Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

Abstract

The exact functionals associated with the (singlet) ground and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as commonly known, a convex function with respect to the density (or, more precisely, the site occupation), the functional associated with the (highest) doubly-excited state is found to be concave. Also, because the density of the first-excited state is non-invertible, its ``functional'' is a partial, multi-valued function composed of one concave and one convex branch that correspond to two separate sets of values of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are ``functions'' whose derivatives are each other's inverse, just as in the ground state formalism. These findings offer insight into the challenges of developing state-specific excited-state density functionals for general applications in electronic structure theory.

Figures (6)

• Figure 1: $E_0$, $E_1$, and $E_2$ as functions of $\Delta v$ for $t = 1/2$ and $U = 1$. Note that $E$ is an even function of $\Delta v$. $E_1$ is concave for $\Delta v < \Delta v_\text{c}$ and becomes convex for larger $\Delta v$ values.
• Figure 2: $\rho$ as a function of $\Delta v$ for $t = 1/2$ and $U = 1$ for the ground-state ($\rho_0$), the singly-excited state ($\rho_1$), and the doubly-excited states ($\rho_2$). $\rho_1$ reaches a critical value, $\rho_\text{c}$, at $\Delta v_\text{c}$. Note that $\rho$ is an odd function of $\Delta v$.
• Figure 3: $f_{--}(\rho,y)$ (red), $f_{-+}(\rho,y)$ (blue), $f_{+-}(\rho,y)$ (yellow), and $f_{++}(\rho,y)$ (green) as functions of $y$ for $t = 1/2$, $U = 1$, and $\rho = 1/5$ (left), $1/2$ (center), and $3/5$ (right). The markers indicate the position of the stationary points on each branch. At $\rho = 3/5$ (right panel), the stationary points of $f_{-+}$ and $f_{+-}$ have disappeared as $\rho > \rho_c$ (see Fig. \ref{['fig:densities']}).
• Figure 4: State-specific exact functionals $F_m(\rho)$ as functions of $\rho$ for $t = 1/2$ and $U = 1$. The ground-state functional $F_0(\rho)$ (red) is convex with respect to $\rho$, the singly-excited state multi-valued functional $F_1(\rho)$ has one convex branch (blue) and one concave branch (yellow), each associated with a separate set of $\Delta v$ values, while the doubly-excited state functional $F_2(\rho)$ (green) is concave. Note that $F$ is an even function of $\rho$.
• Figure 5: Illustration of the Levy constrained-search procedure for $t=1/2$, $U=1$, and $\rho=1/5$. The value of $T+V_{ee}$ is mapped on the surface of the unit sphere that represents the normalized wave functions. The gray parabolas correspond to densities $\rho = z^2 - x^2$. The four branches of $f_{\pm\pm}$ [see Eq. \ref{['eq:fpm']}] are represented as contours and correspond to the intersections of these three-dimensional objects. The dots locate the stationary points on each of these contours.