Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Verify how sensitive the result is with respect to the dimension of the
basis set. Note that the “binding energy” printed on output is calculated
assuming that the isolated H atom has an energy of -1 Ry, but you should
verify what the actual energy of the isolated H atom is for your basis set.
• Make a plot of the ground state molecular orbital at the equilibrium
distance along the axis of the molecule. For a better view of the binding,
you may also try to make a two-dimensional contour plot on a plane
containing the axis of the molecule. You need to write a matrix on a
uniform two-dimensional N x M grid in the following format:
x_0 y_0 \psi(x_0,y_0)
x_1 y_0 \psi(x_1,y_0)
...
x_N y_0 \psi(x_N,y_0)
(blank line)
x_0 y_1 \psi(x_0,y_1)
x_1 y_1 \psi(x_1,y_1)
...
x_N y_1 \psi(x_N,y_1)
(blank line)
...
x_0 y_M \psi(x_0,y_M)
x_1 y_M \psi(x_1,y_M)
...
x_N y_M \psi(x_N,y_M)
and gnuplot commands set contour; unset surface; set view 0, 90
followed by splot ”file name” u 1:2:3 w l
• Plot the ground state molecular orbital, together with a ligand combination
of 1s states centered on the two H nuclei (obtained from codes
for hydrogen). You should find that slightly contracted Slater orbitals,
corresponding to Z = 1.24, yield a better fit than the 1s of H. Try the
same for the first excited state of H 2 and the antiligand combination of
1s states.
• Study the limit of superposed atoms (R → 0) and compare with the results
of codes hydrogen gauss and helium hf gauss, with the equivalent basis
set. The limit of isolated atoms (R → ∞) will instead yield strange
results. Can you explain why? If not: what do you expect to be wrong
in the Slater determinant in this limit?
• Can you estimate the vibrational frequency of H 2 ?
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