Exercise 3: The Lennard-Jones potential The Lennard-Jones potential raims.pratt.duke.edu/sites/aims.pratt.duke.edu/files/MD... · 2014. 1. 31. · 2. Finite Range of the Lennard-Jones

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Exercise 3: The Lennard-Jones potential

1. The Lennard-Jones potential (Analytical task)Calculate the equilibrium distance r0 of the pure Lennard-Jones potential

V = 4ε

[(σr

)12

−(σr

)6], (1)

and show that the Lennard-Jones potential can also be written as

V = ε

[(r0

r

)12

− 2(r0

r

)6]

(2)

2. Finite Range of the Lennard-Jones Potential (Analytical task)For this task we focus on a modified Lennard-Jones Potential with the form

V = 4ε

[(σr

)12

−(σr

)6]

+ A( rσ

)2

+B. (3)

For numeric reasons a cutoff value is introduced which limits the range of the LJ-potentials. Atypical cutoff value is rc = 5

2σ. To avoid steps in the 12-6 potential tail corrections such as

A(rσ

)2+B are used. Determine the analytic values for A and B by forcing the potential smoothly

(zero slope) to zero at rc = 52σ.

3. Mode of the Lennard-Jones dimer (Analytical task)The interaction between two atoms shall be described by a pure Lennard-Jones potential (no cut-off). The potential parameters for specific dimers are given in the table below. Calculate theireigenfrequency within the harmonic approximation. How good is your calculated wavenumber ν incomparison to experimental findings (data taken from NIST )

Molecule H2 N2 O2

r0 [A] 0.741 1.098 1.208ε [eV] 4.478 9.759 5.117µ [u] 0.5 7.0 8.0

ν [cm−1] 4401.2 2358.6 1580.2

−→ Please, turn page.

1

Hint: For this task the following conversions are helpful:

• 1 eVuA2 = 9.6485336 · 1027 1

s2,

• The conversion from ω to ν is given by ν = ω2πc

with the conversion factor1

2πc= 5.3089 · 10−12 s cm−1.

4. Distance correlation function for the (melting) Argon crystal (Tool task)In the last exercise, we performed qualitative molecular dynamics simulations to study the meltingof a crystal. Of course, this can be quantified more exactly by looking at the problem.

The order of a structure can be “measured” by the distance pair correlation function of the atomsg(r), which is defined as

g(r) =1

Natoms

⟨Natoms∑

i

Natoms∑j 6=i

δ (r − |rj − ri|)

⟩t

(4)

with 〈. . .〉t being the time average. This object can be evaluated from the MD geometry output files,e.g., using a simple script that reads the positions of each atom and then computes the distancehistogram of the interatomic distances by placing them into “bins” of the width ∆r.

For this exercise, we have prepared such a script, called distancehistogram-minimol.py which createsthe data for a histogram for a single MD structure.

a) Use the script to plot the distance histogram for the resulting MD geometries of the NVEsimulations at 20, 60, 100 K. Take at least 10 snapshot from the NVE simulations (rerunMinimol starting from the last geometry) and check how the histogram varies. How fastwould g(r) converge?

b) What do you see in g(r)? How would you quantify the phase transition from these curves?

c) If you are not familiar with scripting languages, please take a look at the script itself. Thescript shows (as an example) how to open and close an input file, how to read input data, andhow to manipulate them later. [There is no need to document this task - the reason for askingthis question is simply that this script may serve as a good example. Understanding what itdoes may be helpful for many later computational tasks inside and outside this class.]

2