Efficient Treatment of Local Meta-generalized Gradient Density Functionals via Auxiliary Density Expansion: The Density Fitting J + X Approximation

We report an efficient technique to treat density functionals of the meta-generalized gradient approximation (mGGA) class in conjunction with density fitting of Coulomb terms (DF-J) and exchange-correlation terms (DF-X). While the kinetic energy density τ cannot be computed in the context of a DF-JX calculation, we show that the Laplacian of the density υ can be computed with almost no extra cost. With this technique, υ-form mGGAs become only slightly more expensive (10%-20%) than GGAs in DF-JX treatment - and several times faster than regular τ-based mGGA calculations with DF-J and regular treatment of the density functional. We investigate the translation of υ-form mGGAs into τ-form mGGAs by employing a kinetic energy functional but find this insufficiently reliable at this moment. However, υ and τ are believed to carry essentially equivalent information beyond ρ and ||∇ρ|| (Phys. Rev. B 2007, 75, 155109, DOI: 10.1103/PhysRevB.75.155109), so a reparametrization of accurate mGGAs from the τ-form into the υ-form should be possible. Once such functionals become available, we expect the presented technique to become a powerful tool in the computation of reaction paths, intermediates, and transition states of medium sized molecules.