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exercises:2015_uzh_molsim:nacl_free_energy

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exercises:2015_uzh_molsim:nacl_free_energy [2015/05/26 16:19]
yakutovich
exercises:2015_uzh_molsim:nacl_free_energy [2020/08/21 10:15]
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-====== 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 formula 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> 
exercises/2015_uzh_molsim/nacl_free_energy.txt ยท Last modified: 2020/08/21 10:15 (external edit)