Starting-ABM-algorithm.md

-To start the [ABM algorithm](Integration scheme) one needs to initialize 
+To start the [ABM 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 on the time we obtain:
+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:
 ```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:  
+To use this prescription for the initialization of the ABM algorithm, you need to select in the input file:  
 ```bash
 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 case of `td-wslda` we use a method based Taylor expansion of the evolution:
+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:
 ```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)
 ```