Density Functional Theory & Lieb-Oxfordinequality: conjectures & recent results

Mathieu LEWIN

mathieu.lewin@math.cnrs.fr

(CNRS & Universite de Paris-Dauphine)

collaboration with Elliott H. Lieb (Princeton)

Workshop MOKALIEN, Univ. Paris Dauphine, Nov. 2015

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 1 / 15

Density Functional Theory

I Main idea: describe complicated N-particle system (a proba on R3N ) usingonly its one-particle marginal ρ(x) on R3

I History:

Thomas-Fermi (1920s): simple model

Hohenberg-Kohn-Sham (1963-64): practical method based on semi-empiricalfunctions of ρ

Levy (1979), Lieb (1983): mathematical justification

1970s–: popular in solid state physics, but not so accurate

1990s–: explosion in quantum chemistry, due to increase of computationalresources + discovery of efficient semi-empirical functionals of ρ

1998: Nobel prize to Kohn & Pople

I Example: H20 (N = 10)cost O–H angle

Expensive CCSD(T)-TZ2P(f,d)** ∼ N7 95.89 pm 104.16

Cheap DFT GGA(PBE)-6-311+G** ∼ N≤3 96.90 pm 104.75

Experiment 95.84 pm 104.45

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 2 / 15

A tremendous success

https://www.uni-due.de/akschmuck/wilhelm sicking.shtml

very large molecules like DNA

use about 15% of resources available inscientific computing centers

about 15 000 papers per year with thekeyword ‘density functional theory’

12 of the 100 most cited papers inhistory of science are from DFT (2015)

Rank Authors Journal Year Citations7 Lee, Yang, Parr Phys. Rev. B 1988 46 7028 Becke J. Chem. Phys. 1993 46 145

16 Perdew, Burke, Ernzerhof Phys. Rev. Lett. 1996 35 40525 Becke Phys. Rev. A 1988 26 47534 Kohn, Sham Phys. Rev. 1965 23 059

The top 100 papers, Nature 514, 550–553 (30 October 2014)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 3 / 15

The many-particle Coulomb interactionˆR3N

( ∑1≤j<k≤N

1

|xj − xk |

)dP(x1, ..., xN )

P symmetric = probability density of N indistinguishable electronsin practice P = |Ψ|2 where Ψ = complex-valued quantum wavefunction

one-particle density: ρP(x) = N

ˆR3(N−1)

dP(x , x2..., xN )

I Lowest possible value = multi-marginal optimal transport:

C (ρ) = infρP=ρ

ˆR3N

( ∑1≤j<k≤N

1

|xj − xk |

)dP(x1, ..., xN )

coincides with infimum over Monge states = “strongly correlated electrons”

P =1

N

ˆδy ⊗s δTy ⊗ · · · ⊗s δT N−1y ρ(y) dy (Colombo-Di Marino ’15)

existence and uniqueness of Monge for N = 2 and for N ≥ 2 in 1D(Colombo-De Pascale-Di Marino ’13)

Colombo & Di Marino, Annali di Matematica 194 (2015), Colombo, De Pascale & Di Marino, Canadian J. Math. (2014)Di Marino, Gerolin & Nenna, arXiv:1506.04565 (review)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 4 / 15

Use of optimal transport methods

OT important to devise numerical methods with lower cost

N = 2: classical OT in 3D, with singular cost

N ≥ 3: only Monge radial state, known to be non-optimal in some cases(Colombo-Stra ’15)

2D optimal Monge radial state with ρ = 1B

Rasanen, Seidl & Gori-Giorgi (2011)

3D optimal Monge state for Helium (N = 2)Benamou, Carlier & Nenna (2015)

Seidl, Phys. Rev. A 60 (1999), Seidl, Gori-Giorgi & Savin, Phys. Rev. A 75 (2007), Gori-Giorgi, Seidl & Vignale, Phys. Rev. Lett. (2009), Rasanen, Seidl& Gori-Giorgi, Phys. Rev. B (2011), Buttazzo, De Pascale & Gori-Giorgi, Phys. Rev. A 85 (2012), Mendl & Lin Lin, Phys. Rev. B 87 (2013), Chen,Friesecke & Mendl, J. Chem. Theory Comput. 10 (2014), Cotar, Friesecke & Pass, Calc. Var. Partial Differ. Equ. 54 (2015), Benamou, Carlier & Nenna,arXiv:1505.01136 (2015), Seidl, Vuckovic & Gori-Giorgi, arXiv:1508.01715 (2015) Colombo & Stra, arXiv:1507.08522 (2015),

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 5 / 15

ExtensionsI Quantum case:

C~(ρ) = infΨ antisymm.ρ|Ψ|2 =ρ

~2

2

ˆR3N

|∇Ψ|2 +

ˆR3N

( ∑1≤j<k≤N

1

|xj − xk |

)|Ψ|2

∃Ψ when√ρ ∈ H1(R3) (Lieb ’83)

By scaling ρ 1 same as ~ 1

I Positive temperature classical case:

C T (ρ) = infρP=ρ

ˆR3N

( ∑1≤j<k≤N

1

|xj − xk |

)dP + T

ˆR3N

P logP

I Positive temperature quantum case:

C T~(ρ) = inf

Γ≥0, tr Γ=1ρΓ=ρ

tr

(~2

2(−∆) +

∑1≤j<k≤N

1

|xj − xk |

)Γ + T tr

(Γ log Γ

)Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 6 / 15

Exchange Correlation

I Chemists interested in general properties of C (ρ), valid for all N ≥ 1, thatcould be used to devise clever semi-empirical explicit functionals

C (ρ) =1

2

ˆR3

ˆR3

ρ(x)ρ(y)

|x − y |dx dy︸ ︷︷ ︸

=2π´R3|ρ(k)|2|k|2

dk:=D(ρ,ρ)

+Exc(ρ)

I Exc(ρ) is “almost local” and chemists like to use a local approximation

Exc(ρ) 'ˆR3

e(ρ(x),∇ρ(x), ...) dx with e = −eunif ρ(x)4/3 + corr(ρ(x),∇ρ(x))

Example: Perdew-Burke-Ernzerhof famous functional

EPBE(ρ) = −3

4

(3

π

)1/3 ˆR3ρ(x)

43

1 +µ|∇ρ

13 (x)|2

ρ(x)43 + (µ/ν)|∇ρ

13 (x)|2

dx +

ˆR3ρ(x) εc

(ρ(x)

)dx

+ γ

ˆR3ρ(x) log

(1 +

β

γ|∇ρ

13 (x)|2

ρ(x) + A(ρ(x)

)|∇ρ

13 (x)|2

ρ(x)2 + A(ρ(x)

)ρ(x)|∇ρ

13 (x)|2 + A

(ρ(x)

)2|∇ρ13 (x)|4

)dx

A(ρ(x)

)=β

γ

(e−εc(ρ(x))/γ − 1

)−1

εc = correlation energy of uniform electron gas, = (−eunif + (3/4)(3/π)1/3)ρ1/3 in classical case (ρ 1)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 7 / 15

Exchange Correlation

I Chemists interested in general properties of C (ρ), valid for all N ≥ 1, thatcould be used to devise clever semi-empirical explicit functionals

C (ρ) =1

2

ˆR3

ˆR3

ρ(x)ρ(y)

|x − y |dx dy︸ ︷︷ ︸

=2π´R3|ρ(k)|2|k|2

dk:=D(ρ,ρ)

+Exc(ρ)

I Exc(ρ) is “almost local” and chemists like to use a local approximation

Exc(ρ) 'ˆR3

e(ρ(x),∇ρ(x), ...) dx with e = −eunif ρ(x)4/3 + corr(ρ(x),∇ρ(x))

Example: Perdew-Burke-Ernzerhof famous functional

EPBE(ρ) = −3

4

(3

π

)1/3 ˆR3ρ(x)

43

1 +µ|∇ρ

13 (x)|2

ρ(x)43 + (µ/ν)|∇ρ

13 (x)|2

dx +

ˆR3ρ(x) εc

(ρ(x)

)dx

+ γ

ˆR3ρ(x) log

(1 +

β

γ|∇ρ

13 (x)|2

ρ(x) + A(ρ(x)

)|∇ρ

13 (x)|2

ρ(x)2 + A(ρ(x)

)ρ(x)|∇ρ

13 (x)|2 + A

(ρ(x)

)2|∇ρ13 (x)|4

)dx

A(ρ(x)

)=β

γ

(e−εc(ρ(x))/γ − 1

)−1

εc = correlation energy of uniform electron gas, = (−eunif + (3/4)(3/π)1/3)ρ1/3 in classical case (ρ 1)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 7 / 15

Elementary upper and lower bounds

I Taking P = (ρ/N)⊗N , one finds Exc(ρ) ≤ − 1N D(ρ, ρ) ≤ 0

I Replace Coulomb by smooth potential, still with positive Fourier transform:∑1≤j<k≤N

1

|xj − xk |≥

∑1≤j<k≤N

1− e−a|xj−xk |

|xj − xk |= Da

N∑j=1

δxj ,

N∑j=1

δxj

− aN

2

≥ 2Da

ρ, N∑j=1

δxj

− Da(ρ, ρ)− aN

2

Exc(ρ) ≥ −1

2

ˆR3

ˆR3

e−a|x−y |ρ(x)ρ(y)

|x − y |dx dy − aN

2

Lemma (First order for a spread out density)

For any fixed proba ν ∈ L1(R3) ∩ L6/5(R3),

N1/3C(ν(·/N1/3)

)= C (Nν) = N2D(ν, ν) + o(N2)

Cotar, Friesecke & Pass, Calc. Var. Partial Differ. Equ. 54 (2015)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 8 / 15

Uniform Electron Gas: ρ constant

For ρ = N1Ω where Ω =unit cube, Dirac (1926) found

N1/3Exc(1N1/3Ω) = Exc(N1Ω) ≤ −3

4

(3

π

)1/3

N4/3 + o(N4/3)

using P(x1, ..., xN ) = (N!)−1| det(e i2πk·xj )k∈Z3∩B(0,cN1/3)|2

Theorem (Uniform Electron Gas)

limN→∞

Exc(1N1/3Ω)

N= lim

N→∞

Exc(N1Ω)

N4/3= −eunif > −∞

follows from monotonicity of Exc(1N1/3C )/N and previous lower bound

same limit if Ω replaced by any other domain (e.g. a ball) with volume 1

exact value of eunif is unknown, although everybody thought for decades thateunif ' 1.4441, which is related to the Epstein Zeta function (more on this later)

last week numerics by Seidl-Vuckovic-Gori Giorgi ’15 (N = 50): eunif ≥ 1.3354

Dirac, Proc. Royal Soc. London Ser. A 112 (1926)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 9 / 15

Lieb-Oxford inequality

Theorem (Lieb-Oxford inequality)

For every ρ ∈ L1(R3) ∩ L4/3(R3), we have

Exc(ρ) ≥ −1.64

ˆR3

ρ(x)4/3 dx

best constant increases with N and is exactly known for N = 1

its limit eLO is at least 1.23, but not known rigorously

last week numerics by Seidl-Vuckovic-Gori Giorgi ’15 (N = 50): eLO ≥ 1.401

one parameter in PBE chosen to enforce LO

Chemists’ conjectures

eLO = eunif (Rasanen, Pittalis, Capelle & Proetto ’09)

eunif = 1.4441... (Coldwell-Horsfall & Maradudin ’60, Perdew ’91)related to Epstein zeta function and Jellium

Lieb , Phys. Lett. A (1979), Lieb & Oxford, Int. J. Quantum Chem. (1980), Chan & Handy, Phys. Rev. A (1999), Perdew, in Electronic Structure ofSolids ’91 (1991), Rasanen, Pittalis, Capelle & Proetto, Phys. Rev. Lett. (2009) Coldwell-Horsfall & Maradudin, J. Math. Phys. (1960),

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 10 / 15

Improved Lieb-Oxford with gradient correction

Theorem ([LewLie-15])

Exc(ρ) ≥ − 9

10

(4π

3

)1/3

︸ ︷︷ ︸'1.4508

ˆR3

ρ(x)43 dx

−

0.3270

(ˆR3

|∇ρ(x)| dx

) 14(ˆ

R3

ρ(x)43 dx

) 34

0.9416

(ˆR3

|∇ρ 13 (x)|2 dx

) 13(ˆ

R3

ρ(x)43 dx

) 23

proof uses potential theory, Hardy-Littlewood maximal fns and dirty estimates

eunif ≤ 1.4508, very close to the supposedly optimal 1.4442

discussed by Feinblum-Kenison-Burke ’14, Constantin-Terentjevs-Della Sala-Fabiano’14, Seidl-Vuckovic-Gori Giorgi ’15

Benguria, Bley & Loss (2012): similar with nonlocal gradient term⟨√

ρ, |∇|√ρ⟩

Lewin & Lieb, Phys. Rev. A 91 (2015), Lieb & Narnhofer, J. Stat. Phys. 14 (1976), Benguria, Bley & Loss, Int. J. Quantum Chem. 112 (2012)Feinblum, Kenison & Burke, J. Chem. Phys. (2014), Constantin, Terentjevs, Della Sala & Fabiano, Phys. Rev. B 91 (2014)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 11 / 15

Classical Jellium

I N electrons and a neutralizing background in a domain Ω with |Ω| = N

EJel(xj ,Ω) =∑

1≤j<k≤N

1

|xj − xk |−

N∑j=1

ˆΩ

dy

|xj − y |+

1

2

ˆΩ

ˆΩ

dx dy

|x − y |

eJel := limN→∞

minxj EJel(xj ,N1/3Ω)

N, |Ω| = 1 fixed

Wigner crystallisation conjecture: in limit N →∞, the electrons place themselveson a BCC lattice (hexagonal lattice in 2D)

hexagonal lattice is lowest among all Bravais lattices in 2D (Rankin, Cassels,Ennola, Diananda ’50s-60s)

not yet known that BCC is lowest among all Bravais lattices

large mathematical literature in analytic number theory. Epstein Zeta fn:

ζL (z) =1

2

∑`∈L\0

1

|`|z

If Wigner is right, then eJel = ζBCC(1) ' −1.4441

Wigner Phys. Rev. 46 (1934), Coldwell-Horsfall & Maradudin, J. Math. Phys. 1 (1960), Blanc & Lewin, EMS Surveys in Math. Sci. (2015)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 12 / 15

Classical Jellium and Exc

EJel(xj ,Ω) =∑

1≤j<k≤N

1

|xj − xk |−

N∑j=1

ˆΩ

dy

|xj − y |+

1

2

ˆΩ

ˆΩ

dx dy

|x − y |

Main idea: for Exc(ρ), ρ = ρP plays the role of a background for P

Q

For x1, ..., xN , N distincts points of the lat-tice, define the Monge state

P =

ˆQ

δx1+τ ⊗s δx2+τ · · · ⊗s δxN +τ dτ

which has ρP = 1Ω with Ω = ∪Nj=1(xj + Q)

Exchange-correlation energy of this P

Exc(P) =∑

1≤j<k≤N

1

|xj − xk |−1

2

ˆΩ

ˆΩ

dx dy

|x − y |

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 13 / 15

The controversy

The long range of the Coulomb potential is very dangerous!

Theorem ([LewLie-15])

Q = Wigner-Seitz cell of a lattice L, with no dipole and no quadrupole, |Q| = 1

Exc(P)− EJel(xj ,Ω)

N→

0 (2D)ˆR3

(1

|x |−ˆ

Q

dy

|x − y |

)dx =

2π

3

ˆQ

x2dx (3D)

in contradiction with 20 years of DFT

BCC lattice still has the lowest xc energy: eunif ≥ 0.9507

we do not understand anymore if there is a link between eunif and eJel

in 1D with −|x |, the calculation is exact: eunif = eJel + 1/12

Borwein, Borwein, Shail & Straub, J. Math. Anal. Appl. 143 (1989), 414 (2014)

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 14 / 15

Conclusion

I Many interesting mathematical problems in DFT

I Large-N limit of a multi-marginal OT Coulomb problem

I To date, the best known estimates are:

trial states theory

Lieb-Oxford (general ρ)

(theory) 1.23

(numerics) 1.40≤ eLO ≤ 1.64

Uniform Electron Gas (ρ ≡ct)

(theory) 0.95

(numerics) 1.33≤ eunif ≤ 1.45

I Any improvement welcome in quantum chemistry

I Many other interesting questions:

gradient corrections to uniform case

quantum case and kinetic energy density

...

Mathieu LEWIN (CNRS / Paris-Dauphine) DFT & Lieb-Oxford 15 / 15