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Dispersion Corrections for Density Functional Theory

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Dispersion Corrections for Density Functional Theory Dispersion corrections for density functional theory Stefan Grimme Mulliken Center for Theoretical Chemistry, University of Bonn 13. Oktober 2012 Outline Introduction to quantum chemistry and DFT The dispersion energy Dispersion corrections Examples (molecules and solids) Supramolecular interations Multi-level modeling continuum models exptl. data ’coarse grained’ (grouping of atoms) class. force−fields (no electrons) parameter empirical MO parameter DFT accuracy/generality decreases system size increases testing WFT Ultimate Goals I accuracy <1 % energy error for any chemical process ≈1 kcal/mol (or 0.1 kcal/mol for conformational energies) I robustness I applicability (from molecules to condensed phase) I speed (comp time <days) 'The right answer for the right reason.' The Fact > 80 − 90% of all quantum mechanical electronic structure calculations are nowadays based on DFT (> 95 % in solid-state physics) Why? µ Numerical complexity only weakly dependent on Nel µ Kohn-Sham DFT has an extremely good performance/cost ratio ("biggest bang for the buck") (Kohn-Sham) DFT (for electrons with light nuclei) Electron density: Z Nel = ρ(r)dr Total energy: Ee = Te [ ] + Ven[ρ] + Vee [ρ] + Exc [ρ, ] Exchange-correlation energy: Z 2 Exc = ρ xc [ρ, rρ, r ρ, :::] dr | {z } choice of functional The KS kinetic energy and density of the KS-determinant: Nel Nel X X 2 Te = ni h i jT j i i; ρ(r) = ni j i (r)j i=1 i=1 Big challenges for current KS-DFT 1. Van der Waals (dispersive) interactions non-local correlation effects 2. Self-interaction (delocalization) error (charge-transfer, transition and excited states) 3. Genuine multi-reference cases, strongly correlated systems Dispersion energy The dispersion energy is C6 I a long-ranged ( R6 ) electron correlation effect (London part) I that continously becomes part of the 'normal' correlation energy for short/medium e-e (or inter-atomic) distances I the London part is not included in standard KS-DFT, the short/medium-range part to a very different degree I synonymously (but incorrectly) termed 'van der Waals' (vdW) energy I a time-independent phenomenon (stationary electronic states) A large vdW complex as an example: the C60 dimer 8 TPSS BLYP 6 TPSS-D3 BLYP-D3 exptl. (from solid) 4 R 2 E / -1 0 kcal mol -2 -4 -6 -8 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 R / Angstroem Why is standard KS-DFT wrong? I The fluctuations (excitations to virtual orbitals) are not included I For large R between the fragments ρ ≈ 0 I ρ does not contain significant dispersion signature Eint = EPauli−rep: + EES + Epol: + Edisp:;short + Edisp:;London | {z } | {z } included in KS−DFT missing Dispersion often considered as being 'weak' but: two Coulomb 1 point E=− E =^ 60 kcal/mol + − 10 h charges 10 Bohr (0.5 nm) dispersion two fragments or 10000x10 ^ E=− Eh = 60 kcal/mol molecules 10 6 100 HCNO atoms Dispersion corrections Review: WIREs Comput. Mol. Sci., 1 (2011) 211. correction to standard KS−DFT density based C based effective one−e 6 potential V=VKS +VNL V=VKS E=EKS+Einter−atomic V=VKS +V1e vdW−DF parametrized DFT−D3 DCACP DF DFT−NL LAP asymptotically asymptotically incorrect! incorrect! virt. orbitals perturbative other: RPA/DHDF DFT−SAPT TDDFT (linear response) Short- and long-range behavior interatomic distance DFT−D3(BJ) −k1+k2R2 −C R−6 dispersion/correlation energy medium−range long−range { { thermochemistry vdW−complexes DFT-D3 in a nut-shell 1 X C AB C AB E two−body = − 6 + s 8 disp 2 R6 + f (R0 )6 8 R8 + f (R0 )8 AB AB AB AB AB 0 0 f (RAB ) = a1RAB + a2 RAB ! 0 behavior: A. Koide, J. Phys. B, 9 (1976), 3173, first used by Becke & Johnson (JCP, 122 (2005), 154101). DFT-D3 in a nut-shell Z 1 AB 3 A B C6 = α(i!) α(i!) d! π 0 Atoms-in-molecules coefficients TD-DFT computations of α(i!) for AmHn/BkHl reference molecules (and subtracting H2): 3 Z 1 1 n 1 l AB AmHn H Bk Hl H C6 = d! [α (i!) − α (i!)] [α (i!) − α (i!)] π 0 m 2 k 2 Based on the approximate additivity of polarizabilities (volumes). AB The pre-computed C6 for the reference molecules (≈ 50000 for the whole PSE) are mapped to the real system via geometry dependent coordination numbers (CNs) and interpolation. Molecular dispersion coefficients Comparison of experimen- tal (DOSD) and theoreti- cal molecular C6 coeffi- DFT-D3, MAD = 4.7 % cients (1225 cases, com- Tkatchenko, MAD = 5.3 % pilation by A. Tkatchenko 1000 (FHI Berlin) maximum error: 23% (calc.) [au] 100 ::: 6 (CO CO) C 10 10 100 1000 C6 (exp.) [au] AA Dispersion coefficients (C6 , D3/TDDFT) for the periodic table Examples and Applications 1. Non-covalent interactions (molecules) 2. Molecular crystals and surfaces 3. Supramolecular complexes in solution Non-covalent interaction energies for small systems: typically <5% error with disp.-corr. DFT S66 benchmark set (Hobza et al, JCTC, 7, 2011, 2427) MAD 4 2 0 DFT-D3/NL: -2 kcal/mol DFT-D3 error 0.2-0.5 kcal/mol -4 0 MP2/CBS: -5 0.45 kcal/mol -10 semi-emp.-D3: E / (kcal/mol) 0.7-1 kcal/mol ∆ -15 CCSD(T), reference conventional DFs: -20 1 5 10 15 20 25 30 3540 45 50 55 60 66 >1-3 kcal/mol # molecular complex Examples: ionic liquid components -30 est. CCSD(T)/CBS(aA34,DZ) MP2(aA34) B3LYP B3LYP-D3 B -40 revPBE revPBE-D3 revPBE38 revPBE38-D3 -50 -77 revPBE38-NL -78 -60 -79 + -80 E [kcal/mol] -81 -70 dispersion R -82 effect − -83 -80 -84 -85 cation anion 3,8 4 4,2 4,4 4,6 -90 4 5 6 7 8 9 10 11 12 R [Angstroem] µ very good agreement of DFT-D3 and DFT-NL with CCSD(T) µ about 5 kcal/mol dispersion contribution µ self-interaction error of GGAs (and B3LYP) for large R System-dependent dispersion coefficients Problem I DFT-D3 only partially includes electronic effects on C6 values I Large effects expected for metals with loosely bound valence electrons I C6 coefficients for group 1 & 2 metals change significantly, e.g. Na!Na+ Solution I Calculate α(i!) from bulk-like reference structures similar to the target system, no fitting! I The formula for the C6 has to be generalized: 1 AB 3 R 1 AmXn n Xo 1 Bj Yk k Yl C6 = π 0 m α (i!) − o α (i!) k α (i!) − l α (i!) d! Special dispersion coefficients for ionic solids MM C6 (M=Na, Mg) coefficients and coordination numbers (CN). MM reference CN C6 D3* Na 0 1608 x NaH 0.97 186 Na4Cl4 3.51 32 x Na4Cl4(emb.) 8.90 16 x Mg 0 683 x MgH 0.96 451 MgH2 1.95 176 Mg4O4 5.49 92 x Mg4O4(emb.) 16.6 21 x Ne (comparison) 0 6.3 Results for potential energy curves C2H2 on NaCl CO on MgO Comp. Details : VASP, revPBE/PAW, 400 eV cutoff, 3x3x1 K-points Ehrlich, S.; Moellmann, J.; Reckien, W.; Bredow, T.; SG ChemPhysChem. 12, (2011), 3414-3420. Adsorption of PTCDA on Gold(111) I PTCDA on Au(111) surface 100 PBE PBE-D2 I 3 layers of gold -1 PBE-D3 50 PBE-D3abc revPBE I fixed distances, no revPBE-D3 relaxation 0 E / kcal mol ∆ I VASP: DFT with plane waves (PAW) -50 -100 adsorption Energy exp. -150 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 vertical distance /Å Importance of Dispersion in Molecular Crystals System Me-TBTQ Potential energy curves I Molecule C22H16 I Distance c as coordinate (C3v) I With uncorrected functionals, the I Crystallizes sublimation energy has the wrong trigonal in R3m, 3 sign molecules per unit Different functionals (PBE and cell I RPBE) give similar results Full (internal and cell) geometry 8 optimization (PBE-D3): excellent PBE/PAW 6 PBE-D3(BJ)/PAW agreement with X-ray data (rel. Distance via x-ray at 100 K distance) deviations below 1%! 4 [eV] DFT 2 E ∆ mono EDFT -2 4.5 5 5.5 6 6.5 7 7.5 c [A]˚ A DFT 'failure'? Veiros et al, JCTC 2007, 3, 665-670 isolated molecule crystal structure R method R(Ru-C3) [pm] X-Ray 238 PBE (molecule) 259 PBE-D3 (molecule) 255 PBE-D3 (crystal) 236 µ Don't compare apples with oranges! Supramolecular interactions in solution at T ≈ 298K: the challenge ∆ DFT−D3 ∆ Ea,gas G a,exptl. (kcal/mol) −2.9 0.1 (H 2 O) Huge interaction energies quenched −26 −2 by solvent and entropy. (CH 2 Cl 2 ) Sum of large terms + of opposite sign. −83 −7 (H 2 O,HCOOH 1:1) Method Basic idea: compute all parts non-empirically as accurately as one can In solvent X at temperature T for single structures in the reaction Ka host + guest supramolecular complex T T ∆Ga = ∆E(gas) + ∆GTRV (gas) + ∆δGsolv (X ) | {z } | {z } DFT or semiemp: harm: freq: COSMO−RS low−freq:mode approx: (2) (3) E(gas) = EDFT + Edisp + Edisp | {z } | {z } PW 6B95=QZ== D3−method TPSS−D3=TZ applicable to any semi-rigid (a few conformers) system Investigated complexes 1 2a 2b 5a 5b 3a 3b 6a 6b 4a 4b 7a 7b Results 0 1 2b 3b -2 3a 5b 2a -4 4b -1 -6 4a 6b -8 6a 5a -10 1 2a 2b 5a 5b -12 7b -14 3a 3b 6a 6b (exptl.) / kcal mol a -16 G ∆ -18 -20 7a 4a 4b 7a 7b -22 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 ∆ -1 Ga (theor.) / kcal mol µ Very good agreement between theory and experiment (MAD=2.1 kcal/mol, MD=-0.8 kcal/mol) without any specific empiricism Basic Idea of S12L T T If ∆GTRV (gas) and ∆δGsolv (X ) are accurate one can reverse the equation and estimate 'experimental' ∆E(gas): T T ∆E(exptl:; gas) ≈ ∆Ga(exptl:) − ∆GTRV (gas) − ∆δGsolv (X ) We estimate an error of ±10% for ∆E(exptl; gas), i.e., about 2-3 kcal/mol for neutral and 5-10 kcal/mol for charged complexes.
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