The system I am studying is one molecule comes in close proximity with another molecule, and we are interested in calculating the energy of the resulting induced dipole-dipole interactions. I want to physically understand what $\int \psi^{1*}_0 \psi^{2*}_0 H \psi^{1}_\mu \psi^{2}_\nu dv$ means in the context below.
To do so, we are using perturbation theory. For two molecules having orthonormalized eigenfunctions $\psi^1_\mu$ for molecule 1 and $\psi^2_\nu$ for molecule 2, with energies $E_\mu$ and $E_\nu$. Both molecules are in the ground state such that $\mu=\nu=0$. The dispersion energies as a second order perturbation are given by:
$$ W_{00} = \sum_{\mu \nu}'\frac{(\int \psi^{1*}_0 \psi^{2*}_0 H \psi^{1}_\mu \psi^{2}_\nu dv )^2}{E_{00} - E_{\mu\nu}} $$
where $E_{\mu\nu}=E_{\mu}+E_{\nu}$ and the $\sum'$ denotes summation excluding $\mu\nu=00$, because we are only interested in induced dipole dipole interactions of the excited states.
The Hamiltonian operator here is given as
$$ H = \sum_{ik} e_i^1 e_k^2 [ \frac{\mathbf{r^1_i}\cdot \mathbf{r^2_k} }{R^3} -\frac{3}{R^5}(\mathbf{C} \cdot \mathbf{r_i^1})(\mathbf{C}\cdot \mathbf{r_k^1}) ] $$
where $e_i^1$ is the charge of the i-th charge center of molecule 1, $\mathbf{r^1_i} = [x_i^1,y_i^1,z_i^1]$ is the location of the i-th charge center of molecule 1 with respect to a coordinate system centered at the center of mass of molecule 1, $\mathbf{r^2_k} = [x_k^2,y_k^2,z_k^2]$ and and $\mathbf{C}$ is the vector representing the location of the center of mass of molecule 2 in the coordinate system centered at molecule 1: $\mathbf{C} = [X,Y,Z]$, and R is the distance between two molecule's center of mass.
My difficulty is that given the form of H above, it should give a scalar when it acts on the excited state vectors i.e. $H | \mu \nu \rangle = E_{dipole-dipole}$. Thus, what does it mean mathematically and physically for us to do $\langle 0_1 0_2 | H | \mu \nu \rangle$ as shown in $W_{00}$ above? Typically, I would interpret $\langle a | b \rangle $ as an inner product between vectors/functions a and b, which still makes sense in a case like $\langle a | O | b \rangle$ when $ O | b \rangle$ gives another vector itself. So, shouldn't the inner product like this only be possible between two vectors? Not a vector and a scalar?
I understand that this formula comes from time independent second order perturbation theory generally, and have gone through the derivation of that via Wikipedia, but it does not help me physically interpret what this term means in this context. I understand that the entire term is negative energy correction to the perturbation energy, but that also does not help. Another answer here said that in general these matrix elements can be interpetated as relating to the amplitude of going from the excited states to the ground state in the presence of the perturbation - which kind of makes sense given the intuition that the inner product measures the overlap between two things.
Another answer explained that $\langle e_i | A | e_j \rangle$ is the $ij$th matrix element of the operator $A$, $A_{ij}$ in the (orthonormal complete) basis set spanned by $\{ | e_i\rangle\}$ Thus, the full matrix of the operator is $A = \sum_{ij}A_{ij}|e_i \rangle \langle e_j|$. However, I don't exactly see how this can be true unless $A|e_i\rangle$ equals another vector, and not a scalar. Also, what would the physical interpretation of that matrix element be in this case if $H | \mu \nu \rangle = E_{dipole-dipole}$ already?
