Quantum-Information Measure of Electron Localization

Abstract

Understanding electron localization in molecules and materials plays a central role in electronic structure theory, and will increase in importance with the rise of data driven approaches. The electron localization function (ELF) is widely used to visualize electron organization in molecules and materials, and it remains a central ingredient in modern density functional approximations. Yet its formulation retains highly empirical elements. Here we introduce a fully non empirical measure of electron localization derived from the concurrence of a correlated two spin mixed state. This construction yields a genuine two point localization indicator grounded in quantum information theory, removing the ad hoc steps underlying the ELF. We show that atomic shells, covalent and ionic bonds, lone pairs, molecular dissociation, and charge transfer processes are captured. The method is straightforward to evaluate numerically.

Figures (5)

• Figure 1: $\widetilde{{\mathcal{C}}}(z_1,z_2)$ for H$_2$ at the CASSCF(2,2) level. The coordinates $z_1$ and $z_2$ are along the internuclear axis so that nuclei are at corners. Positions $z_1$ and $z_2$ are sampled along the internuclear axis between the nuclei. Left: near equilibrium ($R=0.5$ Å). Right: stretched configuration ($R=4.0$ Å).
• Figure 2: $\widetilde{{\mathcal{C}}}(z_1,z_2)$ for F$_2$. The coordinates $z_1$ and $z_2$ are along the internuclear axis so that nuclei are at corners. Top: RHF; bottom: CASSCF(2,2). Left: near the equilibrium geometry ($R=1.5$ Å). Right: stretched configuration ($R=5.0$ Å).
• Figure 3: $\widetilde{{\mathcal{C}}}(z_1,z_2)$ for N$_2$. The coordinates $z_1$ and $z_2$ are along the internuclear axis so that nuclei are at corners. Top: RHF; bottom: CASSCF(6,6). Left: near equilibrium ($R=1.0$ Å). Right: stretched geometry ($R=5.0$ Å). Sampling along the internuclear axis.
• Figure 4: $\widetilde{{\mathcal{C}}}(z_1,z_2)$ for N$_2$ at $R=2$ Å. Nuclei are at $z=-1$ and $z=1$ Å. Sampling extends across and beyond the atomic regions Left: RHF; right: CASSCF(6,6).
• Figure 5: $\widetilde{{\mathcal{C}}}(z_1,z_2)$ for LiF from SA‑CASSCF(2,2). The coordinates $z_1$ and $z_2$ are along the internuclear axis so that nuclei are at corners (Li on the left, F on the right). Left: ground state $S_0$. Right: first excited singlet $S_1$. Top: $R=3.8$ Å; bottom: $R=4.8$ Å. $z_1$ and $z_2$ are sampled along the internuclear axis. The avoided crossing occurs at SA-CASSCF(2,2) near $R\approx 4.3$ Å.