Dispersion Interactions from the Exchange-Hole Dipole Moment Erin R. Johnson and Alberto Otero-de-la-Roza Chemistry and Chemical Biology, University of California, Merced E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 1 / 35
Dispersion interactions Biological molecules. Surface adsorption. Molecular crystal packing. Crystal structure prediction. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 2 / 35
Dispersion interactions Long-range non-local correlation effect, not captured by semi-local functionals. Johnson et al. CPL 394 (2004) 334. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 3 / 35
Review of dispersion methods Fluctuation-dissipation theorem/rpa. Non-local correlation functionals (vdw-dfx). Dispersion correcting potentials (DCP). Meta-GGAs fit to binding energies. Post-SCF pairwise energy, fixed C 6 : DFT-D2. Variable C 6 : Tkatchenko-Scheffler, DFT-D3, XDM. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 4 / 35
The XDM method Dispersion arises from interaction of induced dipoles. The source of the instantaneous dipole moments is taken to be the dipole moment of the exchange hole. Becke and Johnson JCP 127 (2007) 154108. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 5 / 35
The exchange hole The exchange hole measures the depletion in probability of finding another same-spin electron in the vicinity of a reference electron. dipole reference electron nucleus hole center An electron plus its exchange hole has zero total charge, but a non-zero dipole moment in general. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 6 / 35
The exchange-hole model The magnitude d X of the exchange-hole dipole moment is obtained using the Becke-Roussel exchange-hole model; PRA 39 (1989) 3761. Ae -ar reference point hole center b Parameters (A,a,b) obtained from normalization, density, and curvature at reference point. Advantages: semi-local (meta-gga) model of the dipole: d x = b. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 7 / 35
The XDM method The dispersion energy comes from second-order perturbation theory E (2) = ˆV 2 int E V int (r A, r B ) = multipole moments of electron + hole at r A interacting with multipole moments of electron + hole at r B E is the average excitation energy, obtained from second-order pertubation theory applied to polarizability. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 8 / 35
The XDM equations The XDM dispersion energy is: E disp = 1 2 ij C 6 f 6 (R ij ) R 6 ij + C 8f 8 (R ij ) R 8 ij The dispersion coefficients are non-empirical. + C 10f 10 (R ij ) R 10 ij +... C 6,ij = α iα j M 2 1 M2 1 j M 2 1 α j + M 2 1 jα i Atomic multipole moment integrals use Hirshfeld atomic partitioning. Ml 2 i = ω i (r)ρ σ (r)[ri l (r i d Xσ ) l ] 2 dr σ E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 9 / 35
Damping function Corrects for the multipolar-expansion error and avoids discontinuities. R n f n (R) = R n + R n vdw R vdw = a 1 R c,ij + a 2 Parameters: a 1 and a 2 R c,ij are proportional to atomic volumes. f damp 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 f 6 0.1 f8 f 10 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 R Ne Ne (Å) E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 10 / 35
Higher-order dispersion coefficients XDM is easily extended to n-body dispersion coefficients to any order. The pairwise dispersion coefficients are: C2n+2 AB = 1 n/2 λ ε A + ε k(n k) Mk 2 A Mn k 2 B B k=1 and the average excitation energies are: ε A = 2 M1 2 A 3 αa 0 where αa 0 is the atom-in-molecule static polarizability. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 11 / 35
Generalization to XDM to higher orders The 3-body dispersion coefficients are: Z la l B l C = ξ la l B l C M 2 l A A M 2 l B B M 2 l C C (ε A + ε B + ε C ) (ε A + ε B )(ε A + ε C )(ε B + ε C ) The triple-dipole Axilrod-Teller-Muto term has coefficient C 9 = Z 111 : E (3) 9 = C 9 3 cos θ A cos θ B cos θ C + 1 R 3 AB R3 AC R3 BC and involves a geometrical factor depending on the atomic positions. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 12 / 35
XDM dispersion coefficients Mean absolute percent errors in XDM dispersion coefficients. X = {H, He, Ne, Ar, Kr, Li, Na, K, Be, Mg, Ca} C 6 X X 12.2% C 9 X X X 12.7% C 9 X He He 12.1% For a set of 178 molecular C 6 coefficients, the MA%E is 10.0%. Otero-de-la-Roza and Johnson, JCP 138 (2013) 054103 E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 13 / 35
3-body damping function The choice of 3-body damping function is not unique. Alternatives are: f (3) 9 (R AB, R AC, R BC ) = f 3 (R AB )f 3 (R AC )f 3 (R BC ) f (3) 9 (R AB, R AC, R BC ) = f 6 (R AB )f 6 (R AC )f 6 (R BC ) The latter damping is chosen to recover the united-atom limit. It gives improved agreement with SAPT for benzene and noble gas crystals. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 14 / 35
Effects on molecular dimer binding energies MAPE (%) 26 24 22 20 18 E disp = 1 2 n ij C n f n (R ij ) R n ij no C 9 with C 9 16 14 6 8 10 12 14 n C 9 and pairwise terms beyond C 10 do not improve performance. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 15 / 35
Applications Our applications of XDM to periodic systems include Crystal geometries Sublimation enthalpies Graphite exfoliation energy Adsorption on metal surfaces Chiral crystals Electrides E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 16 / 35
Implementation for molecules Pure functionals give poor thermochemistry. Can we pair XDM with hybrid or range-separated functionals for molecular calculations? XDM calculations used Gaussian 09 and the postg program. From the wfn file, postg gives: XDM dispersion energy forces for geometry optimization (fixed coefficients) second derivatives for frequencies Download postg from the XDM page at: http://faculty1.ucmerced.edu/ejohnson29 E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 17 / 35
Parametrization set 49 gas-phase dimers from Kannemann and Becke; JCTC 6 (2010) 1081. noble gases dispersion π-stacking dipole - induced dipole mixed dipole - dipole hydrogen-bonding E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 18 / 35
Binding energies XDM with aug-cc-pvtz; mean absolute errors in kcal/mol. Pure functionals: Quantity BLYP PW86 PBE MAE 0.31 0.40 0.50 MA%E 9.8 11.8 14.3 Hybrid and range-separated functionals: Quantity B3LYP BH&HLYP PBE0 CAM-B3LYP LC-ωPBE MAE 0.28 0.37 0.41 0.39 0.28 MA%E 6.7 7.8 10.2 8.3 7.8 E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 19 / 35
Interaction types 1 0.8 PW86 BLYP PBE 0.8 0.6 0.4 PW86 BLYP PBE MAE (kcal/mol) 0.6 0.4 0.2 ME (kcal/mol) 0.2 0 0.2 0.4 0.6 0 ng disp stack d id mixed d d hb Interaction type 0.8 ng disp stack d id mixed d d hb Interaction type For PBE there is a trade-off between interaction types and the parameters will be very dependent on composition of the fit set. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 20 / 35
Interaction types MAE (kcal/mol) 1 0.8 0.6 0.4 B3LYP BH&HLYP PBE0 CAM B3LYP LC ωpbe ME (kcal/mol) 0.8 0.6 0.4 0.2 0 0.2 0.4 B3LYP BH&HLYP PBE0 CAM B3LYP LC ωpbe 0.2 0.6 0.8 0 ng disp stack d id mixed d d hb Interaction type 1 ng disp stack d id mixed d d hb Interaction type LC-ωPBE has the lowest signed errors, followed by BLYP and B3LYP. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 21 / 35
Role of exchange The exact exchange potential decays as 1/r far from a molecule. In terms of the exchange hole, h X remains on the molecule as the reference point moves away from it. The 1/r asymptotic dependence was used to design the B88 exchange functional. Functionals based on B88 or range-separated hybrids with the full exact-exchange limit (LC-ωPBE) give more accurate intermolecular exchange contributions. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 22 / 35
Smaller basis sets 20 15 PW86 BLYP B3LYP LC ωpbe MAPE (%) 10 5 0 6 31+G* aug cc pvdz aug cc pvtz Basis set E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 23 / 35
Benchmark sets Mean absolute errors in XDM binding energies with aug-cc-pvtz (kcal/mol) BLYP PW86 B3LYP LC-ωPBE S22 0.22 0.35 0.31 0.31 S66 0.22 0.29 0.25 0.20 HSG 0.20 0.17 0.12 0.23 S22 and HSG reference data: Marshall et al. JCP 135 (2011) 194102. S66 reference data: Rezac et al. JCTC 7 (2011) 2427. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 24 / 35
Water hexamers Prism Cage Book 1 Book 2 Bag Chair Boat 1 Boat 2 E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 25 / 35
Water hexamers Mean absolute errors with aug-cc-pvtz (kcal/mol) PW86 BLYP B3LYP LC-ωPBE Base functional Error for prism BE 0.61 8.00 4.81 5.16 Relative BE error 1.51 2.99 2.47 1.98 With XDM Error for prism BE -4.40-4.08-2.87-0.57 Relative BE error 0.35 0.21 0.65 0.62 Reference data: Bates and Tschumper, JPCA 113 (2009) 3555. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 26 / 35
Non-equlibrium geometries: S66x8 35 30 25 PW86 BLYP B3LYP LC ωpbe MAPE (%) 20 15 10 5 0 0.9 0.95 1.0 1.05 1.1 1.25 1.5 2.0 Compression/Expansion factor E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 27 / 35
Thermochemistry: Atomization energies Mean absolute errors in G3/99 atomization energies with aug-cc-pvtz (kcal/mol) BLYP B3LYP LC-ωPBE Base functional 11.4 7.8 5.1 With XDM 6.5 4.0 5.5 Reference data: Curtiss et al. JCP 112 (2000) 7374. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 28 / 35
Thermochemistry: Bond energies Mean absolute errors in bond dissociation energies with aug-cc-pvtz (kcal/mol) BLYP B3LYP LC-ωPBE Main group compounds Base functional 7.6 5.7 4.4 With XDM 5.1 4.1 3.6 Transition metal complexes Base functional 5.8 4.5 3.0 With XDM 7.4 3.0 3.1 Main-group reference data: Johnson et al. JPCA 107 (2003) 9953. Transition-metals: Johnson and Becke, CJC 87 (2009) 1369. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 29 / 35
Thermochemistry: Delocalization error Delocalization error affects: Stretched bonds Delocalized radicals HAT transition states Charge transfer HOMO-LUMO gaps Excitation energies Relative energy (kcal/mol) 40 20 0 20 40 60 HF BLYP B3LYP BH&HLYP LC ωpbe 80 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 d H H (Å) Dissociation curve for H + 2 E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 30 / 35
Thermochemistry: Delocalization error Mean absolute errors in H-atom transfer barriers and charge-transfer binding energies with aug-cc-pvtz (kcal/mol) BLYP B3LYP BH&HLYP LC-ωPBE Barriers Base functional 7.8 4.6 2.4 1.3 With XDM 9.1 5.3 2.1 1.4 Charge-transfer complexes Base functional 1.4 4.5 0.7 1.2 With XDM 2.8 3.0 0.1 0.7 Barrier reference data: Lynch and Truhlar, JPCA 105 (2001) 2936. Charge-transfer: Zhao and Truhlar, JCTC 1 (2005) 415. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 31 / 35
XDM summary XDM dispersion coefficients are obtained from first-principles. Computational cost is comparable to semi-local DFT. Excellent binding energies. Improved thermochemistry. E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 32 / 35
Introduction XDM Molecular complexes Thermochemistry Summary Acknolwdgements Group members: Alberto Otero-de-la-Roza Matthew Christian Stephen Dale Joseph Dizon Joel Mallory Electrides and alkalides using the exchange-hole dipole moment model Stephen G. Dale Thursday, 9:00 AM, Quantum Chemistry Session Morial Convention Center, Room: 354 E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 33 / 35
Upcoming meetings Time-Dependent Density-Functional Theory GRS: August 10-11, 2013 GRC: August 11-16, 2013 University of New England, Biddeford, ME E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 34 / 35
Introduction XDM Molecular complexes Thermochemistry Summary Upcoming meetings E. R. Johnson (UC Merced) Dispersion from XDM ACS Meeting (April 2013) 35 / 35