Critical points of molecular charge distributions

The Coulombic potential becomes infinitely negative when an electron and a nucleus coalesce and, because of this, the state function for an atom or molecule must exhibit a cusp at a nuclear position. That is, as shown by Kato (1957), the first derivative of the function is discontinuous at the position of a nucleus. Thus, while the charge density is a maximum at the position of a nucleus, this point is not a true critical point because Vp, like is discontinuous there. However, as discussed in Election E2.1, this is not a problem of practical import and the nuclear positions behave topologically as do (3, — 3) critical points in the charge distribution and hereafter they will be referred to as such. 

This discussion has shown that the principal topological features of a charge distribution can be summarized using the rank and signature classification scheme of its critical points. It has further demonstrated the existence 

Relief map of ihe electronic charge density in the tetrahedrane molecule, C4H4. The plane shown is a 0-4 symmetry plane and contains two carbon nuclei and their associated protons. The charge density at the central critical point is a local minimum with a value of 0.165 au. The two-dimensional maximum in the foreground is the (2, — 2) maximuln in p in the interatomic surface of the out-of-plane carbon nuclei. The value of p at this point is 0.246 au. A contour map of the charge density in the same plane is shown in Fig. 2.10. 

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