DFT: Exchange-Correlation
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1 DFT: Exchange-Correlation Local functionals, exact exchange and other post-dft methods Paul Tulip Centre for Materials Physics Department of Physics University of Durham
2 Outline Introduction What is exchange and correlation? Quick tour of XC functionals (Semi-)local: LDA, PBE, PW91, WC, etc Ab intitio non-local: HF, sx, OEP Empirical non-local (hybrids): B3LYP, HSE DFT+U DFT+D Beyond DFT: GW Appropriate Use Closing statements
3 Density Functional Theory Foundation is Hohenberg-Kohn-Sham approach N-electron system auxiliary KS system n(r) n(r) Auxiliary system has same electronic density as N-electron system Auxiliary system particles are non-interacting Kohn-Sham equations
4 Density Functional Theory Hohenberg-Kohn-Sham approach turns an intractable N-body problem into N coupled one-body problems This is tractable! QM exchange-correlation effects in This is the great unknown in DFT we must approximate Commonly used approximations: LDA, GGA, BLYP, B3LYP..
5 A Formal Definition We can formally define the XC energy through:
6 A Formal Definition We can formally define the XC energy through: Exact kinetic energy
7 A Formal Definition We can formally define the XC energy through: Exact kinetic energy KS kinetic energy
8 A Formal Definition We can formally define the XC energy through: Exact kinetic energy KS kinetic energy Exact e-e interaction
9 A Formal Definition We can formally define the XC energy through: Exact kinetic energy KS kinetic energy Exact e-e interaction Hartree energy
10 A Formal Definition We can formally define the XC energy through: Exact kinetic energy Exact e-e interaction Unknown!
11 Alternative Exact Definition Exact XC interaction is unknown Within DFT we can write the exact XC interaction as E xc [n] = 1 2 n(r) n xc(r,r') r r' drdr' This would be excellent if only we knew what n xc was! This relation defines the XC energy. It is simply the Coulomb interaction between an electron an r and the value of its XC hole n xc (r,r ) at r.
12 Properties of XC functional Although exact form of XC functional unknown, it must satisfy certain properties Property LDA GGA HF *Only for occupied orbitals. Sum rule for XC hole. Self-interaction correction. * Derivative discontinuity. Correct 1/r potential decay. *
13 Properties of XC functional Property LDA GG HF XC hole is the displaced charge A that forms round a point test charge. Sum rule for XC hole. *Only for occupied orbitals. Sum of displaced charge should be the negative of the charge on the test charge. Self-interaction correction. * Derivative discontinuity. Correct 1/r potential decay. *
14 Properties of XC functional Property LDA GG A Sum rule for XC hole. Self-interaction correction. * Derivative discontinuity. Correct 1/r potential decay. * *Only for occupied orbitals. Hartree energy is coulomb interaction of classical charge. Contains interactions between electron and itself which are unphysical. XC functional should correct this. HF
15 Properties of XC functional Property LDA GG A Sum rule for XC hole. Self-interaction correction. * Derivative discontinuity. Correct 1/r potential decay. * XC potential should jump discontinuously when infinitesimal amount of charge added to HF system with integer electrons. *Only for occupied orbitals.
16 Properties of XC functional XC potential should jump discontinuously when infinitesimal amount of charge added to system with integer electrons. Property LDA GG A Sum rule for XC hole. Self-interaction correction. * Derivative discontinuity. Correct 1/r potential decay. * HF For finite systems, XC potential should decay as 1/r at long ranges. Orbitals should decay with individual exponents. This is not the case with LDA and GGA. *Only for occupied orbitals.
17 Generalities All functionals are approximations Reports of Failures of DFT are actually reports of a failure of the XC functional No functional (so far) is accurate(?) for all properties of interest No matter what functional is invented someone will always find a case where it fails Any functional can be applied to any electronic structure problem In this sense it is ab initio but we use experience and intuition to decide which one to use
18 Ladder of functionals (LDA) The simplest XC functional is the local density approximation (LDA) E LDA xc [n(r)] = n(r)ε hom xc [n(r)]dr It was used for a generation in materials science, but is not accurate enough for many chemical purposes Typical errors Over-binds (binding energy too large) Underestimates lattice parameters Phase stability incorrect order Energetics of magnetic materials in error
19 Ladder of functionals (GGA) The generalised gradient approximation (GGA) contains the next term in a derivative expansion of the charge density: E GGA [n(r)] = xc n(r)ε GGA [n(r), n(r)]dr xc Typically (but not always) this is more accurate than the LDA GGA greatly reduce the bond dissociation energy error, and generally improve transition-state barriers But, unlike LDA, there is no single universal form
20 Why the GGA? LDA depends only on one variable (the density). GGAs require knowledge of 2 variables (the density and its gradient). In principle one can continue with this expansion. If quickly convergent, it would characterise a class of many-body systems with increasing accuracy by functions of 1,2,6, variables. How fruitful is this? Depends on parameterisation used, but it will always be semi-local.
21 Ladder of functionals (meta-gga) Next come the meta-ggas Instead of using idea, in principle) typically the kinetic energy is used (same or Example: TPSS (J. Tao, J. P. Perdew, V. N. Staroverov, G. E. Scuseria, Phys. Rev. Lett. 91, (2003) )
22 Various breeds of functionals Non-empirical functionals Such as LDA Some GGAs (PW91, PBE, rpbe, WC, PBEsol, ) Not fitted to any empirical results These come from known exact conditions If these conditions are important to your physical value of interest then you should get a reliable result These functionals often have systematic failures (and successes!) so their reliability can usually be predicted
23 Some GGAs PW91: J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B (1992). PBE: J. P. Perdew, K. Burke and M Ernzerhof, Generalised gradient approximation made simple, Phys. Rev. Lett (1996). RPBE: B. Hammer, L. B. Hansen and J. K. Norskov, Improved adsorption energies within DFT using revised PBE functionals, Phys. Rev. B (1999). WC: Z. Wu and R. E. Cohen, More accurate gradient approximation for solids, Phys. Rev. B 73, (2006)
24 Different fits for ab initio GGAs PW91 is the first reasonable GGA that can be reliably used over a very wide range of materials. PW91 contains much of the known correct physics of the exchange and correlation interactions. PBE is based on PW91 containing the correct features of LDA but the correct (but hopefully not important!) features of PW91 that are ignored are: (1) Correct 2 nd order gradient coefficients of E x and E c in the slowly varying limit. (2) Correct non-uniform scaling of ε x in the limits where s tend to infinity. WC is newer - re-parameterisation of PBE.
25 Charge Differences (LDA-PW91)
26 Empirical XC functionals Good(!) empirical (and non-empirical) functionals are widely applicable None are good at everything, but they are all reasonable for most properties e.g. PBE is not the best functional for any particular property but is probably the best, on average for all properties. Good empirical functionals are usually best for the particular property that they were designed for
27 Various breeds of functionals Over-fitted functionals Contain many fitted parameters usually from a few dozen to hundreds of parameters Usually fitted to a particular set of properties of hundreds of molecules made from atoms of low atomic number There are many of these in the literature. Beware: every one of these (so far) only works for systems for which they are fitted The academic/accelrys version of Castep does not contain any of these
28 Hartree-Fock Exchange is essentially the Pauli exclusion principle Quantum mechanically, electrons are Fermions hence many-particle wavefunction is anti-symmetric E X non local = 1 2 ( ) = Ψ( r 2,r ) 1 Ψ r 1,r 2 which after some generalisation we find the energy is ij,kq * ψ ik ( r)ψ ik r' * ( )ψ jq r r' ( r' )ψ jq ( r) drdr' Note: HF can get properties wrong in the opposite direction from DFT
29 Moving off the ladder: Hybrid Functionals If one method over-estimates your value of interest and another underestimates it, then the answer you want can be obtained by taking a bit of both! (You may notice this is not ab initio) Leads to a zoo of functionals: aesthetically and conceptually unappealing
30 Hybrid functionals Hybrid Functionals: a few empirical parameters These include B3LYP, HSE, B88, PBE0 Generally a is around 20-25%. E c local usually contains parameters Good answers provided one works with systems close to the set of systems for which the functional was fitted.
31 Functionals with Hubbard U DFT+U What this is not: A general method to get the band gap correct for any material What this is: Fixes a very specific problem that LDA/GGA can get wrong in highly correlated materials
32 DFT+U A brief digression the Hubbard model: 2 limits: t >> U : U >> t : H ˆ + = t a iσ a jσ +U ˆ ij conventional band solid. Total energy minimised by minimising kinetic energy. Delocalised Bloch states. n i n i energy minised by minimising potential energy. Avoid double occupancy of sites localised states. At half-filling, each site has one electron Mott insulator i ˆ
33 DFT+U How does this relate to DFT? ˆ H KS DFT = 2 2m 2 + v ext ( r ) + v Hartree ( r ) + v xc ( r ) With v xc from LDA/GGA, we have a mean-field solution This is the same for occupied and unoccupied states LDA/GGA will not be able to predict a Mott insulator LDA/GGA suffers from self-interaction error -excessive electron delocalisation On-site repulsion U not well-treated: important correlations neglected Particularly important for highly localised d and f electrons ij h ij a ˆ i+ ˆ a j
34 DFT+U In DFT+U split electrons into two subsystems: - localised d or f electrons described by - delocalised s and p electrons described by LDA/GGA Introduce new functional: E LDA +U n [ ] = E LDA [ n] + E U σ n i 1 2 U n in j i j [ ] E dc [ σ n ] i
35 DFT+U In DFT+U split electrons into two subsystems: - localised d or f electrons described by - delocalised s and p electrons described by LDA/GGA Introduce new functional: E LDA +U n [ ] = E LDA [ n] + E U σ n i 1 2 U n in j i j [ ] E dc [ σ n ] i Hubbard U term: describes localised electrons
36 DFT+U In DFT+U split electrons into two subsystems: - localised d or f electrons described by - delocalised s and p electrons described by LDA/GGA Introduce new functional: E LDA +U n [ ] = E LDA [ n] + E U σ n i 1 2 U n in j i j [ ] E dc [ σ n ] i Double counting correction: removes contribution from localised orbitals in LDA/GGA functional
37 DFT+U E LDA +U n [ ] = E LDA [ n] + E U [ σ n ] i E dc [ σ n ] i Specifying a U parameter in units of energy completely specifies this interaction Can compute from first principles (various schemes) Or treat as an empirical parameter - Typical value is 4-5 ev. Warning: this method is regularly mis-used (and results published)
38 DFT+U E LDA +U n [ ] = E LDA [ n] + E U [ σ n ] i E dc [ σ n ] i Specifying a U parameter in units of energy completely specifies this interaction In CASTEP specify in.cell file: %block HUBBARD Cu 1 d: 2.5 %endblock HUBBARD
39 Back to non-local functionals For non-local functionals solve a generalised Kohn-Sham problem: Note: Integral over all space adds to complexity of problem Calculations much more expensive than local methods
40 Another Non-local Functional: screened-exchange nl E xc = 1 2 ikjq drdr' ψ * ik ( r)ψ ik r' * ( )ψ jq r r' ( r' )ψ jq ( r) e k s r r' Based on Hartree-Fock Non-local correlation included via screening term The (orbital-dependent) potential is: V xc nl = 1 2 ψ jq jq * ( r)ψ jq r r' ( r' ) e k s r r'
41 Plane wave representation nl E xc = 2π V ikjq G,G',G'' c * * ik ( G)c ik ( G' )c jq ( G'+G'' )c jq ( G + G' ) q k + G'' 2 + k s 2 Note: Double sum over bands Double sum over k-points Triple sum over plane waves Computationally painful Fortunately there s a clever FFT method that reduces this to N plwv log( 2 N plwv )N bands 2 N kpts
42 Non-local: how expensive? Non-local scales with cut-off as:
43 Scaling with k-points Scales as: LDA is linear with k-points
44 Why bother with this expense?
45 Example of non-local functionals: Antiferromagnetic FeO
46 Example of non-local functionals: Antiferromagnetic FeO
47 Optimised Effective Potential (OEP) Recall exchange energy: E x = 1 2 ikjq drdr' ψ * ik * ( r)ψ ik ( r' )ψ jq r r' Hartree-Fock, sx : non-local potential ( r' )ψ jq ( r) OEP: find the optimum local potential arising from the non-local HF E x (i.e. total energy variational w.r.t V) Optimised effective potential can be found from solution of N σ i=1 d σ* r ʹ ψ i ( r ʹ )[ V OEP xc ( r ʹ ) V σ,nl ( ʹ i,xc r )]G σ 0 ( r ʹ, r )ψ σ i ( r ) + cc = 0
48 Optimised Effective Potential (OEP) N σ i=1 d σ* r ʹ ψ i ( r ʹ )[ V OEP xc ( r ʹ ) V σ,nl ( ʹ i,xc r )]G σ 0 ( r ʹ, r )ψ σ i ( r ) + cc = 0 Requires sum over all bands - in principle infinite - computationally impractical - represents a serious convergence issue
49 Optimised Effective Potential (OEP) N σ i=1 d σ* r ʹ ψ i ( r ʹ )[ V OEP xc ( r ʹ ) V σ,nl ( ʹ i,xc r )]G σ 0 ( r ʹ, r )ψ σ i ( r ) + cc = 0 In CASTEP we use DFPT to avoid calculation of unoccupied states (Hollins, Clark, Refson and Gidopoulos, PRB 85, (2012)
50 Optimised Effective Potential (OEP) OEP provides better KS bandgaps than LDA/GGA Superior to HF
51 van der Waals Traditional functionals do not describe van der Waals interactions Recent developments now include semi-empirical dispersion corrections Known as DFT+D Options in Castep include: - OBS [Phys. Rev. B 73, , (2006)] - G06 [J. Comput. Chem. 27, 1787, (2006)] - JCHS [J. Comput. Chem. 28, 555, (2007)] - TS [Phys. Rev. Lett.,102, (2009)] Still open to research and validation tests
52 van der Waals Implementation Implemented for geometry optimisation: - Energies - Forces - Stresses Currently available for a subset of the periodic table Mainly first row elements Not implemented for all functionals Example: - Castep DFT+D on azobenzene on transition metal surface: Phys. Rev. B 80, (2009)
53 Beyond DFT: Quasiparticles Many-body perturbation theory: alternative and accurate means of describing electronic excitations Quasiparticle states and energies determined from.
54 Beyond DFT: Quasiparticles Quasiparticle states and energies determined from - self-energy - non-hermitian - non-local, frequency dependent - incorporates exchange and correlation effects Formally, looks like KS eqns BUT very different How do we solve this?
55 The GW Approximation Take self-energy to be:.
56 The GW Approximation Take self-energy to be: Where - Green s function - screened Coulomb interaction.
57 The GW Approximation What does this describe physically?
58 The GW Approximation Formally, GW is dynamically screened Hartree-Fock Replacing by yields Hartree-Fock In practice, we calculate corrections to the Kohn-Sham eigenvalues: Currently being developed and implemented within CASTEP.
59 Some suggestions LDA Nice covalent systems Simple metals DFT+U Mott insulators and highly correlated materials GGA: Molecules H-bonded materials Highly varying densities (d and f states) Some nasty metals Most magnetic systems Non-local hybrids and sx: Band gaps (with caution) Some nasty magnetic systems (again, with caution) van der Waals Dispersion: DFT+D methods
60 Conclusions Introduced notion of exchange and correlation Introduced various flavours of XC functional in CASTEP XC functional employed should be dictated by physics under consideration There is no golden bullet : all XC functionals have pros and cons New developments: OEP and GW
61 Acknowledgements CDG Tom Hollins, Pete Byrne EPSRC
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