Assessing locally range-separated hybrid functionals from a gradient expansion of the exchange energy density

Locally range-separated hybrid (LRSH) functionals feature a real-space-dependent range separation function (RSF) instead of a system-independent range-separation parameter, which thus enables a more flexible admixture of exact exchange than conventional range-separated hybrid functionals. In particular, the development of suitable RSF models and exploring the capabilities of the LRSH approach, in general, are tasks that require further investigations and will be addressed in this work. We propose a non-empirical scheme based on a detailed scaling analysis with respect to a uniform coordinate scaling and on a short-range expansion of the range-separated exchange energy density to derive new RSF models from a gradient expansion of the exchange energy density. After optimizing a small set of empirical parameters introduced to enhance their flexibility, the resulting second- and fourth-order RSFs are evaluated with respect to atomic exchange energies, atomization energies, and transition barrier heights.