DFT: Exchange-Correlation
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1 DFT: Local functionals, exact exchange and other post-dft methods Stewart Clark University of
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 Empirical non-local (hybrids): B3LYP, HSE DFT+U When should they be used Closing statements 2
3 DFT and exchange-correlation Basically comes from our attempt to map 1 N- body QM problem onto N 1-body QM problems Attempt to extract single-electron properties from interacting N-electron system These are quasi-particles, or Kohn-Sham eigenstates 3
4 Standard Definition Z I i j r i r j i,i r i R I 2 n exact N-body Schrödinger Equation I J Z I Z J R I R J Ψ r 1,r 2,... ( ) = EΨ( r 1,r 2,...) ( r) =... Ψ * ( r 1,r 2,r 3,...)Ψ( r 1,r 2,r 3,...)dr 2 dr i Kohn-Sham Equations n(r') r r' dr' Z I n(r') dr'+ 1 Z I Z J + µ xc n r I r' R I 2 I J R I R J n DFT ( r) = ψ * i ( r)ψ i ( r) i [ ( )] ψ i ( r) = ε i ψ i r ( ) n exact r ( ) = n DFT r ( ) 4
5 Alternative Exact Definition Exact XC interaction is unknown Within DFT we can write the exact XC interaction as E xc = 1 n xc ( r, r') [ n] n( r) drdr 2 r r' ' 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. 5
6 The XC hole From the Pauli exclusion principle we have for a given r n xc ( r, r') dr' = 1 Which is known as the sum rule. 6
7 Non-local Nature of XC Hole 7
8 XC Interaction at Surfaces As an electron approaches a surface (distance x away) then: Coulomb interaction appears to have 1/x singularity. Fixed by properties of XC hole. 8
9 Generalities All functionals are approximations Reports of Failures of DFT is actually a report 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 9
10 Jacob s Ladder 1 The simplest XC functional is the local density approximation (LDA) LDA hom Exc [ n( r)] = n( r) ε xc [ n( r)] dr This is the simplest density functional. It was used for a generation in materials science, but is not accurate enough for many chemical purposes Typical errors o Over-binds (binding energy too large) o Underestimates lattice parameters o Phase stability incorrect order o Energetics of magnetic materials in error 10
11 E GGA xc Jacob s Ladder 2 The generalised gradient approximation (GGA) contains the next term in a derivative expansion of the charge density: GGA [ n( r)] = n( r) ε [ 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 11
12 Why the GGA? LDA depends only on one variable (the density). GGA s 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. 12
13 Jacob s Ladder 3 Next come the meta-gga s Instead of using 2 n typically the kinetic energy is used (same idea, in principle) E xc [ n(r), n(r), 2 n(r) ] or E xc n(r), n(r), 2 ψ(r) [ ] Example: TPSS (J. Tao, J. P. Perdew, V. N. Staroverov, G. E. Scuseria, Phys. Rev. Lett. 91, (2003) ) 13
14 Hartree-Fock Exchange is essentially the Pauli exclusion principle Quantum mechanically, electrons are Fermions hence many particle wavefunction is anti-symmetric Ψ( r 1,r ) 2 = Ψ( r 2,r ) 1 which after some generalisation we find the energy is non local E X = 1 * * ψ ik ( r)ψ ik ( r' )ψ jq ( r' )ψ jq ( r) drdr' 2 r r' ij,kq Note: HF can get properties wrong in the opposite direction from DFT 14
15 Moving off the ladder: Hybrid If a method over-estimates your value of interest and another under-estimates it, then the answer you want can be obtained by taking a bit of both! (You may notice this is not ab initio) This is where things start to go a bit crazy (welcome to the world of making up the answer you want to see apparently some people call this science) E xc = αe x exact + (1 α)e x local + E c local [I m being slightly over-critical here there is some good science being done with these functionals] 15
16 Some properties of E xc The XC interaction for a HEG is known exactly (LDA). For solids, this in a very important limit. The sum rule is required, but not usually implemented in empirical GGAs Density scaling: n l ( x, y, z) = l 3 n( lx, ly, lz) Then, for example lim l E xc [n l ] > E xc [n l ] > le xc [n];l >1 E GGA xc (n, n) E LDA xc [n]; n 0 and many more. However we fit E xc, we don t want to break what we know is correct. There are many GGA s that don t obey known rules. 16
17 Various breeds of functionals: 1 Non-empirical functionals o Such as LDA o Some GGAs (PW91, PBE, rpbe, WC, PBEsol, ) o Not fitted to any empirical results o These come from known exact conditions o If these conditions are important to your physical value of interest then you should get a reliable result o These functionals often have systematic failures (and successes!) so their reliability can usually be predicted 17
18 Some GGA s 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) 18
19 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. 19
20 GGA s (Continued) The revised PBE (known as RPBE) only differs from PBE functional in the choice of mathematical form for ε x. It was chosen only for its simplicity (a simpler F(s) than stated previously). PBE and RPBE contain the same essential physics and also leave out the same (non-essential?) Physics. WC, PBEsol, recently introduced. Somebody will probably get a PhD if they test them thoroughly. 20
21 Charge Differences (LDA-PW91) 21
22 Various breeds of functionals: 2 Hybrid : a few empirical parameters These include B3LYP, HSE, B88, (PBE0) E xc = αe x exact + (1 α)e x local + E c local Generally α is around 20-25%. E c local usually contains parameters You will get good answers so long as your system is close to the systems where functional was fitted 22
23 Various breeds of functionals: 2 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 23
24 Various breeds of functionals: 3 Over-fitted functionals Contains many fitted parameters usually from a few dozen to hundreds of paramters 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 24
25 Various breeds of functionals: 4 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 25
26 DFT+U Standard band theory: in a periodic potential a periodic potential leads to a periodic density The electronic structure conforms to the same symmetry as the crystal V(r)=V(r+L) n(r)=n(r+l) (appears to be a sensible conclusion!) If the material has an odd number of electrons per unit cell then it has to be a metal (highest band will be half filled, implying it must cut the Fermi level, i.e. metallic) 26
27 DFT+U Materials such as CoO and NiO have an odd number of electrons per crystallographic unit cell but are insulators Consider (Ni 2+ O 2- ) 2 Ni 3+ O 2- +Ni + O 2- so that Ni atoms in neighbouring cells have different charges This happens and opens a band gap (unit cell of electronic structure is twice the size of crystallographic cell) LDA/GGA does not get this correct Highly-correlated interaction between electrons in Ni atoms missing in LDA/GGA 27
28 DFT+U We have (slightly simplified here) a new interaction: E LDA +U n( r) ( ) [ ] [ ] = E [ LDA n r ] + E { Iσ U n } m where E U is an additional empirical interaction that mimics the interaction of the highly-correlated d electrons This interaction is fully specified by a U parameter in units of energy. Typical value is 4-5 ev. Warning: this method is regularly mis-used (and results published) 28
29 DFT+U in Castep In the.cell file, add the block: %block hubbard_u <species> <orbital>:<u parameter, in ev> %endblock hubbard_u For example %block hubbard_u Ni d:5.0 %endblock hubbard_u 29
30 Back to non-local functionals ψ i ( r) + V loc ( r)ψ i r ( ) + dr'v nl r,r' ( )ψ i ( r' ) = ε i ψ i r ( ) Note: Integral over all space adds to complexity of problem Calculations much more expensive than local methods 30
31 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 potential is: V xc nl = 1 2 ψ jq jq * ( r)ψ jq r r' ( r' ) e k s r r' 31
32 Plane wave representation nl E xc Note: = 2π V ikjq G,G',G'' Double sum over bands Double sum over k-points c * * ik ( G)c ik ( G' )c jq ( G'+G'' )c jq ( G + G' ) q k + G'' 2 + k s 2 Triple sum over plane waves (aargghh!) Fortunately there s a clever FFT method that reduces this to N plwv log( 2 2 N plwv )N bands N kpts 32
33 Non-local: how expensive? Non-local scales with cut-off as: 33
34 Scaling with k-points Plots shown with and without using symmetry_generate Scales as: LDA is linear with k-points 34
35 Parallel Performance Almost perfect parallel speedup when using k- point parallelism. G-vector parallel speedup highly dependent of computer s interconnect (FFT dominated) 35
36 Why bother with this expense? 36
37 Example of non-local functionals: Antiferromagnetic FeO 37
38 Example of non-local functionals: Antiferromagnetic FeO 38
39 Some suggestions LDA Nice covalent systems Simple metals LDA+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) Note: VdW materials not on this list new functionals claiming vdw to be implemented soon 39
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