Chap 9 Electron-electron interactions • e-e interaction and Pauli exclusion principle (chap 17) • Hartree approximation • Hartree-Fock approximation • Exchange-correlation hole • Density functional theory Dept of Phys M.C. Chang What’s missing with the non-interacting electrons? • exchange effect • screening effect • normalization of band gap (and band structure, FS) • quasiparticle, collective excitation • superconductivity •… Beyond non-interacting electron N 2 ⎛⎞pi G 1 GG HUrVrr=+∑∑⎜⎟ion() i + ee ( i − j ), iij=1,⎝⎠22m ij≠ GG e2 Vrree() i−= j GG , ||rrij− GG GG HrrΨ=Ψ(,12 ,"" ) Err (, 12 , ) This is a differential eq. with N=1023 degrees of freedom. We need approximations. Hartree approximation (1928): G G G G G No quantum correlation assume Ψ(,rr ," )= ψ () rψ () r"ψ ( r ) 12 11 22 NN ～ classical particles GG GG HrrΨ=Ψ(,12 ,"" ) Err (, 12 , ) G ⎡⎤|(')|ψ r 2 2 ∑ j ⎡⎤= 2 G GG⎢⎥2 3 j G ⇒− ∇+Ur()ψψ () r +edr' ()r =εψ ()r ⎢⎥ionii⎢⎥∫ GG i i ⎣⎦2m |'|rr− ⎣⎦⎢⎥ H Hartree (or direct) potential Vee (r) Each electron moves in the potential from all the other electrons, • Need to be solved self-consistently (by iteration). • Self-consistency doesn’t mean the result is correct. What’s wrong with the HA? • The manybody wave function violates the Pauli principle • The calculated total energy is positive (means the electron gas is unstable) Self-consistent Hartree approximation 1. choose initial {ψ i} ↓ G GGG2 nr(') 2. construct nr( )=→= |ψ ( r ) | VH ( r ) e23 dr ' ∑ iee∫ G G i | rr− ' | ↓ 2 ⎡⎤= 2 GGGH G 3. solve ⎢⎥−∇+UrVrion() + ee ()ψεψ '() i r = i '() i r ⎣⎦2m ↓ GG2 4. construct nr'( )= ∑ |ψ 'i ( r ) | i ↓ GG 5. if nr'( )−< nr ( )δ , then STOP. GG else let ψψii(rr )= ' ( ), GOTO 2 E. Kaxiras, Atomic and Electronic Structure of Solids, p.46 Hartree-Fock approximation (1930): • A brief review of variational principle (single particle version) G G Hrψ ()= εψ () r the ground state can be obtained by minimizing ψ H ψ , under the constraint ψψ =1 δ i.e. G ψ H ψλψψ−−=()10 (λε→ ) δψ * ()r () • Hartree-Fock approximation G GG ψψ(,rs ) (, rs )" ψ ( r , s ) 11GG 1 12 2 1 GNN GG 1 ψψ(,rs ) (, rs )" ψ ( r , s ) assume Ψ=(rsrs , , ," ) 211 22 2 2 NN 112 2 N! ##%# GG G ψψNN(,rs11 ) (, rs 2 2 )" ψ NNN ( r , s ) NN22 peG 1 Hartree ⇒ΨHUr Ψ=∑∑ψii +()ψψψψψ + ijijGG iijij==≠1,1()22mrr |'|− energy 1 N e2 − ψ ψψψFock ∑ ijGG ji energy 2|'|ij,1()=≠ i j rr− Variational principle δ ⎛⎞N H 10 * G ⎜⎟Ψ Ψ−∑λψψiii() − = δψ A (,)rs⎝⎠i=1 See hand- written note ⎡⎤=2 G GGGG G G ⇒− ∇+2 UrVr() +H ()ψψεψ (,) rs +dr3 '(,;'')vrsrsF (',') rs = (,) rs ⎢⎥ion ee AAAA∑∫ ee ⎣⎦2m s' Sum over filled G 2 states only ∑ψi ('')rs G H G 23is,' 23nr(') Vr()== edr 'G GGG edr ' Hartree (or direct) potential ee ∫∫|'|rr−−| rr'| * GG ∑ψψii('')(,)rs rs GG i vrsrsF (,; '')=− e2 GG δ Fock (or exchange) potential ee|'rr− | s,' s • The exchange potential exists only between electrons with parallel spins. • The exchange potential is non-local! This makes the HFA much harder to calculate! • Still need self-consistency. • Again, no guarantee on the correctness (even qualitatively) of the self-consistent result! Hartree-Fock theory of uniform electron gas • The jellium approximation Positive Ion charges Uniform background Smooth off • Below we show that plane waves are sol’ns of HFA G G G eik⋅ r ()−en assume ψχ(rs , )=→=− VH edr3 ' G G0 kseeV ∫ |'|rr− G 3 ()+en This cancels with Ur()=− edr 'G G0 ion ∫ |'|rr− G G G ik'(⋅ r− r ') e δ G G GG G e2 ∑ ss,' eik⋅ r ' drv3 '(,;'')F rsrsψ (',')rs= − dr3 'k 's' χ ∑∫ ee ks∫ G G ' s' V |'|rr− V 41πe2 G =(,)− G G ψ r s ∑ 2 k V k ' |'|kk− 22 ⎛⎞= 2 41πe G G ⇒−⎜⎟ ∇−GG ψεψkkk()rr = () 2mV∑ |'|kk2 ⎝⎠kk'< F − ε(k) ∴ HF energy G =22kdk 3'1 επ()ke=− 4 2 GG 2(2)m ∫ π 3 |'|kk− 2 kk< F =22ke2 2 ⎡⎤11−+xx2 1 k =−kxF ⎢⎥+ ln , = 2mkπ ⎣⎦24xx1− F k/kF Lindhard function F(x) “DOS” D(k(ε)) (arb. unit) |F’(x)| Slope at kF has logarithmic div k/kF • If, when you remove one electron from an N-electron system, the other N-1 wave functions do not change, then |ε(k)| is the ionization energy (Koopman’s theo. 1933). • In reality, the other N-1 electrons would relax to screen the hole created by ionization. (called “final state effect”, could be large.) • Total energy of the electron gas (in the HFA) Substrat double counting between k and k’ ⎡⎤G 1 G EkVkHF=−∑ ⎢⎥ε ( ) F ( ) kk< F ⎣⎦2 22 2 22 ⎛⎞33= kekFF⎡ 3= 2 3e ⎤ =−=N⎜⎟ N⎢ 2 () kaFF00 − ka⎥ ⎝⎠52mmaa 4ππ⎣ 5200 2 2 ⎦ EHF 2.21 0.916 4 3 ⇒=−2 ,in which π (rsa0 ) =n N rrss 3 ⎛⎞==222e ⎜⎟rs in a0 =,2 E in Ry= or 2 ⎝⎠me22 a0 ma0 • More on the HF energy 11G GGG2 G G EdrVrrsdrdrrsrs=+333*H ()ψ (,) 'vrsrsF (,; '')ψψ (,) (',') HF∑∑∫∫∫ ee AAee A 22AA,,,'sss * GG* GG GG ψii('')(,)rsψ rsψAA(,)rsψ (',') r s 11nrnr(')() ∑ =edrd233r 'ss' − e 23 drdr 3 ' i,A δ ∑ ∫∫GG ∑ ∫ GG ss' 2||ss,' rr− '2 s,'s |'rr− | GG 1 nrnr(')() GG =edrdr233 ' ss' grrHF (, ') ∫ ∑ GG ss' 2'|s,s' |rr− • Pair correlation function 2 * GG ∑ψψii('')(,)rs rs HF GG i grrssss''(, ')1≡− G δ nrnrss' (')() The conditional probability to find a spin-s’ electron at r’, when there is already a spin-s electron at r. • In the jellium model (ns=N/2V), 2 G GG HFGG 2 ik⋅−(') r r grrss''( , ')≡− 1∑ e δ ss N kk< F 22 GG ⎛⎞⎛⎞sinkr− kr cos kr 3 FF F rrr≡−| ' |, = 1 −⎜⎟⎜⎟ 333 δ ssFss'1' or 1 − jkr ( ) δ ⎝⎠⎝⎠krFF kr • Fock (exchange) potential keeps electrons with the same spin apart (This is purely a quantum statistical effect) exchange hole interaction of the electron with the “hole”: HF 2 en0 (1()− g↑↑ r ) 13ek ε =−()edr3 × =− F X hole ∫ r 24π • Beyond HFA Now there is a hole even if the electrons have different spins! exchange-correlation (xc) hole (named by Wigner) could be non-spherical in real material DOS for an electron gas in HA and HFA HFA HA What’s wrong with HFA? • In the HFA, the DOS goes to zero at the Fermi energy. HFA gets the specific heat and the conductivity seriously wrong. • The band width is 2.4 times too wide (compared to free e) • The manybody wave function is not necessarily a single Slater determinant. Beyond HFA: • Green function method: diagrammatic perturbation expansion • Density functional theory: inhomogeneous electron gas, beyond jellium • Quantum Monte Carlo •… Green function method (diagrammatic perturbation expansion) Correction to free particle energy RPA Random phase approx. (RPA) • The energy correction beyond HFA is called correlation energy (or stupidity energy). EC = EEXACT -EHF • Gell-Mann+Bruckner’s result (1957, for high density electron gas) 2 E/N = 2.21/rS + 0 - 0.916/rS + 0.0622 ln(rS) - 0.096 + O(rS) = EK + EH -EF + EC (E in Ry, rS in a0) • This is still under the jellium approximation. • Good for rS<1, less accurate for electrons with low density (Usual metals, 2 < rS < 5) • E. Wigner predicted that very low-density electron gas (rS > 10?) would spontaneously form a non-uniform phase (Wigner crystal) Luttinger, Landau, and quasiparticles • Modification of the Fermi sea due to e-e interaction (T=0) Perturbation to all orders Z (if perturbation is valid) • There is still a jump that defines the FS (Luttinger, 1960). Its magnitude Z (<1) is related to the effective mass of a QP. • A quasiparticle (QP) ＝ an electron “dressed” by other electrons. A strongly interacting electron gas = a weakly interacting gas of QPs. (Landau, 1956) • It is a quasi-particle because, it has a finite life-time. Therefore, its spectral function has a finite width: This peak sharpens as Free we get closer to the FS electron QP (longer lifetime) Cited 8578 times (by 11/26/2011) Cited 18298 times (by 11/26/2011) 沈呂九 Density functional theory “density functional theory" appears in title or abstract “impact” 1964 “availability of computing power” ⎡⎤ N ⎛⎞p2 GGGGGGG1 ⎢⎥i ++Ur() Vrr ( −Ψ=Ψ ) (,,) rr"" Err (,,) ⎢⎥∑∑⎜⎟ion i ee i j 12 12 iij=1,⎝⎠22m ⎣⎦⎢⎥ij≠ G GG GG Particle density nr()=ΨΨ N dr33 dr""" dr 3* (, r r , ) (, r r , ) ∫ 23∫∫N 2 2 Usually: Uion →Ψ→n DFT: n → Ψ → Uion The 1st Hohenberg-Kohn theorem The potential Uion is a unique functional of the ground state density n. Pf: suppose U, U’ give the same ground state density, n=n’ EHEGG=Ψ Ψ GG;' =Ψ G ''' H Ψ G EHGG''<ΨΨ=Ψ+−Ψ=+Ψ−Ψ G G HUUEUU ' GGG ' G GG G →<+E'()()'() E drnr3 Ur − Ur GG∫ [] same argument also gives GG G EE<+'()()'() drnrUrUr3 − GG∫ []In principle, given n(r), one can uniquely determine U(r). ⇒+<+EEEEGGGG'' →← Since n(r) determines Uion(r), which determines everything else (EG, |G>… etc), one can say that, EG is a functional of n(r): EnGee[]= TnUnVn []++ [] [] The 2nd Hohenberg-Kohn theorem The true ground state density n minimizes the energy functional EG[n], G with the following constraint, ∫ drnr3 ()= N . Pf: If n’ is a density different from the ground-state density n in potential U(r), then the U’(r) (and Ψ’), that produce this n’ are different from the ΨG in U(r). According to the variational principle, En[']=Ψ ' H Ψ ' ≥ΨGGG H Ψ = E [] n Thus, for potential U(r), E [n’] is minimized by the ground-state density n.