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In many Chemistry books, the dipole moment of molecules is calculated through algebraic formulas. However, another definition of the dipole moment is

$$\vec{\mu} = \sum_i q_i \, \vec{r}_i$$

So, how can we calculate the dipole moment of water, for example using this formula? Is there a difference in interpretation of dipole moment in Physics and Chemistry?

Here is the Definition.

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    $\begingroup$ There is no difference in definition, that's for sure. $\endgroup$ – Ivan Neretin Sep 5 at 16:58
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The document the OP references has most of the information: The overall measured dipole moment, the way bond dipoles add up to the molecule dipole, and how to calculate the bond dipole. Also, it states that a proton and an electron at a distance of 100 pm have a dipole moment of 4.80 D. From that information, we should be able to figure everything out.

Bond dipole vs. Molecular dipole

The bond dipoles add up as vectors. In the case of water, we can put a water molecule into an x-y-coordinate system so that oxygen is at the origin and the x-axis divides the H-O-H angle in half. In that orientation, the bond dipoles will add up to a vector pointing in the x-direction. To get the magnitude, we have to take the x-component of the vector. The appropriate formula is given in the textbook:

$$ \text{Total dipole moment} = \text{Bond dipole moment} \cdot 2 \cdot \cos \frac{104^\circ}{2}.$$ The term 2 comes from having two O-H bonds, and the cosine term gives the component of the vector in the direction of the x-axis (i.e. in the direction of the molecular dipole vector. The textbook has a value of 1.84 D for the molecule and 1.5 D for each bond. It is clear that they do not add up like scalars.

Relating the bond dipole moment to the molecular structure

The bond lengths in water are about 95 pm. The textbook gives the bond dipole as 1.5 debye. We can back-calculate what the charge is, given that separating a proton and an electron (with elementary charge $q_e$) by 100 pm accounts for 4.8 debye.

$$\pu{100 pm} \cdot q_\ce{e-} = \pu{4.8 D}$$

$$\pu{95 pm} \cdot q_\ce{OH} = \pu{1.5 D}$$

$$\frac{q_\ce{OH}}{q_\ce{e-}} = \frac{\pu{1.5 D}}{\pu{4.8 D}} \frac{\pu{100 pm}}{\pu{95 pm}} = 0.33$$

So the charge separation along each O-H bond is about a third of an elementary charge.

Could we calculate the charge separation?

Here, we assumed point charges at the location of the atoms. In reality, charge is distributed throughout the electron cloud. You can do quantum mechanical calculations to get the electron density, then express the charge separation as point charges. The easily accessible tool molcalc.org does this calculation (quick and inaccurately), and assigns the hydrogen atoms a partial charge of +0.21 each (so -0.42 for the oxygen atom). This is quite a bit lower than my calculation. On the other hand, the dipole moment is given as 1.88 D, close to the values given in the textbook. I am not quite sure were the discrepancy lies.

So how can we calculate dipole moment of water for example using this formula?

Yes, if you know the structure of the molecule and the partial charges (as point charges in the location of the atomic positions).

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  • $\begingroup$ Thanks for the answer. $\endgroup$ – ado sar Sep 7 at 9:17
  • $\begingroup$ @Karsten Theis, Could you mention which textbook is this? Thanks. $\endgroup$ – M. Farooq Sep 7 at 12:15
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    $\begingroup$ @M.Farooq Now that you mention it, it is not strictly a textbook. It is Libretext supplementary material written by Mike Blaber. Here is the table of contents of that section: chem.libretexts.org/Bookshelves/… $\endgroup$ – Karsten Theis Sep 7 at 16:32
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Is there a difference in interpretation of dipole moment in Physics and Chemistry?

Addressing the above part: The definition of dipole moment is the same in physics and chemistry. However, the chemist's dipole moment has opposite directions as compared to the physicist. Which one is right? Of course, the physicists draw it correctly, which is consistent with electrostatics. There was a nice single paged article in the Journal of Chemical Education on this topic. A must read.

Misconceptions in Sign Conventions: Flipping the Electric Dipole Moment James W. Hovick J. C. Poler

https://pubs.acs.org/doi/10.1021/ed082p889

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