HF (Hartree–Fock)

The HF code solves the Hartree–Fock equation to approximately calculate the ground state density of a one-dimensional finite system of \(N\) like-spin electrons, with Hamiltonian \(\hat{T} + \hat{U} + \hat{V}_{\text{ext}}\), where \(\hat{U}\) is the operator for the softened Coulomb interaction potential given by

\[u(x, y) = (1 + |x-y|)^{-1} \, .\]

A perturbing potential may be applied to the ground state system, and the time-dependent Hartree–Fock equation solved approximately to calculate the system’s time evolution.

Calculating the ground state

Orbitals and corresponding non-degenerate energy eigenvalues \(\{ \varphi_{j}, \varepsilon_{j} \}_{j=1}^{N}\) for a non-interacting system of \(N\) electrons are first calculated from the single-particle Schrödinger equation with Hamiltonian \(\hat{T} + \hat{V}_{\text{ext}}\).

We then proceed to calculate the ground state electron density of this system:

\[n(x) = \sum_{j=1}^{N} \lvert \varphi_{j}(x) \rvert ^{2} \, .\]

From this we find the Hartree potential

\[v_{\text{H}}(x) = \int \! n(y)u(x,y) \, \mathrm{d}y \, ,\]

and the Fock matrix (the self-energy in the exchange-only approximation)

\[\Sigma_{\text{x}}(x,y) = - \sum_{j=1}^{N} \varphi_{j}^{*}(y) \varphi_{j}(x) u(x,y) \, .\]

Defining

\[H(x,y) = \delta(x-y)\hat{T} + \delta(x-y)v_{\text{ext}}(y) + \delta(x-y)v_{\text{H}}(y) + \Sigma_{\text{x}}(x,y) \, ,\]

the Hartree–Fock Hamiltonian \(\hat{H}\) acts via the integral transform

\[\hat{H}\varphi(x) = \int \! H(x,y)\varphi(y) \, \mathrm{d}y \, .\]

When our one-dimensional space is discretized to points on a grid, \(H(x,y)\) itself is a two-dimensional array, and acts as the Hamiltonian on the now one-dimensional arrays \(\{ \varphi_{j} \}_{j=1}^{N}\).

We solve the Hartree–Fock equation

\[\hat{H}\varphi_{j} = \varepsilon_{j}\varphi_{j}\]

to obtain a new set of orbitals and their corresponding eigenvalues.

Using these new orbitals the above process is iterated until a self-consistent solution is reached (i.e. until the change in the new electron density \(n\) is small enough compared to that of the previous iteration).

Time-dependence

A perturbing potential \(v_{\text{ptrb}}\) is added to the ground state system so that it now has Hamiltonian \(\hat{T} + \hat{U} + \hat{V}_{\text{ext}} + \hat{V}_{\text{ptrb}}\).

We construct the Time-Dependent Hartree–Fock Hamiltonian, which again acts via the same integral transform as explained previously, but now with a perturbation potential term added as

\[H^{\text{TD}}(x,y) = H(x,y) + \delta(x-y)v_{\text{ptrb}}(y) \, .\]

Beginning with the ground state (initial time) orbitals and energies \(\{ \varphi_{j}(t=0), \varepsilon_{j}(t=0) \}_{j=1}^{N}\) as already calculated, we apply the Crank–Nicolson method to evolve the system forward in time by a step of size \(\delta t\):

\[\{ \varphi_{j}(t) \}_{j=1}^{N} \rightarrow \{ \varphi_{j}(t=t+\delta t) \}_{j=1}^{N} \, .\]

This process is repeated until the desired duration of the time-evolution has been simulated. The electron density of the system is calculated from the orbitals at each time-step.