H.7 Exercises [atomistic modeling top]

1. Write a MATLAB routine to solve the Ornstein-Zernike equation with hypernetted chain closure to find the radial distribution function for a homogeneous Lennard-Jones fluid.

   Proceed as follows:

   1. Use an iterative technique that, in turn, uses the hypernetted chain closure (Equation H.20) to solve for f_tc and the Ornstein-Zernike equation (Equation H.19) to solve for f_dc.
   2. Start by setting f_tc(r) = f_dc(r) = 0 on a domain that ranges from r = 0 to r = 512σ.
   3. In each step, define a new f_tc by using the hypernetted chain relation:

      (H.32)

      Note that e₁(r) in this case is the Lennard-Jones potential.

   4. In each step, define a new f_dc by Fourier transforming f_tc and f_dc, applying the Fourier-transformed Ornstein-Zernike equation to get a new , and inverse Fourier-transforming  to get a new f_dc. We do this because the Fourier-transformed Ornstein-Zernike equation is much easier to deal with (the spatial integral becomes a product when Fourier-transformed):

      (H.33)

      Here k is the frequency variable and  is the Fourier transform of f. This can be rearranged to give

      (H.34)

      So we Fourier-transform f_tc and f_dc to get  and , apply Equation H.34, and then transform  back.

   5. The two previous steps are repeated until the solutions for f_tc and f_dc are no longer changing. Some attention to numerical stability is needed, especially if ρ^* is high and T ^* is low.

   Plot your results for nine cases as follows: three values of ρ^* (0.1, 0.4, 0.8) and three values of T ^* (0.5, 1.0, 1.5).

2. Given the results from Exercise H.1, calculate the potential of mean force that atoms see in this case. Plot the e_mf for the nine cases from Exercise H.1.

3. Calculate the magnitude of the dipole moment (in Debye) for a water molecule given the geometry of the SPC model.

4. Consider hard spheres of radius a.
   1. What is the closest approach of the centers of two spheres?
   2. What is the doublet potential for the interaction between the two spheres?
   3. Calculate the excluded volume. How does the excluded volume compare with the volume of one of the spheres?
5. Show that the nonintegrability of the Coulomb pair potential guarantees that physical systems must be overall electroneutral.
6. Using the equation for monopole interaction potentials, explain why sodium chloride might be expected to be a crystalline solid when dry but dissolves when exposed to water.

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