Construction of meta-GGA functionals through restoration of exact constraint adherence to regularized SCAN functionals

The strongly constrained and appropriately normed (SCAN) meta-GGA exchange-correlation functional [Sun et al., Phys. Rev. Lett. 115, 036402 (2015)] is constructed as a chemical environment-determined interpolation between two separate energy densities: one describes single-orbital electron densities accurately and another describes slowly varying densities accurately. To conserve constraints known for the exact exchange-correlation functional, the derivatives of this interpolation vanish in the slowly varying limit. While theoretically convenient, this choice introduces numerical challenges that degrade the functional's efficiency. We have recently reported a modification to the SCAN meta-GGA, termed restored-regularized-SCAN (r²SCAN) [Furness et al., J. Phys. Chem. Lett. 11, 8208 (2020)], that introduces two regularizations into SCAN, which improve its numerical performance at the expense of not recovering the fourth order term of the slowly varying density gradient expansion for exchange. Here, we show the derivation of a progression of density functional approximations [regularized SCAN (rSCAN), r++SCAN, r²SCAN, and r⁴SCAN] with increasing adherence to exact conditions while maintaining a smooth interpolation. The greater smoothness of r²SCAN seems to lead to better general accuracy than the additional exact constraint of SCAN or r⁴SCAN does.