Exchange-Correlation Functionals

Max-Planck Institute ofMicrostructure Physics

Halle (Saale)

E.K.U. Gross

The Hamiltonian of Condensed Matter (atoms, molecules, clusters, solids)

)r,R(V)r(W)r(T)R(W)R(TH eneeennn ++++=

with

∑∑∑

∑∑∑

= =ν ν

ν

≠

ν≠µνµ νµ

νµ

==ν ν

ν

−−=

−=

−=

∇−=

∇−=

e ne

nen

N

1j

N

1 jen

N

kjk,j kj

ee

N

,nn

N

1i

2i

e

N

1

2

n

RrZV

rr1

21W

RRZZ

21W

m2T

M2T

Hamiltonian for the complete system of Ne electrons with coordinatesand Nn nuclei with coordinates( ) rrr

eN1 ≡ ( ) RRRnN1 ≡

Stationary Schrödinger equation

)r,R(V)r(W)r(T)R(W)R(TH eneeennn ++++=

with

∑∑∑

∑∑∑

= =ν ν

ν

≠

ν≠µνµ νµ

νµ

==ν ν

ν

−−=

−=

−=

∇−=

∇−=

e ne

nen

N

1j

N

1 jen

N

kjk,j kj

ee

N

,nn

N

1i

2i

e

N

1

2

n

RrZV

rr1

21W

RRZZ

21W

m2T

M2T

Hamiltonian for the complete system of Ne electrons with coordinatesand Nn nuclei with coordinates( ) rrr

eN1 ≡ ( ) RRRnN1 ≡

( ) ( )R,rER,rH Ψ=Ψ

Time-dependent Schrödinger equation

( ) ( ) ( )( ) ( )

( ) ( ) tcostfERZrt,R,rV

t,R,rt,R,rVR,rHt,R,rt

i

e nN

1 j

N

1 jlaser

laser

ω⋅⋅⋅

−=

ψ+=Ψ∂∂

∑ ∑= =ν

νν

)r,R(V)r(W)r(T)R(W)R(TH eneeennn ++++=

with

∑∑∑

∑∑∑

= =ν ν

ν

≠

ν≠µνµ νµ

νµ

==ν ν

ν

−−=

−=

−=

∇−=

∇−=

e ne

nen

N

1j

N

1 jen

N

kjk,j kj

ee

N

,nn

N

1i

2i

e

N

1

2

n

RrZV

rr1

21W

RRZZ

21W

m2T

M2T

Hamiltonian for the complete system of Ne electrons with coordinatesand Nn nuclei with coordinates( ) rrr

eN1 ≡ ( ) RRRnN1 ≡

Born-Oppenheimer approximation

.Rfor each fixed nuclear configuration

( ) ( ) ( ) nn ˆ ˆ ˆ ˆT (r) W (r) W (R) V (r,R) Φ r Φ r+ + + =BO BOe ee en R R( )R BO∈

Neglect nuclear kinetic energy in full Hamiltonian and solve

( ) ( ) ( )RχrR,rΨ BOBO

R ⋅Φ=BO

Make adiabatic ansatz for the complete molecular wave function:

and find best χBO by minimizing <ΨBO | H | ΨBO > w.r.t. χBO :

Nuclear equation

( ) ( ) ( ) ( ) ( )R EχRχ rdrΦRT rΦ BOBOBO

Rn*BO

R =+ ∫

1ˆ ˆT (R) W (R) (-i ) M υ

υ υ

+ + ∇ +

∑n nn ( )RA

BOυ ( )R BO∈

Berry connection

( ) ( ) ( )∫ ∇= rdrΦ)(-i rΦRA BORυ

* BOR

BOυ

( ) ( )∫ ⋅=C

BOBO RdRACγ

is a geometric phase

corresponding to interesting topological features of molecules

expansion of to second order around equilibrium positions yields phonon spectrum

( )R BO∈

Focus on the electronic-structure problem(in Born-Oppenheimer approximation):

.Rfor fixed nuclear configuration

( ) ( ) ( ) nn ˆ ˆ ˆ ˆT (r) W (r) W (R) V (r,R) Φ r Φ r+ + + =BO BOe ee en R R( )R∈BO

This is still an exponentially hard problem!

Why don’t we just solve the MB SE

Example: Oxygen atom (8 electrons)

depends on 24 coordinates

rough table of the wavefunction

10 entries per coordinate: ⇒ 1024 entries1 byte per entry: ⇒ 1024 bytes5×109 bytes per DVD: ⇒ 2×1014 DVDs10 g per DVD: ⇒ 2×1015 g of DVDs

= 2×109 t of DVDs

( )81 r,,r

Ψ

Two fundamentally different classes of ab-initio approaches:

• Wave function approaches

-- Quantum Monte Carlo-- Configuration interaction-- Tensor product decomposition

• “Functional Theories”

Two fundamentally different classes of ab-initio approaches:

• Wave function approaches

-- Quantum Monte Carlo-- Configuration interaction-- Tensor product decomposition

• “Functional Theories”

Write total energy as functional of a simpler quantity and minimize

Motivation

(r, r ') G(r, r ',0 )+γ =

MBPT RDMFT

)'tt,'r,r(G −

“Functional Theories”

DFT

)r,r()r( γ=ρ

Motivation

(r, r ') G(r, r ',0 )+γ =

MBPT RDMFT

)'tt,'r,r(G −

“Functional Theories”

DFT

)r,r()r( γ=ρ

Functional:Φxc[G]

or Σxc[G]

Functional:Exc[γ]

Functional:Exc[ρ]

or vxc[ρ]

Motivation

(r, r ') G(r, r ',0 )+γ =

MBPT RDMFT

)'tt,'r,r(G −

“Functional Theories”

DFT

)r,r()r( γ=ρ

Functional:Φxc[G]

or Σxc[G]easy (e.g. GW)

Functional:Exc[γ]

difficult

Functional:Exc[ρ]

or vxc[ρ]very difficult

Motivation

(r, r ') G(r, r ',0 )+γ =

MBPT RDMFT

)'tt,'r,r(G −

“Functional Theories”

DFT

)r,r()r( γ=ρ

Functional:Φxc[G]

or Σxc[G]easy (e.g. GW)numerically

heavy

Functional:Exc[γ]

difficult

moderate

Functional:Exc[ρ]

or vxc[ρ]very difficult

light

ESSENCE OF DENSITY-FUNTIONAL THEORY

• Every observable quantity of aquantum system can be calculatedfrom the density of the systemALONE

• The density of particles interactingwith each other can be calculated asthe density of an auxiliary system ofnon-interacting particles

ESSENCE OF DENSITY-FUNTIONAL THEORY

• Every observable quantity of aquantum system can be calculatedfrom the density of the systemALONE

• The density of particles interactingwith each other can be calculated asthe density of an auxiliary system ofnon-interacting particles

Hohenberg-Kohn theorem (1964)Kohn-Sham theorem (1965) (for the ground state)

DFT papers per year

Jul 5th, 2017 KITS 18DFT: A Theory Full of Holes, Aurora Pribram-Jones, David A. Gross, Kieron Burke, Annual Review of Physical Chemistry (2014).

HOHENBERG-KOHN THEOREM

1. v(r) ρ(r)one-to-one correspondence between external potentials v(r) and ground-state densities ρ(r). In other words: v(r) is a functional of ρ(r)

2. Variational principleGiven a particular system characterized by the external potential v0(r). There exists a functional, EHK [ρ], such that the solution of the Euler-Lagrange equation

yields the exact ground-state energy E0 and ground-state density ρ0(r) of this system

3. EHK[ρ] = F [ρ] + ρ(r) v0(r) d3r

F[ρ] is UNIVERSAL.

1—1

( ) [ ] 0Er HK =ρ

δρδ

What is a FUNCTIONAL?

E[ρ]

functional

set of functions set of real numbers

ρ(r) R

Generalization:

[ ] [ ]( )rvv r

ρ=ρ

[ ] [ ]( )N1r...r r...rN1

ρψ=ρψ ( )N1 r...r

functional depending parametrically on r

or on

Explicit algorithm to construct the HK map vs ρfor non-interacting particles

Iterative procedure

ρ0(r) given (e.g. from experiment) Start with an initial guess for vs(r) (e.g. LDA potential)

solve (– + vs(r) ) ϕi = ∈i ϕi

vsnew(r) = · Σ (∈iϕi(r)2– ϕi* (- ) ϕi)

1ρ0( r) i = 1

N

h2 ∇ 2

2m

h2 ∇ 2

2m

solve SE with vsnew and iterate, keeping ρ0(r) fixed

(– + vs(r) ) ϕi = ∈i ϕi Σ ϕi* ·

Σ ϕi*(– )ϕi + vs(r)ρ(r) = Σ ∈iϕi(r)2

⇒ vs(r) = · Σ (∈iϕi(r)2– ϕi* (- ) ϕi) 1

ρ(r)

i = 1

N

i = 1

N

h2 ∇ 2

2m

h2 ∇ 2

2m

i = 1

N

ih2 ∇ 2

2m

Consequence: The orbitals are functionals of the density: ϕi[ρ]

KOHN-SHAM EQUATIONS

EHK[ρ] = TS[ρ] + ρ(r) v0(r) d3r + EH[ρ] + Εxc[ρ]

whereis the kinetic energy functional of non-interacting particles

( ) [ ] 0Er HK =ρ

δρδ yields the Kohn-Sham equations:

( ) ( ) ( )( ) ( ) ( )0 H xc/ 2 v v [ ] v [ ]−∇ + + ρ + ρ φ =∈ φ2j j jr r r r r

Rewrite HK functional:

Exc[ρ] is a universal functional of the density which, in practice, needsto be approximated.

Walter Kohn: “The KS equations are an exactification of the Hartree mean-field equation”

The orbitals from these equations yield the true density of the interacting system

[ ] [ ]( )( ) [ ]( )3 2S j jT d r r / 2 r∗ρ = φ ρ −∇ φ ρ∑∫

Approximations to the exchange-correlation functional

xcE Nature 's glue⇒ =

[ ] ( ) ( )( )LDA 3 unifxc xcE d r r rρ = ρ ε ρ∫

Local density approximation (LDA)

[ ]xcE smallest part of total energyρ =

[ ]xcsimplest approximation : E 0 Hartree approachρ ≡ ⇒

Result: lattice constants and bonding distances much too large (20%-50%)

LDA (Kohn and Sham, 1965)

( )unifxcε ρ xc energy per particle of a uniform electron gas of density ρ

(known from quantum Monte-Carlo and many-body theory)

Result: decent lattice constants, phonons, surface energies of metals

Quantity Typical deviation(from expt)

• Atomic & molecular ground state energies

< 0.5 %

• Molecular equilibrium distances

< 5 %

• Band structure of metals, Fermi surfaces

few %

• Lattice constants < 2 %

Quantity Typical deviation(from expt)

• Atomic & molecular ground state energies

< 0.5 %

• Molecular equilibrium distances

< 5 %

• Band structure of metals, Fermi surfaces

few %

• Lattice constants < 2 %

Systematic error of LDA: Molecular atomisation energies too large and bond lengths and lattice constants too small

( )1

32Fk 3

∇ρ<< = π ρ

ρ1

6TFk 4(3 / )

∇ρ<< = ρ π

ρ

( )xcn r, r '

xn 0≤

xc x cn n n= +

[ ] ( ) ( )LDAxcLDA 3 3

xc

n r, r '1E d r r d r '2 r ' r

ρ = ρ−∫ ∫

One would expect the LDA to be good only for weakly inhomogeneous systems, i.e., systems whose density satisfies:

and

Why is the LDA good also for strongly inhomogeneous systems?

Answer: Satisfaction of many exact constraints (features of exact xc fctl)

coupling-constant-averaged xc hole density

( )3xd r 'n r, r ' 1= −∫

Important constraints:

( )3cd r 'n r, r ' 0=∫

are satisfied in LDA

[ ] ( ) ( )( )GGA 3xcE d r f r , rρ = ρ ∇ρ∫

Generalized Gradient Approximations (GGA)

Langreth, Mehl (1983), Becke (1986), Perdew, Wang (1988)PBE: Perdew, Burke, Ernzerhof (1996)

Construction principle: Satisfaction of exact constraints(important lesson from LDA and from gradient expansion of Exc )

Results: GGAs reduce the LDA error in the atomisation energysignificantly (but not completely) while LDA bond lengths areover-corrected (i.e. are in GGA too large compared with expt)

Detailed study of molecules (atomization energies)

32 molecules (all neutral diatomics from first-row atoms only and H2 )

B. G. Johnson, P. M. W. Gill, J. A. Pople, J. Chem. Phys. 97, 7847 (1992)

Atomization energies (kcal/mol) from:

mean deviation from experiment 0.1 1.0 -85.8mean absolute deviation 4.4 5.6 85.8

VWNc

Bx EE + LYP

cBx EE + HF

for comparison: MP2-22.422.4

LIMITATIONS OF LDA/GGA

• Not free from spurious self-interactions: KS potential decays more rapidly than r-1 for finite systemsConsequences: – no Rydberg series

– negative atomic ions not bound– ionization potentials (if calculated from highest

occupied orbital energy) too small

• Dispersion forces cannot be describedWint (R) e-R (rather than R-6)

• band gaps too small: GEgap (LDA/GGA) ≈ 0.5 Egap(expt)

• Energy-structure dilemma of GGAsatomisation energies too largebond lengths too large(no GGA known that gets both correct!!)

• Wrong ground state for strongly correlated solids, e.g. CoO, La2CuO4predicted as metals

[ ] ( )MGGA 3 MGGAxc xcE d r (r) (r), (r) , (r)ρ = ρ ε ρ ∇ρ τ∫

( ) ( )occup 2

,,

1r r2 α σ

α σ

τ = ∇ψ∑ [ ] ( )3sT n d r r= τ∫

Meta Generalized Gradient Approximations (MGGA)

Result: Solves energy-structure dilemma of GGAs

,ρ ∇ρρ

xcE 0=

LDA

GGA

, ,ρ ∇ρ τMGGA

hybrid , , ,ρ ∇ρ τ occupied orbitals

RPA-like , , ,ρ ∇ρ τ occupied & unoccupied KS orbitals + energies

earth (Hartree )

heaven (exact functional)

Jacob’s ladder of xc functionals (John Perdew)

Hybrid functionals

Adiabatic Connection Formula

= Hamiltonian of fully interacting system

( ) ( ) ∑∑≠

== −++=

N

ki1k,i ki

2N

1ii rr

12e rvT H

( ) ( ) ∑∑≠

== −++=

N

ki1k,i ki

2N

1iinuc rr

12e rvT H

λλ

λ

λ=1

0 ≤ λ ≤ 1

Choose vλ(r) such that for each λ the ground- state density satisfies ρλ(r) = ρλ=1(r)

Hence vλ=0(r) = vKS(r) vλ=1(r) = vnuc(r)

Solve many-body Schroedinger eq for each λ, yielding Ψλ

[ ]1

xc0

E d W [ ]λρ = λ ρ∫ [ ]ee HW V Eλ λ λ= ψ ψ − ρ

[ ]

0 0 ee 0 H

HFx i

HF exchange

W V E

E

= ψ ψ − ρ

== ϕ

' GL20 cW 2E= (Correlation energy in 2nd-order Görling-Levy

perturbation theory)

Becke (JCP 1993): W a bλ = + λ

Exact representation of Exc[ρ]:

( )BHH HF DFAxc 0 1 x 1

1 1 1E W W E W2 2 2

= + ≅ +

GGA

B3LYP HF LDA B88 LYP VWNxc x x x c cE 0.2E 0.8E 0.72E 0.8E 0.15E= + + + +

Another way of constructing hybrids: Range separation (Savin, Baer, Kronik)

( ) ( )ij ij

ij ij ij

erf r erfc r1r r r

µ µ= +

long range short range

• Treat long-range part by wavefunction theory, HF• Treat short-range part by density functional approximation (GGA)

PBE0 HF PBE PBExc x x cE 0.25E 0.75E E= + +

For solids, it makes sense to do it the other way around (Heyd, Scuseria, Ernzerhof):

HSE HF,SR PBE,SR PBE,LR PBExc x x x cE aE ( ) (1 a)E ( ) E ( ) E= µ + − µ + µ +

5th-rung functionals (using unoccupied KS orbitals)2

occup unoccupMP2c

ij ab i j a b

ij ab1E2

=ε + ε − ε − ε∑ ∑

RPAc KSE [G ] and beyond RPA-functionals, e.g. plus TDDFTRPA

cE

( )( ) ( ) ( ) ∫ ∫ ∫ ∫∞

λ −δρ+χ−π

λ−=1

0 0

233

xc 'rrriu;'r,r'rr

e'rdrd2dudE

with the response function χ(λ)(r,r';ω) corresponding to H(λ).χ(λ)(r,r';ω) can be obtained from linear-response TDDFT

( ) ( ) ( )λλ χ+λχ+χ=χλ

fW xcClbss

Given an orbital functional for Exc (and hence for Etot), how can one use it in practice?

• Non-selfconsistently:post-LDA, post-GGA, or post-HF

• Self-consistently, the Kohn-Sham way, yielding theOptimized-Effective Potential (OEP) procedure.This determines the variationally best local potential (i.e. that localpotential whose orbitals minimize the given total-energy functional)

• Self-consistently, the Generalised Kohn-Sham (GKS) way.This determines the variationally best orbitals (not restricted to come from a local potential). The resuting GKS potential can benon-local

• Self-consistently, using the Gidopoulos variational principle.This avoids the variational collapse of PT-derived functionals by finding that local potential reproducing the density of a given approximate (derived, e.g., from PT) many-body wave function

• Self-consistently, using the Sham-Schlüter-way (specifically for functionals coming from an approximate selfenergy Σ[GKS]). This yieldsthat local potential whose density reproduces the density of G.

N.I. Gidopoulos, Phys. Rev. A 83, 040502(R), (2011)

From: A.J. Cohen, P. Mori-Sanchez, W. Yang, Chem. Rev. 112, 289 (2012)

Apply HK theorem to non-interacting particles

ρ given ⇒ vs = vs[ρ] ⇒ (– + vs[ρ](r))ϕi (r) = ∈i ϕi (r) ϕi = ϕi[ρ]∈i = ∈i[ρ]

∇2

2

consequence:

Any orbital functional, Exc[ϕ1 ,ϕ2 …], is an (implicit) density functional provided that the orbitals come from a local (i.e., multiplicative) potential.

“optimized effective potential” ≡ KS xc potential

( ) ( ) [ ]N1xcOEPxc ...E

rrv ϕϕ

δρδ

=

( ) ( )( )( )

( )( ) .c.c r

''rv ''rv'r

'r

E ''rd'rd rvj

s

s

j

j

xc33OEPxc +

δρδ

δδϕ

δϕδ

= ∑∫ ∫χKS

-1(r'',r)act with χKS on equation:

( ) ( ) ( )( )( ) .c.c rv'r

'r

E 'rd 'rd 'rv'r,rj s

j

j

xc33OPMxcKS +

δ

δϕ

δϕδ

=χ ∑ ∫∫⇒

OPM integral equationknown functional of ϕj

OEP integral equation

to be solved simultaneously with KS equation:

∑ d3r’ ( Vxc,σ(r') – uxc,iσ(r'))K iσ(r,r')ϕiσ(r)ϕiσ*(r') + c.c. = 0

where K iσ(r,r') = ∑

and uxc,iσ(r) := ·

ϕkσ*(r)ϕkσ(r')∈kσ – ∈iσ

∞k = 1k ≠ i

1 δExc [ϕ1σ ...]ϕiσ*(r) δϕiσ(r)

Nσ

i = 1

( ) ( ) ( )( ) ( ) ( )0 xH c/ 2 v [ ] vv−∇ + + ρ + ϕ =∈ ϕ2j j jr r rr r

By contrast, the GKS equations have a non-local potential:

( ) ( ) ( )2i GKS i i i

1 dr ' v r, r ' r ' r2

− ∇ ψ + ψ = ε ψ ∫

From: A.J. Cohen, P. Mori-Sanchez, W. Yang, Chem. Rev. 112, 289 (2012)

Comparison OEP vs GKS:

• GKS is numerically much simpler than OEP

• Total-energy-related quantities very similar

• Single-particle gap close to experiment (for some hybrids) when used in GKS way

See: J.P. Perdew, W. Yang, K. Burke, Z. Yang, E.K.U. Gross, M. Scheffler, G.E. Scuseria, T.M. Henderson, I. Ying Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Grushin, A. Goerling, PNAS 114, 2801 (2017).

Comparison OEP vs GKS:

• GKS is numerically much simpler than OEP

• Total-energy-related quantities very similar

• Single-particle gap close to experiment (for some hybrids) when used in GKS way

See: J.P. Perdew, W. Yang, K. Burke, Z. Yang, E.K.U. Gross, M. Scheffler, G.E. Scuseria, T.M. Henderson, I. Ying Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Grushin, A. Goerling, PNAS 114, 2801 (2017).

Ionization energy I, electron affinity A, fundamental gap G=I−A and band gap g=εLU-εHO

of an infinite linear chain of H2 molecules, evaluated by extrapolation from finite chains

DFT description of quantum phases:

Magnetism and

Superconductivity

MAGNETIC SYSTEMS

In principle, Hohenberg-Kohn theorem guarantees that m(r) is a functional of the density: m(r) = m[ρ](r). In practice, good approximations for the functional m[ρ] are not known.

Quantity of interest: Spin magnetization density m(r)(order parameter of magnetism)

Strategy: Include m(r) as basic variable, i.e. as an additional density, in the formalism (in addition to the ordinary density ρ(r)).

0ˆ ˆ ˆm(r) (r) (r)+

α αβ βαβ

= µ ψ σ ψ∑

HK theorem

[ ] 1-1 (r),m(r) v(r), B(r) ρ ←→

total energy:

[ ] [ ] ( )∫ ⋅−ρ+ρ=ρ )r(m)r(B)r()r(vrdm,Fm,E 3B,v

universal

( ) ( ) ( ) ( )3 3eev,B

ˆˆ ˆ ˆ ˆH T V r v r d r m r B r d r= + + ρ − ⋅∫ ∫

Start from fully interacting Hamiltonian with Zeeman term:

KS scheme

For simplicity: ,

vxc[ρ,m] = δExc[ρ,m]/δ ρ Bxc[ρ,m] = δExc[ρ,m]/δ m

ρ (r) = ρ+ (r) + ρ- (r) , m (r) = ρ+ (r) - ρ- (r) , ρ± = Σϕ j± 2

B → 0 limitThese equations do not reduce to the original KS equations for B → 0 if,in this limit, the system is magnetic, i.e. has a finite m(r).

( )

0B(r) 0

B r

=

( )

0m(r) 0

m r

=

[ ] [ ] )()( )(B )(v)(vm2 oH

2

rrrrr jjj±±=∈±

−µ±+++

∇− ϕϕvxc(r) Bxc(r)

DENSITY-FUNTIONAL THEORY OF THE SUPERCONDUCTING STATE

• Include order parameter, χ , characterising superconductivity as additional “density”

BASIC IDEA:

L.N. Oliveira, E.K.U.G., W. Kohn, PRL 60, 2430 (1988)

• Include N-body density, Γ, of the nuclei as additional “density”

T. Kreibich, E.K.U.G., PRL 86, 2984 (2001)

General (model-independent) characterization of superconductors: Off-diagonal long-range order of the 2-body density matrix:

( )( ) ( ) ( ) ( ) ( )'yˆyˆxˆ'xˆ'yy,'xx2↓↑

+↑

+↓ ψψψψ=ρ

( ) ( ) ( ) ( )'yˆyˆ xˆ'xˆ YX ↓↑+↑

+↓∞→− ψψ⋅ψψ →

2'xx +

2'yy + ( )'x,x∗χ ( )'y,yχ

( ) ( ) ( )'rˆrˆ'r,r ↓↑ ψψ=χ order parameter of the N-S phase transition

Hamiltonian

( )∫ ∫∫ +∆++= .c.H)'r,r(* 'rdrd- r)d(v WTH 333eeee r( )rρ ( )'r,r χ

S N∆ F PB

χ

x x

m

ANALOGY

“proximity effect”

enne UHHH ++=

Hamiltonian

3 densities:

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ⋅⋅⋅φφ⋅⋅⋅φφ=Γ

ψψ=χ

ψψ=ρ

++

↓↑

=↑↓σσ

+σ∑

2121 Rˆ RˆRˆ Rˆ)R(

'rˆrˆ)'r,r(

rˆrˆr electron density

order parameter

diagonal of nuclear Nn-body density matrix

∫+= )R(W RdTH nNnn )R(Γ

( )∫ ∫∫ +∆++= .c.H)'r,r(* 'rdrd- r)d(v WTH 333eeee r( )rρ ( )'r,r χ

Hohenberg-Kohn theorem for superconductors

Densities in thermal equilibriumat finite temperature

[v(r),∆(r,r’),W(R)] ←→ [ρ(r),χ(r,r’),Γ(R)]1-1

Electronic KS equation

vs[ρ,χ,Γ](r) ∆s[ρ,χ,Γ](r,r’)( ) ( ) ( )rrrr Eu'd'v u 2

32

=+

+µ−

∇− ∫

vs[ρ,χ,Γ](r)∆s*[ρ,χ,Γ](r,r’) ( ) ( ) ( )rrrr Evv 2

'du 2

3 =

+µ−

∇−−∫ '

Nuclear KS equation

)R(E)R( M2

nN

1

2

ψ=ψ

+

∇−∑

=α α

α Ws[ρ,χ,Γ](R)

3 KS potentials: vs ∆s Ws

KS theorem: There exist functionals vs[ρ,χ,Γ], ∆s[ρ,χ,Γ], Ws[ρ,χ,Γ], such thatthe above equations reproduce the exact densities of theinteracting system

No approximation yet!“Exactification” of BdG mean-field eqs.

Electronic KS equation

vs[ρ,χ,Γ](r) ∆s[ρ,χ,Γ](r,r’)( ) ( ) ( )rrrr Eu'd'v u 2

32

=+

+µ−

∇− ∫

vs[ρ,χ,Γ](r)∆s*[ρ,χ,Γ](r,r’) ( ) ( ) ( )rrrr Evv 2

'du 2

3 =

+µ−

∇−−∫ '

Nuclear KS equation

)R(E)R( M2

nN

1

2

ψ=ψ

+

∇−∑

=α α

α Ws[ρ,χ,Γ](R)

3 KS potentials: vs ∆s Ws

KS theorem: There exist functionals vs[ρ,χ,Γ], ∆s[ρ,χ,Γ], Ws[ρ,χ,Γ], such thatthe above equations reproduce the exact densities of theinteracting system

No approximation yet!“Exactification” of BdG mean-field eqs.

Solved in harmonic approximation

CONSTRUCTION OF APPROXIMATE Fxc : 1o HHH +=

[ ]∫ ∑∫

∑∫

+Ω++∆ψψ−

ψ

ψ=

+↓↑

σσ

+σ +µ−

∇−

qqqq

*s

33

3o

23bb.c.H)'r,r()'r(ˆ)r(ˆ'rdrd

rd)r(ˆ)r(ˆH )oR,r(s

v2

2

develop diagrammatic many-body perturbation theory on the basis of the Ho-propagators:

Gs normal electron propagator (in superconducting state)

Fs

Fs*

Ds phonon propagator

Immediate consequence:Fxc = Fxc + Fxc

all diagrams containing Dsall others diagrams

ph el

anomalous electron propagators

Phononic contributions

First order in phonon propagator:

[ ] +=Γχ ,,nFphxc

( )

( )( )

( )( )

Ω−

µ−∈µ−∈−+

Ω

µ−∈µ−∈+ΩαΩ−

Ω−Ω−∆∆

ΩαΩ−=

∑∫

∑∫

),E,E(I EE

1

),E,E(I EE

1 )(Fd21

),E,E(I),E,E(IEE

)(Fd21

jiji

ji

ijji

ji

jiij

2

ijjiji

ji

*ji

ij2

Calculated with Quantum Espresso code

( ) ( )∑λ

λλ Ω−Ωδ=Ωα

qq

2q'k'n,nk'k'n,nk

2 gFFxcphInput to : Full k,k’

resolved Eliashberg function

Purely electronic contributions

RPA-screened electron-electron interaction

]ρ[F,ρF GGAxc

eexc +=χ

Crucial point: NO ADJUSTABLE PARAMETERS

[ ]

Full Green fctn KS Green fctn

Ab-initio prediction of critical temperatures of superconductors