Exchange-Correlation Functional

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1 Exchange-Correlation Functional Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological Sciences University of Southern California How to incorporate many-electron correlations into effective single-electron (Kohn-Sham) equations?

2 Preliminary: Second Quantization (1) Consider a system of N electrons with the Hamiltonian ) H = # h(r ' ) # u(r ', r 1 ) '*+ '31 h r = ħ6 2m 6 + v r ; u r, r ; = e6 r r ; Occupation-number representation: An antisymmetric Fermionic wave function can be expanded as a linear combination of Slater determinants Ψ r +,, r ) = # f n +,, n J Φ LM,,L N r +,, r ) L M,,L N {Q,+} where n? is the occupation number of the κ-th single-electron state ψ? (r), with the constraint? n? = N, and each Slater determinant (which occupies states κ + < κ 6 < < κ ) ) is Φ?M,,? S r +,, r ) = 1 ψ?m (r + ) ψ?m (r ) ) N! ψ?s (r + ) ψ?s (r ) ) = 1 N! #( 1)W ψ?x(m) (r + ) ψ?x(s) (r ) ) W Permutation

3 Preliminary: Second Quantization (2) The quantum-dynamical system is identical to Ψ = # f n +,, n J (az \ + ) L \ M az LN J 0 L M,,L N {Q,+} with the Hamiltonian operator Vacuum H` = # az [ \ m h n az L # az [ \ az L \ mn u pq az d az ] [,L [,L,],d m h n = e drψ [ r h r ψ L (r) mn u pq = e drdr ; ψ [ r ψ L r ; u r, r ; ψ ] r ψ d r ; = [m p u n q] Physicist s notation Chemist s notation and the creation (az \ [ ) & annihilation (az ] ) operators anticommute j \ az?, az?; = δ?,?; az?, az?; = az \ \?, az?; = 0

4 Preliminary: Second Quantization (3) Hamiltonian operator in the coordinate representation H` = Tm + Vm + U ` Tm = ħ6 2m e dr ψ m\ r ψm r Vm = e drψm\ r v r ψm r = e drρz(r)v(r) U` = 1 2 e drdr ψ m\ r ψm\ r ; e 6 r r ψ m(r )ψm(r) See note on second quantization A. L. Fetter & J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, 71) A. Szabo & N. S. Ostlund, Modern Quantum Chemistry (McMillan, 82)

5 Preliminary: Hartree-Fock Approximation Hartree-Fock approximation determines the best single Slater determinant that minimizes the energy Φ = 1 N! # ( 1) W φ W(+) (r + ) ψ W()) (r ) ) W ft r φ ' r = ε ' φ ' r ) h r + # J 1 r K 1 (r) 1*+ J 1 r φ r = e dr Fock operator e 6 r r ; φ 1 (r ; )φ 1 (r ; )φ r K 1 r φ r = e dr e 6 r r ; φ 1 (r ; ) φ r φ 1 (r) See note on Hartree-Fock approximation A. Szabo & N. S. Ostlund, Modern Quantum Chemistry (McMillan, 82)

Energy Functional Exchange-correlation (xc) functional via Kohn-Sham decomposition E ρ r = T { ρ r + e drv r ρ r + 1 2 Kinetic energy of non-interacting electrons External

6 Energy Functional Exchange-correlation (xc) functional via Kohn-Sham decomposition E ρ r = T { ρ r + e drv r ρ r Kinetic energy of non-interacting electrons External potential e drdr ρ r ρ r r r ; Hartree energy (meanfield approximation to the electron-electron interaction energy) + E } ρ r Exchange-correlation energy ρ r Z 1 e Z 2 e R 1 R 2

7 Electron-Electron Interaction Energy ) ) U = 1 2 # # Ψ 1 r ' r 1 Ψ '*+ 13' = e dr e dr ρ 6 r, r r r Two-body density matrix ρ 6 r, r = 1 2 # Ψ ψ m \ ~ (r)ψm \ ~ (r )ψm ~ (r )ψm ~; (r) Ψ ~,~; Creation operator Two-body correlation function g r, r Annihilation operator e = ħ = 1 ρ r = ρ + r, r U = e dr e dr ρ 6 r, r = 1 2 ρ r ρ r; g r, r ; One-body density matrix ρ r ρ r r r ρ + r, r = # Ψ ψm ~ \ (r) ψm ~ (r ) Ψ ~ + e dr e dr ρ r ρ r r r g r, r ; 1 See note on second quantization ƒ

8 Electron Correlation vs. Density Response Information on two-body correlation is encoded in the density response function χ through fluctuation-dissipation theorem; see note on timedependent perturbation H` = H` + Vm(t) H` = Tm + U` = ħ6 2m e dr ψ m\ r ψm r e drdr ψ m\ r ψm\ r ; e 6 r r ψ m(r )ψm(r) Vm(t) = e drψm\ r v r, t ψm r = e drρz(r)v(r, t) χ r r ;, t t ; δ ρz r, t δv r ;, t ; = i ħ Θ t t; Ψ Q ρz r, t, ρz r ;, t ; Ψ Q = i ħ Θ t t; Ψ Q ψm \ r, t ψm r, t, ψm \ r, t ψm r, t Ψ Q Equation-of-motion & functional derivative to derive approximate χ; see A. Nakano & S. Ichimaru, Phys. Rev. B 39, 4930 ( 89); ibid. 39, 4938 ( 89)

9 Pair Correlation: Exchange Hole Radial distribution function g r r ; in a homogeneous system Hartree-Fock (HF) approximation: Ground state is a Slater determinant of plane waves occupied up to the chemical potential μ g r is analytically calculated for homogeneous electron gas with HF Exchange hole g r = k = sin k r k rcos (k r) k r 2mμ ħ = Fermi wave number 6 g r represents Pauli exclusion principle between same-spin electrons embodied in the antisymmetric Slater determinant See note on Hartree-Fock approximation

10 Exchange & Coulomb Holes g(r) of homogeneous electron liquid with various approximations for incorporating the correlation effect, which represents additional Coulomb (or correlation) hole r { = 3 4πρ +/ me 6 P. Vashishta & K. S. Singwi, Electron correlation at metallic densities. V, Phys. Rev. B 6, 875 ( 72) ħ 6 mean electron distance = Boh radius mean el el interaction energy mean kinetic energy

11 Exchange-Correlation Functional Universal functional (of density) that describes many-body effects beyond the mean-field approximation potential energy due to electron-electron interaction v } r = e dr ; e6 ρ r ; r r ; + v } (r) Hartree (mean-field) potential exchange-correlation potential Some commonly used exchange-correlation functionals > LDA (local density approximation): E } = drρ(r)ε ρ(r) > LSDA (local spin density approximation): ε ρ r, ρ (r) > GGA (generalized gradient approximation): ε ρ(r), ρ r PBE: Perdew, Burke & Ernzerhof, Phys. Rev. Lett. 77, 3865 ( 96) > MetaGGA: functional of kinetic-energy density τ ~ (r) = + ψ 6 L~ (r) v } r = δe } δρ(r) L { }}²³ µ } SCAN: Sun, Ruzsinszky & Perdew, Phys. Rev. Lett. 115, ( 15)

12 HK vs. PBE Lesson: Publish something simple that others can use

13 Other Exchange-Correlation Functionals Select an appropriate functional for the purpose & target system of the QMD simulation > LDA+U method for transition metals δe ¹º» ¼ δn ' = ε ¹ + U( + 6 n ') Anisimov et al., Phys. Rev. B 44, 943 ( 91) > DFT-D: van der Waals (vdw) functional for molecular crystals & layered materials E {³ = s ¾ 'Ç1 ÀÁ Â ÀÁ Ã f ÄÅ³ (R ij ) Grimme, J. Comput. Chem. 25, 1463 ( 04); J. Chem. Phys. 132, ( 10) > Nonlocal correlation functional E } ÈÉ = + 6 dr dr ; ρ(r)φ(r, r )ρ(r ) Dion et al., Phys. Rev. Lett. 92, ( 04) Occupation of i-th orbital For comparison of DFT-D & nonlocal correlation functionals, see Shimojo et al., J. Chem. Phys. 132, ( 10)

Validation of XC Functional Comparison with high accuracy methods, such as quantum Monte Carlo (QMC), & experimental data Sensitivity analysis among different exchange-correlation (xc) functionals

14 Validation of XC Functional Comparison with high accuracy methods, such as quantum Monte Carlo (QMC), & experimental data Sensitivity analysis among different exchange-correlation (xc) functionals Consistency of the obtained result with the level of approximation QXMD Example: Atomically thin tellurium (tellurene) Raman spectrum QXMD Experiment QMC Casino QMC ( a (Å) B (N/m) GGA GGA-D Spin GGA Hybrid HSE Hybrid HSE with GGA-D a: Lattice constant B: Bulk modulus

15 Band-Gap Problem Janak s theorem equates the ionization potential (IP) & electron affinity (EA) with the Kohn-Sham (KS) eigenenergies of the highest occupied molecular orbital (HOMO) & lowest unoccupied molecular orbital (LUMO), respectively Janak, Phys. Rev. B 18, 7165 ( 78) Band gap, E gap = IP EA, is usually underestimated with GGA-type exchange-correlation (xc) functional This is partly due to self interaction: Note the Hartree potential v r = dr ; e 6 ρ r ; r r ; includes repulsive interaction of an occupied electron with itself (artifact), but not for an unoccupied electron Self-interaction correction (SIC): A quick fix subtracts the self-interaction of each KS orbital, v,l~ r = dr ; e 6 [ρ r ; ψ 6 L~ ] r r ;, which introduces an orbital-dependent KS potential (expensive & deviates from the DFT principle) Perdew & Zunger, Phys. Rev. B 23, 5048 ( 81) Thorough analysis of the band-gap problem focuses on the discreteness of an electron Cohen et al., Science 321, 792 ( 08) Mori-Sanchez et al., Phys. Rev. Lett. 100, ( 08)

16 Hybrid Exact Exchange Functional Hartree-Fock (HF) approximation, with the underlying antisymmetric Slater determinant, is free from self interaction Hybrid exact-exchange functional incorporates part of the exact exchange using HF approximation & the rest with other xc functional (it is not strictly DFT) B3LYP: e.g., Stephens et al., J. Phys. Chem. 98, ( 94) PBE0: Perdew et al., J. Chem. Phys. 105, 9982 ( 96) Range-separated hybrid exact-exchange mixes HF & other xc functional at different distance HSE: Heyd, Scuseria & Ernzerhof, J. Chem. Phys. 118, 8207 ( 03) For comparison of LSDA, SIC & HF for a 2-electron problem, see Nakano et al., Phys. Rev. B 44, 8121 ( 91) See note on Hartree-Fock approximation

Dept of Mechanical Engineering MIT Nanoengineering group

1 Dept of Mechanical Engineering MIT Nanoengineering group » To calculate all the properties of a molecule or crystalline system knowing its atomic information: Atomic species Their coordinates The Symmetry