EXT (Exact)

The EXT code solves the many-electron time-independent Schrödinger equation to calculate the fully correlated, ground-state wavefunction for a one-dimensional finite system of 2 or 3 spinless electrons interacting via the softened Coulomb repulsion \((|x-x'|+1)^{-1}\). A perturbing potential is then applied to the ground-state system and its evolution is calculated exactly through solving the many-electron time-dependent Schrödinger equation.

Calculating the ground-state

An arbitrary wavefunction \(\Psi\) is constructed (preferably close to the ground-state of the system) and then propagated through imaginary time using the Crank-Nicolson method. Using the eigenstates of the system \(\{\psi_{m}\}\) as a basis:

\[\Psi (x_{1}, x_{2}, \dots, x_{N}, \tau) = \sum\limits_{m} c_{m} e^{-E_{m}\tau}\psi_{m}.\]

Providing the wavefunction remains normalised, the limiting value is the ground-state of the system:

\[E_{m+1} > E_{m} \ \ \forall \ m \in \mathbb{N}^{0} \implies \lim_{\tau \to \infty} \Psi (x_{1}, x_{2}, \dots, x_{N}, \tau) = \psi_{0}.\]

Time-dependence

A perturbing potential is applied to the Hamiltonian, \(\hat{H} = \hat{H}_{0} + \hat{\delta V}_{\mathrm{ext}}\). The system is initially in its ground-state and its evolution is calculated by propagating the ground-state wavefunction through real time using the Crank-Nicolson method.

One-electron systems

EXT also works for systems of 1 electron. Unlike for 2 or 3 electron systems, the ground-state is calculated using an eigensolver. When a perturbation is applied to the system, its evolution is calculated by propagating the ground-state wavefunction through real time using the Crank-Nicolson method (like in the 2 or 3 electron systems).