# What units are used in Gaussian 16 for dipole derivatives output?

I ran a frequency analysis employing Gaussian 16 on MP2/6-31+G(d,p) level of theory with the keyword iop(7/33=1) in order to have access to dipole derivatives with respect to the individual normal mode coordinates. An example line from the output reads

Dipole derivative wrt mode XX:  5.93205D-01 -1.47564D+00  1.93547D-02


Does anybody know in which units the dipole derivatives are actually written and can, ideally, point me at a corresponding documentation?

Up to know I read it should be something like $$\sqrt{\pu{km}/\pu{mol}}$$, but I have no clear evidence that this is the case. I would expect something involving Debye and atomic mass unit, like $$\pu{D}/\sqrt{\pu{u}}\,a_0$$.

• Perhaps atomic units?
– Yoda
May 20 at 8:48
• This is a really good question. The units seems to be mentioned at no place. I tried transforming the cartesian dipole moment derivatives given in the fchk file to normal coordinates and the values differ from the gaussian output. Its not even clear whether the derivative is done with respect to mass weighted normal coordinates or "cartesian" normal coordinates. The difference between these two normal coordinate definitions is a factor of root of reduced mass and the way the unit vectors are normed. The most direct solution would probably be asking the gaussian support. May 21 at 18:34

I recently had the same question. After emailing Gaussian support I was informed that the units are indeed $$\sqrt{\text{km}/\text{mol}}$$. From their reply I also believe that the derivatives are calculated with respect to the 3N-6 vibrational modes obtained through the application of the Sayvetz conditions as detailed in their whitepaper (Paragraph after Eqn. 11).

To understand why Gaussian uses these "odd" units, and how to get away from them, it's useful to look at the definition of the integrated absorption coefficient for the $$i$$th absorption band (peak) which corresponds to the $$i$$th vibrational mode

$$A_i = \frac{1}{nl}\int_{\text{band}}\ln\frac{I_0}{I}\text{d}\tilde{\nu} \ \ \ \ \ \ \ \text{[m mol}^{-1}] \tag{1}$$

which can be obtained directly from an experimental spectrum, and where $$n$$ is the concentration $$[\text{mol m}^{-3}]$$, $$l$$ the path length $$[\text{m}]$$, $$I$$ the transmitted intensity, $$I_0$$ the incident intensity, and $$\tilde{\nu}$$ wavenumber $$[\text{m}^{-1}]$$.

This can be calculated theoretically as

$$A_i = \frac{N_\mathrm{A}d_i}{12\epsilon_0 c^2 }\left|\frac{\partial\mathbf{\mu}}{\partial Q_i}\right|_0^2 \cdot \text{kg}(\text{u}) \cdot \text{D}^{2}\text{(C}^{2}\text{ Å}^{2}) \ \ \ \ \ \ \ \text{[m mol}^{-1}]\tag{2}$$

Where $$N_\mathrm{A}$$ is Avogradro's constant $$[\text{mol}^{-1}]$$, $$d_i$$ is the degeneracy of the $$i$$th mode, $$\epsilon_0$$ is the permittivity of free space $$[\text{F m}^{-1}]$$, $$c$$ is the speed of light $$[\text{m s}^{-1}]$$, while the "derivative term" contains a vector of the derivatives of the electric dipole with respect to the $$x$$, $$y$$, and $$z$$ components of the $$i$$th (mass-weighted) normal mode coordinate (evaluated at the equilibrium geometry) $$[\text{D Å}^{-1}\text{u}^{-1/2}]$$. I've included the final two terms as they explicitly detail the conversion factors required to arrive at the final units of $$[\text{m mol}^{-1}]$$

$$\text{kg}(\text{u}) = (1.661\times10^{-27})^{-1} \text{u kg}^{-1}\tag{3}$$

$$\text{D}^{2}\text{(C}^{2}\text{ Å}^{2}) = (3.336\times10^{-20})^{2} \text{ C}^{2} Å^{2} \text{ D}^{-2} \tag{4}$$

Evaluating the purely constant terms and conversion factors (precision truncated for brevity):

$$\frac{N_\mathrm{A}}{12\epsilon_0c^2} \cdot \text{kg}(\text{u}) \cdot \text{D}^{2}\text{(C}^{2}\text{ Å}^{2}) \tag{5}$$

$$\!\!\!\!\!\! = \frac{6.022\times10^{23}[\text{mol}^{-1}] \times (1.661\times10^{-27})^{-1} [\text{u kg}^{-1}] \times (3.336\times10^{-20})^{2} [\text{ C}^{2} Å^{2} \text{ D}^{-2}]}{12\times8.854\times10^{-12} [\text{F m}^{-1}]\times (2.998\times10^8 [\text{m s}^{-1}])^2}$$

$$\!\!\!\!\!\!\!\!\!\!\!\!= 42.2561 \text{ km mol}^{-1} \text{u } Å^{2} \text{ D}^{-2} \tag{7}$$

The Gaussian help pages give this exact same conversion factor, which (I believe) means that the dipole derivatives reported by Gaussian include the constant terms. To get the true derivatives in $$[\text{D Å}^{-1}\text{u}^{-1/2}]$$, you need to divide the reported derivatives in $$[\text{km}^{1/2}\text{ mol}^{-1/2}]$$ by $$\sqrt{42.2561} \text{ km}^{1/2}\text{ mol}^{-1/2} \text{u }^{1/2} Å \text{ D}^{-1}$$

This then means that the Intensity values printed by Gaussian $$[\text{km mol}^{-1}]$$ make a bit more sense, they're the sum of the squares of the dipole derivatives $$[\text{km}^{1/2} \text{ mol}^{-1/2}]$$ along $$x$$, $$y$$, and $$z$$.

Many textbooks and articles will state the $$42.2561 \text{ km mol}^{-1} \text{u } Å^{2} \text{ D}^{-2}$$ factor verbatim, and will often exclude the relevant conversion factors (even in explicit numerical evaluations), so be warned if you venture into the literature! As an example see Bernath's Spectra of Atoms and Molecules (3rd Ed.) Eqn. 7.246.

The above equations will need to be reworked if Beers Law is applied using the common logarithm rather than the natural logarithm, this amounts to a $$1/\ln(10)$$ term in the theoretical definition of $$A_i$$.

As has been said, you can get the derivatives without having to include that specific iop command, by looking in the formatted checkpoint (.fchk) file which can be obtained by running Gaussian's formchk utility on the checkpoint (.chk) file. However, these numbers don't match those in the .log file, as they are the derivatives with respect to single atom displacements along the x, y, and z axes. To obtain the derivatives with respect to displacements along the normal mode vectors, you first must generate the matrix $$\mathbf{L}_{\text{CART}}$$ detailed in the whitepaper - these are the displacement vectors printed by Gaussian (though in the .log file they are scaled by the square root of the reduced mass and so this scaling must be undone). Then for each mode, multiply each element of the displacement vector $$\mathbf{L}_{\text{CART}}$$ (corresponding to the motion of a single atom in x, y, or z in a given mode) by the the associated per-atom dipole derivative vector. Finally sum up all of the contributions to the derivative from each atom in each direction for a given mode and multiply by 31.2231 to convert from $$[\text{e u}^{1/2}]$$ to $$[\text{ km}^{1/2}\text{ mol}^{-1/2}]$$ and obtain the same values given by the IOp command.

• +1 Do you have a reference for the equation of the integrated absorption coefficient $A_i$ ? May 26 at 9:01
• Sure, See "Introduction to Infrared and Raman Spectroscopy, 3rd Ed. by Colthup and Daly" Page 102 May 26 at 9:08