Multiparticle Bonding and Antibonding states

Physics Q&A
1

Hi, I have a question regarding the LCAO lecture, in which was stated (by the course notes):

“Therefore if each atom has a single electron in the outermost shell, these atoms attract because the bonding orbital hosts two electrons with opposite spins. On the other hand, if each atom has 0 or 2 electrons in the outermost shell, the net force from the bonding and antibonding orbitals cancels out, but Coulomb repulsion remains.”

I have quite a bit of trouble following the reasoning towards this statement, mainly because I assumed that the wavefunctions we used were single electron wave functions (the notes talk about kinetic energy K of the electron), which thus does not seem the case.

This is what I think I get:

  • |1\rangle and |2\rangle represent an electron being in an orbital of atom 1 respectively 2.
  • The Hamiltonian expressed in the basis of these states defines an eigenvalue problem (the Schrödinger equation H|\psi\rangle = E|\psi\rangle) which has eigenstates |\psi_\pm\rangle = \frac{1}{\sqrt{2}}(|1\rangle \pm |2\rangle) with corresponding energies E_\pm = E_0 \mp t.
  • As t>0 and t larger for smaller internuclear distance d, E_- increases with smaller d and E_+ decreases, so |\psi_+\rangle is a bonding/attracting state because in this state energy is minimized for smaller d (limited of course by the nuclear repulsion).
  • |\psi_+\rangle is an even/symmetric spatial orbital, which comes with an odd/anti-symmetric spin orbital (following the definition of fermions), for |\psi_-\rangle is this the other way around. This only goes if |\psi_\pm\rangle would be a state describing 2 indistinguishable (non-interacting?) particles.

I assume that with

Therefore if each atom has a single electron in the outermost shell, these atoms attract because the bonding orbital hosts two electrons with opposite spins.

is meant that here the state is |\psi_+\rangle and thus an anti-symmetric spin state makes that the 2 electron have opposite spin. I don’t see however how |\psi_+\rangle, and thus also |1\rangle and |2\rangle I guess, can describe a multiparticle system (the notes talk about 0, 1 and 2 electron per atom, so |\psi\rangle should be able to describe a 0, 2 and 4 electron state?).

Also I don’t get why the latter part of the statement holds true:

On the other hand, if each atom has 0 or 2 electrons in the outermost shell, the net force from the bonding and antibonding orbitals cancels out, but Coulomb repulsion remains.

2

Hey Leon, welcome!

You are right that we are solving a single particle problem, and that many-particle states are only approximated by single-particle ones. If, however, we neglect electron-electron interaction, then every single particle state can be occupied by 2 electrons with opposite spins. So |\psi_+\rangle is a single particle wave function, and it doesn’t describe a many-particle system. The many-particle system—if, once again, we neglect electron-electron interaction—is described by telling which orbitals are occupied by how many electrons. The mathematical formalism for this description is second quantization, and it is beyond the scope of our bachelor program as far as I know.

Hope that explains things.

3

Thx for your response, this really helped clear things up!
If I understand correctly you would rewrite your state’s to the Fock basis if you would want a mathematical multi-particle description, so a two atom case with 2 electrons in |1\rangle and 0 electrons in |2\rangle would correspond with |2,0\rangle in Fock space? (Or I guess |1,1,0,0\rangle as it seems that Fock states contain spin orbitals and not spatial orbitals only) (It is indeed not part of the bachelor program but I did encounter second quantisation and Fock states in my minor, but just never really connected them back to these single particle states, so connecting those concepts here does help in my understanding, eventhough it isn’t something that is exam-material).

4

Yes, exactly.

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5

I have have questions regarding exactly that text.

From reading the replies, is it just to say that the antibonding state |\psi_- ⟩ only serves for when there is an event amount of electrons in the outer orbitals of both atoms, then this state cancels out the bonding state ?

what would happen if there were 2 electrons on the first atom and only 1 electron on the second atom.
Would then there be 2 electrons at E_+ but only 1 in E_- ?

6

That would result in a net attractive force: the bonding orbital contributes 2x and antibonding 1x.

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