HYB (Hybrid)

The HYB code solves the Hybrid DFT equation (containing a linear combination of the LDA exchange-correlation potential and Fock operators) 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 Hybrid DFT 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

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

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

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

and the LDA exchange-correlation potential \(v_{\text{xc}}^{\text{LDA}}\) (see iDEA LDA).

Let \(\alpha \in \mathopen[ 0, 1 \mathclose]\) be a fixed parameter which determines the linear mixing of Hartree-Fock and LDA. Then defining

\[H_{\alpha}(x,y) = \delta(x-y)\hat{T} + \delta(x-y)v_{\text{ext}}(y) + \delta(x-y)v_{\text{H}}(y) + \alpha\Sigma_{\text{x}}(x,y) + (1-\alpha)\delta(x-y)v_{\text{xc}}^{\text{LDA}}(y) \, ,\]

the Hybrid Hamiltonian \(\hat{H}_{\alpha}\) acts via the integral transform

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

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

We solve the Hybrid equation

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

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).

The parameter \(\alpha\) may be specified manually, however it is also possible to determine an “optimal” value for \(\alpha\) by considering the generalized Koopmans’ theorem, which gives conditions under which the model best describes a physical system.

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 Hybrid Hamiltonian, which again acts via the same integral transform as explained previously, but now with a perturbation potential term added as

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

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.