Posts Tagged ‘excess kinetic energy’

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

Wednesday, July 21st, 2010

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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Looking at bonds in a different way: the Laplacian.

Tuesday, July 6th, 2010

The Cheshire cat in Alice’s Adventures in Wonderland comes and goes at will, and engages Alice with baffling philosophical points. Chemical bonds are a bit like that too. In the previous post, we saw how (some) bonds can be tuned to be strong or weak simply by how a lone pair of electrons elsewhere in the molecule is oriented with respect to the bond. Here I explore another way of looking at bonds. To start, we must introduce a quantity known as ∇2ρ(r), henceforth termed the Laplacian of the electron density ρ(r).

Firstly, a recipe: obtain a description of the electron density distribution in the molecule; we will call this the wavefunction (and programs such as Gaussian can write this out in something called a wavefunction file, or .wfn). In a cube of space enclosing the molecule, at each point obtain the second derivatives of ρ(r) with respect to the x, the y and the z coordinate of the point, and populate a (3,3) matrix with the values. Diagonalize the matrix, and add the three eigenvalues of the matrix at that point together to get ∇2ρ(r). Repeat this procedure at regular intervals for all the other points in the cube of space (typically ~200 points in each of the three directions). You will end up with a cube of (in this case 8 million) Laplacian values for the molecule.

Typically (in atomic units), any one value may range from ~-1.0 to ~+1.0, but more meaningful insight is obtained by a (local-virial theorem) expression which relates the Laplacian to a sum of the potential and kinetic energy densities (see. eg here for more detail). A negative Laplacian is dominated by a lowering of the (negative) potential energy at that point in space, whereas a positive Laplacian arises by a domination of the (positive) excess kinetic energy. Measured at the ~mid-point of a (homonuclear) bond, the former indicates an attractive covalent bond, whereas the latter will indicate either an ionic bond or a third type known as charge-shift in which the covalent term (in the valence-bond description of the bond) is repulsive rather than attractive (the actual bond binding energy arises from resonance terms between the covalent and ionic structures). A -ve Laplacian is describing local accumulations or concentrations of (bonding) electron energy densities, whereas a +ve value is describing local depletions. The former can also be used to identify a Lewis base or nucleophilic region, and the latter a Lewis acid or electrophilic region.

Now that we have a cube of points describing the Laplacian for the molecule, we can look at the surface defined by any particular (positive or negative) value of the function to see what insight, if any, can be obtained. Time for some pictures.

Ethane. Laplacian isosurface +/- 0.3 Click for 3D

The above is ethane, contoured at a Laplacian isosurface value of either -0.3 (red surface) or +0.3 (blue surface). Interpreted simply, all seven bonds in this molecule coincide with the red components, which can be taken as typical covalent interactions. The blue spheres represent the valence atomic orbital regions, which have been depleted at the expense of the bond. Nicely intuitive thus far. Let us contour the Laplacian at a rather lower value of +/- 0.2.

Ethane, Laplacian isosurface +/- 0.2 Click for 3D

New blue features have appeared which correspond to +ve Laplacian values. Close inspection reveals them to coincide with what we might describe as the anti-bonding regions of each bond (eight in all). They have been named σ-holes.  Indeed, one might reasonably expect a depletion from just those regions in favour of the bonding regions (one might also regard it an electrophilic region, susceptible to eg nucleophilic attack). Well, we could explore both lower and higher values of the Laplacian (for example, a value of either -0.511 or -0.869 happens to have special significance for the C-C or C-H bonds of ethane) but to keep this blog short, I will move on to (and conclude with) benzene, another iconic molecule.

Benzene. Laplacian isosurface +/- 0.3 Click for 3D

Benzene. Laplacian isosurface +/- 0.2 Click for 3D

Again, the +/- 0.3 isosurface has the expected red bonds, and at the lower value, further blue regions (it is tempting, but we really should not call them anti-bonds!) materialize. Look at the central region of the ring, where depletion seems to have happened.

I close with a musing. Firstly, it is noteworthy that the Laplacian can actually be measured, it is not merely a theoretical concept (although the experiments are in fact pretty difficult, and need very specialised apparatus) but a real observable. Secondly, (at certain values) the Laplacians do seem to recover the simple picture of covalent bonding. The issue really is how far to push the analogy and whether in fact it results in any significant additional insight compared to more conventional ways of representing bonds. At least the pictures are pretty!

Postscript: One can use  a sub-set of electrons to calculate the Laplacian.  Shown below is benzene calculated for just the σ and π-electrons.

Benzene, σ-manifold

Benzene. π-manifold

Notice how the σ set does not differ much from the total set, but the π-set shows accumulation above and below the plane, at the expense of depletion in the plane (one must be aware that integration of the  Laplacian over all space should yield the value of zero). This explains the unusual features of the total set at the  0.2 theshold above.

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