2nd Quantization states

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SUMMARY

The discussion focuses on the application of the Pariser-Parr-Pople model to a benzene ring, specifically addressing the ground state wave function |Psi_0> when the Hamiltonian H is not diagonal. Participants conclude that the ground state is a superposition of eigenstates of the number operator ##\widehat{n}_s##, indicating that electrons are not localized solely to individual carbon atoms. The interaction terms in the Hamiltonian lead to a non-diagonal matrix, necessitating diagonalization to obtain the appropriate eigenvectors representing the ground state.

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I'm currently reading a textbook that is introducing second quantization, and it is using the pariser parr pople model as an example. I realized I was confused on what form the states of such a model would take.
As a heads up, I'm really sorry but I can never get latex to work when I hit preview on physicsforums for some reason. Haven't been able to figure it out across multiple browsers.

So for a bit of context, this question is for the pariser parr pople model applied to something such as a benzene ring, with one orbital per carbon atom. Towards the end of this section, the textbook says the following:

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Where n_s gives the number of electrons at site s, and Z_s is the effective nuclear positive charge of atom s.


If the ground state isn't an eigenstate of H, ie h_ss' =/= 0, what |Psi_0> look like? My intuition tells me that seeing as the electron won't be soley localized to each atom, there'd be some kind of superposition where the electron at s = 1 could also be at s = 2 or s = N (and soforth for all the other electrons) but I wasn't quite sure. The textbook has unfortunately been very scant with examples.
 
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Yup, it should most likely be a superposition of the eigenstates of ##\widehat{n}_s##. If the Hamiltonian contains interaction terms such that its matrix is non-diagonal, then after the diagonalization its appropriate eigenvectors (in particular, the ground state with the lowest energy) will be given by linear combinations of the basis vectors which labeled the rows and columns of this original non-diagonal Hamiltonian matrix.
 
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