# 407: Numerical Solutions for the Lennard-Jones Potential

- Page ID
- 137731

Merrill (Am. J. Phys. **1972**, *40*, 138) showed that a Lennard-Jones 6-12 potential with these parameters had three bound states. This is verified by numerical integration of Schrödinger's equation. The integration algorithm is taken from J. C. Hansen, ** J. Chem. Educ. Software, 8C2**, 1996.

Set parameters:

- n = 200
- \(x_{min} = 0.75\)
- \(x_{max} = 3.5\)
- \(\Delta = \frac{xmax - xmin}{n-1}\)
- \( \mu\) = 1
- \( \sigma\) = 1
- \( \varepsilon\) = 100

Numerical integration algorithm:

i = 1 .. n j = 1 .. n x_{i} = xmin + (i - 1) \( \Delta\)

\[ V_{i,~j} = if \bigg[ i =j,~4 \varepsilon \bigg[ \left( \frac{ \sigma}{x_i} \right)^12 - \left( \frac{ \sigma}{x_i} \right) ^6 \bigg] ,0~ \bigg] \]

\[ T_{i,~j} = if \bigg[ i=j, \frac{ \pi ^{2}}{6 \mu \Delta ^{2}}, \frac{ (-1)^{i-j}}{ (i-j)^{2} \mu \Delta^{2}} \bigg] \]

Hamiltonian matrix: H = T + V

Find eigenvalues: E = sort(eigenvals(H))

Display three eigenvalues: m = 1 .. 4

E_{m} =

\( \begin{array}{|r|}

\hline \\

-66.269 \\

\hline \\

-22.981 \\

\hline \\

-4.132 \\

\hline \\

1.096 \\

\hline

\end{array} \)

Calculate eigenvectors:

k = 1 .. 3

\[ \psi (k) = eigenvec (H, E_k)\]

Display results: