Electronic Structure with DFT: GGA and beyond

1. Electronic Structure with DFT:GGA and beyond Jorge Kohanoff Queen’s University Belfast United Kingdom j.kohanoff@qub.ac.uk

2. density functional theory (dft):kohn-sham equations • Kohn-Sham equations: • This PDE must be solved self-consistently, as the KS potential depends on the density, which is constructed with the solutions of the KS equations.

3. Exchange and correlation in dft: the local density approximation (lda) • The inhomogeneous electron gas is considered as locally homogeneous: • LDA XC hole centred at r, interacts with the electron also at r. The exact XC hole is centred at r’ • This is partially compensated by multiplying the pair correlation function with the density ratio (r)/(r’) XC energy density of the HEG

4. Exchange and correlation in dft: the local density approximation (lda) • Location of the XC hole (Jones and Gunnarsson, 1982)

5. lda-lsda: trends and limitations • Favors more homogeneous electron densities • Overbinds molecules and solids (Hartree-Fock underbinds) • Geometries, bond lengths and angles, vibrational frequencies reproduced within 2-3% • Dielectric constants overestimated by about 10% • Bond lengths too short for weakly bound systems (H-bonds, VDW) • Correct chemical trends, e.g. ionization energies • Atoms (core electrons) poorly described (HF is much better) • XC potential decays exponentially into vacuum regions. It should decay as –e2/r. Hence, it is poor for dissociation and ionization • Poor for metallic surfaces and physisorption • Very poor for negatively charged ions (self-interaction error) • Poor for weakly bound systems: H-bonds (), VDW (non-local) • Band gap in semiconductors too small (~40%) • Poor for strong on-site correlations (d and f systems, oxides, UO2)

6. beyond the lda • Inhomogeneities in the density • Self-interaction cancellation • Non-locality in exchange and correlation • Strong local correlations • Gradient expansions • Weighted density approximation • Exact exchange in DFT (OEP local vs HF non-local) • DFT-HF hybrids • Self-interaction correction • Van der Waals and RPA functionals • LSDA+U • Multi-reference Kohn-Sham • GW approximation (Many-body)

7. gradient expansions:generalized gradient approximation • EXC expanded in gradients of the density where  is the spin polarization s=||/2kF is the density gradient And FXCis the enhancement factor • First-order term is fine, but higher-order terms diverge. Only by some re-summation to ∞-order the expansion converges. • GGA: FXC is designed to fulfil a number of exactly known properties, e.g. Perdew-Burke-Ernzerhof (PBE) • Exchange: uniform scaling, LSDA limit, spin-scaling relationship, LSDA linear response, Lieb-Oxford bound • Correlation: second-order expansion, hole sum rule, vanishes for rapidly varying densities, cancels singularity at high densities (r+dr) r+dr

8. Properties of the gga • Improves atomization and surface energies • Favors density inhomogeneities • Increases lattice parameters of metals • Favors non-spherical distortions • Improves bond lengths • Improves energies and geometries of H-bonded systems • There is error cancellation between X and C at short range • XC potential still decays exponentially into vacuum regions • Some improvement in band gaps in semiconductors • What was correct in LDA is worsened in GGA • Still incorrect dissociation limit. Fractionally charged fragments • Inter-configurational errors in IP and EA • Error cancellation between X and C is not complete at long-range. X hole is more long-ranged than XC hole

9. hybrid functionals • Combine GGA local exchange with Hartree-Fock non-local exchange: • Parameter  fitted to experimental data for molecules (~0.75), or determined from known properties. • PBE0, B3LYP, HSE06 • Properties: • Quite accurate in many respects, e.g. energies and geometries • Improve on the self-interaction error, but not fully SI-free • Improve on band gaps • Improve on electron affinities • Better quality than MP2 • Fitted hybrids unsatisfactory from the theoretical point of view

10. self-interaction correction (sic) • Self-interaction can be removed at the level of classical electrostatics: • Potential is state-dependent. Hence it is not an eigenvalue problem anymore, but a system of coupled PDEs • Orthogonality of SIC orbitals not guaranteed, but it can be imposed (Suraud) • Similar to HF, but the Slater determinant of SIC orbitals is not invariant against orbital transformations • The result depends on the choice of orbitals (localization) Perdew-Zunger 1982 Mauri, Sprik, Suraud

11. Dynamical correlation: vdw • Van der Waals (dispersion) interactions: are a dynamical non-local correlation effect • Dipole-induced dipole interaction due to quantum density fluctuations in spatially separated fragments • Functional (Dion et al 2004): • Expensive double integral • Efficient implementations (Roman-Perez and Soler 2009) • Good approximations based on dynamical response theory • Beyond VDW: Random Phase Approximation (Furche) 2(r,t) 1(r,t) 1(t) E1(t).2(t)  = VDW kernel fully non-local. Depends on (r) and (r’)

12. dynamical correlation: vdw and beyond • Empirical approaches: • With f a function that removes smoothly the singularity at R=0, and interferes very little with GGA (local) correlation. • Grimme (2006): C6 parameters from atomic calculations. Extensive parameterization: DFT-D. • Tkatchenko and Scheffler (2009): C6 parameters dependent on the density. • Random Phase Approximation (RPA): captures VDW and beyond. Can be safely combined wit exact exchange (SIC). Infinite order perturbation (like Coupled clusters in QC). Furche (2008); Paieret al (2010); Hesselmann and Görling (2011)

13. Strong static correlation: lsda+u • Strong onsite Coulomb correlations are ot captured by LDA/GGA • These are important for localized (d and f) electronic bands, where many electrons share the same spatial region: self-interaction problem • Semi-empirical solution: separate occupied and empty state by an additional energy U as in Hubbard’s model: • This induces a splitting in the KS eigenvalues: fi=orbital occupations

14. summary of dft approximations

15. electronic structure of UO2 Using the quantum-espresso package (http://www.quantum-espresso.org/) • Pseudopotentials • Plane wave basis set

16. Properties • fluorite structure • fcc, 3 atoms un unit cell • Lattice constant = 10.26 Bohr • Electronic insulator. Eg=2.1 eV • Electronic configuration of U: [Rn]7s26d15f3 • U4+: f2 • 5f-band partially occupied (2/7) • UO2: splitted by crystal field:t1u(3)+t2u(3)+ag(1) • Still partially occupied (2/3) • Jahn-Teller distortion opens gap.

17. Pseudopotential

18. Convergencewith energy cutoff

19. Energy-volumecurve

20. GGA(PBE) density of states

21. GGA+U density of states

22. GGA+U density of states: distorted

23. Van der waals for imidazolium salts • Animportantfamily of RoomTemperatureIonicLiquids (Green solvents) • Competingelectrostatic vs dispersioninteractions • Largesystemsstudiedwith DFT, within LDA or GGA • [Del Popolo, Lynden-Bell and Kohanoff, JPCB 109, 5895 (2005)] • Forcefieldsfittedto DFT-GGA calculations • [Youngs, Del Popolo and Kohanoff, JPCB 110, 5697 (2006)] • Electrostaticswelldescribed in DFT (LDA or GGA) • Dispersion (van derWaals) interactions are absent in both, LDA and GGA

24. DFT imidazoliumsalts

25. Resultsfor simple dimers M. Dion et al, PRL 92, 246401 (2004) (C6H6)2 Ar2 and Kr2 Bond lengths: 5-10% too long Binding energies: 50-100% too large

26. Resultsforsolids Polyethylene Silicon • Reasonable results for molecular systems • Keeps GGA accuracy for covalent systems •  General purpose functional

27. SolidfccArgon (E. Artacho) Some overbinding, and lattice constant still 5% too large … but much better than PBE (massive underbinding and lattice constant 14% too large)

28. Thedouble integral problem • (q1,q2,r12) decays as r12-6 • Ecnl = (1/2)   d3r1d3r2(r1) (r2)(q1,q2,r12) can be truncated for r12 > rc ~ 15Å • In principle O(N) calculation for systems larger than 2rc ~ 30Å • But... with x ~ 0.15Å (Ec=120Ry) there are ~(2106)2 = 41012 integration points • Consequently, direct evaluation of vdW functional is much more expensive than LDA/GGA

29. Factoring(q1,q2,r12) G. Román-Pėrez and J. M. Soler, Phys. Rev. Lett. 2009 Expand  in a basis set of functions p(q) FT

30. Interpolation as anexpansion • Basis functions p(q) interpolate between grid points • Lagrange polynomials: grid given by zeros of orthogonal polynomials • Cubic splines: grid points defined on a logarithmic mesh 20 grid points are sufficient Smoothening of  required at small q

31. O(NlogN) algorithm do, for each grid point i find i and i find qi=q(i ,i ) find i = i p(qi )  end do Fourier-transformi  k  do, for each reciprocal vector k find uk =  (k) k end do Inverse-Fourier-transform uk ui  do, for each grid point i find i , i , and qi find i , i /i, and i / i find vi end do • Implemented into SIESTA, but not SIESTA-specific: • Input:i on a regular grid • Output: Exc , vixc on the grid • No need for supercells in solids • No cutoff radius of interaction

32. If you can simulate a system with LDA/GGA, you can also simulate it with vdW-DFT Algorithmefficiency Message

33. Imidazoliumcrystals: Volumes

34. Imidazoliumcrystals: Hexafluorophospates emimpf6 bmimpf6 ddmimpf6

35. Imidazoliumcrystals: Hexafluorophospates [bmim][PF6] [ddmim][PF6]

36. Imidazoliumcrystals: Chlorides [mmim][Cl] [bmim][Cl]-m

37. [bmim][cl]: polymorphism [bmim][Cl]-Monoclinic [bmim][Cl]-Orthorhombic Energy difference per neutral ion pair

38. Imidazoliumclusters: Triflates [bmim][Tf] [bmim][Tf]2 Geometry Dimer association energy

39. Conclusions The non-empirical van der Waals functional of Dion et al. (DRSLL) improves significantly the description of the geometry of imidazolium salts. Volumes are improved respect to PBE, but still overestimated by 5%. Energetics is also improved. It is similar to that of empirical force fields such as CLaP. The cost of calculating the van der Waals correlation correction is 10 times that of PBE. However, in a self-consistent calculation for 100 atoms the overhead is only 20%.

40. The theoretical landscape 100 1000 10000 100000 1000000 High QM/MM Classical Semi-empirical Tight-binding Accuracy Ab initio Low Size (number of atoms)

41. QM/MM • Treat relevant part of the system quantum-mechanically, and the rest classically. • The problem is how to match the two regions. Easy for non-bonded interactions, more difficult for chemical bonds • One can also treat part of the system as a polarizable continuum, or reaction field (RF)

42. Quantum Mechanics in a local basis

43. Tight binding

44. Tight binding

45. Tight binding models for water A. T. Paxton and J. Kohanoff, J. Chem. Phys. 134, 044130 (2011) • Ground-up philosophy • Water molecule • Minimal basis. On-site energies to reproduce band structure • 1s orbital for H: Hs • 2s & 2p orbitals for O: Os, Op • O-H hopping integrals: • Values at equilibrium length to reproduce HOMO-LUMO gap: tss, tsp • GSP functional form. Cut-off between first and second neighbours • Charge transfer: Hubbard terms fitted to reproduce dipole moment: UO, UH • O-H pair potential: GSP form, fitted to reproduce bond length and symmetric stretching force constant • Crystal field parameter sppselected to reproduce polarizability

46. Tight binding models for water • Ground-up philosophy • Water dimer • O-O hopping integrals: tss, tsp , tpp, tpp • GSP form. Cut-off after first neighbours • O-O pair potential • Various forms (GSP, quadratic) to • reproduce binding energy curve • Fitting procedure • By hand (intuitive) • Genetic algorithm • This is the end of the fitting • All the rest are predictions

47. Ice-XI • DFT ice sinks in water! • Polarizable model marginal • Point charge model is fine

48. Tight-binding liquid waterA. T. Paxton and J. Kohanoff, J. Chem. Phys. 134, 044130 (2011)

49. Tight binding model for TiO2 Band structure: A. Y. Lozovoi, A. T. Paxton and J. Kohanoff DFT TB Rutile Anatase O on-site energies Os, Op and O-O hopping integrals: tss, tsp , tpp, tpp

50. Water/TiO2 interfaces (Sasha Lozovoi) Single water adsorption: dissociation