Converting operator matrix (Quantum Chemistry question)

[HAYAO]

Dear all,

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 matrix D which I am going to use the following one:
D = er
Where e is charge of electron, r 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.

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 interaction E_n and the wavefunction vector V_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 strength f_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/3he²)(E_n-E₀)<V₀|D|V_n>²
However, it is obvious that basis set used for D in this particular equation is the same as the basis set used for Vn and therefore I have no idea how they are supposed to look like and how the matrix element of D can be solved so that I can calculate the oscillator strength (I can only imagine D operator in three-dimension space).

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.

 

[DrClaude]

D has a simple representation in the position basis. Conceptually, you would need to express the basis functions ##|V_n\rangle## in terms spatial wave functions, ##V_n(\mathbf{r}) = \langle \mathbf{r} | V_n \rangle##. Once this is done, you get the matrix elements of D as
$$
\langle V_0 | \hat{D} | V_n \rangle = \int V_0(\mathbf{r}) e\mathbf{r} V_n(\mathbf{r}) d^3\mathbf{r}
$$
I don't know how easy it is going to be to get the position representation of the basis functions.

 

[HAYAO]

Thanks.

I see. That seems to be rather difficult. How do you derive ##\langle \mathbf{r} |## for doing the following?

##\langle \mathbf{r} | V_n \rangle##