Some time ago we found that the first excited state in our O-Cu-O cluster with no phonons had a simple representation and didn’t depend on the parameter U (representing the on-site Coulomb repulsion between two electrons on the same site). So, it’s not evident that the base state will always depend on U, perhaps at high energies there is a level crossing and the U-independet first excited state becomes the base state. In this post I explore such a posibility.
My tool of choice for this is, again, Sage (if you don’t know it, give it a try! it’s awesome). First, I define some variables, write the hamiltonian and calculate the eigenvalues
H = matrix([
[U + 2*epsilon,t,0,t,0,0,0,0,0],
[0,t,0,t,U - 2*epsilon,t,0,t,0],
eigval = H.eigenvalues()
Using the parameters we’ve been working on, we can have a look at the numerical values of the eigenvalues
for e in eigval:
print real(n(e.subs(epsilon=0.5, t=0.5, U=7)))
As we can see, the base state is eigval and the first excited state is eigval. Furthermore, the energy of the first excited state, in cm-1 is
real((eigval.subs(epsilon = 0.5, t = 0.5, U = 7)-eigval.subs(epsilon = 0.5, t = 0.5, U = 7))*8056)
Which is exactly what I expected. To make a plot of the base state and the first excited state I use this simple loop
p_list_2 =  # The base state
p_list_0 =  # The first excited state
for u in [0.0, 0.1.., 10]:
x2 = real(n(eigval.subs(epsilon = 0.5, t = 0.5, U = u)))
x0 = real(n(eigval.subs(epsilon = 0.5, t = 0.5, U = u)))
And now we can make a plot
plot2 = list_plot(p_list_2,gridlines=True,legend_label='base state', rgbcolor='black')
plot0 = list_plot(p_list_0,gridlines=True,legend_label='first excited state', rgbcolor='red')
p = plot2 + plot0
p.axes_labels(['U (eV)','eigenvalue (eV)'])
So, it seems that there is no level-crossing between these two states but they seem to overlap when . In fact, altough the expression for the base state is complicated we can calculate the limit when . Altough it takes a long time, Sage can perform this calculation simbolically
eigval.limit(U = oo).show()
So, indeed, they converge.
Just to be sure there is not any other state crossing eigval I also made a plot of all 9 energy levels:
I think it’s interesting that there are degeneracies when and and also that there are three states which seem to be completely independent of . As I mentioned earlier, some eigenvalues have a simple analytic representation, in fact, 5 of them do
The other 4 states have a much more complicated expression.
In conclusion, we don’t have to worry about an energy crossing and the base state will always depend on , altough much less at large values.