[sdfsfgfdgdfdf]

I think there is a mistake in the materials the professor gave us and I just feel bad asking him, so i figured maybe someone here could help me. He writes the following expression for the wave function of a charge-transfer complex:

ψ_AB = ψ₀(AB) + ψ₁(A^-B⁺)

and then he writes the following expression for the energy:

E = E₀ - (β₁₀ - E₀×S₁₀)/(E₁-E₀)

and then he goes on to say that the first term represents the electrostatic and the second term, the covalent contribution to the energy.

That would mean that the electrostatic contribution comes from ψ_o(AB), which I think should be the wave function for the covalent contribution. I think the electrostatic contribution should come from ψ₁(A^-B⁺), and the first term should be denoted as E₁. Now, I know that this seems irrelevant, but it isn't because then he goes on to explain the two contributions to the energy in detail, and it's confusing and it only makes sense if the expression for the energy is written as:

E = E₁ - (β₁₀ - E₀×S₁₀)/(E₁-E₀)

Am I right?

[vex]

It's hard to tell just based off the two equations you've posted. Could you fill in a bit more detail? Is there a specific system you're looking at? Are A and B separate complexes?

[sdfsfgfdgdfdf]

The complex is made up of A and B, where A is a Lewis acid and B is a Lewis base. Its a charge-transfer complex, and its actual state is somewhere between AB (in which covalent bonding is dominant) and the ionized state (A^-B⁺). Other than that, it's not any specific complex, he just explains the theory in general terms.

[vex]

Hmm. Yeah, this is puzzling.

I think you might be misunderstanding the problem. The Marcus theory of charge transfer postulates a system in a state somewhere between two diffusing complexes (AB, before electron transfer) and two datively bound complexes (A⁺B^-, after electron transfer). Therefore, the wavefunction of the system would be a linear combination of the two "possibilities" as represented in the equation for ψ_AB that you posted.

Where your instructor draws the distinction between the two energetic terms may be in the expression for E. The first term E₀ represents the energy of the unperturbed Hamiltonian for your system. If that has been discussed in your class, it might be helpful to post that, but I'm willing to bet that the expression for potential energy in your Hamiltonian only accounts for the electrostatic potential energy, which, compared to covalent bonding, is very easy to describe. Therefore, E₀ accounts for all of the energy of your system when covalency is "turned off."

However, you and I both know that's not how it really works. So, we have to add to, or perturb, the Hamiltonian by adding correction factors for the covalency of the bond. That's what the second term represents. S10 is probably an overlap integral that is a measure of how well the orbitals from one species interact with the other species.

It's hard for me to be certain without having actually been in the lecture, but you've just gotten caught up in the subscripts (which is very easy to do!) and that you don't necessarily want to connect the 0s and 1s written in the wavefunction equation to the 0s and 1s in the energy equation.

I'd certainly take it up with your prof, though, and I'm kind of interested to know how it goes!

[sdfsfgfdgdfdf]

I think you might be misunderstanding the problem. The Marcus theory of charge transfer postulates a system in a state somewhere between two diffusing complexes (AB, before electron transfer) and two datively bound complexes (A+B-, after electron transfer). Therefore, the wavefunction of the system would be a linear combination of the two "possibilities" as represented in the equation for ψAB that you posted. - I understood this part

So, if in this case, E₀ is the non perturbed state, or electrostatic state, that means that it does actually refer to the term ψ₁(A⁺B^-), but it is denoted as E₀ and not E₁  because it is the energy of a non perturbed state? Or there is no connection between the terms in the expression for the wave function and those in the expression for the energy?

I understand that its silly to get confused by subscripts, but they are actually very important if you're going to understand what is what and what you're doing with the problem, and some people just use the arbitrarily, like everyone is going to know intuitively what they're referring to

We did use perturbed orbitals, but further on in the lectures, in the context of perturbed wave functions of the molecular orbitals when the conditions change, when a reagent is approaching for example, and yes we did use ψ₀ to denote the non perturbed state in that case. I didn't consider that that could be the case here also, because the expression for the energy wasn't derived, it was just given, just like that... But he does explain it, saying that E₀ represents the electrostatic contribution, and that the second term in the expression for the energy represents the covalent contribution to the energy, which is dependent on the properties of the atomic orbitals, their orientation, overlap etc.