(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Have a H2 molecule in a constant E field. This produces a shift in the on site energies of the 2 H atoms so they have energies ε-δ,ε+δ. δ is proportional to field strength.

1)Calculate the tight binding density matrix when the hopping integral is -1. (γ=-1) δ is positive and the molecule contains 2 electrons.

2)Show the trace is 2, independent of δ.

3)Calculate the density matrix for vanishing electrix field, and compare to the previous matrix for δ→0

4)Polarizability is the derivative of the electrical dipole w.r.t. the electric field, at E=0. Given the electrical dipole operator of the H2 molecule P=1/2[|1><1|-|2><2|], where |j><j| is the projector over the 1s atomic orbital centered on the jth atom, and using the previous density matrix, calculate the polarizability of the H2 molecule. (Assume E=cδ, with c just a constant.)

2. Relevant equations

Some included below.

3. The attempt at a solution

I think I've everything right up to the last part. I don't really want to put up the whole density matrix as it's a bit messy and long, but I'm fairly sure I have it right since I get trace 2 and it agrees with the δ=0 case.

For the last part I'm wondering whether for |1> and |2> I use (1,0) and (0,1), or if I use the vectors I got when I solved Hψ=Eψ, with the H matrix entries being ε+δ, -1, -1, ε-δ. I tried calculating the polarizability both ways, by finding P, then finding trace(Pρ), then differentiating this w.r.t. E then setting E to zero. Using the (1,0),(0,1) vectors I get an answer of zero which I presume isn't right. Using the second set of vectors I get a final answer of -4/c. Of course I may just be using a completely wrong method for the last part, I'm not too sure. Does it sound like I'm on the right track?

Oh, and in part 2, what does trace=2 mean? I'm guessing that this is the number of electrons, but I couldn't find why.

Thanks alot for any help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Desnity matrix method to calculate polarizability of H2 molecule.

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**