Strictly-correlated-electron approach to excitation energies of dissociating molecules

In this work we consider a numerically solvable model of a two-electron diatomic molecule to study a recently proposed approximation based on the density functional theory of so-called strictly correlated electrons (SCE). We map out the full two-particle wave function for a wide range of bond distances and interaction strengths and obtain analytic results for the two-particle states and eigenenergies in various limits of strong and weak interactions, and in the limit of large bond distance. We then study the so-called Hartree-exchange-correlation (Hxc) kernel of time-dependent density functional theory which is a key ingredient in calculating excitation energies. We study an approximation based on adiabatic SCE (ASCE) theory which was shown to display a particular feature of the exact Hxc kernel, namely, a spatial divergence as function of the bond distance. This makes the ASCE kernel a candidate for correcting a notorious failure of the commonly used adiabatic local density approximation (ALDA) in the calculation of excitation energies of dissociating molecules. Unlike the ALDA, we obtain nonzero excitation energies from the ASCE kernel in the dissociation regime but they do not correspond to those of the true spectrum unless the interaction strength is taken to be very large such that the SCE theory has the right regime of validity, in which case the excitation energies become exact and represent the so-called zero-point oscillations of the strictly correlated electrons. The commonly studied physical dissociation regime, namely, large molecular separation at intermediate interaction strength, therefore remains a challenge for density functional approximations based on SCE theory.

Figure 1

The ground-state density $n$ as a function of the dimensionless coordinate $z=2\pi x/L$. The densities for various $L$ values are denoted by $n_L$ in the plot. The corresponding KS potential $v_s/L^2$ [Eq. (13)] is plotted in arbitrary units for comparison and indicated by a dashed line. For large $L$, we obtain two peaks of fixed width while for small $L$ the system becomes homogeneous.

Figure 2

Selected KS orbitals as a function of the dimensionless coordinate $z=2\pi x/L\in [-\pi ,\pi ]$ plotted for the bond distance $L/2=10.5$. We display the ground state and first few excited states corresponding to the bonding and antibonding orbital pairs represented by the pair of Mathieu functions $C_0$ and $S_1$ as well as the pair $C_1$ and $S_2$. For this bond distance, the bonding and antibonding orbitals coincide for positive $z$.

Figure 3

The ground- and first-excited-state wave functions for interaction strength $\lambda =1$ plotted for $L=9$ and 21. The rightmost panels display the corresponding ground-state densities.

Figure 4

The potential $v_{\lambda }$ for $\lambda =1$ for $L=21$ compared to $v_s$.

Figure 5

The ground- and excited-state wave functions at large interaction strength $\lambda =1000$ for the bond distances $L/2=3$ and 10.5. We note that the wave function localizes in narrow strips along the lines $|x_1-x_2|=L/2$. The rightmost panels display the corresponding ground-state densities.

Figure 6

The potential $v_{\lambda }$ for $\lambda =1000$ for $L=21$ compared to $v_s/4$.

Figure 7

The ASCE kernel for $L=1,10,20$. We see that with growing $L$ plateaus develop the heights of which grow exponentially with $L$.

Figure 8

The function $\mathcal{P}\left(x\right)/\mathcal{P}\left(0\right)$ for $L=10$ (left panel) and $L=20$ (right panel) where we clearly see a step structure appearing for increasing $L$.