Chap 9 -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 ? • exchange effect • screening effect • normalization of (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 ⎡⎤|(')|r 2 2 ∑ j ⎡⎤= 2 G GG⎢⎥2 3 j G ⇒− ∇+Ur()ψψ () r +edr' ()r = ()r ⎢⎥ionii⎢⎥∫ GG i i ⎣⎦2m ψ|'|rr− ⎣⎦⎢⎥

H Hartree (or direct) potential Vee (r) εψ 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 ( )=→= | ( r ) | VH ( r ) e23 dr ' ∑ iee∫ G G i ψ | rr− ' | ↓

2 ⎡⎤= 2 GGGH G 3. solve ⎢⎥−∇+UrVrion () + ee () '() i r = i '() i r ⎣⎦2m ↓ GG2 ψεψ 4. construct nr'( )= ∑ |ψ 'i ( r ) | i ↓ GG 5. if nr '( )−< nr (ψψ ) , thenδ STOP. GG else let ii (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 H , under the constraint = 1 δ ψψ i.e. * G ()H −−=()10 ( → ) δψ()r ψ ψλψψ ψ • Hartree-Fock approximation ψ G GG ψψ(,rs ) (, rs ) ψ" ( r , s ) 11GG 1 12 2 1 GNN GG 1 ψψ(,rs ) (, rs )λε ψ" ( r , s ) assume Ψ= (rsrs , , ," ) 211 22 2 2 NN 112 2 N! ##%# GG G (,rs ) (, rs )" ( r , s ) NNψψ11 2 2 ψ NNN

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 Ψ H Ψ− −10 = δψ* G ⎜⎟∑ iii() A (,)rs⎝⎠i=1

See hand- λψψ 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 ψ G 2 states only ∑ i ('')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 Fock (or exchange) potential ee|'rr− | s,' s δ • 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− δ G G G ik'(⋅ r− r ') e G G GG G e2 π ∑ ss,' eik⋅ r ' drv3 '(,;'')F rsrsψ (',')rs= − dr3 'k 's' χ ∑∫ ee ks∫ G G ' s' V |'|rr− V 41e2 ψG =(,)− G G r s ∑ 2 k V k ' |'|kk− 22 ⎛⎞= 2 41πe G G ⇒−⎜⎟ ∇−GG kkk()rr = () 2mV∑ |'|kk2 ⎝⎠kk'< F − ψεψε(k) ∴ HF energy

G =22kdk 3'1 επ()ke=− 4 2 GG 2(2)m ∫ 3 |'|kk− 2 kk< F =22ke2 2 ⎡⎤11−+xx2 1 k =−kxF ⎢⎥+ ln , = 2mk⎣⎦24π xx1− F k/kF π 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 4⎣ 5200 2 2 ⎦ E 2.21 0.916 4 HF ππ3 ⇒=−2 ,in which (rsa0 ) =n N rrss 3 ⎛⎞==222e r in a =, E in Ry= or ⎜⎟s 0 2 2 π ⎝⎠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 ii('')(,)rs rs AA(,)rs (',') r s 11nrnr(')() ∑ ψ =edrd233r 'ss' − e 23 drdr 3 ' i,A δ ∑ ∫∫GG ∑ ∫ GG ss' 2||ss,' rr− '2 s,'s |'rr− | ψ GG 1 nrnr(')() GG =edrdr233 ' ss' grrHF (, ') ψ ∫ ∑ GG ss' 2'|s,s' |rr− ψ • 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 • •… 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 () 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. • The energy functional

En[]= Tn[ ]+ V[] n+ Un [] G ee G G = Fn[ ]+Un [ ] Un[]= ∫ drnrUr3 () () The F [n] functional is the same for all electronic systems.

“No ones knows the true F [n], and no one will, so it is replaced by various uncontrollable approximations.” (Marder, p.247) Fn[]=+ Tn [] VHXC [] n + V [] n π • Kinetic energy functional dk322= k =25k Tn[]== V V F ∫ 3 2 For free electron gas ()2 210mmπ

2 G 1/3 Thomas-Fermi approx. knrF = (3()π )

(good for slow density variation) 222/3 33= ( π ) G ∴ TnVTF [] nr ()5/3 10π 2m G G • Hartree energy functional enrnr2 ()(') Vn[]= drdr33 ' G G (exact) H 2|'|∫∫ rr−

• Exchange-correlation functional Vxc[n] calculated GG with QMC methods Local density approx. LDA 3 Vxc[] n drnr ()ε xc (()) nr (Ceperley & Alder) (LDA) ∫

where εxc[n] is the xc-energy (per particle) for free electron gas with local density n(r). 2 2 For example, 33ek e G 1/3 επ()nnr=−F =− () 32 () x 44 Generalized ππGGG VLDA[] n drnr3 ()ε nr (),∇ nr () gradient approx. xc∫ xc [ ] (GGA) Kohn-Sham theory No approx. yet G G G G enrnr2 ()(') En[]=+ Tn [] drnrUr333 () () + drdr 'GG + V [] n ∫∫∫2|'|rr− XC

1. KS ansatz: Parametrize the particle density in terms of a set of one-electron orbitals representing a non-interacting reference system G G 2 nr()= ∑ φi () r i

2. Calculate non-interacting kinetic energy in terms of the φi 's G ⎛⎞=2 G Tn[]→= T [] n dr3*φφ () r −∇ 2 () r 0 ∑∫ ii⎜⎟ i ⎝⎠2m if you don't like this approximation, then keep

Tn[]=+− T00 [] n() Tn [] T [] n

3. Determine the optimal one-electron orbitals using the variational method under the constraint φijφδ= ij ⎧⎫ δλφφ⎨⎬En[]− ∑ iii()−= 1 0 ⎩⎭i From J. Hafner’s slides not valid for Kohn-Sham equation excited states

2 G ⎧⎫= 223G nr(') δVntxc [ ] G G → ⎨⎬−∇++Ur() e dr 'GG +G iii () r = () r 2|'|mrr∫ − nr( ) ⎩⎭δ φλ

where VnVnTnTntxc[]=+− xc [][ ]0 [ ] φ

• Similar in form to the Hartree equation, and much simpler than HF eq.

However, here everything is exact, except the Vtxc term. (exact but unknown)

• Neither KS eigenvalues λi , nor eigenstates, have accurate physical meaning. N • However, G G 2 nr()= ∑ φi ( r) ← the density is physical i=1

Also, the highest occupied λi relative the vacuum is the ionization energy.

• If one approximates T～T0, and use LDA, LDA 3 G G G Vnxc[] drnrnr ()ε xc [ ()] then δVnr[ ()] G ∫ txc G = []nr() nr() xc δ ε G ⎧⎫=2 G nr(') GG G −∇++223Ur() e dr ' +εφλφ nr () () r = () r ⎨⎬∫ GG xc[] i i i ⎩⎭2|'|mrr− Self-consistent Kohn-Sham equation, an ab initio theory

1. choose initial {φi} Parameter free ↓ G G GG2 nr(') G 2. construct nr ( )=→= | ( r ) | V ( r ) e23 dr ' + nr ( ) ∑ iKS∫ GG xc[] i φε|'|rr− ↓ (for LDA)

2 ⎡⎤= 2 GGGG 3. solve ⎢⎥−∇++Ur () VKS () r '() i r = i '() i r ⎣⎦2m ↓ GG2 φλφ 4. construct nr '( )= ∑ |φ 'i ( r ) | i ↓ GG 5. if nr '( )−< nr (φφ ) , thenδ STOP. GG else let ii (rr )= ' ( ), GOTO 2 GG enrnr2 ()(') • Total energy Edrdr=−λ 33' ∑ i ∫∫ GG i 2|'|rr− Double-counting correction. Recall similar correction in HFA. Strength of DFT

From W.Aulbur’s slides Weakness of DFT • Band-gap problem: HKS theorem not valid for excited states. Band-gaps in semiconductors and insulators are usually underestimated.

LDA DFT Calcs. (dashed lines)

Metal in “LDA” calculations

Can be fixed by GWA…etc

Rohlfing and Louie PRB 1993 • Neglect of strong correlations - Exchange-splitting underestimated for narrow d- and f-bands. - Many transition-metal compounds are Mott-Hubbard or charge-transfer insulators, but DFT predicts metallic state. LDA, GGAs, etc. fail in many cases with strong correlations. From J. Hafner’s slides