Practical auxiliary basis implementation of Rung 3.5 functionals

Approximate exchange-correlation functionals for Kohn-Sham density functional theory often benefit from incorporating exact exchange. Exact exchange is constructed from the noninteracting reference system's nonlocal one-particle density matrix γ(r^-vector,r^-vector′). Rung 3.5 functionals attempt to balance the strengths and limitations of exact exchange using a new ingredient, a projection of γ(r^-vector,r^-vector ′) onto a semilocal model density matrix γ_SL(ρ(r^-vector),∇ρ(r^-vector),r^-vector−r^-vector ′). γ_SL depends on the electron density ρ(r^-vector) at reference point r^-vector, and is closely related to semilocal model exchange holes. We present a practical implementation of Rung 3.5 functionals, expanding the r^-vector−r^-vector ′ dependence of γ_SL in an auxiliary basis set. Energies and energy derivatives are obtained from 3D numerical integration as in standard semilocal functionals. We also present numerical tests of a range of properties, including molecular thermochemistry and kinetics, geometries and vibrational frequencies, and bandgaps and excitation energies. Rung 3.5 functionals typically provide accuracy intermediate between semilocal and hybrid approximations. Nonlocal potential contributions from γ_SL yield interesting successes and failures for band structures and excitation energies. The results enable and motivate continued exploration of Rung 3.5 functional forms