Need for a subsystem variation principle

An understanding of chemistry requires a regional description of a system. The notion that a molecule can be viewed as a collection of atoms linked by a network of bonds, a notion that has already been shown to be rooted in the topological properties of the charge distribution, is the operational principle 

Since electrons are indistinguishable and P is antisymmetrized, the average of / a sum of N one-electron operators can be replaced by N times the average of i-one of the operators. If the sum of the operators is replaced by N times a single operator in each term in the above equation and is then summed over all the spin coordinates and integrated over the space coordinates of all electrons but one (operations denoted by the symbol Jdr, see eqn (1.4)), the result is 

This equation or, equivalently, eqn (5.47) can be used to define, respectively, the quantities (Bader and Preston 1969)J 

the kinetic energy of the topologically defined atom T( 2) is a well-defined quantity and one has 

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