Uniform scaling, electronic charges

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).

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. 

For an actual surface the positive charge is not uniformly distributed over a half-space. For steps at the surface or corrugated surfaces (on an atomic scale), the contours of constant electronic density have ridges and valleys corresponding to those of the distribution of positive ions at the surface but they are smoother and less bumpy. 

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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