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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.

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  • $\begingroup$ I am assuming that you are referring to the transition dipole moment. In that case, the expression given in the wiki page (en.wikipedia.org/wiki/Transition_dipole_moment) is accurate. Here is a journal that uses it to calculate the permanent dipole: journals.aps.org/pra/abstract/10.1103/PhysRevA.68.022501 $\endgroup$
    – CoffeeIsLife
    Commented Oct 31, 2017 at 5:16
  • $\begingroup$ I believe that the dipoles are real. There is ab initio software that predicts IR spectra. I am not sure if they use another form to calculate the transition dipole moment. $\endgroup$
    – CoffeeIsLife
    Commented Oct 31, 2017 at 5:20
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    $\begingroup$ I think you misunderstand. I'm referring to the idea that one that determine the total dipole moment of a molecule by adding up small contributions due the polarity of all the individual bonds. $\endgroup$
    – jheindel
    Commented Oct 31, 2017 at 5:58
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    $\begingroup$ If you look that deep, then the bonds themselves are not really real. $\endgroup$
    – Ivan Neretin
    Commented Oct 31, 2017 at 8:18
  • $\begingroup$ @IvanNeretin I don't think this is looking that deep really. I think it makes sense that there should be a way of calculating the total dipole approximately by summing up individual small dipoles which are found based on properties of the atoms themselves. I have never heard of this, but I don't see why it shouldn't be possible even if it doesn't give a good answer. Don't you think there's some sense to the idea? $\endgroup$
    – jheindel
    Commented Oct 31, 2017 at 17:00

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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".

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  • $\begingroup$ I appreciate the answer. I am aware of this. I'm thinking of something different. Namely when one looks at a molecule we tend to determine whether it has a dipole by looking for the absence of certain symmetries and whether the bonds themselves will be polarized towards a certain atom. The electric dipole moment is a property of the whole charge distribution while the qualitative approach looks at polarity in specific regions (namely bonds). I am wondering if this qualitative view (bond polarity as small dipoles) can be quantified and related to the electric dipole which you describe. $\endgroup$
    – jheindel
    Commented Oct 31, 2017 at 18:56
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    $\begingroup$ The total dipole moment is the trace of the charge distribution (density matrix) dotted into the dipole moment integrals along each Cartesian axis. Since the trace is a sum, the total dipole moment can be thought of as a sum of smaller dipole moments (sum of MO dipole moments, for example). The trace is invariant to change of basis, so the partitioning is not unique, so you can choose any basis of "small dipoles" you want. $\endgroup$
    – jjgoings
    Commented Oct 31, 2017 at 19:07
  • $\begingroup$ Ya that's what I was thinking would be the case and mention in the edit I just added. Ok so I guess one could localize the orbitals however one pleases and choose these small dipoles to be the bond dipoles. I suppose this could be useful even if it is arbitrary. Ok well that's what I was asking about. Those small dipoles. $\endgroup$
    – jheindel
    Commented Oct 31, 2017 at 19:13

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