DFT & Lieb-Oxford - Inria

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DFT & Lieb-OxfordMathieu LEWIN
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 ) using only its one-particle marginal ρ(x) on R3
I History:
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 computational resources + 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
A tremendous success
use about 15% of resources available in scientific computing centers
about 15 000 papers per year with the keyword ‘density functional theory’
12 of the 100 most cited papers in history of science are from DFT (2015)
Rank Authors Journal Year Citations 7 Lee, Yang, Parr Phys. Rev. B 1988 46 702 8 Becke J. Chem. Phys. 1993 46 145
16 Perdew, Burke, Ernzerhof Phys. Rev. Lett. 1996 35 405 25 Becke Phys. Rev. A 1988 26 475 34 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
( ∑ 1≤j<k≤N
|xj − xk |
) dP(x1, ..., xN )
P symmetric = probability density of N indistinguishable electrons in practice P = |Ψ|2 where Ψ = complex-valued quantum wavefunction
one-particle density: ρP(x) = N
C (ρ) = inf ρP=ρ
{ˆ R3N
P = 1
N
ˆ δy ⊗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
Extensions I Quantum case:
{ ~2
2
By scaling ρ 1 same as ~ 1
I Positive temperature classical case:
C T (ρ) = inf ρP=ρ
{ ˆ R3N
C T ~(ρ) = inf
{ tr
( ~2
2 (−) +
Exchange Correlation
I Chemists interested in general properties of C (ρ), valid for all N ≥ 1, that could be used to devise clever semi-empirical explicit functionals
C (ρ) = 1
dk:=D(ρ,ρ)
+Exc(ρ)
I Exc(ρ) is “almost local” and chemists like to use a local approximation
Exc(ρ) ' ˆ R3
Example: Perdew-Burke-Ernzerhof famous functional
1 3 (x)|2
( ρ(x)
) dx
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, that could be used to devise clever semi-empirical explicit functionals
C (ρ) = 1
dk:=D(ρ,ρ)
+Exc(ρ)
I Exc(ρ) is “almost local” and chemists like to use a local approximation
Exc(ρ) ' ˆ R3
Example: Perdew-Burke-Ernzerhof famous functional
1 3 (x)|2
( ρ(x)
) dx
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) ≤ − 1 N D(ρ, ρ) ≤ 0
I Replace Coulomb by smooth potential, still with positive Fourier transform:∑ 1≤j<k≤N
1
|xj − xk | = Da
N∑ j=1
2
For any fixed proba ν ∈ L1(R3) ∩ L6/5(R3),
N1/3C ( ν(·/N1/3)
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 = N1 where =unit cube, Dirac (1926) found
N1/3Exc(1N1/3) = Exc(N1) ≤ −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)
follows from monotonicity of Exc(1N1/3C )/N and previous lower bound
same limit if replaced by any other domain (e.g. a ball) with volume 1
exact value of eunif is unknown, although everybody thought for decades that eunif ' 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
Exc(ρ) ≥ −1.64
ˆ R3
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
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 of Solids ’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
) 1 3 (ˆ
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 with || = N
EJel(xj ,) = ∑
minxj EJel(xj ,N 1/3)
N , || = 1 fixed
Wigner crystallisation conjecture: in limit N →∞, the electrons place themselves on 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
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
ˆ
ˆ
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 =
which has ρP = 1 with = ∪N j=1(xj + Q)
Exchange-correlation energy of this P
Exc(P) = ∑
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 ,)
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 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) 0.95
I Many other interesting questions:
gradient corrections to uniform case
quantum case and kinetic energy density
...