Methods for van der Waals Interactions
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1 Methods for van der Waals Interactions Alexandre Tkatchenko Theory Department, Fritz Haber Institut der MPG Berlin, Germany berlin.mpg.de Haber Institute FHI DFT and Beyond Workshop, Jul. 15, 2011
2
3 Van der Waals (dispersion) energy A B
4 Van der Waals (dispersion) energy B A + B A + + B A R
5 Van der Waals (dispersion) energy B A + B A + + B A R Ubiquitous interatomic and intermolecular interaction Scales linearly with system size Weak compared to hydrogen or covalent bonds... but is a significant component of intermolecular binding energies!
6 VdW Interactions Are Ubiquitous...
7 Current state of the art of modeling Full CI Accuracy, Reliability and Predictive Power Post Hartree Fock (MP2, CCSD(T), RPA) Computational Cost Hybrid DFT (Semi) local DFT: LDA and GGA Semi empirical methods (based on DFT or Q.C.) Empirical potentials ( force fields )
8 Current state of the art of modeling Full CI Accuracy, Reliability and Predictive Power Post Hartree Fock (MP2, CCSD(T), RPA) Computational Cost Density functional Density functionaltheory theory approximatefunctionals functionals with with approximate Semi empirical methods (based on DFT or Q.C.) Empirical potentials ( force fields )
9 Correlated methods: MP2, RPA, CCSD(T) Compute the correlation energy based on a Hartree Fock or DFT wavefunction Include different many body diagrams for the correlation energy MP2: Minimal level for correlation energy RPA: Doubles to infinite order, no singles CCSD(T): Quantum chemisty gold standard
10 DFT and vdw interaction: Rare gas dimers HF Ar Ar DFT CCSD(T)
11 Why existing methods are not sufficient? Current state of the art Polymorphism in drugs Ritonavir (HIV protease inhibitor) 1996 (launch) 1998 (recall) 2001 (trans form found) Polymorph energies are typically within 1 kcal/mol per molecule
12 Why existing methods are not sufficient? Current state of the art Polymorphism in drugs Molecular switches at surfaces Ritonavir (HIV protease inhibitor) 1996 (launch) 1998 (recall) 2001 (trans form found) Standard DFT yields wrong adsorption structure. Polymorph energies are typically within 1 kcal/mol per molecule No available method gives quantitative energetics! Yu, Nakano, Ikeda, Nature (2003).
13 Accurate First Principles Modeling of vdw Interactions is Challenging A RAB B
14 Accurate First Principles Modeling of vdw Interactions is Challenging r1 A RAB r2 B Extremely expensive!
15 Accurate First Principles Modeling of vdw Interactions is Challenging r1 1 Accurate Microscopic Modeling of Coulomb Screening A RAB 2 r2 B Full (All Order) Many Body van der Waals Energy
16 VdW corrected DFT methods
17 Concepts and methods for dispersion in DFT
18 Concepts and methods for dispersion in DFT Non local functionals (depend explicitly on r and r') (Langreth, Lundqvist et al.). Modified pseudopotentials (von Lilienfeld et al.) Highly empirical (hybrid) meta GGA functionals (Truhlar et al.) Interatomic (pairwise or beyond) dispersion corrections (Many people) Wu and Yang JCP (2002); Grimme J. Comp. Chem. (2004,2006); Dion et al. PRL (2004); Zhao and Truhlar JCP (2006); von Lilienfeld et al. PRL (2004); Johnson and Becke JCP ( ); Tkatchenko and Scheffler PRL (2009); and many others...
19 S22 benchmark database Hydrogen bonding P. Jurecka, J. Sponer, J. Cerny, and P. Hobza, Phys. Chem. Chem. Phys. (2006)
20 S22 benchmark database VdW bonding P. Jurecka, J. Sponer, J. Cerny, and P. Hobza, Phys. Chem. Chem. Phys. (2006)
21 S22 benchmark database Mixed bonding P. Jurecka, J. Sponer, J. Cerny, and P. Hobza, Phys. Chem. Chem. Phys. (2006)
22 Langreth Lundqvist functional (vdw DF 04 and vdw DF 10)
23 Langreth Lundqvist functional Dion, Rydberg, Schroeder, Langreth, Lundqvist, PRL (2004). Lee, Murray, Kong, Lundqvist, Langreth, PRB (2010).
24 Langreth Lundqvist functional (vdw DF 04 and vdw DF 10) vdw DF 04 vdw DF 10 Exchange: revpbe Exchange: PW86 Local corr.: LDA Local corr.: LDA No free parameters 2 parameters C6 error: ~ 20% (*) Vydrov and van Voorhis, PRA (2010). C6 error: ~ 60% (*)
25 Approximations for in vdw DF r1 A RAB r2 B 1) Local approximation for the response function 2) Only pairwise point point interaction, ignoring non additive many body energy
26 Performance of vdw DF on the S22 database
27 Performance of vdw DF on the S22 database
28 Limitations of the Langreth Lundqvist functional vdw DF 04 is a general purpose functional, however its performance is not optimal (errors significantly larger than chemical accuracy) vdw DF 10 is a special purpose functional with good performance for molecules. Asymptotic vdw interactions are significantly underestimated (~60%), so no good performance should be expected for larger systems The vdw DF concept does not have flexibility for the coupling to semi local DFT functional. This is being improved by Vydrov and van Voorhis (PRL 2009, JCP 2010)
29 Empirical Minnesota functionals (Zhao Truhlar, M06 family)
30 Empirical Minnesota functionals (Zhao Truhlar, M06 family) Highly flexible (and empirical) form for a functional (20 40 parameters) trained on a broad set of benchmark (theory and experiment) data A family of functionals: M06: Hybrid meta GGA functional M06 L: GGA (non hybrid) version of M06 M06 2X: M06 with twice the amount of HF exchange M06 HF: M06 with 100% HF exchange
31 The description of vdw in M06 Flexible meta GGA form allows for an approximate modeling of vdw interactions at short range whenever there is a non negligible electron density overlap M06 converges to zero interaction at ~ 5 Å separation between atoms, in the same way as LDA and GGA Thus, M06 does not include the correct physics of vdw interactions, typically leading to underbinding of vdw bound systems
32 The description of vdw in M06 Marom, Tkatchenko, Gobre, Rossi, Hod, Scheffler, Kronik, to be submitted.
33 Performance of M06( L) for S22
34 Interatomic vdw correction methods
35 Interatomic vdw methods Two parameters per atomic pair: (1) VdW C6 interaction coefficient and (2) vdw radius. Clearly, if (1) and (2) are empirical, this leads to many fitting parameters. This was frequently the case before 2008.
36 Damping function for vdw corrected DFT Steep damping function required to separate the short range from vdw energy About 5 different functional forms proposed Fermi function (above) among the most popular ones Works for molecules, but the functional form is arbitrary!
37 Evolution of interatomic vdw methods Grimme's D1,D2 ( ): Parameterization for many elements in the periodic table Jurečka et al. (2007): Accurate parameterization for organic molecules Reduced empiricism, errors of ~ 20% 40% in C6 coefficients Tkatchenko and Scheffler (2009): C6 coefficients and vdw radii from ground state electron density Better theoretical ground, but still very empirical Johnson and Becke ( ), Silvestrelli (2008): C6 and vdw radii from HF or DFT orbitals Highly empirical, some approximations unfounded. First principles C6 accurate to 5%
38 Highlights of TS vdw method VdW asymptotics with an accuracy of 5% Transparent partitioning of vdw coefficients over atoms Reduced empiricism (1 parameter in the damping function) Polarizability is directly proportional to volume Computationally efficient Valid for the whole periodic table Tkatchenko and Scheffler, PRL (2009)
39 Performance of TS vdw method for molecules Mean absolute error of 5.5% for 1225 molecular C6 from reference DOSD data of W. J. Meath et al.
40 Performance of TS vdw method for molecules Results depend negligibly (1% deviation) on the employed xc functional
41 C6 is a functional of the density 6 (Carbon Carbon C6 coefficient / Hartree Bohr ) sp3 sp3 sp2
42 DFT+vdW Leading dispersion term is added to DFT total energy, damped at short interatomic distance, Effective vdw parameters are functionals of the electron density: C6 = C6[n(r)], RvdW = RvdW[n(r)] DFT+vdW: Tkatchenko and Scheffler, PRL (2009)
43 Performance of DFT+vdW for S22
44 Performance of DFT+vdW for S22 (Mean Absolute Error)
45 Performance of DFT+vdW for S22: Different functionals
46 What is missing in DFT+vdW? r1 A RAB r2 B 1) Long range electrostatic screening among fluctuating dipoles 2) Non additive many body vdw energy beyond two body
47 SCS+MB DFT+vdW : Screening and Many Body vdw Energy PBE+vdW 690 mev/molecule PBE+vdWSCS+MB 565 mev/molecule Experiment mev/molecule Benzene crystal Tkatchenko, DiStasio Jr., Car, Scheffler, to be submitted. 2.9 kcal/mol due to screening and many body VdW effects
48 DFT+vdW: Challenges DFT+vdW treats intra and intermolecular interactions accurately. Recent extension permits the treatment of insulators, ionic, and semiconductor solids Coupling of (short range) DFT and (long range) vdw remains empirical, and one of the biggest issues to address No satisfactory model for metallic bulk systems yet Interatomic potentials may not work for highly anisotropic low dimensional systems (Dobson, Rubio, et al.)
49 Applications
50 Unraveling the Role of vdw Interactions for Peptide Secondary Structure Ala15LysH+ forms stable helices in vacuo up to ~ 700 K (in solution only up to ~ 340 K) Direct first principles folding simulations are not feasible, but unfolding dynamics could provide similar insight! Experiments: Kohtani, Jones, Schneider, Jarrold, JACS (2004)
51 DFT PBE+vdW Tkatchenko, Rossi, Blum, Ireta, Scheffler, PRL (2011) DFT PBE
52 Van der Waals and Inorganic/Organic Interfaces perylene 3,4,9,10 tetracarboxylic 3,4,9,10 dianhydride (PTCDA) Ag(111)
53 Van der Waals and Interfaces: Which theoretical method? PTCDA Experiment: A. Hauschild et al., PRL (2007) Ag(111)
54 Van der Waals and Interfaces: Which theoretical method? PBE LDA/PBE: (semi )local DFT. Can be applied to > 1000 atoms LDA PTCDA Experiment: A. Hauschild et al., PRL (2007) Theory review: A. Tkatchenko et al., MRS Bulletin (2010) Ag(111)
55 Van der Waals and Interfaces: Which theoretical method? vdw DF: Non local Langreth Lundqvist functional. Can be applied to > 1000 atoms vdw DF PTCDA Experiment: A. Hauschild et al., PRL (2007) vdw DF: L. Romaner et al., New J. Phys. (2009) Ag(111)
56 Van der Waals and Interfaces: Which theoretical method? PBE D: PBE+Pairwise correction by Grimme. Can be applied to > 1000 atoms PBE D PTCDA Experiment: A. Hauschild et al., PRL (2007) PBE D: E. McNellis, PhD thesis (2010) Ag(111)
57 Van der Waals and Interfaces: Which theoretical method? PBE+vdW: PBE+Pairwise correction by Tkatchenko Scheffler. Can be applied to > 1000 atoms PBE+vdW Experiment: A. Hauschild et al., PRL (2007) PBE+vdW: E. McNellis, PhD thesis (2010) PTCDA Ag(111)
58 Van der Waals and Interfaces: Which theoretical method? EX+cRPA: Exact exchange + approximate many body correlation energy. EX+cRPA Can be applied to < 200 atoms PTCDA Experiment: A. Hauschild et al., PRL (2007) EX+cRPA: M. Rohlfing and T. Bredow, PRL (2008) Ag(111)
59 PBE+vdW(ZK) PBE+vdW(ZK) PBE+vdW
60 PBE+vdW(ZK) Ag Au Cu Joint work with V. Ruiz, E. Zojer, M. Scheffler.
61 VdW Interactions Are Ubiquitous...
62 Thank you! M. Scheffler R. Car O. A. von Lilienfeld M. Rossi V. Blum L. Kronik N. Marom
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