Time Evolution of the Hartree-Fock Wave Function

Question

The Hartree-Fock Hamiltonian is given by $$ \hat{H} = \sum_i \hat{h}_i + \frac{1}{2}\sum_i\sum_j \left[\hat{J}_{ij} - \hat{K}_{ij}\right] $$ and by choosing a basis, the Hamiltonian can be written in matrix form, where the entries are given by $\langle \phi_i|\hat{H}|\phi_j\rangle$.

The time evolution for a time independent Hamiltonian is in turn given by $|\Psi(t)\rangle = e^{-iHt/\hbar}|\Psi(0)\rangle$ and for time dependent Hamiltonians we can do e.g. a Magnus expansion to calculate the time evolution.

In our case, however, the Hamiltonian depends on the wave function at the next time step and a Magnus expansion is therefore only possible up to the 0-th order.

What is a better way to determine the time evolution of a Hartee-Fock state? I have already heard about TDHF, but I have not found an introduction including derivation that explains everything from scratch. So if TDHF is the right approach I would be interested in a good reference.

Answer 1

In order to determine the time evolution of a Hartree-Fock state, the Time-Dependent Hartree-Fock (TDHF) method is indeed a suitable approach. TDHF is a mean-field method used to study the dynamics of many-body quantum systems, particularly in the context of time-dependent processes.

In TDHF, the time evolution of the wave function is approximated by propagating the single-particle orbitals (ϕi) according to the time-dependent Hartree-Fock equations. These equations are derived from the time-dependent Schrödinger equation and can be solved iteratively to obtain the time-dependent wave function.

Here's an outline of the derivation for TDHF:

Starting with the time-dependent Schrödinger equation for a many-body system:

$$ i \hbar \partial / \partial t \ | \Psi(t) \rangle = \hat{H} |\Psi (t)\rangle$$

Expand the wave function in terms of the time-dependent single-particle orbitals:

$$ |\Psi(t)\rangle = \sum_{i} C_i(t) |\Phi_i(t)\rangle $$

Insert this expansion into the time-dependent Schrödinger equation and project onto each single-particle orbital |ϕk⟩:

$$ i\hbar \partial / \partial t \ C_k (t) = \sum_i \langle \Phi_k | \hat{H} | \Phi_i \rangle C_i (t) $$

Express the Hamiltonian in terms of creation and annihilation operators (second-quantization) and the Fock matrix elements.

Assuming that the wave function remains normalized during time evolution, apply the closure relation for single-particle states:

$$ \sum_i C_i^* (t) C_i(t) = 1 $$

By solving these TDHF equations, you can determine the time evolution of the coefficients $C_i(t)$ and hence obtain the time-dependent Hartree-Fock wave function $|\Psi (t) \rangle$.

For a detailed and comprehensive introduction to TDHF, including the derivation and practical implementation, you can refer to the following references:

"Time-Dependent Density Functional Theory" by Carsten A. Ullrich (ISBN-13: 978-0199563029)

"Many-Particle Physics" by Gerald D. Mahan (ISBN-13: 978-1475710435)

[Collected and summarized]