Anomalous scaling and breakdown of conventional density functional theory methods for the description of Mott phenomena and stretched bonds

Density functional theory provides the most widespread framework for the realistic description of the electronic structure of solids, but the description of strongly correlated systems has remained so far elusive. We consider a particular limit of electrons and ions in which a one-band description becomes exact all the way from the weakly correlated metallic regime to the strongly correlated Mott-Hubbard regime. We provide a necessary condition a density functional should fulfill to describe Mott-Hubbard behavior in this one-band limit and show that it is not satisfied by standard and widely used local, semilocal, and hybrid functionals. We illustrate the condition in the case of few-atom systems and provide an analytic approximation to the exact exchange-correlation potential based on a variational wave function which shows explicitly the correct behavior, combining in a neat way lattice and continuum methods.

Figure 1

Vanishing of screening effects at large atomic number in the atomic limit. The $1s$ Hubbard $U$ is shown vs the atomic number. In red, we show the bare value $U_0$ from Coulomb integrals [Eq. (A3)], while the blue dots are the “screened” value obtained as $U=E_I^{Z,Z}-E_I^{Z,Z-1}$ where $E_I^{Z,n}$ is the $n\mathrm{th}$ ionization energy [36] for atomic number $Z$. ($E_I^{1,0}\equiv E_A$ is the affinity energy for hydrogen.) The horizontal line is the cost $\mathrm{\Delta }$ of a charge fluctuation $1s^{1}1s^1\to 1s^{1}2s^1$.

Figure 2

Hopping reduction factor. (a) Exact results (full lines) as a function of $U/U_{\mathrm{BR}}$ for the Hubbard model in dimension $D$. Here, $U_{\mathrm{BR}}$ is a measure of the effective noninteracting bandwidth [40], $U_{\mathrm{BR}}\equiv \frac{16}{N}{\sum }_{k\in \mathrm{occ}.}{\epsilon }_k$ where ${\epsilon }_k$ are the single-particle noninteracting energies and the sum is restricted to the occupied states. We show results for a two-site system (black), for an infinite chain (red), and for the Bethe lattice in infinite dimensions (blue). The Bethe lattice data were derived from numerical [41] and analytic [42] results (full blue line). The cusp signals $U_{\mathrm{c}2}$ where the Mott-Hubbard transition occurs from the metallic phase ($U<U_{\mathrm{c}2}$) to the insulating phase ($U>U_{\mathrm{c}2}$). The dashed line was obtained with the Gutzwiller wave function for the same Bethe lattice. (b) Hopping reduction factor obtained with a Gutzwiller-type wave function in a hydrogenic molecule as a function of atom separation $a$ for different atomic numbers. The dots (squares) correspond to the cases considered in Figs. 5 and 5 [Figs. 5, 5, and 3].

Figure 3

Charge along the bond for interatomic distance $a=4.5$ ($a_B{Z}^{-1}$) and various values of $Z$. The inset shows the interacting charge minus the nointeracting charge ($Z=\infty$). Charges where computed within full CI (see Appendix pp8).

Figure 4

Schematic plot of different scalings of the charge density in atomic units as a function of the position in units of the Bohr radius. The black dots indicate the position of the atoms. (a) The scaled distance is fixed at $a=6$ ($a_B{Z}^{-1}$). As $Z$ grows, the atoms approach each other in atomic units and the Hubbard interaction decreases as $U/t\sim 1/Z$. In scaled units, this panel will be qualitatively similar to Fig. 3. (b) The distance in atomic units is fixed at $\tilde{a}=6a_B$ the scaled distance is shown in the legend. Here, $U/t\sim e^Z$ due to the exponential suppression of tunneling. (c) The scaled distance is fixed at $a=6$($a_B{Z}^{-1}$) for $Z=1$ and then it is changed as $a\sim lnZ$ so that $U/t\sim 70$ is kept constant (see legend). As $Z$ grows, the atoms approach each other in atomic units and the density increases in the bond region, but both more slowly than in (a). Notice that in cases (a) and (c) the density in the bond is driven to the high-density limit where conventional functionals converge to the noninteracting limit despite $U/t\sim 70$ in case (b) corresponds to a highly stretched bond. Case (a) is the usual DFT scaling [46] to high density [Eq. (10)] and case (b) is the extreme stretched case, while case (c) is the scaling considered in this work.

Figure 5

Hartree-exchange-correlation potential for a two-site system. The potential is plotted along the bond direction $x$ for $z=y=0$ and different values of $a$ and $Z$. Panels (a) and (c) show the potential obtained inverting the full CI density (dots) and in the L+REP approximation using orthogonalized atomic orbitals (lines). Open circles indicate the positions of the ions. Panels (b) and (d) show the LDA [43] results corresponding to panels (a) and (c). The strength of the correlations for each case is defined by the value of $q$ in Fig. 2.

Figure 6

Hartree-exchange-correlation potential for the stretched hydrogen molecule. We show $v_{\mathrm{Hxc}}$ as a function of $x$ for $z=y=0$ for different values of $R$. The dots in (a) were obtained inverting the full CI ground-state density as explained in Appendix pp8 while the lines are the L+REP results. Open circles indicate the position of the ions. Panel (b) shows the LDA results. The cases $a=1.4$ and 3($a_B{Z}^{-1}$) were shifted by 0.5 and 0.25 ($Z^2$ Ha), respectively, for clarity. The first two cases are in the weak and intermediate correlation regimes while the case $a=6$($a_B{Z}^{-1}$) is in the strong-coupling regime [cf. Figs. 2 and 7].

Figure 7

Height of the Hartree exchange-correlation potential at the bond midpoint for the two-site system as a function of internuclear separation for different $Z$. In (a) full lines are obtained using the L+REP approximation subtracting a small spurious term at infinity [Eqs. (D21) and (D22)]. Dots are obtained inverting the full CI charges. For $Z=1$, we include also data obtained with an accurate variational wave function (Appendix pp8). Diamonds are from Ref. [55]. (b) Shows separately the anomalously scaling Mott barrier height contribution, Eq. (D21), and the contribution scaling as $1/\left(Za\right)$ (cond) from Eq. (D22). The dashed lines are the LDA results.

Figure 8

Mott barriers for a four-atom chain. Panel (a) shows $v_{\mathrm{Hxc}}$ along the path shown in the inset for four H atoms arranged in a square with interatomic distance $a=6a_B$. Panel (b) compares the interacting charge computed with an accurate quantum chemistry method (CC) and the charge from the solution of the Kohn-Sham potential obtained with the L+REP method. Both are displayed as the difference between the interacting and the noninteracting charge ($Z=\infty$) along one side of the square with the origin at the bond midpoint.