Exercise: Geometry Optimisation of a Lennard-Jones Potential

Contents

Exercise: Geometry Optimisation of a Lennard-Jones Potential#

The potential energy surface for a pair of &lqduo;Lennard–Jones” is given by

\[U_\mathrm{LJ} = \frac{A}{r^{12}} - \frac{B}{r^6}\]

1.#

Plot this function for \(A\) = 1 × 105 eV Å12 and \(B\) = 40 eV Å6 for the range \(3.6 \leq r \leq 8.0\).

2.#

The first derivative of this potential energy function is given by

\[U^\prime_\mathrm{LJ} = -12 \frac{A}{r^{13}} + 6\frac{B}{r^7}\]

Write code to find the lowest-energy interatomic separation for a pair of Lennard–Jones atoms (with the potential parameters given above) using gradient descent. Start with \(r = 5.0\) and \(\alpha=100\). How does your optimisation code perform with different values of \(\alpha\) and different starting values of \(r\) (try \(r\) = 3.2 Å, \(r\) = 4.4 Å, and \(r\) = 6.0 Å)?

3.#

Write code to perform geometry optimisation of your Lennard-Jones potential using the Newton-Raphson method.

The second derivative of \(U_\mathrm{LJ}\) is

\[U^{\prime\prime} = 156\frac{A}{r^{14}} - 42\frac{B}{r^8}\]

Investigate how the Newton-Raphson method performs with starting points of \(r\) = 3.2 Å, \(r\) = 4.4 Å, and \(r\) = 6.0 Å.

Contents