Covalent bonding - Energy gain

[RicardoMP]

Homework Statement

I'm considering a molecule made by three atoms, each a vertex of an equilateral triangle. Each atom has a covalent bond with its neighbours, sharing their only valence electron. I must estimate the energy gain when creating the molecule, using tight binding theory.

Homework Equations

How much is the energy gain when the molecule is created?

The Attempt at a Solution

I used tight binding theory and named each atom's orbital |1>,|2> and |3>. Assuming that $$<i|H|j>=-t (hopping)$$ if $$i\neq j$$ and $$<i|H|j>= \epsilon _0$$ if $$i=j$$. I wrote my 3x3 matrix and calculated the eigenvalues which are-2t, t and t. The solutions tell that the energy gain is -2t-2t+t = -3t and I don't understand why.

 

[TSny]

I wrote my 3x3 matrix and calculated the eigenvalues which are-2t, t and t.

Shouldn't each eigenvalue of $H$ also contain $\epsilon_0$?

The solutions tell that the energy gain is -2t-2t+t = -3t and I don't understand why.

How many electrons can simultaneously occupy the lowest energy state?

 

[RicardoMP]

Shouldn't each eigenvalue of $H$ also contain $\epsilon_0$?
How many electrons can simultaneously occupy the lowest energy state?

The eigenvalues calculated are actually $$\tilde{E}=\epsilon _0 - E$$
I guess only 2 electrons can occupy the same energy state, each with opposite spins.

 

[RicardoMP]

Shouldn't each eigenvalue of $H$ also contain $\epsilon_0$?
How many electrons can simultaneously occupy the lowest energy state?

Oh, since I have 3 electrons to distribute over the the orbitals, the lowest energy state (bonding orbital) is filled with 2 electrons, each one with energy -2t (smallest energy eigenvalue from the hamiltonian) and the other electron stays in a higher energy orbital (anti-bonding orbital), raising its energy by t, therefore -2t-2t+t = -3t.
Is that it?

 

[TSny]

Oh, since I have 3 electrons to distribute over the the orbitals, the lowest energy state (bonding orbital) is filled with 2 electrons, each one with energy -2t (smallest energy eigenvalue from the hamiltonian) and the other electron stays in a higher energy orbital (anti-bonding orbital), raising its energy by t, therefore -2t-2t+t = -3t.
Is that it?

Yes, that sound's right to me.