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.
