Electronic density Taylor series expansion

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

Then, they proposed to examine the Taylor series expansion of the spherical average conditional pair density in the vicinity of the point s = 0 where one is measuring the short-range behaviour of the electron at point r2 approaching the reference point rl. The leading term of the Taylor series expansion is given by... 

Each topological feature of p(r), whether it be a maximum, a minimum, or a saddle, has associated with it a critical point (abbreviated to cp), a point denoted by the coordinate where Vp(rc) = 0. Such points are evident in the relief maps of the electron density given in Figures 1 and 2 for two planes of the diborane molecule. The behavior of the density in the neighborhood of a cp is obtained via a Taylor series expansion... 

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