Covalent bonding - Energy gain

  • #1
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2

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 [tex] <i|H|j>=-t (hopping) [/tex] if [tex] i\neq j [/tex] and [tex] <i|H|j>= \epsilon _0 [/tex] if [tex] i=j [/tex]. 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.
 

Answers and Replies

  • #2
TSny
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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?
 
  • #3
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2
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 [tex] \tilde{E}=\epsilon _0 - E [/tex]
I guess only 2 electrons can occupy the same energy state, each with opposite spins.
 
  • #4
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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?
 
  • #5
TSny
Homework Helper
Gold Member
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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.
 

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