# Bonding in alkali metal complexes

How exactly is the bonding described in complexes involving alkali metals? In particular, how much of a covalent component is there? Take three examples:

$\ce{[Na(H2O)6]+}$: This previous answer by @tschoppi suggests that the bonding in this complex is purely electrostatic in nature due to the lack of available d-orbitals for covalent bonding. This seems like it might be true because it would explain why the water complexes are not strongly bound.

$\ce{Na2(edta)}$: This previous question suggests that the sodium here is not complexed with the EDTA but rather that it again forms an essentially ionic (i.e electrostatic) bond with little covalent character.

$\mathrm{[Na(\text{15-crown-5})]^+}$: This is very stable, suggesting a strong bond of some sort, probably with some covalent character. However, it doesn't seem to be radically different from the previous ligands, particularly EDTA, which has similar oxygens available for bonding.

• Well d-orbitals are there, they’re just not accessable. I would assume that this is a similar case to $\ce{Zn^2+}$, where complexes do exist but are often kinetically very labile. There is simply no compelling reason for sodium to stay in a certain complex environment. – Jan May 4 '16 at 16:29
• QTAIM analysis of $[\ce{Li}(\text{12-crown-4})]^+$ indicates pretty strongly that the interaction between the $\ce{Li+}$ and the crown oxygens is almost purely ionic in nature (very low $\rho$, with $\nabla^2\rho > 0$. I would assume $[\ce{Na}(\text{15-crown-5})]^+$ is similar. One key difference b/w carboxylic and etheric oxygens is charge: the excess electron at the carboxyl probably results in an appreciably more diffuse electron cloud, making it less favorable for tight ionic binding. – hBy2Py May 4 '16 at 16:39
• Binding doesn't have to be covalent in nature for it to be strong - it is just because of the chelate effect (for both edta and crown ethers) and macrocyclic effect (for crown ethers). – orthocresol May 8 '16 at 11:10

The bonding of alkali cations to water, to EDTA, and to crown ethers is strictly electrostatic in all three cases, at least based upon QTAIM analysis$^{[1]}$ of representative systems.

For this analysis, I ran gas-phase quantum chemical optimizations on $[\ce{Li}(\text{12-crown-4})]^+$, $[\ce{Na}(\text{EDTA})]^{3-}$, and $[\ce{Na}(\ce{H2O})_6]^+$ using ORCA v3.0.3. (I can post the computation inputs/outputs if anyone is interested.) I also optimized $\ce{NaCl}$ as an additional system with uncontroversial ionic bonding. While the gas-phase results will poorly represent a variety of aspects of the dissolved species, I think they should capture well enough the characteristics of the wavefunction necessary to study the types of bonding involved. Other gas-phase calculations I've run on, e.g., $[\ce{Cu}(\ce{H2O})_6]^{2+}$ have optimized smoothly to the expected octahedral ligand arrangement, lending credence to this assumption.

Some notes on the optimizations:

• Geometry optimization of $[\ce{Li}(\text{12-crown-4})]^+$ converged readily.

• For $[\ce{Na}(\text{EDTA})]^{3-}$, despite starting in a prototypical octahedral chelation geometry, the EDTA molecule immediately began to 'loosen its grip' on the $\ce{Na+}$, clearly heading toward a less compact configuration. In fact, one of the oxygen atoms actually discarded its bonding interaction with the $\ce{Na+}$ in favor of an intramolecular hydrogen bond. I performed the QTAIM analysis on the last geometry obtained before ORCA reached its iteration limit in the geometry optimzation procedure.

• Similarly, for $[\ce{Na}(\ce{H2O})_6]^+$, despite starting the optimization in a prototypical octahedral coordination geometry, the water ligands 'wandered' throughout the optimization process, in a fashion loosely reminiscent of a molecular dynamics run. As with the EDTA complex, I just chose the final geometry after the optimizer halted as the one on which I ran the QTAIM analysis.

For illustration, I generated GIF animations of the $[\ce{Na}(\text{EDTA})]^{3-}$ and $[\ce{Na}(\ce{H2O})_6]^+$ optimizations. The individual frames were produced with the 'Movie Maker' module of VMD 1.9.1 and the final GIFs were generated using EZgif.com.

Several metrics generated by QTAIM analysis argue for strictly ionic bonding of the alkali metal atoms in these systems:

1. Low electron density $(\rho \substack{<\\\sim}0.1)$ and positive density Laplacian $(\nabla^2\rho>0)$ at the relevant line critical points between the alkali cation and the coordinating oxygen (or nitrogen) atoms

2. High localization index $(\mathrm{LI}\substack{>\\\sim}0.9)$ in the alkali atom basin

3. Low delocalization index $(\mathrm{DI}\substack{<\\\sim}0.75)$ between the alkali atom basin and those of the coordinating atoms

Strictly speaking, in addition to $\#1$ above, the terms of the Laplacian perpendicular to the bond path must also be small in magnitude in order to diagnose ionic bond character$^{[2]}$, but I have omitted those values here.

The table below presents these metrics for the above representative chemical systems, as well as some reference molecules possessing uncontroversial bonding types. Due to the fairly regular geometry of $[\ce{Li}(\text{12-crown-4})]^+$, the metrics didn't vary much throughout the system and I averaged them for each bond and atom type. Despite the lack of a deep energy well, the metrics were also similar for the six oxygens in $[\ce{Na}(\ce{H2O})_6]^+$, so I averaged the values here as well. For the irregular geometry of $[\ce{Na}(\text{EDTA})]^{3-}$, the values differed enough that I chose to report them indivdually; the numbering of the coordinating atoms is arbitrary. All non-literature data were generated by MultiWFN v3.3.7 on Windows 7 using the default settings and the "Medium" grid for generation of the atomic basins for the $\mathrm{DI}$ and $\mathrm{LI}$ calculations. All values are in atomic units: $\rho\equiv{e^-\over\mathrm{Bohr}^3}$ and $\nabla^2\rho\equiv{e^-\over\mathrm{Bohr}^5}$ ($\mathrm{DI}$ and $\mathrm{LI}$ are dimensionless).

$$~ \\ \textbf{QTAIM Results} \\ \begin{array}{ccccccccc} \hline \mathbf{Species} & ~ & \mathbf{Bond} & \rho_\mathbf{LCP} & \mathbf{\left(\nabla^2\rho\right)_\mathbf{LCP}} & \mathbf{DI} & ~ & \mathbf{Atom} & \mathbf{LI} \\ \hline & & \ce{Li-O} & 0.036 & +0.26 & 0.077 & & \mathrm{Li} & 0.922 \\ [\ce{Li}(\text{12-crown-4})]^+ & & \ce{O-C} & 0.25 & -0.50 & 0.856 & & \ce O & 0.870 \\ & & \ce{C-C} & 0.26 & -0.70 & 0.949 & & \ce C & 0.656 \\ & & \ce{C-H} & 0.28 & -1.01 & 0.901 & & \ce H & 0.420\\ \hline & & \ce{Na-O}1 & 0.015 & +0.086 & 0.064 & & \ce{Na} & 0.974 \\ & & \ce{Na-O}2 & 0.021 & +0.131 & 0.096 & & \ce{O}1 & 0.899 \\ [\ce{Na}(\text{EDTA})]^{3-} & & \ce{Na-O}3 & 0.020 & +0.121 & 0.089 & & \ce{O}2 & 0.900 \\ & & \ce{Na-N}1 & 0.014 & +0.074 & 0.056 & & \ce{O}3 & 0.900 \\ & & \ce{Na-N}2 & 0.018 & +0.094 & 0.065 & & \ce{N}1 & 0.766 \\ & & & & & & & \ce{N}2 & 0.769 \\ \hline [\ce{Na}(\ce{H2O})_6]^+ & & \ce{Na-O} & 0.014 & +0.086 & 0.057 & & \ce{Na} & 0.975 \\ & & & & & & & \ce{O} & 0.916 \\ \hline [\ce{NaCl}]^0 & & \ce{Na-Cl} & 0.035 & +0.197 & 0.321 & & \ce{Na} & 0.978 \\ & & & & & & & \ce{Cl} & 0.975 \\ \hline [\ce{LiF}]^{0\,\ddagger} & & \ce{Li-F} & 0.079 & +0.749 & 0.179 & & \ce{Li} & 0.957 \\ & & & & & & & \ce{F} & 0.991 \\ \hline [\ce{CH4}]^{0\,\ddagger} & & \ce{C-H} & 0.291 & -1.168 & 0.980 & & \ce{C} & 0.661 \\ & & & & & & & \ce{H} & 0.469 \\ \hline ^\ddagger~_{\text{Values from Ref. [2]}} \end{array}$$

As can clearly be seen, the interactions between the alkali metals and the ligating atoms are uniformly of ionic character: low electron density and positive Laplacian at the LCP; low $\mathrm{DI}$ between the paired basins, and quite high $\mathrm{LI}$ for the alkali metal basins. This is consistent with the ionic species $\ce{LiF}$ and $\ce{NaCl}$ in the table, and is in marked contrast to the covalent systems included.

$^{[1]}$ More information can be found on Wikipedia and at RFW Bader's page at McMaster University. A more readable (but unfortunately paywalled) description is provided in Ref. [2].
$^{[2]}$Bader & Matta, Found Chem 15: 253 (2013). doi:10.1007/s10698-012-9153-1.

• Very nice analysis! Is there a plus sign missing in the $\nabla^2\rho$ column of sodium chloride? All other values are signed. And also: Do you know how zinc-edta and zinc hydrates compare, or should I ask that in a separate question? ;) – Jan May 8 '16 at 15:02
• @Jan Yup. Fixed - thanks! I don't know specifically how $\ce{Zn}$ complexes compare, but what QTAIM analysis I've done on aquo-$\ce{Cu}$ complexes indicates strictly ionic character. I would assume $\ce{Zn}$ complexes would be even more ionic. As I noted, the aquo-$\ce{Cu}$ complexes optimize to a stable configurational minimum - I don't know if the $\ce{Zn}$ complexes might "wander" more like the sodium. I'd think $\ce{Zn}$ would compare better to, say, $\ce{Ca}$, though, what with them both being divalent. – hBy2Py May 8 '16 at 15:17
• NaEDTA complex is very poorly chosen since, your know, it has rather high charge that will undoubtedly force chelation rings to open (i'm surprised it didn't dissociate) . Calling LiF in gas phase ionic is an interesting and novel idea. – permeakra May 8 '16 at 17:07
• @permeaka Would you prefer "non-covalent" for $\ce{LiF}$, then? I did think about optimizing, e.g., $[\ce{Na2H2EDTA}]^0$ instead. But, e.g., $[\ce{CuEDTA}]^{2-}$ presumably stays chelated... is there something more problematic about the triply negative charge? – hBy2Py May 8 '16 at 17:28
• @permeakra That being said, I'm not particularly interested in the stability or macroscopic physical "existenceness" of these systems, just the properties of the electron density between the alkali atoms and the ligating oxygens/nitrogens. I presume you don't have the same complaint about the aquo complex...given how similar the various metrics are between the aquo and EDTA complexes, I would argue that, indeed, the high negative charge matters little for the present purpose. – hBy2Py May 8 '16 at 17:31