Metric measures of interparticle interaction in an exactly solvable two-electron model atom

The exact ground-state solutions for the model of two particles in a confining harmonic oscillator potential interacting through a repulsive harmonic oscillator force are used from the standpoint of geometric distances [Phys. Rev. Lett. 106, 050401 (2011).] between wave functions and densities. The distances from the noninteracting reference state are calculated at a specified confinement by increasing the coupling of the interparticle interaction. Based on the analytic expressions for coupling-dependent geometric measures, a discussion of the Hohenberg-Kohn mapping is given.

Figure 1

Exact ${\left(D_{\psi }\right)}^{1/2}$ and the approximate ${\left(D_{\psi }^i\right)}^{1/2}$ dimensionless quantities as a function of the interparticle coupling constant $\Lambda \in [0,0.5]$. They are calculated from Eqs. (18) and (19), respectively. The solid curve refers to the exact result. The dashed, dotted, and dash-dotted curves are based on the Löwdin-Shull ($i=\text{LS}$), the Kohn-Sham ($i=\text{KS}$), and the energetically optimized ($i=\text{OP}$) approximate product wave functions, respectively. All distances are calculated with respect to the noninteracting reference state. See the text for further details.

Figure 2

Density distance $D_{\rho }$ as a function of the wave-function distance $D_{\psi }$. Both functions are based on the exact analytical solutions for the interacting Hamiltonian given by Eq. (3). See the text for further details.

Figure 3

Wave-function distances $D_{\psi }$ and $D_{\psi }^{\text{KS}}$ as a function of the unique density distance $D_{\rho }$. The solid curve refers to the exact result and the dashed curve is based on the auxiliary Kohn-Sham product state. See the text for further details.