Determination of the parameter q_k without hybrid-functional supercell calculations

Identify a practical approach for determining the fractional charge parameter q_k—defined by the condition that the polaron energy level equals its piecewise-linear value—without resorting to hybrid-functional calculations on the polaron supercell, for example by using semilocal density functional theory or alternative probes.

Background

In the unified hybrid-functional formulation, the many-body self-interaction correction depends on parameters α_k and q_k. While α_k can be estimated nonempirically (e.g., via dielectric constants or hydrogen probe techniques), evaluating q_k typically requires tracking how the polaron level changes with both functional parameter and fractional charge, which the authors note generally involves hybrid-functional supercell computations.

The authors present a semilocal formulation that avoids dependence on q_k, but they explicitly state that finding q_k without hybrid calculations remains unclear, indicating a methodological gap for users who wish to retain the hybrid-based framework without incurring its computational cost.