1. E LECTRONIC S TRUCTURE WITH DFT: GGA AND BEYOND Jorge Kohanoff Queen’s University Belfast United Kingdom j.kohanoff@qub.ac.uk

2. 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. DENSITY FUNCTIONAL THEORY ( DFT ): KOHN - SHAM EQUATIONS

3. 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’) E XCHANGE AND CORRELATION IN DFT : THE LOCAL DENSITY APPROXIMATION ( LDA ) XC energy density of the HEG

4. Location of the XC hole (Jones and Gunnarsson, 1982) E XCHANGE AND CORRELATION IN DFT : THE LOCAL DENSITY APPROXIMATION ( LDA )

5. 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 –e 2 /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, UO 2 ) LDA - LSDA : TRENDS AND LIMITATIONS

6. 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) BEYOND THE LDA

7. E XC expanded in gradients of the density where  is the spin polarization s =|  |/2k F  is the density gradient And F XC is the enhancement factor First-order term is fine, but higher-order terms diverge. Only by some re-summation to ∞-order the expansion converges. GGA: F XC is designed to fulfil a number of exactly known properties, e.g. Perdew-Burke-Ernzerhof (PBE) 1. Exchange : uniform scaling, LSDA limit, spin-scaling relationship, LSDA linear response, Lieb-Oxford bound 2. Correlation : second-order expansion, hole sum rule, vanishes for rapidly varying densities, cancels singularity at high densities GRADIENT EXPANSIONS : GENERALIZED GRADIENT APPROXIMATION r+dr  (r+dr)

8. 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 I P and E A Error cancellation between X and C is not complete at long-range. X hole is more long-ranged than XC hole P ROPERTIES OF THE GGA

9. 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 : 1. Quite accurate in many respects, e.g. energies and geometries 2. Improve on the self-interaction error, but not fully SI-free 3. Improve on band gaps 4. Improve on electron affinities 5. Better quality than MP2 6. Fitted hybrids unsatisfactory from the theoretical point of view HYBRID FUNCTIONALS

10. 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) SELF - INTERACTION CORRECTION ( SIC ) Perdew-Zunger 1982 Mauri, Sprik, Suraud

11. 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) D YNAMICAL CORRELATION : VDW  1 (r,t)  2 (r,t)  1 (t)E 1 (t).  2 (t)  = VDW kernel fully non-local. Depends on  (r) and  (r’)

12. Empirical approaches: With f a function that removes smoothly the singularity at R=0, and interferes very little with GGA (local) correlation. Grimme (2006): C 6 parameters from atomic calculations. Extensive parameterization: DFT-D. Tkatchenko and Scheffler (2009): C 6 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); Paier et al (2010); Hesselmann and Görling (2011) DYNAMICAL CORRELATION : VDW AND BEYOND

13. 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: S TRONG STATIC CORRELATION : LSDA + U f i =orbital occupations

14. SUMMARY OF DFT APPROXIMATIONS

15. ELECTRONIC STRUCTURE OF UO 2 Using the quantum-espresso package (http://www.quantum-espresso.org/)http://www.quantum-espresso.org/ Pseudopotentials Plane wave basis set

16. P ROPERTIES fluorite structure fcc, 3 atoms un unit cell Lattice constant = 10.26 Bohr Electronic insulator. E g =2.1 eV Electronic configuration of U: [Rn]7s 2 6d 1 5f 3 U 4+ : f 2 5f-band partially occupied (2/7) UO 2 : splitted by crystal field: t 1u (3)+t 2u (3)+a g (1) Still partially occupied (2/3) Jahn-Teller distortion opens gap.

17. P SEUDOPOTENTIAL

18. C ONVERGENCE WITH ENERGY CUTOFF

19. E NERGY - VOLUME CURVE

20. GGA(PBE) DENSITY OF STATES

21. GGA+U DENSITY OF STATES

22. GGA+U DENSITY OF STATES : DISTORTED

23.  An important family of Room Temperature Ionic Liquids (Green solvents)  Competing electrostatic vs dispersion interactions  Large systems studied with DFT, within LDA or GGA [Del Popolo, Lynden-Bell and Kohanoff, JPCB 109, 5895 (2005)]  Force fields fitted to DFT-GGA calculations [Youngs, Del Popolo and Kohanoff, JPCB 110, 5697 (2006)]  Electrostatics well described in DFT (LDA or GGA)  Dispersion (van der Waals) interactions are absent in both, LDA and GGA V AN DER WAALS FOR IMIDAZOLIUM SALTS

24. DFT IMIDAZOLIUM SALTS

25. R ESULTS FOR SIMPLE DIMERS Ar 2 and Kr 2 (C 6 H 6 ) 2 Bond lengths: 5-10% too long Binding energies: 50-100% too large M. Dion et al, PRL 92, 246401 (2004)

26. R ESULTS FOR SOLIDS Polyethylene Silicon Reasonable results for molecular systems Keeps GGA accuracy for covalent systems  General purpose functional

27. S OLID FCC A RGON (E. A RTACHO ) Some overbinding, and lattice constant still 5% too large … but much better than PBE (massive underbinding and lattice constant 14% too large)

28. T HE DOUBLE INTEGRAL PROBLEM   (q 1,q 2,r 12 ) decays as r 12 -6  E c nl = (1/2)   d 3 r 1 d 3 r 2  (r 1 )  (r 2 )  (q 1,q 2,r 12 ) can be truncated for r 12 > r c ~ 15Å  In principle O(N) calculation for systems larger than 2r c ~ 30Å  But... with  x ~ 0.15Å (E c =120Ry) there are ~(2  10 6 ) 2 = 4  10 12 integration points  Consequently, direct evaluation of vdW functional is much more expensive than LDA/GGA

29. F ACTORING  ( Q 1, Q 2, R 12 ) G. Román-Pėrez and J. M. Soler, Phys. Rev. Lett. 2009 Expand  in a basis set of functions p  ( q) FT

30. I NTERPOLATION AS AN EXPANSION 20 grid points are sufficient Smoothening of  required at small q 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

31. O(N LOG N) ALGORITHM do, for each grid point i find  i and  i find q i =q(  i,  i ) find   i =  i p  (q i )  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 q i find   i,   i /  i, and   i /   i  find v i end do Implemented into SIESTA, but not SIESTA-specific: Input:  i on a regular grid Output: E xc, v i xc on the grid No need for supercells in solids No cutoff radius of interaction

32. A LGORITHM EFFICIENCY Message If you can simulate a system with LDA/GGA, you can also simulate it with vdW-DFT SystemAtoms CPU time in GGA-XC CPU time in vdW-XC vdW/GGA overhead Ar 2 20.75 s (44%)7.5 s (89%)400% MMX polymer1241.9 s (2%)10.6 s (16%)17% DWCN16811.9 s (0.6%)109 s (5.2%)4%

33. I MIDAZOLIUM CRYSTALS : V OLUMES LDAPBEVDWEXP [emim][PF6]877.41088.31059.11023.9 [bmim][PF6]513.4636.6620.0605.0 [ddmim][PF6]1710.92258.02095.52000.6 [mmim][Cl]607.9750.5728.6687.6 [bmim][Cl]-o843.81039.91019.8961.1 [bmim][Cl]-m836.31051.91024.6966.7 Error-13.5%+8.5%+5%

34. I MIDAZOLIUM CRYSTALS : H EXAFLUOROPHOSPATES bmimpf6 emimpf6 ddmimpf6

35. [bmim][PF6] [ddmim][PF6] C 4 -C 4 C 1 ”- C 1 ” C 12 - C 12 ’ C 4 ’-C 4 ’C 2 -C 4 ’C 4 ’- C 1 ” P-PC 2 -PC 1 ”-PC 4 ’-PC 12 ’-PError LDA 5.184.284.834.524.236.006.663.553.954.274.35 6.0% PBE 5.624.245.445.125.106.566.894.004.754.964.78 7.0% VDW 5.474.295.234.904.796.446.803.824.224.684.65 3.3% EXP 5.474.365.014.854.616.256.454.234.014.624.63 C5-C5C1”-C1”C4’-C4’C2-C4’C4’-C1”P-PC2-PC1”-PC4’-PError LDA 4.324.24 3.734.613.645.053.703.884.75 7.0% PBE 4.644.853.995.073.975.694.164.214.95 2.6% VDWE 4.604.554.054.804.075.644.214.204.91 2.6% VDW 4.544.604.044.803.955.623.904.114.87 2.0% EXP 4.564.524.055.083.865.543.954.294.94 I MIDAZOLIUM CRYSTALS : H EXAFLUOROPHOSPATES

36. I MIDAZOLIUM CRYSTALS : C HLORIDES [bmim][Cl]-m C5-C5C1’-C1’C2-C1’C2-ClC1’-Cl LDA3.073.543.373.343.81 PBE3.943.863.633.483.74 VDW3.343.823.56 3.77 C5-C5C1”-C1”C4’-C4’C2-C4’C4’-C1”C2-ClC1”-ClC4’-Cl LDA4.874.505.173.253.833.283.513.70 PBE5.265.145.723.633.813.383.863.89 VDW5.214.975.423.533.843.463.843.94 EXP5.194.965.333.483.623.393.823.80 [mmim][Cl]

37. [ BMIM ][ CL ]: POLYMORPHISM [bmim][Cl]-Monoclinic[bmim][Cl]-Orthorhombic Energy difference per neutral ion pair LDAPBEVDWFF(CLaP)  E(M-O) [eV] -0.05-0.06-0.010.08

38. I MIDAZOLIUM CLUSTERS : T RIFLATES [bmim][Tf][bmim][Tf] 2 LDAPBEVDWFF(CLaP)  E [eV] -1.411-1.053-1.453-1.457 Dimer association energy LDAPBEVDWFF(CLaP) Error7.2%2.3%1%-- Geometry

39. C ONCLUSIONS 1.The non-empirical van der Waals functional of Dion et al. (DRSLL) improves significantly the description of the geometry of imidazolium salts. 2.Volumes are improved respect to PBE, but still overestimated by 5%. 3.Energetics is also improved. It is similar to that of empirical force fields such as CLaP. 4.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. T HE THEORETICAL LANDSCAPE Size (number of atoms) 1000100001000000100000100 High Low

41. 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) QM/MM

42. Q UANTUM M ECHANICS IN A LOCAL BASIS

43. T IGHT BINDING

45. T IGHT BINDING MODELS FOR WATER Ground-up philosophy Water molecule 1.Minimal basis. On-site energies to reproduce band structure a) 1s orbital for H:  Hs b) 2s & 2p orbitals for O:  Os,  Op 2.O-H hopping integrals: a) Values at equilibrium length to reproduce HOMO-LUMO gap: t ss , t sp  b) GSP functional form. Cut-off between first and second neighbours 3.Charge transfer: Hubbard terms fitted to reproduce dipole moment: U O, U H 4.O-H pair potential: GSP form, fitted to reproduce bond length and symmetric stretching force constant 5.Crystal field parameter  spp selected to reproduce polarizability A. T. Paxton and J. Kohanoff, J. Chem. Phys. 134, 044130 (2011)

46. T IGHT BINDING MODELS FOR WATER Ground-up philosophy Water dimer 6.O-O hopping integrals: t ss , t sp , t pp , t pp  GSP form. Cut-off after first neighbours 7.O-O pair potential Various forms (GSP, quadratic) to reproduce binding energy curve 8.Fitting procedure a) By hand (intuitive) b) Genetic algorithm This is the end of the fitting All the rest are predictions

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

48. T IGHT - BINDING LIQUID WATER A. T. P AXTON AND J. K OHANOFF, J. C HEM. P HYS. 134, 044130 (2011)

49. T IGHT BINDING MODEL FOR T I O 2 Band structure: A. Y. Lozovoi, A. T. Paxton and J. Kohanoff Rutile Anatase DFTTB O on-site energies  Os,  Op and O-O hopping integrals: t ss , t sp , t pp , t pp 

50. W ATER /T I O 2 INTERFACES (S ASHA L OZOVOI ) Single water adsorption: dissociation

51. W ATER /T I O 2 INTERFACES (S ASHA L OZOVOI ) A water layer on TiO 2

52. W ATER /T I O 2 INTERFACES (S ASHA L OZOVOI ) Bulk water on TiO 2 4 TiO 2 layers (480 atoms) 184 water molecules (552 atoms) 1032 atoms TBMD 1 ps (one week)

53. W ATER /T I O 2 INTERFACES (S ASHA L OZOVOI ) z-density profile of bulk water between TiO 2 surfaces

54. P ROTON TRANSFER IN WATER Chemical bonds broken and formed

55. O RGANIC MOLECULES (T ERENCE S HEPPARD ) Benzylacetone in water

56. B ENZYLACETONE AT W ATER /T I O 2 INTERFACE Sasha Lozovoi and Terence Sheppard

57. S UMMARY In developing a force field there are 3 main aspects to consider. In order of relevance: Designing the model Choosing the properties to be reproduced Choosing a methodology to fit those properties Fitting procedures cannot work out their magic if the model is not good enough To become model-independent we need to introduce electrons explicitly: ab initio and DFT methods These are expensive. Simplifications like semi-empirical, tight-binding and QM/MM methods bridge the gap Further simplifications lead to extremely efficient bond- order potentials (BOP)