Natural coordinate system for an atomic basin

A gradient path g(r ) through a given point r e P is defined by 

In the following, eqn (E3.3b) will be regarded as defining a coordinate transformation within 1. The basic ideas behind the construction of this transformation are as follows. The parameter s will be one of the new coordinates to obtain a proper coordinate transformation from to R, wc need two additional coordinates to specify the initial value of eqn (E3.3a). Thus r(s = 0) = To must belong to a two-dimensional differentiable manifold, and the two additional coordinates mentioned above are the local coordinates of Fq in the manifold. We make the following general observations let M be a two-manifold contained in 1, with an atlas (C/ , ), where the open sets t/ cover M, and the one-to-one, continuous maps - R  

Correspondingly, we obtain a single coordinate system over illustrated in Fig. E3.1, through the mapping h defined by 

Equation (E3.5) implies that, through the one-to-one map h, we can uniquely associate with a point r contained in the basin Q, a triple (s, 0, 4 ) where 0, 0) determines the path through r and s is the coordinate of r along that path. 

This natural coordinate system is of particular use in the determination of atomic properties. As discussed in Chapter 6, the atomic value of a property F is given by the average over the atomic basin of an effective single-particle density /(r). Thus the value of the property F for atom 2 is 

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