Starting-ABM-algorithm.md

@@ -5,7 +5,7 @@ f(y,t) \equiv \dfrac{1}{i}(\hat{H}(\{\varphi_n\},t)-\langle H\rangle_n)\varphi_n
 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
 ```
@@ -17,11 +17,11 @@ 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:
+In this case, we need to generate the first five steps using the single-step method. In the case of `td-wslda`, we use a method based on the 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)
 ```
-where we expanded the evolution operator into the Taylor series up to the same order as the precision of the ABM algorithm. 
+where we expanded the evolution operator into a Taylor series up to the same order as the ABM algorithm's precision. 
 
 To use the Taylor expansion method for the generation of the first five steps, you need to select in the input file:  
 ```bash