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I want to know how to convert operator matrix when using Dirac Bra-Ket notation when it must be converted into a new dimension.

I am currently working on transition dipole moment operator matrixwhich I am going to use the following one:D

=Der

Where e is charge of electron,is position operator. Bold letters mean vector or matrix while non bold letters are scalars. This position operator was conveniently taught in my school from basis sets under three-dimensional space (like polar coordinates and Cartesian coordinate). I am assuming that there would be three elements for each basis vector. However, I wish to convert this operator matrix into 16-dimensional molecular coordinate.r

The reason I need to do this is the system I am working on consists of 16 identical molecules that shows excitonic interaction and needed to know the splitting and their strength of the degenerate 16 energy state because of this interaction. The Hamiltonian for this system was created under the basis set of the subsystems of 16 molecules without any interaction and did the characteristic polynomial problem to get the eigenvalue and eigenfunction (which corresponds to the energy WITH the excitonic interactionE_{n}and the wavefunction vectorV_{n}for corresponding new states).

Now that I know the energy of each of the new states where excitonic interactions are considered, I wished to know the oscillator strengthf_{0→n}for each of these new states so I can predict the absorption spectra of the system. In order to do this, I must solve the following problem:

f_{0→n}= (8πm/3he^{2})(E_{n}-E_{0})<V_{0}||DV_{n}>^{2}

However, it is obvious that basis set used forin this particular equation is the same as the basis set used forDand therefore I have no idea how they are supposed to look like and how the matrix element ofVncan be solved so that I can calculate the oscillator strength (I can only imagineDoperator in three-dimension space).D

I am extremely confused about quantum physics (chemistry) since I am in an experimental Lab where most of the people have no idea about actual calculation in quantum chemistry. I do not even know if I am understanding even a bit of quantum chemistry right. So I absolutely have no idea where to start except speculation based on my knowledge of high-school mathematics.

If I am making a terrible mistake up there, please bash me and correct me.

I also apologize for my English because I am barely a native speaker (I'm Japanese).

Thank you very much.

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# Converting operator matrix (Quantum Chemistry question)

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