Optical bounds on many-electron localization

We establish rigorous inequalities between different electronic properties linked to optical sum rules, and organize them into weak and strong bounds on three characteristic properties of insulators: electron localization length \ell <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℓ</mml:mi> </mml:math> (the quantum fluctuations in polarization), electric susceptibility \chi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>χ</mml:mi> </mml:math> , and optical gap E_{\mathrm{G}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>E</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>G</mml:mi> </mml:mstyle> </mml:msub> </mml:math> . All-electron and valence-only versions of the bounds are given, and the latter are found to be more informative. The bounds on \ell <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℓ</mml:mi> </mml:math> are particularly interesting, as they provide reasonably tight estimates for an ellusive ground-state property – the average localization length of valence electrons – from tabulated experimental data: electron density, high-frequency dielectric constant, and optical gap. The localization lengths estimated in this way for several materials follow simple chemical trends, especially for the alkali halides. We also illustrate our findings via analytically solvable harmonic oscillator models, which reveal an intriguing connection to the physics of long-ranged van der Waals forces.