We formally classify density functionals that utilize the full density Hessian as Hessian-level meta-generalized gradient approximations (HL-MGGAs), distinguishing them from standard meta-GGAs that rely on the orbital-dependent kinetic energy density or the density Laplacian. Within this broader class, we introduce a simplified, nonempirical functional, $\vartheta \text{−}\mathrm{PBE}$, and detail the general self-consistent implementation of HL-MGGAs within the projector-augmented-wave (PAW) method. By utilizing the complete set of spatial second-order density derivatives, the functional's underlying descriptor successfully distinguishes between distinct one-electron density limits, such as single-center atomic densities and two-center bonds, that standard iso-orbital indicators often conflate. Benchmarks across molecular and solid-state datasets reveal that while $\vartheta \text{−}\mathrm{PBE}$ delivers accurate chemisorption energies and molecular properties, challenges remain in predicting bulk lattice constants. Ultimately, this work demonstrates the physical utility and feasibility of designing orbital-independent, Hessian-based exchange-correlation functionals.

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• Density functional theory
• Electron-correlation calculations
• GGA
• Meta-GGA