Horizontal and Vertical Displacements of the Electronic Structure

In the thermodynamic treatment of molecular systems [23,33,113-115] one aims at describing changes in the electronic structure accompanying the displacement of the molecule under consideration from one equilibrium (ground-state) density, 

Pi(r) = p[A. V r. to another, P2(r) = p[iV2,v2 r], both uniquely determined by the two state parameters determining the current electronic hamiltonian the overall number of electrons Nt and the external potential due to the nuclei v,(r), i= 1,2. We call such shifts in the system electronic structure the horizontal displacements [7,33] on the ground-state density surface , p[N, v r] = p0(r). A displacement from one v-representable electronic density to another gives rise to the associated change in the generalized density functional for the ground-state energy  

It should be emphasized, that this density functional for the ground-state energy differs from the (HK) functional for the energy, Ev[p, in which the external potential is fixed (not related to the trial density p). Only for the exact ground-state density Ev[p0] = E[p0. The variational principle of equation (16), also defined for the fixed v(r), determines the minimum of Ev[p subject to the density normalization constraint iV[p] = J p(r)dr = N. The associated Euler equation for the optimum (ground-state) density p(r) = p /V0. v r] = p0(r) reads  

The horizontal character of changes in the ground-state electronic structure is in contrast to a search for the equilibrium partitioning of the fixed molecular ground-state density p(r) into the subsystem densities p(r) = pa(r) = pa[p r] (a row vector), e.g., those of m constituent AIM or two reactants, which sum up to this given density, p(r) = PJr)- The latter is vertical in character, i.e., performed for the fixed molecular density [20-24,33], as is also the case in the Levy constrained search construction of F[p] [124]. As argued elsewhere [33], such a 

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