====== Profiles of potential energy and free energy ======

We are going to start with the simplified example of isolated $\text{Na}^+$ and $\text{Cl}^-$ in the //gas phase//, where we can directly compare the results of our computer simulation against the analytical formulae used to describe the interaction potential.

We have provided an input file ''NaCl_pot.in'' and a script ''potential_energy.sh'' that uses this input file to calculate the potential energy as a function of Na-Cl distance.
<note>**TASK 1**

  - Look into ''NaCl_pot.in'' and write down the formula used for the potential energy of the interaction between $\text{Na}^+$ and $\text{Cl}^-$ in Hartree atomic units. (2P)
  - Use ''./potential_energy.sh'' to calculate the potential energy as a function of Na-Cl distance. Create a plot of the resulting potential energy profile in ''pot_profile'' and the mathematical formula.
  - What do you observe, when the distance approaches 1/2 of the simulation box? How might the finite size of the simulation box have impacted the MD simulation in the [[nacl_md|previous exercise]]? (2P)
</note>

For the next task, we remain with our simple system, but now perform molecular dynamics at $T=1\,\text{K}$.

We have prepared a script ''free_energy.sh'', which runs MD simulations with //constrained// Na-Cl distance at $1\,\text{K}$.
It then integrates the average value of the Shake Lagrange multiplier to calculate the (low-temperature) free energy profile.
<notes>**TASK 2**

  - What is a Lagrange multiplier? How can we obtain the free energy profile as a function of the Na-Cl distance using the associated Lagrange multiplier? (2P)
  - Run the simulation. What kind of motion does the NaCl dimer perform?
  - Compare the low-temperature free energy profile in ''fe_profile'' with the potential energy profile. Do the two profiles agree? //Note:// The profiles are shifted with respect to each other. What would be a reasonable reference point for both profiles? (2P)
  - What effects would you expect at higher temperature? //Hint:// If you like, you can adapt the temperature in the input file and give it a go.
</note>

Now, we are ready to move to a more realistic system -- NaCl in water.
We have performed constrained MD of NaCl in water and saved the trajectory of the corresponding Lagrange multipliers (ask your teaching assistant).

The script ''./integrate.sh'' computes the average values of the Shake Lagrange multipliers and uses them to perform the free energy integration.
<note>**TASK 3**
  - Perform the free energy integration and plot the free energy profile.
  - In the [[nacl_md|previous exercise]], you determined the average time required for dissociation of Na-Cl. Is the free energy barrier consistent with the time scale determined before? //Hint:// Use the Arrhenius equation. You can obtain an estimate for the attempt frequency from the high-frequency oscillations in the Na-Cl distance in the previous exercise. (2P)
</note>

Another way to gain access to the free energy is through the radial distribution function (rdf) of the //unconstrained// system. 
The rdf $g(r)$ is related to the free energy $F(r)$ through the following set of equations
$$\begin{eqnarray} 
g(r)4\pi r^2 &\propto& \int \delta(r-r') \exp(-\beta H(r'))\,dr  \\
P(r) &\propto& \int \delta(r-r') \exp(-\beta H(r'))\,dr  \\
F(r) &=& -k_BT \ln\,P(r)
\end{eqnarray}$$


We have performed a trajectory spanning 50 ns of unconstrained molecular dynamics of NaCl in water (ask your teaching assistant). The individual frames are spaced by 1 ps in order to reduce correlation between subsequent frames.

<note>**TASK 4**
  - In the [[h2o_md|previous exercise]], we computed the O-O radial distribution function for water with acceptable statistics using just 20 ps of simulated time. Give two reasons, why collecting enough statistics for the Na-Cl radial distribution function requires much longer simulation times (with our setup).
  - Compute the radial distribution function for the provided trajectory and plot it as a function of Na-Cl distance.
  - Use the equations above to compute the free energy profile. Does it agree with the one constructed from the Shake Lagrange multipliers?
</note>