CHEM 938: Density Functional Theory

1. CHEM 938: Density Functional Theory exchange-correlation functionals January 28, 2010

2. Exchange-Correlation Functionals the accuracy of a DFT calculation hinges on the exchange-correlation functional that is used before we begin: • functionals are usually split into exchange parts and correlation parts • in general, you can mix and match functionals, but this isn’t always advisable • remember, we are after the true exchange-correlation hole, which depends on a cancellation of the exchange and correlation holes in some region

3. Exchange-Correlation Functionals consider an H2 molecule Fermi hole Coulomb hole total hole 0.7 Å 1.1 Å 2.6 Å RH-H total hole gives correct behaviour: neither the Fermi nor Coulomb holes alone are sufficient

4. Exchange-Correlation Functionals the accuracy of a DFT calculation hinges on the exchange-correlation functional that is used before we begin: • functionals are usually split into exchange parts and correlation parts • in general, you can mix and match functionals, but this isn’t always advisable • remember, we are after the true exchange-correlation hole, which depends on a cancellation of the exchange and correlation holes in some region • improper combinations may not provide adequate cancellation and provide an inaccurate description of the total hole • the point is to use functionals that are designed (or known) to work well together

5. Exchange-Correlation Functionals the accuracy of a DFT calculation hinges on the exchange-correlation functional that is used before we begin: • functionals depend on the density, ρ, but may also depend on ρ, 2ρ, or |ρ|2 • spin is introduced through spin polarization parameter • functionals are often named after their developers and sometimes the year • SVWN = Slater, Vosko, Wilks, Nusair • BP86= Becke, Perdew (86) • BLYP= Becke, Lee, Yang, Perdew • TPSS= Tao, Perdew, Staroverov, Scuseria

6. Jacob’s Ladder of Functionals John Perdew suggested the following hierarchy of functionals to reach chemical accuracy in DFT calculations chemical accuracy • successively adding ‘ingredients’ yields better approximation to the true functional (in theory) generalized random phase approximation unocc. {} • functionals at higher levels depend on those at lower levels hyper-GGA X • complexity (and computational effort) increases at higher levels |ρ|2 or meta-GGA 2ρ • provides a direction for the improvement of functionals, but not nearly as straightforward as in ab initio methods ρ GGA ρ LSDA Hartree world

7. Strategies for Developing XC Functionals there is no clear rationale for developing XC functionals 1. Select ingredients for functional: • corresponds to selecting a rung on Jacob’s ladder 2. Determine which properties of the XC hole and energy can be met: • there are known constraints that must be satisfied by the true XC hole and energy • only some of these constraints can be met at different rungs of Jacob’s ladder 3. Pick a mathematical form that can meet the constraints: • this is a black art 4. Select parameters that enter into this mathematical form: • non-empirical functionals select parameters to exactly satisfy constraints • empirical functionals select parameters to reproduce experimental data

8. Constraints certain properties of the XC hole and energy are known Limit of uniform spin densities: • the uniform spin density corresponds to a homogeneous electron gas • this constraint is satisfied by all non-empirical functionals, but can be violated by many empirical functionals Lieb-Oxford bound: • this is satisfied by all non-empirical functionals • this constraint is violated by most empirical functionals, but for unrealistic densities

9. Constraints certain properties of the XC hole and energy are known Scaling with uniform spin densities: • introduce scaled uniform electron density: • higher γ yields a more compactdensity • with scaling, EX and EC must satisfy: • exchange energy scales with the density • all functionals satisfy this • correlation energy becomes constant as density becomes infinite • violated by many functionals • follows from previous two constraints

10. Constraints certain properties of the XC hole and energy are known Limits for one-electron systems: • this cancels out exactly the self-interaction error • constraint can only be satisfied using exact exchange with hyper-GGAs • the electron should not be correlated with itself • this constraint can only be satisfied at the third rung of Jacob’s ladder

11. Constraints certain properties of the XC hole and energy are known Behaviour of exchange energies: • exchange only affects electrons of identical spin • total exchange energy should be sum of exchange energies of α-α and β-β interactions • exchange energy is always negative • these conditions are satisfied by all functionals

12. Parameter Selection many modern XC functionals contain parameters selected to reproduce experimental data examples: B88 exchange: β = 0.0042 by least squares fit to exchange energies of rare gas atoms B3LYP: a = 0.20, b = 0.72, c = 0.81 from fit to atomization energies from the G2 data set of molecules fitting to experimental data can yield accurate results, but parameters may not transfer to other systems

13. Parameter Selection John Perdew and coworkers have promoted the view that any parameters in functionals should be selected to satisfy constraints ‘...if an approximation fails to be essentially exact for the limited class of systems where it can be, it is a self-contradiction and should not underpin any major area of science.’ Perdew et al., J. Chem. Phys. (2005), 123, 062201. P86 exchange: • all numbers in this functional were selected to satisfy constraints on the exchange hole non-empirical parameters are more meaningful, but may yield poorer results than empirical functionals

14. Local Density Approximation first rung on Jacob’s ladder employs only the density • functionals based on the uniform electron gas model • electrons move on a positive background charge distribution • number of electrons, N, and volume, V approach infinity, and density (N/V) remains constant • good representation of a metal • assume that the exchange-correlation energy can be expressed as: exchange and correlation energy per electron of uniform electron gas with density ρ(r) probability that there is an electron at r

15. Local Density Approximation first rung on Jacob’s ladder employs only the density • assume that the exchange-correlation energy can be expressed as: • further assume that: • X and C are exchange and correlation functionals • they are local functionals of the density: they only depend on ρ(r) and not on combinations of ρ(r) and ρ(r’) • since these functionals depend only on the local value of the density, this is called the local density approximation: LDA • if the different spin densities are considered, then this is called the local spin density approximation: LSDA

16. Local Density Approximation what is the local exchange functional? • recall, that the exchange energy can be expressed as the interaction between the density and the exchange hole: • in a uniform electron gas, hX is spherically symmetric around the reference electron and has a uniform density • furthermore, hX contains exactly 1 electron • the radius of the holes is thus: • this radius is approximated as the average distance between electrons in the system

17. Local Density Approximation what is the local exchange functional? • from basic electrostatics, the potential associated with a uniform spherical charge distribution will scales as 1/rS • which gives the exchange energy of a uniform electron gas as: sometimes this is treated as an empirical parameter instead of using the ideal value • it can be shown that: this is called Slater exchange (although it was also developed earlier and independently by Dirac and Bloch)

18. Local Density Approximation what is the local correlation functional? • derivation of the local exchange hole benefitted from the fact that the hole integrates to one electron (that let us easily determine a radius) • correlation hole integrates to zero, so we can follow a similar first-principles approach in principle • instead, the development of the correlation hole was achieved by generating parameterized functional forms that reproduce highly accurate correlation data obtained through quantum Monte Carlo calculations

19. Local Density Approximation what is the local correlation functional? • most common form is from Vosko, Wilks and Nusair (VWN) • there are five different forms of these functionals • usually VWN1, VWN3 or VWN5 are used in practice • they are also combined with other types of functionals, so you should be careful when reporting and comparing results • other local exchange-correlation functionals are in use: • Perdew-Zunger (PZ or PZ81) • Perdew-Wang (PW)

20. Local Density Approximation how does it do? • LDA functionals do surprisingly well considering the huge difference between a uniform electron gas and a molecule • errors on atomization energies (32 molecules): 36 kcal/mol • (Johnson, Gill, Pople, J. Chem. Phys.98, 5612 (1993)) • mean absolute errors on bond lengths (108 species): 0.016 Å • (Scheiner, Baker, AndzelmJ. Comput. Chem.18, 775 (1997)) • mean absolute errors on vibrational frequencies (32 molecules): 75 cm-1 • (Johnson, Gill, Pople, J. Chem. Phys.98, 5612 (1993)) of course, there are many cases where it does not perform well, and improvements are possible

21. Generalized Gradient Approximation how do we move beyond LDA? density in a molecule is not homogeneous: • we need to account for how the density changes with space • this is done by using the gradient of the density in addition to the density itself • where s is the reduced density gradient: • exponent of 4/3 ensures this is a dimensionless quantity including the gradient of the density into EXC is the generalized gradient approximation (GGA)

22. Generalized Gradient Approximation including the gradient of the density into EXC is the generalized gradient approximation (GGA) GGA functionals are generally split into exchange and correlation parts: and in each part, the gradient terms are treated as corrections to LDA there is no paradigmatic system on which to base GGAs, as a result their development has relied heavily on general constraint satisfaction, mathematical intuition and fitting • development of GGAs is often governed more by yielding accurate results than providing a meaningful physical picture

23. Generalized Gradient Approximation early GGA functionals were based on mathematical intuition and empirical parameter fitting Becke developed the first successful GGA exchange functional (B or B88) • mathematical form was selected to accurately reproduce the exchange hole sum rules and the exchange energy asymptotically far from a finite system • β is an empirical parameter set to 0.0042 to reproduce the known exchange energies of rare gas atoms • Becke exchange is used extensively at the GGA level by combining it with GGA correlation functionals. Popular combinations include: BP86: B88 with Perdew’s 1986 correlation functional BLYP: B88 with the correlation functional of Lee, Yang and Parr

24. Generalized Gradient Approximation John Perdew and coworkers have made significant efforts to develop non-empirical GGAs Perdew, Burke and Ernzerhof developed the ‘best’ non-empirical GGA (PBE): • actually a set of exchange and correlation functionals that should be used together • satisfies many constraints on the exchange-correlation hole, and exchange-correlation energy • all parameters entering into the mathematical forms used are selected to satisfy constraints – not to reproduce experimental data • very simple from the standpoint of implementation other non-empirical GGA functionals include: • PW91: precursor to PBE by Perdew and Wang

25. Generalized Gradient Approximation do GGAs offer any significant improvement? SVWN BLYP BPW91 BP86 PBE 0.016 Å 0.021 Å 0.017 Å 0.022 Å 0.012 Å bond lengths 75 cm-1 73 cm-1 39 cm-1 frequencies 36 6 12 7 9 atomization energies (kcal/mol) 15.2 15.0 note these data represent various molecule sets and should not really be compared too critically NMR chemical shifts (ppm, C, N, and O) GGAs offer some improvement in terms of energies and frequencies

26. Meta-GGAs the next step beyond GGAs is to include more terms in the expansion is the kinetic energy density where: • one could also include Laplacians of the density, 2ρ, but they are hard to calculate and provide the same information as τ for densities relevant to molecules • using τ also permits the satisfaction of more constraints than 2ρ • meta-GGAs are fairly new, but have been implemented into many programs recently • the TPSS functional is the current state-of-the-art meta-GGA • results are very good in terms of energies and structures • slightly more effort than GGAs, but comparable to hybrid functionals like B3LYP

27. Hyper GGAs fourth rung of Jacob’s ladder includes ‘exact exchange’ • exchange energies dominate the exchange-correlation term – so we have to try to get that correct • fortunately, we know how to calculate exchange using orbitals: • this can be expressed as the following functional: unfortunately, we can’t just use the exact exchange functional because we don’t have the exact correlation functional

28. Hyper GGAs unfortunately, we can’t just use the exact exchange functional because we don’t have the exact correlation functional Fermi hole Coulomb hole total hole 0.7 Å 1.1 Å 2.6 Å RH-H

29. Hyper GGAs hyper GGAs include some exact exchange • the most general form builds off of meta-GGAs • in practice, available hyper GGA exchange functionals take the following form: • where a is a parameter that determines the amount of exact exchange that is used • functionals of this sort are often termed hybrid functionals because they are a hybrid between Hartree-Fock and DFT exchange • they were popularized by Axel Becke, who justified their use through the adiabatic connection method

30. Hyper GGAs let’s revisit the adiabatic connection method • recall, the adiabatic connection method let us incorporate the error in the kinetic energy of the Kohn-Sham orbitals into EXC • when λ=0, the system is composed of non-interacting electrons represented by a single Slater determinant • the only non-classical interaction is the exchange energy arising from the anti-symmetry of the determinant • we can calculate this exactly using Hartree-Fock exchange • when λ=1, the system is composed of fully interacting electrons • the energy includes all exchange and correlation interactions • we can calculate this approximately using exchange-correlation functionals

31. Hyper GGAs let’s revisit the adiabatic connection method • recall, the adiabatic connection method let us incorporate the error in the kinetic energy of the Kohn-Sham orbitals into EXC • this integral connects Hartree-Fock exchange to DFT exchange-correlation put in Figures from the text

32. Hyper GGAs including HF exchange yield hybrid functionals • in an early work, Becke assumed a 50-50 mix of Hartree-Fock and DFT exchange • this is called the Becke half-and-half functional • note that EXHF means exact exchange using the Hartree-Fock exchange energy expression, but we use Kohn-Sham orbitals – not Hartree-Fockorbitals • this functional performs quite well: 6.5 kcal/mol error on a large set of atomization energies • seems to work well for weakly bonded systems (e.g. benzene dimers) • however, this is probably fortuitous • we will visit this point later because weakly bonded systems are a serious problem for DFT

33. Hyper GGAs including HF exchange yield hybrid functionals • the most widely used functional, B3LYP, is given by: • a = 0.20, b=0.72, 0.81 • be careful, some programs use different sets of parameters due to different implementations or mistakes • the parameters used here were selected to reproduce experimental data • as such, this empirical functional does not actually satisfy many constraints • nonetheless, it performs quite well: ~2.0 kcal/mol error on atomization energies of small molecules • however, it is not a panacea and does fail in many well-known circumstances: we will look at these further later in the course • many other hybrid functionals have been developed

34. Fifth Rung the ultimate goal is to develop functionals that provide accuracy that rivals experiments • to do this, it is necessary to incorporate the unoccupied Kohn-Sham orbitals • these are no where close to being ready • they will be very computationally-intensive when they are available

35. Jacob’s Ladder of Functionals John Perdew suggested the following hierarchy of functionals to reach chemical accuracy in DFT calculations chemical accuracy • not even close to being ready generalized random phase approximation unocc. {} • available as hybrid functionals, offer pretty good performance hyper-GGA X |ρ|2 or meta-GGA 2ρ • only recently developed, but promising ρ • improvement over LSDA, often used for molecular and solid state calculations GGA ρ LSDA • not often used in chemistry, but often good for solid state calculations Hartree world

36. Combinations of Functionals numerous functionals are implemented in various software packages Gaussian 03: exchange functionals: • S = Slater exchange with coefficient = 2/3, LDA • Xalpha = Slater exchange with coefficient = 0.7, LDA • B = Becke’s 1988 exchange, GGA • PW91 = Perdew-Wang 1991 exchange, GGA • MPW = modified PW91, GGA • G96 = Gill’s 1996 exchange, GGA • PBE = Perdew, Burke and Ernzerhof exchange, GGA • O = Handy’s OPTX exchange, GGA

37. Combinations of Functionals numerous functionals are implemented in various software packages Gaussian 03: correlation functionals: • VWN = Vosko, Wilks, Nusair correlation III, LDA • VWN5 = Vosko, Wilks, Nusair correlation V, LDA • LYP = Lee, Yang, Parr correlation, GGA • PL = Perdew’s local correlation, LDA • P86 = Perdew’s GGA correlation, GGA • PW91 = Perdew Wang correlation, GGA • B95 = Becke’s correlation, GGA • PBE = Perdew, Burke, Ernzerhof correlation, GGA

38. Combinations of Functionals numerous functionals are implemented in various software packages Gaussian 03: hybrid functionals: • B3LYP = Becke’s exchange, HF exchange, LYP correlation • B3P86 = Becke’s exchange, HF exchange, P86 correlation • BHandH = Slater exchange, HF exchange, LYP correlation • BHandHLYP = Becke’s exchange, HF exchange, LYP correlation • others standard combos: • SVWN = Slater exchange, VWN correlation • PW91PW91 = PW91 exchange and correlation • BPW91 = Becke exchange, PW91 correlation • PBEPBE = PBE exchange and correlation • BLYP = Becke exchange, LYP correlation • BP86 = Becke exchange, P86 correlation • you can specify others

39. Combinations of Functionals numerous functionals are implemented in various software packages NWChem:

40. Implementation of XC Functionals XC functionals are complicated mathematically and do not lend themselves to simple analytical evaluation

41. Implementation of XC Functionals XC functionals are complicated mathematically and do not lend themselves to simple analytical evaluation consider SVWN: XC functionals are usually evaluated numerically

42. Numerical Quadrature quadrature let’s us estimate an integral as a sum of discrete areas example: f(x) f(x) on grid • integration with trapezoid rule f(x) x

43. Numerical Quadrature quadrature let’s us estimate an integral as a sum of discrete areas for more complicated integrals: • introduce a grid, map the functional onto it, and then perform quadrature weight of grid point p • grid should be dense around nucleus where density is high and become more sparse at large distances • generally achieved using set of overlapping spherical grids centered on each atom (a few hundred grid points/atom) • integrals are broken down into angular and radial parts

44. Software numerous density functional theory codes exist (the following list is nowhere near complete) ADF: • many functionals available • uses Slater basis functions instead of Gaussians • can treat molecular and periodic systems • support for a large number of properties • commercial, but available on hpcvl CPMD: • many functionals available • uses plane-wave basis sets and pseudopotentials • can treat molecular and periodic systems • designed primarily for molecular dynamics simulations • free to academics

45. Software numerous density functional theory codes exist (the following list is nowhere near complete) Gaussian: • general quantum chemical package • many methods available including DFT • wide range of functionals • uses Gaussian basis sets • support for many properties • primarily for calculations on molecules • commercial, but available on hpcvl NWChem: • general quantum chemical package • many methods available including DFT • wide range of functionals • uses Gaussian basis sets • support for many properties • primarily for calculations on molecules • free to academics

46. Software numerous density functional theory codes exist (the following list is nowhere near complete) Quantum-ESPRESSO: • many functionals available • uses plane-wave basis sets and pseudopotentials • can treat molecular and periodic systems • designed primarily for molecular dynamics simulations • free to academics and fully open-source Turbomole: • DFT program – very fast • many methods available including DFT • wide range of functionals • uses Gaussian basis sets • support for many properties • primarily for calculations on molecules • commercial

47. Software numerous density functional theory codes exist (the following list is nowhere near complete) VASP: • many functionals available • uses plane-wave basis sets and pseudopotentials • can treat periodic systems • designed with solid state calculations in mind • commercial

48. DFT in Gaussian our department has a license for Gaussian, so we’ll use it to do a few calculations you can access Gaussian through: • department webmo server (http://130.15.98.195/cgi-bin/webmo/login.cgi) • department license allowing you to install it on departmental computers • hpcvl, or other clusters what Gaussian does: • quantum chemical calculations with wide range of ab initio and DFT methods • calculates energetics • optimize geometries of minimum energy structures and transition state • calculate vibrational frequencies • analysis of electronic structure • excited state calculations • NMR, etc. • not so great for dynamics, condensed phase or surface calculations there are other software packages that do these calculations, too

49. Gaussian Input File example input for water link 0 commands route line charge and multiplicity geometry in cartesian coordinates • route line specifies the type of calculation to perform • this example performs a geometry optimization at the Hartree-Fock level using a 3-21G basis set • as we look at different properties one may wish to calculate, I’ll point out different keywords you may want to include

50. DFT in Gaussian Input: • b3lyp on the route line indicates this is a DFT calculation using the B3LYP functional charge and multiplicity geometry in cartesian coordinates