Density double-hybrid functionals

In order to make improvements over the LSDA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not only the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA) and is the second rung on Jacob s Ladder. The third rung, meta-GGA functional, includes a dependence on the Laplacian of the density (V p) or on the orbital kinetic energy density (t). The fourth row, the hyper-GGA or hybrid functionals, includes a dependence on exact (HF) exchange. Finally, the fifth row incorporates the unoccupied Kohn-Sham orbitals. This is most widely accomplished within the so-called double hybrid functionals. 

In [170] the authors obtain a test set of ten molecules of specific atmospheric interest in order to evaluate the performance of various Density Functional Theory (DFT) methods in (hyper)polarizability calculations as well as established ab initio methods. The authors make their choice for these molecules based on the profound change in the physics between isomeric systems, the relation between isomeric forms and the effect of the substitution. In the evaluation analysis the authors use arguments chosen from the information theory, the graph theory and the pattern recognition fields of Mathematics. The authors mentioned the remarkable good performance of the double hybrid functionals (namely B2PLYP and mPW2PLYP) which are for the first time used in calculations of electric response properties. 

S. Kofimann, B. BCirchner, F. Neese. Performance of modern density functional theory for the prediction of hyperfine structure meta-GGA and double hybrid functionals. Mol Phys., 105 (2007) 2049-2071. 

Because of its computational efficiency and good results for molecular properties, the MP2 method is one of the two most commonly used methods for including correlation effects on molecular ground-state equilibrium properties. (The other widely used method is the density-functional method—Section 16.5.) Also, in perhaps the most accurate form of density-functional theory, the MP2 correlation energy (16.13) is multiplied by an empirical constant and added to the density functional energy to give improved results see the discussion of double-hybrid functionals in Section 16.5. 

J.M.L. (2008) Highly accurate first-principles benchmark data sets for the parametrization and validation of density functional and other approximate methods. Derivation of a robust, generally applicable, double-hybrid functional for thermochemistry and thermochemical kinetics.. Phys. Chem. A, 112, 12868-12886. 

T. Schwabe and S. Grimme, Double hybrid density functionals with long range dispersion corrections Higher accuracy and extended applicability. Phys. Chem. Chem. Phys. 9, 3397 3406 (2007). 

Goerigk L, Grimme S (2009) Calculation of electronic circular dichroism spectra with time-dependent double-hybrid density functional theory. J Phys Chem A 113 767-776... 

Grimme s third major methodological development is the notion of double hybrid density functionals. This development came when he recognized that the principle of perturbation theory, expressed through MP2, could be appfied to DFT. He characterizes this realization as not a big step. ... 

Kozuch, S. Gruzman, D. Martin, J. M. L. DSD-BLYP a general purpose double hybrid density functional including spin component scaling and dispersion correction, J. Phys. Chem. C2010,114, 20801-20808. 

Grimme extended the double hybrid density functional formalism to be able to compute BCD. For a test suite of six molecules, performance of B2PLYP is notably better than for B3LYP. 

Goerigk L, Grimme S (2011) Efficient and accurate double-hybrid-meta-GGA density functionals-evaluation with the extended GMTKN30 database for general main group thermochemistry kinetics and noncovalent interactions. J Chem Theory Comput 7 291-309... 

Neese F, Schwabe T, Grimme S (2007) Analytic derivatives for perturbatively corrected Double hybrid density functionals theory implementation and applications. J Chem Phys 126 124115... 

Goerigk L, MoeUmann J, Grimme S (2009) Computation of accurate excitation energies for large organic molecules with double-hybrid density functionals. Phys Chem Chem Phys 11 4611-4620... 

Chai, J., and Head-Gordon, M. (2009). Long-range corrected double-hybrid density functionals, / Chem. Phys. 131, p. 174105, doi 10.1063/l. 3244209. 

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