Starting-ABM-algorithm.md

+To start the [AMB algorithm](Integration scheme) one needs to initialize 
+```math
+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 we obtain:
+```math
+f_{k-1}=f_{k-2}=f_{k-3}=f_{k-4}=0
+```
+
+To use this prescription for initialization of ABM algorithm you need to select in the input file:  
+```bash
+selfstart               0        # use Taylor expansion for 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 case of `td-wslda` we use a method based Taylor expansion of the evolution:
+```math
+\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 Taylor series up to the same order as the precision of the ABM algorithm. 
+
+In order to use the Taylor expansion method for the generation of the first five steps you need to select in the input file:  
+```bash
+selfstart               1        # use Taylor expansion for first 5 steps? 
+```
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