Starting ABM algorithm

To start the [ABM algorithm](Integration scheme), one needs to initialize

$$f(y,t) \equiv \dfrac{1}{i}(\hat{H}(\{\varphi_n\},t)-\langle H\rangle_n)\varphi_n(\bm{r},t)$$

for four time steps back: $f_{k-1}$, $f_{k-2}$, $f_{k-3}$, $f_{k-4}$.

Starting from a stationary state

In the case of a stationary state, $\varphi_n$ are eigenstates of the Hamiltonian, and the instantaneous quasi-particle energies are equal to quasiparticle energies $E_n=\textrm{const}$. Then, as long as the Hamiltonian does not depend on the time we obtain:

$$f_{k-1}=f_{k-2}=f_{k-3}=f_{k-4}=0$$

To use this prescription for the initialization of the ABM algorithm, you need to select in the input file:

selfstart               0        # use Taylor expansion for the first 5 steps? 

Starting from a non-stationary state

In this case, we need to generate the first five steps using single-step method. In the case of td-wslda, we use a method based on Taylor expansion of the evolution operator:

$$\varphi_n(\bm{r},t+\Delta t) = e^{-i(\hat{H}(\{\varphi_n\},t)-\langle H\rangle_n)\Delta t}\varphi_n(\bm{r},t)\approx \sum_{k=0}^{4}\frac{(-i\Delta t)^k}{k!}(\hat{H}(\{\varphi_n\},t)-\langle H\rangle_n)^k\varphi_n(\bm{r},t)$$

where we expanded the evolution operator into the Taylor series up to the same order as the precision of the ABM algorithm.

To use the Taylor expansion method for the generation of the first five steps, you need to select in the input file:

selfstart               1        # use Taylor expansion for first 5 steps?