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Theoretical Chemistry – Quantum ChemistryNew density functionals

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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

Figure 1: Principle of 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.

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[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.

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