deletions | additions       diff --git a/Forces and geometry optimization.tex b/Forces and geometry optimization.tex  index ab10fb4..097a21c 100644  --- a/Forces and geometry optimization.tex  +++ b/Forces and geometry optimization.tex 
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\right]\ .  \end{equation}  %  This previous transformation has two advantages for numerical implementations:  First, it is not necessary to implement the derivative of the ionic  potential, that can be quite complex and difficult to code, especially  when relativistic corrections are included. Thesecond  advantageis  that the representation  of the derivative of the ionic potential  requires a better discretization than the potential itself, so the this  alternative expression of the force formula is that it  is more precise when discretized. discretized on a grid, as the orbitals are smoother than the ionic potential.  We illustrate thissecond  point in Fig.~\ref{fig:n2forces} where the forces obtained with the two methods  are compared. While taking the derivative of the atomic potential  gives forces with a considerable oscillation due to the discretization  (this point is discussed in detail in the following section), discretization,  using the derivative of the orbitals gives a force that is smooth and  equivalent to the one obtained by finite-differences with half the  spacing.  Additionally, eq.~\ref{eq:forcesgrad} is easier to implement than eq.~\ref{eq:forcespot} as it does not require the implementation of the derivatives of the potential, that can be quite complex and difficult to code, especially when relativistic corrections are included.