Henry Rzepa's blog

Tag: kinetic energy density

  • Are diazomethanes hypervalent molecules? Probably, but in an unexpected way!

    A recently published review on hypervalency[cite]10.1039/C5SC02076J[/cite] introduced a very simple way of quantifying the effect. One of the molecules which was suggested to be hypervalent using this method was diazomethane. Here I take a closer look.

    The new method is called the valence electron equivalent γ. It is defined as “the formal shared electron count at a given atom, obtained by any combination of valid ionic and covalent resonance forms that reproduces the observed charge distribution”. These atom charge maps can be obtained from various kinds of quantum mechanical calculation; the one adopted in the review was Bader’s QTAIM analysis.

    Three resonance forms used to estimate γ for the atoms in diazomethane are shown below. According to Durrant[cite]10.1039/C5SC02076J[/cite], “if γ(X) > 8, neither form of the octet rule is obeyed and the atom is hypervalent”. His procedure gives γ(N) = 10 (table 3 and also structure 27a[cite]10.1039/C5SC02076J[/cite]), suggesting that the third resonance form shown below is the most appropriate.

    As a result, the nitrogen is to be considered hypervalent, with five formal covalent bonds and hence a ten shared-electron valence shell. If such hypervalence is to be considered as more than just a convenient representation and to have a deeper underlying foundation, similar conclusions should ideally emerge from other ways of analysing the wavefunctions for such species. Here I try the NBO (Natural Bond Orbital) and ELF (Electron localization Function) analyses of two types of wavefunction using both DFT (density functional theory) ωB97XD/Def2-TZVPP and multi-reference CASSCF(12,12)/Def2-TZVPP Hamiltonians.‡

    Firstly, the NBO method, which localizes electron pairs. There are four bonding NBOs associated with the central nitrogen and a modest contribution from a fifth, the terminal nitrogen lone pair (bottom).

    The carbon has four NBOs comprising one “non-bonding lone pair”, two C-Hs, a C-N and one partial C-N (bottom). This partitioning can be quantified using Wiberg atom bond indices: C, 3.716; N 3.802; N 2.907 (the C-N and N=N bond orders are 1.425 and 2.368). Note that the “non-bonding carbon lone pair” would not contribute to the C bond index, which is already 3.72. Is numerical evidence perhaps emerging that the carbon may exceed the octet in its valence shell? In contrast, the central nitrogen, with a bond index of 3.802, but not having any associated “non-bonding lone pair”, does not look to exceeding the octet. It seems well short of achieving γ(N) = 10, equivalent to a bond index of 5 as shown in the resonance form above.

    Now for ELF, which is derived from the electron density and the basin locations from its kinetic energy density. The values are the integrations for the individual ELF basins, of which 0.498*2 = 0.996 is the monosynaptic basin for the carbon lone pair noted above. The total for this carbon comes to 8.16, the adjacent nitrogen 6.59 and the terminal nitrogen 7.52e. 

    The CASSCF(12,12) values are respectively are 8.18, 6.50 and 7.50e, which are very similar. So these two types of electron partitioning show no evidence that the central nitrogen has γ(N) = 10; it’s actually ~6.5, which is a long way short of 10! However, γ(C) = ~8.2, which does exceed the octet, if only very modestly. Perhaps this can best be summarised with the following representation based on the bond orders, which indicates three shared covalent bonds to carbon, a partial (dashed) C-N interaction and a lone pair implied by the minus sign, the total of which exceeds γ(C) = ~8. It all boils down really to whether that “lone pair” of electrons on the carbon is truly a non-bonding electron pair or whether it should be considered a fully covalent pair associated also with the nitrogen. Certainly, there appears to be no evidence from NBO or ELF that this is the case.

    The next logical question to ask is whether the effect can be “optimised” such that γ(C) > 8 or even >> 8. I will address this in the next post.

    ‡FAIR data for all calculations is available at DOI: 10.14469/hpc/3476

  • Ammonide: an alkalide formed from ammonia and resembling an electride.

    Alkalides are anionic alkali compounds containing e.g. sodide (Na–), kalide (K–), rubidide (Rb–) or caeside (Cs–). Around 90 examples can be found in the Cambridge structure database (see DOI: 10.14469/hpc/3453  for the search query and results). So what about the ammonium analogue, ammonide (NH4–)? A quick search of Scifinder drew a blank! So here I take a look at this intriguingly simple little molecule.‡

    It can be formed by adding two electrons to the ammonium cation; NH4+ + 2e ↠ NH4–. One might be encouraged to do this since the LUMO (lowest unoccupied molecular orbital, below) of the ammonium cation has A1 symmetry and so can accept two electrons without the penalty of Jahn-Teller distortions. These electrons will apparently expand the valence electron “octet” around the nitrogen from 8 to 10; a hypervalent species then?

    So what are the (calculated) properties of NH4–? The energy of the now HOMO (highest occupied molecular orbital) at the ωB97XD/Def2-TZVPPD/solvent=water level is -3.6eV, a respectable electron affinity (the additional electrons are said to be bound). More insight can be obtained from the NBO analysis, which produces localized versions of the molecular orbitals. There are four equivalent NBOs, one of which is shown below.

    Each is bonding along one H-N bond, mildly anti-bonding along the other three N-H bonds, but again bonding in the H-H regions! This matches the observations made earlier that when more electrons are pumped into normally valent main group molecules, they will occupy the antibonding levels. This is accompanied by a reduction in the bond orders associated with the central atom. In this case, the N-H bond orders are reduced from 0.79 to 0.602 and the total bond index at the nitrogen is reduced from 3.16 to 2.408. The bond index at hydrogen is at first sight increased from 0.79 to a surprising 1.0003, but this is explained because the H-H bond orders are 0.1328 (three for each H), which brings the H index up to 1.0. The N-H vibration (A1 symmetric) is 3466 cm-1 for NH4+  which is reduced to 2659 for NH4–.†

    So it appears that adding two electrons to the ammonium cation induces H-H bonding! More insight can be obtained from an ELF analysis of the electron density basins.

    The above shows four attractors (as they are called) centered at the hydrogen nuclei, with 2.053e each (4*2.053 = 8.212e). The remaining ~2e are located in basins (green) centered at two different types of attractors. One is along the axis of each N-H bond and exo to it (0.316e) and the other sits on top of any set of three hydrogens (0.103e), 1.68e in total. The value of the ELF function at the attractor is shown above. You should realize that ELF=1.0 corresponds to perfectly localized electrons (for which the kinetic energy density is zero) and ELF=0.5 would correspond to a free-electron gas. The ELF value of e.g. 0.77 is getting close to an electron gas, and in fact corresponds to what we call an electride.

    So, the nitrogen valence shell electron octet is not actually exceeded! The additional two electrons in ammonide sit beyond the nitrogen, in H-H regions. Whether or not it is a viable species for detection remains to be established, but even its computed bonding properties have proved interesting and it deserves to join the alkalide family. 

    Postscript

    Ammonide exists in a shallow well in the potential energy surface, shown below, with the dissociation to ammonia and hydride anion being exothermic.

    The intrinsic reaction coordinate shows one interesting feature at  IRC ~-1.1 which corresponds to repulsion between the hydride and the lone pair of the nitrogen atom resulting in inversion of configuration during the latter stages of the IRC.

    ‡FAIR data collection; 10.14469/hpc/3455. †Perhaps unsurprisingly, these values are somewhat basis set dependent. Thus a ωB97XD/Def2-QZVPPD/Water calculation gives the following values: bond index at N, 1.998, N-H bond index, 0.4995, H-H bond index 0.1689, H bond index 1.0062, total Rydberg population, 0.2738, ν(A1) 2686 cm-1. The ELF basins are H, 2.039, exo-basins 0.282 and 0.141 (total 1.692). The improved basis set better describes the diffuse regions beyond the N-H bonds.

  • The weirdest bond of all? Laplacian isosurfaces for [1.1.1]Propellane.

    In this post, I will take a look at what must be the most extraordinary small molecule ever made (especially given that it is merely a hydrocarbon). Its peculiarity is the region indicated by the dashed line below. Is it a bond? If so, what kind, given that it would exist sandwiched between two inverted carbon atoms?

    1.1.1 Propellane

    One (of the many) methods which can be used to characterize bonds is the QTAIM procedure. This identifies the coordinates of stationary points in the electron density ρ(r) (at which point ∇ρ(r) = 0) and characterises them by the properties of the density Hessian at this point. At the coordinate of a so-called bond critical point or BCP, the density Hessian has two negative eigenvalues and one positive one. The sum, or trace of the eigenvalues of the density Hessian at this point, denoted as ∇2ρ(r), provides in this model a characteristic indicator of the type of bond, according to the following qualitative partitions:

    1. ρ(r) > 0, ∇2ρ(r) < 0; covalent
    2. ρ(r) ~0, ∇2ρ(r) > 0; ionic
    3. ρ(r) > 0, ∇2ρ(r) > 0; charge shift

    The third category of bond was first characterised by Shaik, Hiberty and co. using valence-bond theory1 and they went on to propose [1.1.1] propellane (above, along with F2) as an exemplar of this type.2 Matching the conclusions drawn from VB theory was the value of the Laplacian. As defined above, for the central C-C bond, both ρ(r) and  ∇2ρ(r) have been calculated to be positive, supporting the identification of this interaction as having charge-shift character.3

    The Laplacian represents one of those properties where quantum mechanics meets experiment, in that its value (and that of ρ(r) itself) can be measured by (accurate) X-ray techniques.4 This was recently accomplished for propellane,5 with the same conclusion that the Laplacian in the central C-C region has the significantly positive value of +0.42 au. The electron density ρ(r) at this point was measured as 0.194 au. Calculations5 at the B3LYP/6-311G(d,p) level report ρ(r) as ~0.19 and ∇2ρ(r) as +0.08 au. Whilst the former is in good agreement with experiment, the latter is calculated as rather smaller than expected. This was originally interpreted as indicating that the “the experimental bond path has a stronger curvature [in ρ(r)] than the theoretical” although more recent thoughts are that both experimental and theoretical uncertainty may account for the discrepancy.5,6 An experiment worth repeating?

    A hitherto largely unexplored aspect of characterising a bond using the Laplacian is whether the value at the bond critical point is fully representative of the bond as a whole. The Laplacian is related to two components of the electronic energy by the Virial theorem;

    2G(r) + V(r) = ∇2ρ(r)/4; H(r) = V(r) + G(r)

    where G(r) is the kinetic energy density, V(r) is the potential energy density and H(r) the energy density. Charge-shift bonds exhibit a large value of the (repulsive) kinetic energy density, a consequence of which is that ∇2ρ(r) is more likely to be positive rather than negative. The relationships above hold not just for the specific coordinate of a bond critical point, but for all space. Accordingly, another way therefore of representing the Laplacian ∇2ρ(r) is to plot the function as an isosurface, including both the negative surface (for which |V(r)| > 2G(r)) and the positive surface [for which |V(r)| < 2G(r)].

    Such an analysis is the purpose of this post, using wavefunctions evaluated at the CCSD/aug-cc-pvtz level (see DOI: 10042/to-5012). The values of ρ(r) and ∇2ρ(r) at the bcp for the central bond are 0.188 and +0.095 au, which compares well with previous calculations. The values for the wing C-C bonds are 0.242 and -0.491 respectively (and were measured5 as 0.26 and -0.48). Laplacian isosurfaces corresponding to ± 0.49 (the value at the wing C-C bcp), ± 0.47 and ± 0.2 (which reveals prominent regions of +ve values for the Laplacian) can be seen in the figures below (and can be obtained as rotatable images by clicking).


    Laplacian isosurface contoured at ± 0.49
    		 –

    Laplacian isosurface contoured at ± 0.47. Red = -ve, blue= +ve.

    Laplacian isosurface contoured at ± 0.20

    A significant feature is the isosurface at -0.47, which corresponds to the lowest contiguous Laplacian isovalued pathway connecting the two terminal carbon atoms (and which coincidentally is similar in magnitude to that reported5 as measured for these two atoms). Three such bent pathways of course connect the two carbon atoms. The energy density H(r) shows a minimum value of -0.21 au along any of these pathways. It is significantly less negative (-0.13) for the direct pathway taken along the axis of the C-C bond.

    Energy density H(r) @-0.21

    Energy density H(r) @-0.13

    ELF isosurface @0.7

    A useful comparison with this result is the ELF isosurface. This too is computed at the correlated CCSD/aug-cc-pVTZ using a new procedure recently described by Silvi.7 Contoured at an isosurface of +0.7, the ELF function is continuous between the two terminal atoms, much in the manner of Laplacian. Significantly, the ELF function at the bcp appears at the very much lower threshold value of 0.54, and forms a basin with a tiny integration for the electrons (0.1e). Since both methods provide a measure of the Pauli repulsions via the excess kinetic energy, the similarity of the Laplacian to the ELF function is probably not coincidental.

    The issue then is whether a bond must be defined by the characteristics of the electron density distribution along the axis connecting that bond, or whether other, non-least-distance pathways can also be considered as being part of the bond.8 The former criterion defines a pathway involving a positive Laplacian (+0.095) and would be interpreted as indicating charge shift character for that bond. The latter involves three (longer) pathways for which the Laplacian is strongly -ve, and which would therefore per se imply more conventional covalent character for the interaction. Considered as a linear (straight) bond, it has charge shifted character; considered as three “banana” bonds, it may be covalent. Weird!

    1. Shaik, S.; Danovich, D.; Silvi, B.; Lauvergnat, D. L.; Hiberty, P. C., “Charge-Shift Bonding – A Class of Electron-Pair Bonds That
      Emerges from Valence Bond Theory and Is Supported by the Electron Localization Function Approach,” Chem. Eur. J., 2005,
      11, 6358-6371, DOI: 10.1002/chem.200500265 and references cited therein.
    2. W. Wu, J. Gu, J. Song, S. Shaik, and P. C. Hiberty, “The Inverted Bond in [1.1.1]Propellane is a Charge-Shift Bond,” Angew. Chem. Int. Ed., 2008,
      DOI: 10.1002/anie.200804965; 10.1002/cphc.200900633
    3. S. Shaik, D. Danovich, W. Wu & P. C. Hiberty, “Charge-shift bonding and its manifestations in chemistry”, Nature Chem, 2009, 1, 443-3439. DOI: 10.1038/nchem.327
    4. P. Coppens, “Charge Densities Come of Age”, Angew. Chemie Int. Ed., 2005, 44, 6810-6811. DOI: 10.1002/anie.200501734
    5. M. Messerschmidt, S. Scheins, L. Grubert, M. Pätzel, G. Szeimies, C. Paulmann, P. Luger. “Electron Density and Bonding at Inverted Carbon Atoms: An Experimental Study of a [1.1.1]Propellane Derivative, Angew. Chemie Int. Ed., 2005, 44, 3925-3928. DOI: 10.1002/anie.200500169
    6. L. Zhang, W. Wu, P. C. Hiberty, S. Shaik, “Topology of Electron Charge Density for Chemical Bonds from Valence Bond Theory: A Probe of Bonding Types”, Chem. Euro. J., 2009, 15, 2979-2989. DOI: 10.1002/chem.200802134
    7. F. Feixas , E. Matito, M. Duran, M. Solà and B. Silvi, submitted for publication. See also this abstract.
    8. See for example the work of R. F. Nalewajski

    Rzepa, Henry S. The weirdest bond of all? Laplacian isosurfaces for [1.1.1]Propellane. 2010-07-21. URL:http://www.ch.ic.ac.uk/rzepa/blog/?p=2251. Accessed: 2010-07-21. (Archived by WebCite® at http://www.webcitation.org/5rOFp6EuM)

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