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Is there a way to determine the angle between non-bonding electron pairs that aren't obvious? Take $\ce{H2S}$ for example:

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The two non-bonding electron pairs would be at the yellow end of this molecule, but I am curious as to what the angle is between them? Is there even a way to tell?

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    $\begingroup$ It's quite difficult to tell chemistry.stackexchange.com/questions/50906/… $\endgroup$ – Mithoron Mar 12 '18 at 21:51
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    $\begingroup$ There are some ways one could claim to calculate such angles, but any calculations of that sort would be ... highly controversial. Meaning, naturally, I will happily perform such a calculation (when I get a chance) to stoke the fires of such controversy. :-D $\endgroup$ – hBy2Py Mar 12 '18 at 22:09
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Yes, there are ways one could claim to calculate an angle between two non-bonding electron pairs.

BUT: As Mithoron points out, this Chem.SE question illustrates how quantum chemical calculations and photoelectron spectroscopy both demonstrate the non-equivalence of the lone pairs of $\ce{H2O}$, an analysis which presumably applies equally well to the analogous $\ce{H2S}$. Thus, the methods used for calculating such an angle will be controversial, and the results may or may not be of any particular practical value given that they're at odds with PES data. That being said, I'll show here one way that a putative 'non-bonding electron pair angle' can be calculated.

The calculation below is based on application of the quantum theory of atoms in molecules (QTAIM) to the quantity called the electron localization function (ELF), which is a scalar function in $\mathbb R^3$. QTAIM is useful for identifying intrinsic features of various three-dimensional fields that arise in quantum chemical calculations, with a significant focus placed on the 'critical points,' where the field gradient is zero. It was originally developed for analysis of the electron density distribution, but has been extended to the ELF and other quantities. One of my favorite papers illustrating coupled QTAIM/ELF analysis is a 2009 review by Matito and Solà (Coord Chem Rev 253: 647); it also discusses the localization and delocalization indices (see here and here), which I won't go into in any detail here.


I'm going to start out by emphasizing the distinction between electron density and electron localization. The electron density at a point is fairly straightforward: it's a measure of, on average, how many electrons (or fraction thereof) are located at that point, per unit volume. Due to its waveform nature, each electron's position is a continuous distribution function, and the electron density of the system represents the collective distribution function of all the electrons present.

The electron localization at a point can be harder to get a grasp of: it's a measure of how "spread out" the "location distribution" is, of the electrons that contribute to the density at that point. Regardless of the magnitude of the electron density, the electron localization at a point is high when the electrons contributing to the density at that point overall contribute relatively little to the electron density in other parts of the system. That is to say, these electrons associated with the point of interest don't "travel around very far" from that point, in their quantum-mechanical meanderings about the system. Conversely, the electron localization is low when the electrons associated with the point do contribute significantly ("wander") to "distant" regions of the system.

My answer here provides a more concrete example of how the electron localization can differ significantly among systems whose electron density distributions are otherwise relatively similar. It also anticipates the discussion below, noting features of the three-dimensional ELF field that suggest the locations of electron lone pairs on N, O and F atoms. It is this ELF field that I will be focusing on in the following analysis.


I performed DFT calculations of $\ce{H2O}$, $\ce{H2S}$, $\ce{H2Se}$, $\ce{H2Te}$ and $\ce{H2Po}$ for this analysis using ORCA v3.0.3, and I carried out the subsequent QTAIM/ELF analysis in Multiwfn v3.3.7. The ORCA input for $\ce{H2S}$ was

! RKS PBE0 def2-TZVP def2-TZVP/J RIJCOSX
! OPT GRID4 GRIDX5 PRINTBASIS

* xyzfile 0 1 H2S.xyz

H2S.xyz was an initial geometry file generated by Avogadro, overwritten with the optimized geometry at the end of the ORCA run. The inputs for $\ce{H2O}$ and $\ce{H2Se}$ were identical except for the identity of the central atom; I added relativistic effects for $\ce{H2Te}$ via the ! ZORA simple keyword, and for $\ce{H2Po}$ via the use of a 60-electron effective core potential by adding ! ECP{def2-TZVP,def2-TZVP/J}. (While there is an argument to be made that I perhaps should've included relativistic effects for $\ce{H2Se}$ also, I expect them to minimally affect the geometry and overall valence-electronic structure of that system, which is what is of interest here.)

I generated a MOLDEN wavefunction file for each computation, to serve as inputs to Multiwfn. Using $\ce{H2S}$ as an example, the shell command was:

$ orca_2mkl H2S -molden

I then renamed the resulting H2S.molden.input to H2S.molden, which is necessary for Multiwfn to correctly identify it as an ORCA-generated MOLDEN file.

After loading H2S.molden into Multiwfn, I used the following series of commands to locate the 3-D maxima ("attractors") of the ELF distribution:

  • Main menu: [17] Basin analysis
  • [1] Generate basins
  • [9] ELF
  • [3] High quality grid

Multiwfn's search routine found six ELF attractors, one of which was identified as a cluster of 'degenerate attractors'. The figure below plots these attractors (via Multiwfn sub-command [0] and some MS Paint manipulation) atop two different views of the $\ce{H2S}$ molecule (click to enlarge):

H2S molecule with attractors

As can be seen, there are two attractors located right where chemical intuition would expect lone pairs to sit, along with two more attractors located at the hydrogen atoms. The final non-degenerate attractor is right in the center of the sulfur atom, and could be interpreted as representing the $n=1$ core electron shell. The degenerate attractor is distributed around the $n=1$ attractor, and could similarly be interpreted as the $n=2$ core electron shell. Interestingly, the two $n=3$ valence electrons of the sulfur that are involved in bonding to the $\ce{H}$ atoms do not exhibit independent attractors---each shares an attractor with the electron originating from its respective hydrogen atom.

The QTAIM method provides a way to subdivide the space occupied by a molecule based solely on the properties of the ELF distribution, and associate portions of the electron density to each of these attractors. Integrating these subdivisions of the electron density then provides the number of electrons associated with each attractor:

$$ \begin{array}{cccc} \hline \text{lp} & \text{H} & \text{S 1s} & \text{S 2sp} \\ 2.114 & 1.855 & 2.143 & 7.865 \\ \hline \end{array} $$

All of these values seem pretty reasonable---the attractors corresponding to single orbitals $(\ce{lp}$, $\ce{H}$, $\ce{S 1s})$ have about $\ce{2 e-}$ associated with them, and the $\ce{S 2sp}$ attractor has about $\ce{8 e-}$. In my experience, the signs and magnitudes of these deviations from $2.0$ and $8.0$ are typical for ELF basins.

So, leaving the question of actual physical significance aside, I would argue that these ELF attractors provide a reasonable representation of the (non-)bonding structure of the molecule. Thus, to the main question: what is the structure? Conveniently, angles and distances among the attractors and atoms can be calculated via Multiwfn sub-command [-2]:

$$ \begin{array}{cccc} \hline \angle\,\ce{H-S-H} & 92.3^\circ & r_\ce{S-H} & 2.535\,\mathrm{Bohr} \\ \angle\,\ce{lp-S-lp} & 127.8^\circ & r_\ce{S-lp} & 1.838\,\mathrm{Bohr} \\ \angle\,\ce{H-S-lp} & 108.5^\circ \\ \hline \end{array} $$

(In the above, the actual nuclear positions of the atoms were used where relevant, not the positions of the associated attractors. While each such attractor generally falls close to its associated nucleus, it's usually displaced by a tiny amount, $<0.1\,\mathrm{Bohr}$. I've always attributed this to numerical precision limits of the calculation methods, but I don't know for sure if that's what causes it.)

Therefore: By this QTAIM/ELF method, the angle between the two lone pairs of $\ce{H2S}$ is calculated to be $127.8^\circ$. This is considerably greater than the $\ce{H-S-H}$ angle, consistent with the intro-chem conception of the increased 'steric bulk' of a lone pair.

For comparison, I used the same procedure to generate results for $\ce{H2O}$, $\ce{H2Se}$, $\ce{H2Te}$ and $\ce{H2Po}$:

$$ \begin{array}{c|c|ccccc} \text{Quantity} & \text{Units} & \ce{H2O} & \ce{H2S} & \ce{H2Se} & \ce{H2Te} & \ce{H2Po} \\ \hline \int_\text{lp}{\rho} & \ce{e-} & 2.265 & 2.114 & 2.206 & 2.265 & 2.390 \\ \int_{\ce{x}\ce{-H}}{\rho} & \ce{e-} & 1.667 & 1.855 & 1.861 & 1.807 & 1.806 \\ \int_\text{core}{\rho} & \ce{e-} & 2.129 & 10.008 & 27.273 & 44.567 & 77.609 \\ \hline r_{\ce{x}\ce{-H}} & \mathrm{Bohr} & 1.813 & 2.535 & 2.772 & 3.129 & 3.289 \\ r_{\ce{x}\ce{-lp}} & \mathrm{Bohr} & 1.103 & 1.838 & 2.526 & 3.216 & 3.399 \\ \hline \angle \ce{H-x}\ce{-H} & ^\circ & 105.1 & 92.3 & 91.2 & 90.9 & 89.6 \\ \angle \ce{lp-x}\ce{-lp} & ^\circ & 114.9 & 127.8 & 139.5 & 156.8 & 156.8 \\ \angle \ce{H-x}\ce{-lp} & ^\circ & 109.1 & 107.8 & 104.0 & 98.1 & 98.2 \\ \hline \end{array} $$

Due (presumably) to numerical precision limitations, the calculated values of the four different $\ce{H-x}\ce{-lp}$ angles for each system varied by $1^\circ$ or so; I've reported the means of the obtained values above.

Those at home who are doing their math carefully will note that there is some electron density missing in the systems with the heavier central atoms. This is because at the 'high' grid quality used, the deep core electrons are poorly captured by the numerical integration involved. (Indeed, the deep core electrons may not be accurately captured at just about any computationally reasonable grid size!) I think the $\int_\text{lp}{\rho}$ and $\int_{\ce{x}\ce{-H}}{\rho}$ values should be pretty reliable, though.

Naturally, $\ce{H2O}$ does not exhibit a degenerate attractor for the $n=2$ valence electron shell, though it does have a non-degenerate attractor for the $n=1$ core shell. The two heaviest chalcogenides examined possess degenerate attractors representing their $n\geq 2$ core electrons, as with sulfur.

In general, the $\ce{x}\ce{-H}$ and "$\ce{x}\ce{-lp}$" bond lengths increase as the central atom is varied down the group, which is unsurprising ("bigger atoms are bigger"). Interestingly, while the $\ce{x}\ce{-lp}$ distance starts out considerably shorter than the $\ce{x}\ce{-H}$ bond length for $\ce{x}=\ce{O}$, by the time one reaches $\ce{x}=\ce{Te}$ the relative magnitudes are reversed. I don't have a particularly good explanation for this trend; presumably it's due to some sort of counterbalance between the nucleus-nucleus repulsion and nucleus-electron attraction forces.

Where these results are particularly dramatic is in the clear trends in the bond angles: as the central atom is made heavier, the $\ce{lp-x}\ce{-lp}$ bond angle increases markedly while the $\ce{H-x}\ce{-H}$ and $\ce{lp-x}\ce{-H}$ angles steadily decrease. The lone pair 'steric bulk' apparently increases uniformly down the group for this series of analogous systems, at least as determined by this QTAIM/ELF approach. The answers to this question provide more detail into the physical reasons for this decreasing trend in $\ce{H-x}\ce{-H}$ bond angles.

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