Electron-density distribution Laplace concentration

If the second derivative, and hence the curvature of/, is negative at x, then / at x will be larger than the average of / at all neighbouring points, i.e. / concentrates at point x73. Therefore - V2p(r), which is the second derivative of a function depending on three coordinates x, y and z, has been called the Laplace concentration of the electron density distribution. Furthermore, the Laplacian of pir) provides the link between electron density p(r) and energy density Hir) via a local virial theorem (equation 8)67,... 

The analysis of p(r) leads to useful information about chemical bonding. Additional information on the electroiiic structure of Ng molecules can be gained by analyzing not only p(r) but also V p(r), the Laplacian of the electron density distribution. The Laplacian of a scalar hinction f indicates where this function concentrates (V f < 0) and where it is depleted (V f > 0) [33]. For f = p(r), the Laplace concentration — V p(r) reveals where the electrons lump together in the molecule [34]. 

Analysis of the electron density distribution confirms that Ng,C bonding in HeCCHe " is due to s-electron donation from He while in NeCCNe and ArCCAr it is due to pcr-electron donation from Ng. Again, this is nicely reflected by the calculated Laplace concentrations (Fig. 19). 

When calculating the elements of the RPH, it does not require much additional effort to calculate also other properties of the reaction complex that depend on s. An example can be found in the investigation of the isomerization reaction of the methoxy radical to the hydroxymethyl radical by Colwell and Handy, where the authors also determined the dipole moment and the components of the polarization tensor as a function of s. A systematic step in this reaction has been made by Konkoli, Kraka, and Cremer, who analyzed the electron density distribution p(r.s) of the reaction complex along the RP, calculating difference density distribution Ap(r,s) (equation 39), Laplace concentrations V-plr.s),... 

The investigation of the electron density di.stribu-tion p r, s) along the path asing difference density distributions Ap r, s), Laplace concentrations —V /o(r,j), and other density properties helps to explain changes in the forces and the overall energy along the RP. 

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