Minor typo fixes

Author: EfremBraun

paper/basic_training.tex

@@ -1051,7 +1051,7 @@ \subsubsection{ Ewald Summation}
 However, here, because we use point charge electrostatics, $\rho$ is a set of delta functions.
 
 The Ewald method is based on (temporarily) replacing the point charge distributions by smooth charge distributions in order to apply existing numerical techniques to solve this partial differential equation (PDE).
-The most common smooth function used in Ewald method is the Gaussian distribution, although other distributions have been used as well.
+The most common smooth function used in the Ewald method is the Gaussian distribution, although other distributions have been used as well.
 Thus the overall charge distribution is divided into a short-range or ``direct space'' component ($\rho^{sr}$) involving the original point charges screened by the Gaussian-distributed charge of the same magnitude (Figure~\ref{fig:screening}) but opposite sign, and a long-range component involving Gaussian-distributed charges of the original sign ($\rho^{lr}$).
 
 \begin{figure}[h]
@@ -1076,7 +1076,7 @@ \subsubsection{ Ewald Summation}
 
 The potential due to long-range charge interactions does not decay rapidly, and thus requires consideration of all periodic copies.
 This would pose severe problems if calculated via direct summation, but the smoothness of the charge $\rho^{lr}$ (and hence potential ($\phi^{lr}$) allows the use of fast PDE solvers. 
-Specifically, while the sum is long-ranged in real space, taking the Fourier transform converts it into a sum in reciprocal space which is short-ranged in reciprocal space, damped by a factor $\exp{-k^2 \sigma^2/2}$ where $k$ is the reciprocal space vector and $\sigma$ is the width of the Gaussian. 
+Specifically, while the sum is long-ranged in real space, taking the Fourier transform converts it into a sum in reciprocal space which is short-ranged in reciprocal space, damped by a factor $\exp\left(-k^2 \sigma^2/2\right)$ where $k$ is the reciprocal space vector and $\sigma$ is the width of the Gaussian. 
 
 The final term in Ewald summation is a so-called self term which gets subtracted out of the overall sum; it is calculated only once at the beginning of the simulation as it depends only on the charge magnitudes and not their positions.
 It also does not contribute to the force.
@@ -1148,7 +1148,7 @@ \subsubsection{Grid based Ewald summation}
 \begin{checklist}{Determine handling of cutoffs}
 \begin{itemize}
 \item As a general rule, electrostatics are long-range enough that either the cutoff needs to be larger than the system size (for finite systems) or
-periodicity is needed along with full treatment of long-range electrostatics (Section~\ref{sec:classical_electrostatics} % changed section ref since the folloing does not currently exist (Section~\ref{lr_electrostatics})
+periodicity is needed along with full treatment of long-range electrostatics (Section~\ref{sec:classical_electrostatics}) % changed section ref since the folloing does not currently exist (Section~\ref{lr_electrostatics})
 \item Nonpolar interactions can often be safely treated with cutoffs of 1-1.5 nm as long as the system size is at least twice that, but long-range dispersion corrections may be needed (Section~\ref{sec:force_fields})
 \end{itemize}
 \end{checklist}