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
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