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Development and Implementation of Modern Density Functional Methods for Property Calculations

Development and Implementation of Modern Density Functional Methods for Property Calculations Martin Kaupp , Alexei V. Arbuznikov Universität Würzburg. SPP1145 Antragskolloquium, Bonn, 4.6.2005. Development and Implementation of Modern Density Functional Methods for Property Calculations

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Development and Implementation of Modern Density Functional Methods for Property Calculations

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  1. Development and Implementation of Modern Density Functional Methods for Property Calculations Martin Kaupp, Alexei V. Arbuznikov Universität Würzburg SPP1145 Antragskolloquium, Bonn, 4.6.2005

  2. Development and Implementation of Modern Density Functional Methods for Property Calculations Martin Kaupp, Alexei V. Arbuznikov Universität Würzburg Report on completed projects: -shortcomings of standard functionals in calculations of magnetic resonance parameters -„localized hybrid potentials“ (LHPs) -performance of LHPs for: nuclear shieldings of main-group systems electronic g-tensors of TM complexes difficult hyperfine couplings SPP1145 Antragskolloquium, Bonn, 4.6.2005

  3. Development and Implementation of Modern Density Functional Methods for Property Calculations Martin Kaupp, Alexei V. Arbuznikov Universität Würzburg Planned work and work in progress: -„double local hybrid“ potentials (position-dependent EXX admixture) -work towards a self-consistent implementation of Becke‘s nondynamical correlation model -implementation of OEP-based potentials into a relativistic non-collinear 2-component ansatz SPP1145 Antragskolloquium, Bonn, 4.6.2005

  4. The Reasonable Performance of Hybrid Functionals: A Starting Point -Ex covers about 85-95% of Exc -exact exchange is self-interaction-free -based on 100% exact exchange, inclusion of nondynamical correlation is difficult; typical Ec cover only dynamical correlation -approximate LDA- or GGA-type Ex functionals include spurios self-interactions but also significant non-dynamical correlation • appropriate mixing of GGA and exact exchange provides a reasonable compromise between minimizing self-interaction and including non-dynamical correlation

  5. – angular momentum operator - PSO operator – spatial part of spin-orbit operator Calculation of second-order magnetic response properties Electronic g-tensors: Nuclear shielding tensors: We need vxc!

  6. Slope RC SD SVWN 0.568 0.986 79 BPW91 0.546 0.966 121 B3LYP 0.945 0.993 93 FT98 0.381 0.887 163 PKZB 0.561 0.980 93 SAOP 0.334 0.884 143 ideal 2000 1750 1500 1250 1000 750 500 250 d(57Fe), calcd. / ppm 0 -250 -500 -750 -1000 -750 -500 -250 0 250 500 750 1000 1250 1500 1750 d(57Fe), exptl. / ppm Performance of DFT for δ(57Fe) Chemical Shifts of Organoiron Complexes /ppm vs. Fe(CO)5 1 - Fe(C4H4)(CO)3 2 – Fe(CO)5 3 – Fe(CO)3(CH2CHCHCH2) 4 - Fe(CO)4(CH2CHCN) 5 – Fe(CO)2Cp(CH3) 6 – Fe(CO)3(CH2CHCHO) 7 – FeCp2 Phys. Chem. Chem. Phys.2002, 4, 5467. See also: M. Bühl Chem. Phys. Lett.1997, 267, 251.

  7. Performance of different functionals in calculating g-shift components of 3d complexes 300 BP86 300 200 250 200 100 B3PW91 g (calc.) /ppt 150 g (calc.) /ppt 100 0 D 50 D slope 0.40, R = 0.972 -100 0 slope 0.58, R = 0.981 -50 -200 -100 -200 -100 0 100 200 300 -150 Dg (exp.) /ppt -200 -100 0 100 200 300 Dg (exp.) /ppt improved slope by exact-exchange admixture but: potential problems with spin contamination TiF3, CrOF4-, MnO3, Fe(CO)5+, Co(CO)4, Ni(CO)3H, Cu(acac)2, Cu(NO3)2 J. Comput. Chem.2002, 23, 794. 300 200 BHPW91 g (calc.) /ppt 100 D 0 slope 0.99, R = 0.956 -100 -200 -200 -100 0 100 200 300 Dg (exp.) /ppt

  8. Calculated Isotropic Hyperfine Coupling Constant for the4P GS of the Phosphorus Atom hybrid 50% nx,HF hybrid ca. 20% nx,HF GGA Uncontracted 20s15p4d2f Partridge basis set.

  9. problem with self-consistent implementation: nxexact (nxHF) is non-local  violation of Kohn-Sham framework!  unnecessary coupling terms in magnetic property calculations often too large linear response for main group species solution: 1) require nxexact to be local and multiplicative 2) use optimized effective potential (OEP) framework (LHF/CEDA)  localized hybrid potentials  higher accuracy, no coupling terms possible for any hybrid functional! localized hybrid potentials in ReSpect: AVA, MK Chem. Phys. Lett.2004, 386, 8. AVA, MKChem. Phys. Lett.2004, 391, 16 (open-shell). AVA, MKInt. J. Quantum Chem.2005, 102, 261. See also, e.g.: W. Hieringer, F. Della Sala, A. Görling Chem. Phys. Lett.2004, 383, 115. A. M. Teale, D. J. Tozer Chem. Phys. Lett.2004, 383, 109. Localized Hybrid Potentials: A New Class of Exchange-Correlation Potentials „classical“ hybrid functionals, e.g. B3LYP:

  10. nonlocal CDFT local UDFT B3PW91 a0=0.2 B-EXX-PW91 a0=0.5 B3(L)-PW91 B-EXX(L)-PW91 B-PW91 (GGA) a0= 0.5 a0= 0.6 a0= 0.7 intercept B -60.7 -95.6 -26.4 -5.4 0.8 6.8 -38.8 (ppm) slope A 1.176 1.350 1.051 1.013 1.001 0.990 1.074 regression 0.9923 0.9753 0.9983 0.9988 0.9987 0.9984 0.9977 coefficient standard deviation 60.2 125.5 24.9 20.5 21.3 22.9 30.3 (ppm) Localized Hybrid Potentials for Nuclear Shieldings (linear regression analysis for 22 main group compounds) A. V. Arbuznikov, M. Kaupp Chem. Phys. Lett.2004, 386, 8. See also: W. Hieringer, F. Della Sala, A. Görling Chem. Phys. Lett.2004, 383, 115. A. M. Teale, D. J. Tozer Chem. Phys. Lett.2004, 383, 109.

  11. ideal 1200 800 400 0 local scalc in ppt -400 -800 -1200 non-local -1600 -2000 -2400 -1400 -1000 -600 -200 200 600 1000 sexp in ppt Localized Hybrid Potentials with 50% Exact Exchange Nuclear shieldings Chem. Phys. Lett.2004, 386, 8. Nuclear shielding tensors for 22 main group molecules (32 values)

  12. nonlocal CDFT local UDFT B3-PW91 a0=0.2 B-EXX-PW91, a0=0.5 B3(L)-PW91 B-EXX(L)-PW91 B-PW91 (GGA) a0= 0.4 a0= 0.5 a0= 0.6 intercept B 5.9 17.0 3.2 0.3 -2.3 -7.5 5.1 (ppm) slope A 0.606 0.988 0.537 0.649 0.733 0.864 0.449 regression 0.978 0.932 0.977 0.985 0.983 0.961 0.966 coefficient standard deviation (ppm) 10.4 30.8 9.4 9.2 11.1 20.1 9.6 Localized Hybrid Potentials with 50% Exact Exchange g-Tensors for 3d Transition Metal Complexes (19 values) A. V. Arbuznikov, M. Kaupp Chem. Phys. Lett.2004, 391, 16.

  13. Localized Hybrid Potentials with 50% Exact Exchange g-Tensors for 3d Transition Metal Complexes 300 ideal 250 non-local 200 150 100 Dgcalc in ppt 50 local 0 -50 -100 -150 -150 -100 -50 0 50 100 150 200 250 300 Dgexp in ppt Chem. Phys. Lett.2004, 391, 16. 9 complexes, 19 values: Co(CO)4, CrOF4-, CrOF4-, Cu(NO3)2, Fe(CO)5+, Mn(CO)5, MnO3, Ni(CO)3H, TiF3

  14. 2p 2s 1s A Challenge: The Hyperfine Coupling of the Nitrogen Atom Ground State (4S) mixture of local exchange and exact exchange, no vc Things are even more complicated for phosphorus!

  15. The next step: Use more flexible hybrids or start from 100% exact exchange?

  16. Essential properties of g(r): g(r) = 0in the homogeneous region g(r) = 1 in the one-electron (asymptotic) region balance non-dynamic correlation is taken into account complete elimination of self-interaction Extending the Concept of Hybrid Functional: Position-Dependent EXX Admixture Local hybrid functional:J. Jaramillo, G. E. Scuseria, M. Ernzerhof J. Chem. Phys.2003, 118,1068.

  17. 1.2 1.0 0.8 0.6 tw(r)/t tw(r)/t 4p2r(r)/N 4p2r(r)/N 0.4 0.2 1.0 0.0 0.8 0 1 2 3 4 5 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 Typical radial behaviour of the ratio W /  - Cl … or for open-shell systems: W, /  , ( = , ) P atom r, Bohr r, Bohr

  18. Spatial behavior of local mixing functions 1.0 0.5 0.0 (tw/t)2 (tw/t)3 (tw/t)4 tw/t Example: O3 molecule We need to examine different LMFs, both analytically and numerically. Collaboration with S. Kümmel, Dresden.

  19. Traditional hybrid functionals and potentials Localized hybrid potentials (local and multiplicative) OEP a0g(r) "Double local hybrid" (DLH) potentials: (localizedpotentials derived from local hybrid functionals) Local hybrid functionals: Self-consistent implementation of local hybrid functionals: "double local hybrid" (DLH) potentials

  20. Self-consistent implementation of local hybrid functionals: "double local hybrid" (DLH) potentials The challenging part is the derivation of the FDO expression:

  21. Resolution of the identity Step from FDO to OEP treatment is analogous to localized hybrid potentials. Initial approximation of "slowly varying LMF“ (already implemented) 3D integration by parts Self-consistent implementation of local hybrid functionals: "double local hybrid" (DLH) potentials

  22. Self-consistent implementation of Becke’s real-space model of nondynamical correlation* The model: Starting from 100% exact exchange, we need to correct for the too delocalized nature of the exchange hole in certain places. nondynamical correlation energy Normalization factor f(r) (derived from inverted BRx89 machinery): x defined implicitly: Slater potential self-consistent implementation related to DLH work exchange hole curvature *A. D. Becke J. Chem. Phys.2003, 119, 2972; J. Chem. Phys. 2005, 122, 064101.

  23. Acknowledgments, Collaborations A. V. Arbuznikov, H. Bahmann Cooperations: A. Görling, W. Hieringer (Erlangen) S. Kurth, E. K. U. Gross (Berlin) S. Kümmel (Dresden) A. D. Becke (Kingston) M. Ernzerhof (Montreal) Development of ReSpect: V. G. Malkin (Bratislava) O. L. Malkina (Bratislava) R. Reviakine Funding Deutsche Forschungsgemeinschaft

  24. Theory and Applications A comprehensive treatment, Wiley-VCH 2004. With 36 chapters on methodology and applications.

  25. Magnetic Resonance Parameters: Various Hamiltonians Effective spin Hamiltonian provides link between magnetic resonance experiment and quantum mechanical treatment: Quantum-mechanical Hamiltonian including perturbations (e.g. Breit-Pauli, Douglas-Kroll, ZORA): Property may often be expressed as second derivative:

  26. Thank you for your attention! Würzburg

  27. Hybrid functionals (traditional implementation) • Attractive points: • yield very accurate thermochemical properties and molecular structures (B3LYP, …); • the only way (currently) to get reasonable magnetic-resonance parameters for transition-metal complexes • Disadvantages: • step outside the Kohn-Sham formalism in an uncontrolled way; • lead to a computationally demanding iterative procedure in the perturbational calculation of the properties like nuclear shielding constants and electronic g-tensors (coupled-perturbed equations); • clear relation between the quality of the orbitals and orbital energies, and the quality of the NMR parameters is lost; • yield significantly too deshielded values for nuclear shieldings of main-group molecules.

  28. V. G. Malkin, O. L. Malkina, M. E. Casida, D. R. Salahub, JACS116, 5898 (1994). DFT calculations of the NMR chemical shieldings with “pure” density functionals (LDA, GGA, 2-dependent meta-GGA):

  29. Density Functional Theory (n3-n4) Semi-Empirical MO-Theory Hartree-Fock (HF), SCF, MO-LCAO n4 Coupled Cluster Methods (CC) Configuration Interaction (CI) Perturbation Theory (MBPT) (e.g. CCSD:n6, CCSD(T):n7 ) (e.g. MP2:n5, MP3:n6, MP4:n7) (e.g. SDCI: n6) “Dressed CI Methods” CPF, ACPF, MCPF, CEPA typically n6 Multiconfiguration SCF (MCSCF, CASSCF, GVB) Multireference Perturbation Theory (CASPT2, MR-MP2, MR-MPn?) Multireference Configuration Interaction (e.g. MR-SDCI, MR-ACPF) Multireference Coupled Cluster Theory nm: formal scaling factors relative to system size n. Note that linear pre-factors are also important Treatment of unperturbed systems: a hierarchy of post-Hartree-Fock methods

  30. "easily manageable" functionals orbital - dependent functionals Classification of the exchange-correlation (xc) functionals (potentials) Abbreviations:S - Slater, 1951; VWN - Vosko-Wilk-Nusair, 1980; P - Perdew, …; B - Becke, LYP - Lee-Yang-Parr, 1988; (P)W - (P) - Wang, 1991; FT - Filatov - Thiel, 1998; (P)KZB - (P) - Kurth - Zupan - Blaha, 1999; T(P)SS - Tao-(P)-Staroverov-Scuseria, 2003; KLI - Krieger-Li-Iafrate, 1992; LHF - "Localized Hartree-Fock", 2001; SAOP - "Statistical Average of Orbital-depending Potentials", 2000 ; GRAC - "GRadient Asymptotic Correction, 2001; B3 - Becke's 3-parametric hybrid scheme, 1993; BH - Becke's "Half-and-half " scheme, 1992.

  31. KLI approximation • J. B. Krieger, Y. Li, G. J. Iafrate, Phys Rev. A45, 101 (1992); 46, 5453 (1992) • not invariant with respect to the unitary transformations of the orbitals Localized Hartree-Fock (LHF) method F. Della Sala, A. Görling, J. Chem. Phys.115, 5718 (2001) Common energy denominator approximation (CEDA) O.V. Gritsenko, E. J. Baerends, Phys Rev. A64, 042506 (2001) …other approaches …(e.g., W. Yang, Q. Wu, Phys. Rev. Lett.89, 143002 (2002)) OEP integral equation: • its numerical solution is computationally expensive; • expansion methods for molecules lead to severe numerical instabilities; • construction of the linear response function is computaionally demanding as well; • any bypass of its direct solving would therefore be strongly appreciated. Solutions:

  32. Nuclear shieldings (ppm) for main-group compounds(IGLO-IV basis set, GIAO) A. V. Arbuznikov, M. Kaupp, Chem. Phys. Lett.386, 8 (2004) Molecule Nucleus Nonlocal coupled-perturbed calculations Local uncoupled DFT calculations Exptl. B3-PW91 B-EXX-PW91, a0=0.5 B3(L)-PW91 B-EXX(L)-PW91 B-PW91 (GGA) a0=0.5 a0=0.6 a0=0.7 C2H4 13C 47.3 51.5 49.5 56.3 58.2 60.1 45.9 64.5 CO 13C -18.0 -21.6 -12.0 -7.3 -6.0 -4.8 -14.1 1 17O -82.7 -85.7 -64.2 -41.9 -34.8 -28.0 -79.2 -62.3 H2CO 13C -24.1 -18.3 -24.5 -19.1 -17.6 -16.3 -26.5 -4.43 HCN 13C 69.9 70.0 72.6 76.1 77.0 77.9 71.1 82.1 15N -47.8 -48.8 -36.1 -21.5 -17.1 -12.7 -44.6 -20.4 H2S 33S 713.3 722.6 719.6 738.6 744.1 749.5 709.0 752.0 N2 15N -91.7 -100.0 -78.6 -68.1 -64.9 -61.8 -84.8 -61.6 0.5 N2O 15Nterm 83.6 75.7 94.7 102.8 105.2 107.6 90.1 99.5 15Ncentr -7.0 -15.6 5.3 13.9 16.3 18.7 0.9 11.3 17O 172.5 171.9 183.3 197.6 202.1 206.5 173.1 180.5 F2 19F -246.9 -214.6 -258.0 -241.6 -237.0 -232.4 -265.5 -234.2 O3 17Oterm -1691 -2050 -1404 -1297 -1265 -1234 -1470 -1310 17Ocentr -1113 -1540 -862.9 -816.4 -802.9 -789.7 -889.2 -744 P2H2 31P -304.7 -314.9 -257.1 -205.0 -190.4 -176.2 -284.6 -166 PN 31P -50.3 -69.1 -13.1 19.8 29.5 38.9 -32.4 53 15N -428.3 -450.0 -383.3 -343.3 -330.8 -318.6 -409.1 -349 SO2 33S -256.7 -289.6 -211.4 -172.8 -164.3 -139.0 -226.0 -125.9 17O -284.6 -289.7 -245.1 -185.7 -172.0 -133.4 -276.3 -225.1

  33. Nuclear shieldings for main-group compounds(IGLO-IV basis set, GIAO): Linear regression analysis for comparison between theory and experiment: 22 molecules - 32 shieldings: C2H2, C2H4, CH4, CH3F, CHF3, CO, CO2, F2, H2CO, H2O, H2S, HCl, HCN, HF, N2, N2O, NH3, O3, P2H2, PH3, PN, SO2 B-EXX-PW91,exact-exchange admixture a0 = 0.6: local vs. non-local exchange-correlation potentials

  34. g-shift components (ppt) for transition-metal complexes A. V. Arbuznikov, M. Kaupp, Chem. Phys. Lett.391, 16 (2004)

  35. B-EXX-PW91,exact-exchange admixture a0 = 0.5: local vs. non-local exchange-correlation potentials g(calc.) = A g (expt.) + B, with g (expt.) and B in ppt; the Standard Deviation is . Perfect agreement with experiment corresponds to A = 1, B = 0, zero Standard Deviation, and Regression Coefficient equal to 1. g-shift components for transition-metal complexes. Linear regression analysis for comparison between theory and experiment: 9 complexes - 19 g-shifts: Co(CO)4, CrOF4-, CrOCl4-, Cu(NO3)2, Fe(CO)5+, Mn(CO)5, MnO3, Ni(CO)3H, TiF3

  36. B-EXX-PW91,exact-exchange admixture a0 = 0.5: local vs. non-local exchange-correlation potentials EPR g-tensor components for some 3d transition metal complexes

  37. perturbationally, based on 1-comp. approach efficient approximations to SO integrals - include spin-orbit coupling variationally, in 2-comp. DKH approach A Relativistic Density-Functional Machinery to Compute NMR and EPR Parameters - mainly DFT used to incorporate electron correlation - include scalar relativistic effects (all-electron-DKH or PP approaches) Gaussian basis set property program ReSpect (Version 1.2), 2004.; V. G. Malkin, O. L. Malkina, R. Reviakine, A. V. Arbuznikov, M. Kaupp, B. Schimmelpfennig, I. Malkin, T. Helgaker, K. Ruud

  38. Performance of DFT for g-Shift Tensors of Light Main-Group Radicals 15000 10000 H2O+ Dg33 5000 Dgcalc. (a.u.´106) 0 -5000 -10000 -10000 -5000 0 5000 10000 15000 Dgexp. (a.u.´106) UDFT-BP86, IGLO-III basis; H2O+, CO+, HCO, C3H5, NO2, NF2, MgF, 2,4,6-tris-t-Bu-C6H2O, tyrosyl O. L. Malkina, J. Vaara, B. Schimmelpfennig, V. G. Malkin, M. KauppJ. Am. Chem. Soc.2000, 122, 9206.

  39. gx-Component of Semiquinone Radical Anions in 2-Propanol Dgx 5000 4800 UQ-M.- (iPrOH)6 4600 BQ.- (iPrOH)4 DMBQ.- (iPrOH)4 4400 4200 DQ.- (iPrOH)4 4000 NQ.- (iPrOH)4 Dgx calculated /ppm ideal agreement 3800 DMNQ.- (iPrOH)4 DMEQ.- (iPrOH)4 3600 3400 3200 3000 3200 3400 3600 3800 4000 4200 Dgx experimental /ppm (in 2-propanol) UDFT/BP86 results: J. Am. Chem. Soc.2002, 124, 2709. High-field EPR data in 2-propanol: Stehlik et al., 1993, 1995.

  40. Modelling g-Tensors for Semiquinones in their Protein Environment Photosystem I (QK-. in Synechococcus Elongatus) Purple Bacteria Reaction Center (QA-. in Rhodobacter Sphaeroides R-26) Purple Bacteria Reaction Center (QB-. in Rhodobacter Sphaeroides R-26) model: UQ-M-.(hist)(SIG) model: DMNQ-.(NMF)(indole) model: UQ-M-.(NMF)(imidazole)(indole) Dgx Dgy Dgz calc. 4280 3012 -17 exp. 43003100 -100 Dgx Dgy Dgz calc. 3866 2902 -67 exp. 39402950 -190 Dgx Dgy Dgz calc. 3889 2707 14 exp. 3930 2710 -49 UDFT/BP86, Dgx scaled by 0.92. S. Kacprzak, M. Kaupp J. Phys. Chem. 2004, 108, 2464; EPR data: MacMillan et al., 1997; Feher et al., 1995.

  41. g-Tensor Dynamics for Aqueous Benzosemiquinone Car-Parrinello ab initio molecular dynamics simulations average of snapshots for 6.3 ps trajectory (calc. 4.5 Å cluster, 300 K, exp. 80 K): Dgx= 4488 ppm (scaled) exp. 4300 ppm Dgy= 2992 ppm exp. 2980 ppm g-shift tensor calc., 5.0 Å cluster J. Asher, N. Doltsinis, M. Kaupp J. Am. Chem. Soc. 2004, 126, 9854.

  42. Performance of DFT for the calculation of isotropic hyperfine coupling constants in transition metal complexes [Mn(CN)4]2- calc. [Cr(CO)4]+ calc. -0.08 rNa-b/2S (in a.u.) -0.10 [Cr(CO)4]+ exp. -0.12 [Mn(CN)4]2- exp. BLYP BPW91 B3PW91 BHP86 -0.14 BP86 B3LYP BHLYP BHPW91 Functional -0.16 HF exchange admixture causes problems with spin contamination! -0.18 improved core-shell spin polarization with HF exchange admixture  more negative spin density Spin density rNa-bat the metal nuclei, normalized to the number of unpaired electrons. M. Munzarová, M. Kaupp J. Phys. Chem. A1999, 103, 9966.

  43. Attractive properties of exact exchange: • self-interaction free ( is cancelled completely • for one-electron systems) • correct asymptotic behaviour (at ) Localized hybrid exchange-correlation potentials: theory and implementation Motivation:search for better exchange-correlation potentials for the description of properties (in particular, NMR and EPR parameters): self-consistent potential needed! Ex is ca. 85-95% of the entire Exc Ex = Exexact ? • (Exexact + EcDFT ): • reasonable for atoms; • very poor for molecules (non-dynamical correlation is missing).

  44. Calculated Isotropic Hyperfine Coupling Constant for the4P GS of the Phosphorus Atom uncontracted 17s12p4d Partridge basis set.

  45. Averaged mixing coefficient a0 = Closed-shell main-group molecules . Open-shell 3d-transition-metal complexes ( , andW, are used;  =,  )

  46. Traditional hybrid functionals and potentials: - non-local and non-multiplicative Localized hybrid potentials: - local and multiplicative Local hybrid functionals: „Double local hybrid“ exchange-correlation potentials? application of the OEP method replacement of a0=const by ø[g(r)] Double local hybrid potentials: 1. Rigorous (planned): 2. Model potentials (preliminary work):

  47. Local hybrid functional: (J. Jaramillo, G. E. Scuseria, M. Ernzerhof J. Chem. Phys.2003, 118,1068) Let us average to see what a constant a0 would be! Cf.: Average g(r): (A. V. Arbuznikov, M. Kaupp Int. J. Quantum Chem. 2005, 105) a0is within 0.49 - 0.62 for both main-group molecules(NMR shieldings) and transition-metal complexes(g-tensors) A preliminary theoretical justification of the observed optimum a0 values …

  48. Still other choices of local mixing functions, g(r), are possible • Normalization of the „projected“ exchange hole: • D. Becke J. Phys. Chem.2003, 119, 2972. • suggested as part of a model of nondynamical • correlation. (inverted machinery of the BRx89 functional A. D. Becke, M. R. Roussel Phys. Rev A 1989, 39, 3761). curvature of exchange hole Example: O3 molecule Slater potential

  49. Next things to do: • -self-consistent implementation of „double local hybrid potentials“ • self-consistent implementation of Becke‘s „coordinate-space“ • model of nondynamical correlation • -combination of localized hybrid potentials with two-component • relativistic calculations of MR parameters, based on the • Douglas-Kroll-Hess Hamiltonian (non-collinear)

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