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## New Density Functionals

In many areas of application, Kohn-Sham density functional theory is indispensable, as it provides a unique combination of computational efficiency and accuracy. Nevertheless, the existing exchange-correlation functionals have clear accuracy limitations, in particular when it comes to universal applicability to different systems and different properties. For example, standard hybrid functionals like B3LYP, which include a spatially constant fraction of exact (Hartree-Fock-type) exchange, lack this universality. Relatively low fractions of exact exchange appear optimum for thermochemistry, larger amounts appear necessary to get good reaction barriers, magnetic properties and bonding in transition-metal systems or certain classes of excitations in TDDFT calculations. There is thus a need to construct improved functionals that still are computationally efficient yet more accurate than existing ones. We study several classes of "occupied-orbital-dependent" functionals in this context.

### Development of local hybrid functionals with position-dependent exact-exchange admixture

In so-called "local hybrid functionals" [1] (see also [2]), the exact-exchange admixture is done in a position-dependent rather than spatially uniform way (Figure 1). The position dependence is governed by a "local mixing function" (LMF). We have proposed two classes of LMFs that have provided for the first time accurate results for thermochemistry and reaction barriers [3,4,5]. The first and currently most successful class uses a scaled ratio* t* of von Weizsäcker kinetic energy density and noninteracting local kinetic energy density ("*t*-LMFs [3,5]), the second class depends on the dimensionless density gradient *s* (*s*-LMFs [4,5]). The two types of LMFs may also be combined to provide excellent accuracy for both thermochemistry and barriers, and at the same time the correct long-range asymptotic behavior. More recent developments include a) LMFs including spin-polarization as an additional variable [6], b) a combination of local hybrids with Grimme’s DFT-D3 dispersion correction terms [7], and c) a new generation of local hybrids based on a common LMF for both spin channels and improvements to the correlation functional of local hybrids involving range separation and self-interaction corrections to the short-range part [8]. Another notable direction is the construction of LMFs within a first-principles framework, based on a local version of the adiabatic connection [9].

Self-consistent implementations are already available based on either the fully nonlocal exact-exchange potential [10] or with the LHF/CEDA approximation to the optimized effective potential (OEP) [11]. The latter has been used to compute nuclear shielding constants within an uncoupled Kohn-Sham framework [12], whereas g-tensors have been implemented within a coupled-perturbed KS framework based on the former implementation [13]. Our current efforts concentrate on a development version of the Turbomole package.

### References

[1] J. Jaramillo, G. E. Scuseria, M. Ernzerhof *J. Chem. Phys.* **2003**, *118*, 1068.

[2] J. P. Perdew and K. Schmidt in V. van Doren and C. van Alsenoy (Eds.), *Density Functional Theory and its Application to Materials*, AIP Conference Proceedings, Vol. 577 (AIP, Melville, New York, 2001). F. G. Cruz, K.-C. Lam, K. Burke *J. Phys. Chem. A* **1998**, *102, *4911.

[3] *A thermochemically competitive local hybrid functional without gradient correction* H. Bahmann, A. Rodenberg, A. V. Arbuznikov, M. Kaupp *J. Chem. Phys.* **2007**, *126,* 011103/1-4.

[4] *Local hybrid exchange-correlation functionals based on the dimensionless density gradient *A. V. Arbuznikov, M. Kaupp *Chem. Phys. Lett.* **2007**, *440*, 160-168.

[5] *Local hybrid functionals: An assessment for thermochemical kinetics,* M. Kaupp, H. Bahmann, A. V. Arbuznikov *J. Chem. Phys.* **2007**, *127*, 194102/1-12.

[6] *Local hybrid functionals with an explicit dependence on spin polarization* A. V. Arbuznikov, H. Bahmann, M. Kaupp *J. Phys. Chem. A* **2009,** *113*, 11891-11906.

[7] *Evaluation of a combination of local hybrid functionals with DFT-D3 corrections for the calculation of thermochemical and kinetic data* K. Theilacker, A. V. Arbuznikov, H. Bahmann, M. Kaupp *J. Phys. Chem. A ***2011**, *115*, 8990-8996.

[8] *Importance of the correlation contribution for local hybrid functionals: range separation and self-interaction corrections* A. V. Arbuznikov, M. Kaupp *J. Chem. Phys.* **2012**, *136*, 014111/1-13.

[9] *What Can We Learn from the Adiabatic Connection Formalism about Local Hybrid Functionals?* A. V. Arbuznikov, M. Kaupp *J. Chem. Phys.* **2008**, *128*, 214107/1-12.

[10] See, e.g. a) *On occupied-orbital dependent exchange-correlation functionals. From local hybrids to Becke’s B05 model* A. V. Arbuznikov, M. Kaupp *Z. Phys. Chem.* **2010**, *224*, 545-567; b) *Advances in local hybrid exchange-correlation functionals: From thermochemistry to magnetic-resonance parameters and hyperpolarizabilities* A. V. Arbuznikov, M. Kaupp *Int. J. Quantum Chem.* **2011**, *111*, 2625-2638, and references cited therein.

[11] *From local hybrid functionals to "localized local-hybrid" potentials: Formalism and thermochemical tests* A. V. Arbuznikov, M. Kaupp, H. Bahmann *J. Chem. Phys.* **2006**, *124*, 204102/1-15.

[12] *Coupled-Perturbed Scheme for the Calculation of Electronic g-Tensors with Local Hybrid Functionals* A. V. Arbuznikov, M. Kaupp *J. Chem. Theory Comput.* **2009**, *5*, 2985-2995.

[13] *Nuclear shielding constants from localized local hybrid exchange-correlation potentials* A. V. Arbuznikov, M. Kaupp *Chem. Phys. Lett.* **2007**, *442*, 496-503.

### Work on Becke's real-space model of nondynamical correlation (B05)

Another attractive "occupied-orbital-dependent" functional is the B05 model by Becke et al.. Here nondynamical correlation is modelled in coordinate space. This functional is rather complicated, and a main challenge consisted in the self-consistent implementation. After some preliminary steps into this direction [14], this has been achieved recently [10a,15]. We have also used insight from the study of the B05 functional in designing our most recent local hybrids (see above [6,8]).

[14] *Normalization of the effective exchange hole in Becke’s nondynamical correlation model: Closed-form analytic representation* A. V. Arbuznikov , M. Kaupp *J. Mol. Struct., Theochem* **2006**, *762*, 151-153.

[15] *On the self-consistent implementation of general occupied-orbital dependent exchange-correlation functionals with application to the B05 functional* A. V. Arbuznikov, M. Kaupp J. *Chem. Phys.* **2009**, *131*, 084103/1-12.