sum of Hartree products with modal SPFs and its variational equations.

Transfer fails when one merely changes basis after propagation or truncates singular values without coupled variational time evolution. Similar numerical economy does not make every tensor method MCTDH.

Relationships to Other Abstractions¶

Current abstraction Multi-Configuration Time-Dependent Hartree Domain-specific

Parents (1) — more general patterns this builds on

Multi-Configuration Time-Dependent Hartree is a kind of Approximation Prime

Approximation is the proposed minimal parent: MCTDH is a strict, controlled approximation by an adaptive variational manifold.

Hierarchy path (1) — routes to 1 parentless root

Multi-Configuration Time-Dependent Hartree → Approximation → Representation → Abstraction

Neighborhood in Abstraction Space¶

Multi-Configuration Time-Dependent Hartree sits in a sparse region of the domain-specific corpus (93^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08