# Covalent bonding - Energy gain

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1. May 5, 2017

### RicardoMP

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
How much is the energy gain when the molecule is created?

3. 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.

2. May 5, 2017

### TSny

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

3. May 5, 2017

### RicardoMP

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.

4. May 5, 2017

### RicardoMP

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?

5. May 5, 2017

### TSny

Yes, that sound's right to me.