# Bond length from infrared spectra?

It's been taught by my lecturer that infrared (IR) spectra are widely used for chemical bonding determination.

My question is: is it possible to calculate bond lengths according to the IR spectra?

If it is yes, let me know how to determine it. If it is no, let me know another interpretation method which is more reliable for it.

• You can get some estimates on bond strengths (in simple cases), which might be useful for ordering bond orders/strengths (the "classical" literature on metal-carbonyl complexes has used that extensively). This does not, however, assign a bond length, but might be used to estimate it from other known compounds based on the bond strength order. Nonetheless, with today's computers, @Tristan Maxson's suggestion on performing calculations works best, in my opinion. Jul 26 '20 at 15:29

This is a task suited to single crystal XRD. This does require you to be able to form a crystal of your substance which can be difficult at times but this allows for potentially high accuracy of bond lengths/angles to be determined.

In the spirit of this community though, if you do know the bonding structure from something like NMR, you could model the molecule in a code such as NWChem or Gaussian and with an appropriate level of theory (accounting for solvation etc) get fairly accurate bond lengths.

### A partial list of bond lengths I have determined from IR spectra over my career:

$$\begin{array}{ccc} \rm{Molecule} & \rm{Bond ~Length ~ (picometers)} & \rm{References}\\ \hline \ce{Li_2}(1^1\Sigma_g^+) & 267.298 74(19)& \href{}\textrm{2009 JCP, 2013 PRA} \\ \ce{Li_2}(1^3\Sigma_u^+) & 417.000 6(32)& \href{}\textrm{2011 JMS, 2013 PRA (2)}\\ \ce{Li_2}(1^1\Sigma_u^+) & 310.792 88(36)& \href{}\textrm{2009 JCP, 2013 PRA}\\ \ce{Li_2}(1^3\Sigma_g^+) & 306.543 6(16)& \href{}\textrm{2011 JMS, 2013 PRA (2)}\\ \ce{Li_2}(1^3\Pi_u) & 258.9 848(23)& \href{}\textrm{2015 arXiv}\\ \ce{BeH}(1^2\Sigma^+) & 134.2396(3)& \href{}\textrm{2015 JMS}\\ \ce{BeD}(1^2\Sigma^+) & 134.1713(3)& \href{}\textrm{2015 JMS}\\ \ce{BeT}(1^2\Sigma^+) & 134.1485(3)& \href{}\textrm{2015 JMS}\\ \vdots & \vdots & \vdots \\ \hline \end{array}$$

### The same methodology was used for the following molecules since 2009:

Several electronic states and isotopologues of: Cs$$_2$$, Sr$$_2$$, ArXe, LiCa, LiNa, MgH, Br$$_2$$, Mg$$_2$$, HF, HCl, HBr, HI, Be$$_2$$, NaH, and many more (see here for references).

### There are several ways to do this:

• The way I've been doing it is to fit a potential energy surface such that when fed through the vibrational Schrödinger equation, the eigenvalue differences match the experimental IR spectra.
• Philip Morse did this for 56 different molecules in 1929 and got $$r_0$$ values with "calc-obs" (calculation minus observation) values within $$\pm 0.1\require{mediawiki-texvc}\,\AA$$ every time.
• But the Morse potential has two problems: (1) it has only 3 parameters ($$D_e$$, $$r_e$$, and $$k_e$$), so it's not very flexible, and (2) it decays exponentially to dissociation, when in fact we know it should decay with an inverse power-law, like $$C_6/r^6$$.
• For several decades, any attempt to solve these two problems would cause us to lose the crucial property that the Morse potential is exactly solvable. So people abandoned "direct potential fits" from the 1930s until recently.
• The Morse/Long-range (MLR) potential that I developed with Bob LeRoy in the mid-to-late 2000s, solves both shortcomings of the Morse potential: It is a Morse potential at the bottom, but if you work out the calculus of $$r\rightarrow \infty$$ you will discover that the potential literally becomes $$C_6 / r^6$$ or whatever you want (you can choose your own long-range function $$u(r)$$) toward dissociation (and we have more parameters and hence more flexibility to fit to lots of data). By this time we had computers so we did not care about "exact solvability" and we could do iterative non-linear least-squares-fitting of numerically obtained eigenvalues of the vibrational Schrödinger equation, to 17,477 spectroscopic lines (many of them from FTIR: Fourier Tranform IR spectra). Jim Mitroy called our 2009 paper a "landmark in diatomic spectral analysis".
• Prior to the revival of "direct potential fits", you could use Bob LeRoy's software dParFit (diatomic parameter fit) to fit IR spectra to band constants or Dunham constants, which will give you $$B_0$$ (the rotational constant for the first vibrational level), from which you can estimate $$r_e$$ very accurately (but not directly as in a direct-potential-fit). Alternatively one can build a potential without numerically solving the Schrödinger equation (as necessary in the "direct potential fits" described above) via the semi-classical RKR method which will result in extremely accurate (with about as many digits as shown in my above table) bond lengths as well.

### Polyatomics:

I generalized the MLR potential to polyatomics and in principle it can be used to directly fit IR spectra of polyatomics to equilibrium bond lengths, angles, force constants and atomization energies, but writing the code to do this non-linear least-squares-fitting is something I don't feel like doing now that I've found new enjoyment in working on quadratization, quantum computing, quantum dynamics, electronic structure, bioinformatics, and other areas. If I could afford a student, we could get this working for polyatomics in a month.

### Side note:

Fitting an entire potential energy surface when there's no IR (infrared / vibrational) data available, and there's only MW (microwave / rotational) data available, was considered a bit more special, but we did it for ZnO in 2014. We obtained $$r_e = 1.704682(2)$$ and the energy difference between $$v_0$$ and $$v_1$$: $$728.395\pm0.007$$ cm$$^{-1}$$ when the best available value was $$726\pm 20$$ cm$$^{-1}$$ at the time (we were about 3 orders of magnitude more accurate!). To obtain a vibrational energy difference so precisely, without any vibrational spectra, is another strength of fitting potentials to spectroscopic data.

### References for the table:

• I was wrong. one can get bond lengths from IR Jul 26 '20 at 18:42
• This is definitely much further than we learn in physical chemistry for the limits of IR. What would be the maximum size of a molecule that would be practical? As in, what size becomes so large that experimentally the data would be unresolvable? Jul 26 '20 at 20:37
• @TristanMaxson I think no matter what the size of the molecule, things like "the OH stretch" tend to show up in spectra, so you could probably get the bond length for the OH bond this way. For years now I've wanted to apply my PolyMLR (polyatomic MLR potential) to larger molecules, but I would have to put a lot of my other interests aside (such as running this site, taking care of my family, and publishing things that are more "new" and "exciting" for me) or get a student or post-doc that would be interested in helping do some of the work (ex. derivatives of the PolyMLR for least-squares fit) Jul 26 '20 at 21:00
• What do the values in brackets in the middle column of the table signify? Dec 7 '20 at 16:27
• @RoniSaiba When you see something like 156.442(31) in science, usually it means 156.442 +/- 0.031. This is the convention I used in my answer. Dec 7 '20 at 18:11

I agree with all answers provided so far: you cannot quantitatively deduce bond lengths from infrared spectra. However, see the answer by Nike Dattani about the inverse, predicting IR spectra from theory.

IR (and Raman) spectra can be very useful tools to understand properties associated with bond lengths. An example I really like comes from the high pressure solid hydrogen literature. Studying high pressure hydrogen is extremely challenging, and standard structure determination techniques such as x-ray diffraction do not work very well because (i) samples are placed in diamond anvil cells to achieve the high pressures, so are very small, (ii) hydrogen is the lightest of all elements, so it only scatters x-rays very weakly, and (iii) hydrogen has no core electrons, so x-rays don't probe the position of the protons but instead probe the position of the bond.

IR and Raman measurements are possible, and in fact are the main probes used to study this system. Hydrogen at pressures of about 200 GPa is in the so-called phase III, which is made of H$$_2$$ molecules of a unique bond length. These lead to a single vibron Raman and IR peak. At higher pressures above about 230 GPa (although somewhat dependent on temperature), hydrogen undergoes a phase transition to so-called phase IV. This transition was identified by the appearance of a second vibron peak in the Raman spectrum at a different frequency to the original one as explained in this paper. The fact that there are two distinct peaks indicates that there are now two types of H$$_2$$ molecules, of different bond lengths. These phases are now called "mixed" phases, and are believed to be a stepping stone to atomic hydrogen. So although one cannot obtain quantitative bond lengths from an IR spectrum, they can still provide extremely useful information.

Addendum: While the method described below is commonly used by experimentalists, I was mistaken that IR spectrum could not be used to obtain bond lengths. Nike's answer does a great job explaining how the entire potential energy surface can be determined by fitting using some model calculations. The important thing to learn from all this is that what you can compute is more often than not limited mainly by how complicated a model you are willing to use.

IR can not be used in general to predict bond lengths, but under the right circumstances it is at least related. For example, the CO stretch frequencies of metal carbonyl compounds are related to the strength of the bond. If you know the bond lengths of few such compounds, you can derive a very accurate linear correlation between the bond length and the frequency.

So while you can't directly predict bond lengths from IR alone, you can likely develop a correlation between a particular IR frequency and bond length for a series of related compounds. Depending on what you have available, these bond lengths could be experimental or could be obtained from quantum mechanical calculations.

I am surprised no-one above mentions the classic textbook example of the IR spectrum of HCl gas, which shows rather beautiful rotational structure superimposed on the vibrational band. The spacing of the rotational peaks, of course gives direct access to the angular momentum and the bond length.

I say "directly" and "of course" - but you have to know the mass of the respective atoms and trust the selection rules. Another charm of the method is that you can see apparent bond lengthening when the molecule is raised to a high rotational state - which is sufficient to prompt thought about the real significance of bond length.