Scaling subsystem electronic charges

Examples of a non-uniform scaling of the subsystem electronic charges ame represented by the paths 2-6 in Fig. 2 we have summarized in the figure Eqs. (44) and (53)-(55) identifying the subsystem scaled external potentials corresponding to the four points defining these scaling trajectories. 

In other words, the two scaled subsystem external potentials are defined to give rise to the true ground state densities of interacting subsystems, irrespectively of the current value of the the scaled electronic charge, which we indicate by the following mapping relation ... 

Figure 2. Uniform (1) and nonuniform (2-6) scaling paths of electronic charges in the complementary subsystems of M = (i4 B). The subsystem external potentials of the scaled hamiltonians for the starting and end points of each path are also listed, with the upper (lower) entries corresponding to ti (B).

The above definition of the external potentials coupled to densities of the nonuniformly scaled electronic charges of subsystems in - (A, A g) again implies for the two extreme imiform scaling points [see Fig. 2 and Eq.(44)] ... 

Consider now a case when the charges of electrons in one subsystem, say d, are scaled to zero A = 0. Since then electrons in d do not interact both among themselves, A = 0, and with the other, -group of electrons,... 

The Hellmann-Feynman theorem for the uniform, A-sca-ling scheme of electronic interactions/charges in the scaled ground state 4 Ip, P ] = of the two subsystems in M(, A) reads ... 

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