How to interpret the relation between CO stretch frequencies and backbonding strength for polycarbonyl compounds?

Question

I am working on a series of octahedral complexes of the type $\ce{[M(CO)2(CH3)X3]-}$ where $\ce{M} = \ce{Co}, \ce{Rh}, \ce{Ir},$ and $\ce{X}$ is a range of ligands (halide, cyanide, nitrosyl, etc.). Now, the complexes all have either a mirror plane or a $C_2$ axis which makes the two $\ce{CO}$ ligands equivalent by symmetry (This is a crucial point).

I know the usual rule that the more electron density the metal has, the stronger the backbonding to the $\ce{CO}$ ligand will be, and this will make the $\ce{C#O}$ bond weaker and will therefore reduce the $\nu_\ce{CO}$ stretch frequency in the infrared region.

However, in this case there are more than two $\ce{CO}$ ligands which are symmetrically equivalent. This means there are two different $\ce{CO}$ stretch bands. I have encountered cases where changing the $\ce{X}$ ligand causes one band to go up and the other band to go down. How do I interpret the change in back-bonding in these cases?

My first thought was that, since they are equivalent, we are sort of taking the linear combination of the bond stretch vectors ($\vec{r}_1 +\vec{r}_2$ and $\vec{r}_1 -\vec{r}_2$) so the higher frequency band is the one we want to look at to see if overall the $\ce{CO}$ stretches have gone down. But this seems a bit unscientific and not rigorous, and the stretch frequency values are certainly not linear combinations, they are both near $\pu{2000 cm-1}$.

My question is—how should I interpret the $\ce{CO}$ stretch data and the backbonding strength when there are more than one $\ce{CO}$ ligand?

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Example data ($\ce{M} = \ce{Rh}$):

$$\begin{array} {|r|r|}\hline & \mathrm{CO\;stretch\;freq. (cm^{-1})} \\ \hline \ce{X} = \ce{F} & 1980,2067 \\ \hline \ce{X}= \ce{Cl} & 1992,2062 \\ \hline \end{array}$$

Here one band goes up and the other goes down when changing $\ce{F}$ to $\ce{Cl}$. So what can I infer about the backbonding to $\ce{CO}$?

Note that these data (i.e. $\ce{CO}$ stretches, geometries) are all from computations. But I am not just looking for computational approaches; I am also curious as to how the $\ce{metal-CO}$ $\pi$-backbonding in a polycarbonyl compound will be analysed, if the $\ce{CO}$ stretches are coupled to each other due to symmetry (even in experimental IR spectrum).

Answer 1

Disclaimer: This is not going to be an answer to the question. This is just a collection of thoughts on it, which hopefully helps to form a better understanding.

First and foremost, and this should go without saying: If you have experimental data, use it. If you do not have experimental data, go find it. No calculation is good enough to be used without calibration, especially when using DFT (see also DFT Functional Selection Criteria).
However, calculations can give you some insight into the molecule's behaviour.

The standard methodology to calculating the spectrum is to use the harmonic approximation for the force constants. That won't be good enough for anything else than a qualitative interpretation of an experimental spectrum; for example if you need to identify signals in your spectra. It won't be good enough for something as sensitive as the carbon monoxide stretch.
I wouldn't be surprised if you found a couple of different opposing trends when using different calculation techniques.
From your example data it seems quite obvious that the values for the two structures are too close together to get any meaningful interpretation.

I also have doubts that the molecule actually is symmetric. If you have a $C_3$ moiety in the mirror plane this is always going to be challenging. It might be symmetric enough to approximate this and to analyse it within these constraints, but that is a point that should be carefully checked. Especially when using calculation techniques, this is something that needs to be calibrated, too.

Assuming that the complexes are symmetric and therefore the carbonyl ligands are equivalent, one would indeed expect two frequencies: one where the stretches are symmetrical, one where they are asymmetrical. Denoting the equatorial plane with $\ce{M^*}$, this would be \begin{gather} \ce{ \overset{\leftarrow}{O}\bond{-}\overset{\rightarrow}{C} \bond{-}M^*\bond{-} \overset{\leftarrow}{C}\bond{-}\overset{\rightarrow}{O} }\tag{symmetrical}\\ \ce{ \overset{\rightarrow}{O}\bond{-}\overset{\leftarrow}{C} \bond{-}M^*\bond{-} \overset{\leftarrow}{C}\bond{-}\overset{\rightarrow}{O} }\tag{asymmetrical}\\ \end{gather}

The key to the relationship between the carbonyl stretch frequency and the backbonding is the force constant of this bond in comparison to "free" carbon monoxide. In simple and only phenomenological terms, the electron density of the metal pushes into the antibonding orbital of the ligand, hence weakening the bond. There is probably some data that relates the frequencies with some other properties of the metal and some orbital interpretations of that, and linking that to backbonding.
Within this framework you can probably form some differential equations to find the force constant and doe similar maths and come to similar conclusions and you can possibly generalise them to form trends.

This is typically the same mathematical foundation with which you would go about analysing experimental spectra. I'm sorry to not be of more help here; but since you're clever enough to come up with this question, I'm confident you'll find a way to do the maths.

If you have the calculations already and you are confident enough that they describe your molecule, there are probably better methods to estimate backbonding.

From the top of my head, I can think of Energy Decomposition Analysis (EDA), possibly coupled with natural orbitals (search ETS-NOCV). Similar approaches come with Natural Bond Orbital (NBO) analyses. Backbonding is a description based on molecular orbital theory, so it would be quite odd to not be looking at orbitals when analysing it.

I wouldn't be surprised if there were models that employ the Quantum Theory of Atoms in Molecules (QTAIM), too. There is probably a way to find an option within Multiwfn to analyse backbonding. In these cases, you'll probably find a better audience at Matter Modeling SE.

Again apologies for that long post with little to no actual information.