Dispersion Interactions in Density-Functional Theory
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1 Dispersion Interactions in Density-Functional Theory 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 GRC: TD-DFT (Aug 2013) 1 / 37
2 Dispersion interactions Biological molecules. Surface adsorption. Molecular crystal packing. Crystal structure prediction. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 2 / 37
3 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 GRC: TD-DFT (Aug 2013) 3 / 37
4 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 GRC: TD-DFT (Aug 2013) 4 / 37
5 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) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 5 / 37
6 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 GRC: TD-DFT (Aug 2013) 6 / 37
7 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) 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 GRC: TD-DFT (Aug 2013) 7 / 37
8 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 GRC: TD-DFT (Aug 2013) 8 / 37
9 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 GRC: TD-DFT (Aug 2013) 9 / 37
10 Damping function Corrects for the multipolar-expansion error and avoids discontinuities. f n (R) = R n R n + R n vdw R vdw = a 1 R c,ij + a 2 a 1 and a 2 are parameters fit for use with a particular XC functional. R c,ij are proportional to atomic volumes. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 10 / 37
11 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 GRC: TD-DFT (Aug 2013) 11 / 37
12 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. Otero-de-la-Roza and Johnson, JCP 138 (2013) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 12 / 37
13 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 GRC: TD-DFT (Aug 2013) 13 / 37
14 Effects on molecular dimer binding energies MAPE (%) E disp = 1 2 n ij C n f n (R ij ) R n ij no C 9 with C n C 9 and pairwise terms beyond C 10 do not improve performance. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 14 / 37
15 Implementation for molecules Pair XDM with GGA, hybrid, and range-separated functionals. 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 E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 15 / 37
16 Parametrization set 49 gas-phase dimers from Kannemann and Becke; JCTC 6 (2010) dispersion π-stacking dipole - induced dipole mixed dipole - dipole hydrogen-bonding E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 16 / 37
17 Binding energies XDM with aug-cc-pvtz; mean absolute errors in kcal/mol. Pure functionals: Quantity BLYP PW86 PBE MAE MA%E Hybrid and range-separated functionals: Quantity B3LYP BH&HLYP PBE0 CAM-B3LYP LC-ωPBE MAE MA%E E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 17 / 37
18 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 GRC: TD-DFT (Aug 2013) 18 / 37
19 Thermochemistry Mean absolute errors in XDM thermochemical benchmarks. (kcal/mol) BLYP B3LYP LC-ωPBE G3/99 atomization energies Base functional With XDM Main group bond energies Base functional With XDM Charge-transfer complexes Base functional With XDM JCP 138 (2013) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 19 / 37
20 Implementation for solids Quantum ESPRESSO, plane waves/psuedopotentials d xσ, valence τ, ρ. ω i, all-electron ρ, ρ at. E disp fast compared to E DFT. Errors in calculated XDM binding energies (kcal/mol) PBE PW86 B86b BLYP MAE MA%E JCP 136 (2012) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 20 / 37
21 Graphite E ex (mev/atom) c (Å) B86b PW86 PBE revpbe Expt. Expt. ACFDT RPA LDA GGA+vdw QMC VDW DF E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 21 / 37
22 Sublimation enthalpies E crys el Eel mol H sub (V, T) = Eel mol + E trans + E rot + Evib mol + pv ( E crys el + E crys ) vib DFT + dispersion DFT + dispersion, supercell E trans + E rot + pv 4RT (7/2RT) Rigid molecules: E mol vib = Ecrys vib for intramolecular modes Dulong-Petit limit: E crys vib Zero-point vibrational contributions neglected 6RT (5RT) for intramolecular modes E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 22 / 37
23 Sublimation enthalpies Mean absolute errors in sublimation enthalpies, in kj/mol Quantity XDM DFT-D2 TS vdw-df B86b PW86 PBE PBE PBE v1 v2 MAE MA%E molecular crystals, low polymorphism, varied interaction types. Well known sublimation enthalpies at or below room temperature. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 23 / 37
24 Crystal structures Vibrational Helmholtz free energy: Thermal pressure: 3n [ ωj F vib (V, T) = 2 + k BT ln (1 )] e ω j/k B T p th = F vib V Equilibrium condition: j= E V = p th = p sta 0 Relax the crystal under negative pressure, p th DOS (1/cm -1 ) Frequency (cm -1 ) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 24 / 37
25 Crystal structures Mean absolute errors in cell lengths, in atomic units. XDM DFT-D2 TS vdw-df Quantity B86b PW86 PBE PBE PBE v1 v2 MAE MA%E JCP 137 (2012) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 25 / 37
26 Chiral crystals E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 26 / 37
27 Enantiomeric excess Enantiomers have the same solvation energies. Assume same temperature effects. Predicted ee: E = E dl E l ee = β2 1 β β = e E/RT Enantiomeric excess values (%): Amino acid DFT Expt. Serine Histidine Leucine Alanine Cysteine Tyrosine Valine Proline Aspartic acid Glutamic acid E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 27 / 37
28 Enantiomeric excess 1 Enantiomeric Excess E (kcal/mol) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 28 / 37
29 Electrides An electride is an ionic substance in which a localized electron acts as an anion. Existing electrides require a cage like structure to stabilise the cation: crown ethers and cryptands. High magnetic susceptibilities, variable conductivities, very strong reducing agents. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 29 / 37
30 NCI analysis Reduced density gradient: s = 1 ρ 2(3π 2 ) 1/3 ρ 4/3 Maps intermolecular contacts (i.e. Pauli repulsion zones). Simpler to calculate than Bader s topology Color-mapping and region extent: interaction strength. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 30 / 37
31 NCI analysis Implemented in nciplot for molecules and critic2 for solids. JACS 132 (2010) 6498 and PCCP 14 (2012) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 31 / 37
32 Electrides Use the NCI index to visualize the electrons - non-nuclear attractors will have low density and reduced gradient. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 32 / 37
33 Electrides J. L. Dye used van der Waals radii to generate approximate channels and vacancies of electrons - JACS 1996, 118 (1996) E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 33 / 37
34 Introduction XDM Molecules Solids Summary Graphite surfaces 2 2 f p p s h e d. o e - Applications FIG. 2. The two-layer finite DFT model for the step edge is shown, together with the methane molecule representing the tip. graphite two-dimensional cells (containing two carbon atoms) were enough to represent a step. However, in order to spurious self-interactions between adjacent FIG. 1. avoid Illustration of the molecular dynamics simulation of molecules, at tip least 4 cells required the amethane 5.6 nm diameter AFM sliding overwere graphite with in a step edge. Details are given in the exceeding text. complementary direction, our computational FIG. 2. The two-layer finite DFT model for the step edge is resources at present. Because of this limitation, we deshown, together with the methane molecule representing the cided to use a finite model. Several finite graphite steps R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 34 / 37 tip. weree.tested and we decided to use a model containing two
35 Introduction XDM Molecules Solids Applications Summary Graphite surfaces Appl. Phys. Lett. In press. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 35 / 37
36 XDM summary XDM dispersion coefficients are obtained from first-principles. Computational cost is comparable to semi-local DFT. Excellent binding energies. Improved thermochemistry. Applications for molecules and the solid state. E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 36 / 37
37 Acknolwdgements Group members: Alberto Otero-de-la-Roza Matthew Christian Stephen Dale Joseph Dizon Joel Mallory E. R. Johnson (UC Merced) Dispersion from XDM GRC: TD-DFT (Aug 2013) 37 / 37
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