Suppressing the gauge problem in local hybrid functionals without a calibration function: The choice of local mixing function

Modern functionals based on the exact-exchange (EXX) energy density like local hybrid functionals (LHs) or range-separated LHs have recently received additional attention due to their advantages over established functionals when it comes to the local balance between self-interaction errors and static-correlation errors. A possible theoretical drawback of such functionals over the years has been the so-called gauge problem due to the inherent ambiguity of exchange-energy densities. Modern LHs like LH20t or more sophisticated functionals based thereon have been constructed using suitably optimized calibration functions (CFs) to minimize the mismatch of the semi-local and EXX energy densities. Here, we show that the unphysical contributions arising from the gauge problem may also be reduced significantly without a CF by tailoring the position-dependence of the EXX admixture (local mixing function, LMF) in a way to suppress spurious positive energy-density contributions locally in space. This is achieved by building the so-called x-LMFs upon the ratio between EXX and semi-local exchange-energy densities. The resulting LH24x functional provides similar accuracy, e.g., for the GMTKN55 test suite, as LH20t, but without introduction of a CF! We provide detailed comparative analyses of integrated energies and spatially resolved energy densities. The good performances of LHs for chemically relevant energy differences are to some extent due to the core nature of unphysical artifacts that cancel out efficiently.