How real are bond dipoles? In general chemistry, it is common to teach students to determine a molecular dipole by having them first determine "bond dipoles" which are just based on electronegativity. Then, by adding up these vectors, one determines the direction of the overall dipole.

In quantum mechanics, however, the molecular dipole is found by calculating the expectaion value $\langle\psi|\hat{\mu}|\psi\rangle$. This can be decomposed into $x$, $y$, and $z$ components if one so desires. Regardless, the actually quite useful idea of a bond dipole is nowhere to be found.

I assume that there is not a rigorous way of defining a bond dipole so that a sum of these bond dipoles equals the expectation value above. This is because there is not a well-defined way to partition electron density which is exact. I suppose one could do this using the theory of atoms in molecules.

Nonetheless, other ideas like local mode vibrations are approximate in the sense that a local mode is not an eigenfunction of the polyatomic vibrational Hamiltonian, yet local modes are still very useful and even more accurate in some contexts.

The same is sort of true of bond dipoles in that if one imagines a very long molecule which is polar at one end, a nearby molecule which moves along this large molecule will experience a changing electric field because the dipole of the molecule is only a true point-dipole when the molecule is very far away. So, the field experienced by nearby molecules is more akin to the sum of bond dipoles in the region nearby.

So, is there a rigorous way to determine bond dipoles from first principles? Is it as simple as projecting the total dipole onto a bond axis? What is the form of this projector? Also, in cases where the partitioning of atoms is well-defined, such as the theory of atoms in molecules, are bond dipoles well-defined and do they relate to the total dipole as we expect?

Answers to any of these questions would be welcome.



To be more clear, I am talking about the bond dipoles described on this wikipedia page. They do not provide any formal way of actually calculating the bond dipoles. They mention one can get these dipoles after calculating the total dipole, but I am not sure of the uniqueness of this under unitary transformations.
If by "bond dipole moment" you mean "electric dipole moment", then the electric dipole moment is the first term in a multipole expansion of the molecular charge distribution. In that respect, an electric dipole moment is an approximation. 

For a static charge distribution, we are interested in the resulting electric fields or potentials. For a general charge distribution, this does not have a closed form expression, but can be increasingly approximated by higher order terms in a multipole expansion. The dipole approximation is good enough for many cases. Depending on the experimental setup, the only surviving term might be the dipole term. So in that sense it maps to an experimental value and can be considered "real".