Many electrons: Density functional theory
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1 Many electrons: Density functional theory Bedřich Velický V. NEVF 54 Surface Physics Winter Term 0-0 Troja, 8 th November 0
2 This class is devoted to the many-electron aspects of the solid state (computational theory explores the many-electron features of the electronic structure of solids and solid surfaces discusses in some detail the jellium model of the bulk and of a planar surface explains in detail the original Kohn-Sham Density functional theory (DFT and the local density approximation (LDA outlines its various improvements and generalizations
3 General comments on the many-electron problem
4 Basic formulation. Adiabatic approximation Positions and charges of nuclei (ions... fixed parameters { RJ, Z J}. DOGMA (simplest version quantum treatment of the assembly of electrons and nuclei solves all problems of atomic systems. External field (for el. Coulomb field of nuclei specifies the system electron potential energy υ( r Z J e r R =, e = J J e 4πε 0 4. Total electron charge NEUTRALITY N = ZJ (or... ±, ±, TASK OF THE EQUILIBRIUM THEORY electron I. Solve the SR... electron ground state and energy structure equilibr. energy geometry 4 ( e Hˆ Ψ = E Ψ, Hˆ = Tˆ + Uˆ ˆ ee + V II. Minimize the total energy w. resp. to R at fixed N ( e E ( e = 0; E = [ R, N ] + J TOT N TOT E e ZJZK J R R J J K J K ( e
5 Qualitative picture (~ for simple metals. Quantum effects classical elst. system... unstable. Stabilized by Planck constant x p kinetic energy of quantum fluctuations at the surface leaking of electrons into vacuum. Pauli principle I. Aufbau principle... atoms Fermi sea... solids at the surface states from both continuous and discrete spectra. Hartree field 5 υ ( r... (Coulomb mean field of electron clouds H compensation of diverging fields and energies screening -- suppresion of long range Coulomb fields most of the Coulomb energy absorbed into the mean field part at the surface electrostatic surface dipole, work function
6 Qualitative picture (~ for simple metals cont d 4. Exchange (Pauli II. and correlation kinematic (exchange/pauli and dynamical (Coulomb correlation quantum fluctuations around the Hartree mean field exchange and Correlation hole (Hartree-Fock /exact electron digs around itself a ditch ( energy < 0... cohesion, chemical bond spin arrangement... Hund 's rules, magnetic ordering ate t h surface electron and its hole detached and spatially separated... classical image potential 6
7 A neat expression for the total electron energy Hamiltonian ˆ ee = e = e δ( i δ( r j i j i r r r j r r r r r r i j 7 ˆ ee r r e ( i ( r r r r r rj i j ( e Hˆ = Tˆ + Uˆ + Vˆ Tˆ = pˆ i= m ˆ V = υ( ri = d rυ( r δ( r ri d rυ( r nˆ ( r i ee U d d U = d d δ δ δ( r r Total energy E = Ψ Hˆ Ψ Hˆ n( r Hˆ = Tˆ + Vˆ + U = T + d r ( r n( r + { ˆ ˆ ee υ ˆ e r r + d rd r nˆ( r{ nˆ( r δ ( r r } n (, rr N e i pair correlation function δ( r ri δ( r ri = δ( r r δ( r r i } self-interaction avoided particle density
8 A neat expression for the total electron energy Final form of the total energy external field potential energy Hartree mean field Hˆ = Tˆ + d rυ( r n( r + d rd r n( r n( r e r r exchange and correlation e r r + d rd r ( n( rr, n( r n( r d rd r e ( ˆ( ( ( ˆ( ( ˆ( ( r n n n n n δ r r r r r r r r quant. fluct. about mean-field + excl. SI Instead of the unwieldy wave function Ψ it is enough to know particle density n dual to the external field pair correlation function n dual to the Coulomb interaction 8
9 A neat expression for the total electron energy Final form of the total energy external field potential energy Hartree mean field Hˆ = Tˆ + d rυ( r n( r + d rd r n( r n( r e r r exchange and correlation e r r + d rd r ( n( rr, n( r n( r d r d r e ( nˆ( r n( ( nˆ ( n( nˆ( δ ( r r r r r r r r quant. fluct. about mean-field + excl. SI Instead of the unwieldy wave function Ψ it is enough to know particle density n dual to the external field pair correlation function n dual to the Coulomb interaction 9
10 Exchange and correlation hole Suggestive transformation ZJδ ( r R ˆ ˆ ( e + ( H = T d rn r d r n r r r e r r + d rn( r d r n( r hole + d rn( r d r e n( n( r r r r h (, r r ( n ( r, r xc J ˆ ˆ ( υ( H = T + d + d + d r n r r rn( r υ ( r H rn( r ε ( r external field Hartree mean field energy per particle 0
11 Properties of the exchange and correlation hole h xc ε (, ( n (, rr n n( r e d r r r r = r ( r = r h ( r, r xc energy per particle Properties of the hole parametric dependence on the electron position for r sum rule r, it holds h ( r, r (decay of correlations d r h (, r r = xc that includes elimination of SI hole decay may be slow: Friedel oscillations xc Any reasonable theory must contain these elements and obey these rules
12 Jellium
13 Jellium: definition Jellium model of extended electron systems pedigree: Sommerfeld model... and its justification central problem of an interaction electron gas: Long range of Coulomb forces Definition neutral crystal ionic charge evenly spread Properties + J + = Jδ J = Ω Ω n Z ( r R n d rn ( r in equilibrium -- homogeneous and neutral n = n n+ = 0 TOT compensation: υ + υ = + = 0 H
14 in Jellium: properties cont d k-space 4 single parameter n = n determines all equilibrium properties WIGNER RADIUS = 4π n + Bohr radius n 0 =.6 0 r s m simple metals r =... 6 volume per electron ( ar FERMI SPHERE one-el. distr. s 0 N n k s ( e F = dimensionless parameter diagonal in k indep. of interaction = n Ω = π = k F π ( 9 / 4 a0rs 4π kf ( π Ω
15 Jellium: energetics Energetics E for any kind of energy E Ω = Ω = N E N thermodynamics E Ω = n ε microscopic picture Atomic 5 units εk + υh ε εtot = εh + εx + ε C ε Fermi energy and average kinetic energy E ε E ε F K F K = = h m 5 E k F h ma HF F ( a k 0 = = Ry.68 E F 0 s 5 F Ry. r = = s r
16 Jellium: some results Jellium is TOTAL ENERGY/ELECTRON atomic units εtot = εh + εx + εc WIGNER FORMULA = r r r s s s stabilized by exchange and further by the correlations r = n hole n ( gr ( h xc (, r r + n( r 6
17 Correlation energy in jellium -- crucial for LDA 0 εc Ry r 0 s Correlation energy in jellium Wigner... historical, approx. compared with QMC numerical, heavy computation in practice -- Perdew-Zunger fit by analytic expressions range of previous slide 7
18 Density functional theory (DFT
19 Kohn - Hohenberg theorem Density functional theory is an exact many-electron theory in principle true interacting electrons effective problem of fictitious non-interacting eletrons yields: n( r, m( r, total energy E (+ other things, perhaps repeatedly employs the variational principle for the ground state E= Ψ Hˆ Ψ < Ψ Hˆ Ψ iff ΨΨ < theorem Kohn - Hohenberg the total energy of a ground state of interacting electrons is a unique functional of the electron density Valid chain υ( r Hˆ Ψ, E n( r KH theorem claims that the correspondence is unique: υ( r n( r (proof by the variational principle skipped 9
20 Variational principle for the energy functional KH theorem claims the unique correspondence υ( r n( r Then n( r Ψ, E functionals of Ψ Hˆ Ψ = Ψ TˆΨ + Ψ Uˆ ˆ ee Ψ + Ψ V Ψ E[ n] = n( r F[ n] + dr υ( r n( r universal functional υ( r enters ( e Variational principle (K. - H. N = dr n ( r fixed ˆ ˆ ˆ ˆ E[ n] = Ψ H Ψ < Ψ H Ψ = Ψ T + U Ψ + dr υ( r n ( r E[ n] < F[ n ] + dr υ( r n ( r N [ n] = N [ n ] = N ( e ( e ( e ee 0
21 Orbital theory of Kohn and Sham What i s F[ n] F[ n] = T[ n] + U [ n] F[ n] = Ts[ n] + { T Ts + Uee}[ n] G Tacit adiabatic postulate 0 e n υ n = 0 υ0 T ee T natural decomposition s KS decomposition NEW kinetic energy of fictitious non-interacting electrons e n n T having the same n s We develop two cases in parallel, one is clear and serves as an aid for the other: e = 0 non-interacting el 's e 0 Kohn-Sham theory 0 = s + υ0 = s + + E [ n] T [ n] dr ( r n( r E[ n] T [ n] G[ n] dr υ( r( n r
22 Exact Kohn - Sham equations Write Euler-Lagrange equations e = 0 non-interacting el 's e 0 0[ ] = s[ ] + υ0( ( [ ] = s[ ] + [ ] + ( e ( e E0 N 0 E N δt s δt s δg δ n( r + υ 0( r µ = 0 n( r n( r EQUATION FOR Kohn-Sham theory E n T n dr r n r E n T n G n dr υ(( r n r δ µδ = δ µδ = 0 µ n( r Here, the solution is known m 0 α α α m eff α α α 0 = Ts + dr υ0 δ + δ + υ ( r µ = 0 υ ( r eff n( r ψα r Lagrange multiplier Use the eff. potential as a real one ( + υ ( r ψ = E ψ ( + υ ( r ψ = E ψ e n( r = ψ ( r = ( E E 0 µ = ( e α n e E = T + dr υn + G s ( υ υ = E E = E dr n + G δe δn α α eff This is the complete set of equations of the Kohn-Sham theory
23 Exchange and correlations in KS theory Exact decomposition H rn r d r e e n d n d r r r r KS G[ n] = d ( ( + ( hxc (, r r r r r r U + E defines KS energy KS hole different from the true one, but similar properties sum rule d (, r r = KS r hxc energy per particle Effective potential ε KS KS n = d e hxc (, r r ( r, r r r eff δg δe δn( r H( δn( r υ ( r = υ( r + = υ( r + υ r + Final expression for the total energy H r υxc E = E U d n++ E α
24 Exchange and correlations in KS theory Exact decomposition H rn r d r e e n d n d r r r r KS G[ n] = d ( ( + ( hxc (, r r r r r r U + E defines KS energy KS hole different from the true one, but similar properties sum rule d (, r r = KS r hxc energy per particle Effective potential eff 4 KS KS n = d e hxc δg δe δn( r H( δn( r υ ( r = υ( r + = υ( r + υ r + H r υxc (, r r ( r, r r r Final expression for the total energy is not known E = E U d n++ E α ε The formidable E functional... resort to approximations
25 LDA Local density approximation For jellium, KS theory is exact ε KS J (, r n = ε ( n CRUCIAL STEP LDA Ansatz ε υ KS LDA xc δe δ n( r J ( r,[ n] = ε ( n( r J ( nε n r ( r = ( ( LDA KS equations n defines the LDA energy ( ( m r e + υ + υ ( r + υ H LDA xc ( r ψ = E α α α ( As easy to solve as Hartree or Slater equations ( Of course, by iteration in a self-consistent cycle ( LDA really works. Two questions Why ψ 5 Could we do better
26 Why is LDA so good General properties LDA eqs. have the structure of the exact KS eqs. Kinetic and Hartree parts are included in full, "only" LDA is a parameter-free theory, no fitting, no adjustments 6 is approximate Numerically, LDA is comparatively undemanding, solutions fully converged Altogether : a well structured Why is LDA so good. exact in the limit of jellium and the. it has a variational principle qualitative criteria v xc AB - INITIO METHOD. the LDA hole satisfies the sum rule d r xc and similar B ONUS c high density limit h = Altogether : typical COMPREHENSIVE THEORY : ε is not sensitive to the shape of the hole interpolation scheme
27 Why II: ε c is not sensitive to the shape of the hole One reason for the success of LDA is that the correlation energy depends only on gross features of the hole ε ε ε ε KS e KS (,[] r n = d r r hxc (, r r r radial symmetry of the kernel KS KS n d h xc KS KS (,[] r = u e (, r r+ u u ( r,[ n] ( r,[ n] KS = e udu sin ϑdϑdϕh (, xc r r+ u spherical average hks xc (, r u = e u ( r, u KS du hxc SUMMARY Correlation energy per particle of any kind (true, exact DFT, LDA is completely specified by a single global characteristic of the respective hole obtained in two steps ( by the spherical average of ( as the first moment of the resulting radial function 7
28 Exact and LDA hole in neon atom compared 8
29 Exact and LDA hole in silicon crystal compared [0] plane in Si structure bond zigzags density contours of valence electrons Si nuclei and bond connecting lines + centers of holes in the main plot 9
30 LDA and beyond Ways to go beyond LDA present choice in green do something entirely else: GW, QMC, DMFT Several forks augment LDA by a non-system correction: LDA+ U do something within DFT relativistic with elmg fields extensions Within DFT time dependent finite temperatures improvements within non-relativistic DFT 0
31 LDA and beyond Ways to go beyond LDA present choice in green do something entirely else: GW, QMC, DMFT Several forks augment LDA by a non-system correction: LDA+ U do something within DFT relativistic with elmg fields extensions Within DFT time dependent finite temperatures improvements within non-relativistic DFT n a n 0 n
32 LDA and beyond Ways to go beyond LDA present choice in green do something entirely else: GW, QMC, DMFT Several forks augment LDA by a non-system correction: LDA+ U do something within DFT relativistic with elmg fields extensions Within DFT time dependent finite temperatures improvements within non-relativistic DFT n towards realistic solids & molecules a n 0 n
33 Improving LDA within DFT Classification according to John Perdew Improving LDA -- Jacob' s ladder hybrid functionals meta-gga GGA LSDA LDA This is an oversimplified scheme of the vast list of approximations to DFT. Try to look at
34 Improving LDA within DFT Classification according to John Perdew Improving LDA -- Jacob' s ladder hybrid functionals meta-gga GGA LSDA LDA hybrid here between DFT and HF Greek µετα means 'beyond' or 'after' here Generalized gradient approximation Local spin density approximation Local density approximation This is an oversimplified scheme of the vast list of approximations to DFT. Try to look at 4
35 Improving LDA within DFT Classification according to John Perdew Improving LDA -- Jacob' s ladder "exact" theory hybrid functionals meta-gga GGA LSDA LDA Hartree Genesis 5 8: in reality chemically accurate exchange partly non-local HF, example BLYP orthodox DFT E no parameters, local = d rε ( r n( r I viděl ve snách, a aj, žebřík stál na zemi, jehožto vrch dosahal nebe; a aj, andělé Boží vstupovali a sstupovali po něm. And he dreamed, and behold a ladder set up on the earth, and the top of it reached to heaven: and behold the angels of God ascending and descending on it.
36 Improving LDA within DFT Classification according to John Perdew Improving LDA -- Jacob' s ladder hybrid functionals exchange partly non-local meta-gga GGA local LSDA LDA E = d rε ( r n( r TWO RIVALS semi-empirical parametrization orthodox DFT no parameters 6
37 Improving LDA within DFT Classification according to John Perdew Improving LDA -- Jacob' s ladder local kinetic energy hybrid functionals meta-gga GGA LSDA LDA example BLYP see below ε ( r = φ( n, n, n, τ ε ( r = φ( n, n ε ε ( r = φ( n, n ( r σ σ = φ( n σ σ σ σ τ σ = ψ ασ E = E + a ( E E + a ( E E + a ( E E BLYP LDA HF LDA GGA LDA GGA LDA xc xc 0 x x x x x c c c a, a, a... empirical parameters 0 x c 7
38 Epilogue At surfaces, LDA works all too well, considering the fast drop of the electron density from the bulk value to zero Next time, we will have a look at truncated jellium.
39 The end
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