Electronic Structure Calculation - thebackbone of computational material science

Reinhold Schneider, MATHEON TU Berlin

JvNeumann colloquium TU Munich 2012

Aim: Advanced MaterialsNumerical simulation of material properties on atomic and

molecular scale

In 24-06-2011 US presidentObama launched theMaterial Genome Initiativefor global competitiveness(100 Mio. $ p.a.)

... Accelarating the pace of discovery and development of advanced material systems

will crucial to achieving global competivenes in 21st century. ... to discover, to develop,

manufacture and deploy advanced materials at least as twice as fast as possible today,

at a fraction of cost ..

... Rapid advances in computational modeling .. and moreadvanced algorithms for modeling materials behaviour must bedeveloped to supplement physical experiments...

Material engineering – Molecular engineering

Physics of material systems —– atomic and molecularproperties,

almost correct (nonrelativistic) phyiscs is quantum mechanics⇔ N particle Schrodinger eqn.

(Hege et al. ZIB Berlin)

Ab initio electronic structure calculation: predict the behaviour of molecular systemsfrom first principles of quantum mechanics, i.e. El. Schr. equation

Physically valid basic model, except relativistic and non Born Oppenheimer effects

Electronic Schrodinger EquationAcknowledgement: Thanks to H.J. Flad

Electronic Structure Calculation

Electronic Schrodinger eqn.

N′ stationary nonrelativistic electrons +

Born Oppenheimer approximation

HΨ = EΨThe Hamilton operator (in atomic units)

H = −12

∑i

∆i −N′∑i

K∑ν=1

Zν|xi − aν |

+12

N′∑i 6=j

1|xi − xj |

acts on anti-symmetric wave functions Ψ ∈ H1((R3 × ±12)

N′),

Ψ(x1, s1, . . . , xN′ , sN′) ∈ R , (xi , si) ∈ R3 × ±12 .

Output: ground-state energyE0 = min〈Ψ,Ψ〉=1〈HΨ,Ψ〉 , Ψ = argmin〈Ψ,Ψ〉=1〈HΨ,Ψ〉

I most quantities for molecules (chemistry) andcrystals (solid state physics) can be derived from E0, e.g.bonding and ionization energies or energy potential surfaces,atomic forces and optimal molecular geometry,

Fj = −∇aj

(E0(a1, . . . ,ak ) +

∑l<m

ZlZm

|al − am|

)I usually these quantities are (small) differences E0,a − E0,b

beyond ground state calculationI excited statesI linear response and material propertiesI electron-dynamics thermo dynamics open quantum systemsI ...

Facts to know

The ES has been welll studied in Analysis or MathematicalPhysics

I Kato, ... (.. 60..), The energy space is : V : H1((R3×±12)

N′)i.e. the Hamilton operator maps H : V → V ′ boundedly.

I HVZ-Theorem ( ..60..), E0 is an eigenvalue of finitemultiplicity: −∞ < E0 < infσess(H) if N ′ ≤ Z :=

∑Kν=1 Zν

I Agmon ( .. 70 .. ), exponential decay at infinity:Ψ(x) = O(e−a|x|) if |x| → ∞.

I Kato, T.- von Ostenhoff & T. Soerensen ... (98),cusp-singularities:e.g. electron-nucleon (e-N) cusp O(|xi − aν |)and electron-electron (e-e) cusp O(|xi − xj |)

I Yserentant (03) mixed regularity Ψ ∈ H1,s, s ≤ 12 , resp. 1

Basic Problem - Curse of dimensionalityI linear eigenvalue problem, but extremely high-dimensionalI + anti-symmetry constraints + lack of regularity.I only single and two particle operators

Curse of dimensionality: complexity n = O(eαN)

Traditional approximation methods, eg. FEM, Fourier series, polynomials, wavelets

etc.)I approximation error in R1: . n−s , s- regularity

I , R3N′ : app. error . n−s3N′ , (s < 5

2 ) with n DOFsFor large systems N′ >> 1 ( N′ > 1) the electronic Schrodinger equation seemsto be intractable!

70 years of impressive progress has been awarded by the Nobel price 1998 in

Chemistry: Kohn, Pople

Dilemma: Accuracy – versus – Complexity ( macroscopicN ∼ 1023 !!!)

Example: small energy scalesO2 binding to hemoglobin modeled by a Fe-porphyrin complex(heme)

| | | | |6.3e-2 mEh6.3e-16.3Rea tion barrierBinding energy

6.3e46.3e5O2heme

1Eh ≈ 27,2114eV Hartree

Wave function methods versus DFT

The computation of the electronic structure is extremelyexpensive, but NOT impossible,

I wave function, direct numerical solution of ES equation,Complexity O(N5−7) instead of O(eN)

I Quantum Monte Carlo methodsI density functional theory : reduction to nonlinear 3d PDE’s,

Complexity O(N2dimVh) ∼ O(N3)

Roughly 100 to 10.000 times more expensive than e.g. computational mechanics

0.001 eV

Intermolecular interactions

Magnetic couplings

Atomic cores

1 eV 10 eV0.1 eV0.01 eV

Chemical bonds

CC QMC DFT

Conclusions

Dilemma: Accuracy – versus – Complexiiy ( N >> 10 !!!)

Consequence for material science:I If possible, electronic structure computation should be

repaced by simplified modelsBUT:

I neverthelss, the electronic Schrodinger equation is thephysically valid basic model

I all scales (e..g of accuracy, system size etc.) are importantI multi-scale modeling,I combination of ab initio computation with modern statistical

tools, e.g. machine learning (TUB + FHI + Argonne Nat.Lab.)

Challenges in electronic structure calculation

I higher accuracy, in particular for strongly correlatedsystems

I time-dynamics and spectral propertiesI accuracy of wave function methods or Monte Carlo, with

costs ≈ DFTI large systems, i.e. low order scalingI systematic multi-scale modeling

in natural sciences, improvement of numerical approximation,whis is essential for further success is often NOT considered asscience!

Low dimensional models

There are low dimensional quantities characterizing thesolution of ES

1. electron density: r 7→ n(r), t ∈ R, Kohn Hohenberg:Density Functional Theory

2. reduced density matrices: p = 1 : γ(r, r′) reduced densitymatrix functional theory

3. (single particle) Greens functions G(r, r′, ω), (Greensfunctions functional theory), many body perturbationtheory (MBPT), e.g GW or dynamical mean field theory

in between DFT and MBPT:I linear response theory: TDDFT (time-dependent DFT) or

RPA (random phase approximation)

Full electron DFT calculation

with the LAPW (linearized augumentedplane waves)

joint work with H. Chen,

Effective single particle models - DFT and

I Closed Shell Restr. HF (RHF) or Density Functional theoryN := N′

2 number of electron pairs (spinfree formulations)

I minimization of the energy functional JKS(Φ)

JKS(Φ) =

∫

12

N∑i=1

|∇φi |2 +

∫nVcore +

12

∫ ∫n(x)n(y)

|x − y |dx dy − αExc(n)

I w.r.t. orthogonality constraintsΦ = (φi)

Ni=1 ∈ H1(R3)N and

⟨φi , φj

⟩= δi,j

I φi ∈ H1(R3), electron density n(x) :=∑N

i=1 |φi(x)|2

Canonical Kohn Sham equation→ Effective

Schrodinger equation

Necessary condition→ Kohn Sham equations

HΦφi = λiφi , λ1 ≤ λ2 ≤ · · · ≤ λN ,

where Φ = φ1, · · · , φN for λ1 ≤ . . . λN < . . ., with

HΦ = −12

∆+Vext +VH +Vxc = −12

∆+Veff : H1(R3)→ H−1(R3) .

Periodic systems - bulk crystals

Simple Periodic system - unit cell Ω = [−L2 ,

L2 ]3, cubic lattice

R = LZ3, and reciprocal lattice R∗ = 2πL Z3.

For k ∈ R∗, we set ek(x) = |Ω|−1/2eik·x (plane wave)L2

#(Ω) = u ∈ L2loc(R3) : u is R−periodic.

Bloch - Floquet theory

φ(x) = ek(x)ϕk(x) , ϕk ∈ H1#(Ω) periodic boundary conditions

satisfying ∀ ν = 1, . . . ,N, k ∈ Z3

e−ik·xHΦeik·xϕk,ν =(− 1

2(∇− ik)2 + Vext (x)

)ϕk,ν = λk,νϕk,ν = 0

Results: Numerical Analysis

Basis functions: plane waves (e.g. ABINIT, VASP, etc.), wavelets (e.g. BigDFT), FVMor FEM etc. ....require smooth pseudo potentials - valence electron computationa priori analysisI plane waves: Cances, Cahkir, Maday: Numerical analysis of the planewave

discretization of orbital-free and Kohn-Sham models (2011)I FEM: A. Zhou , X. Gong, H. Chen: Numerical approximations of a nonlinear

eigenvalue problem and applications to a density functional model. see talk A.Zhou

I orbital free DFT: B. Langwallner, C. Ortner, E. Sueli (2011)

All electron calculation and LAPW

All electron calculations are difficult due to singular Coulomb potentials:⇒ ϕ ∼ e−ax ∼ (1− ax + . . .), x→ 0Basis functions: Gaussian basis function (standard in quantum chemistry- bulk systemCRYSTAL), numerical basis functions (SIESTA, FHI-AIMS) and LAPW (bulk crystal)Reasons all electron calculationI for high accuracyI for certain elements, e.g. oxygen and transition metalsI for post DFT or post HF methods: e.g. GW (bulk crystal) , RPA or Coupled

Cluster in quantum chemistry

LAPW history and codesI APW (augumented plane waves) Slater (1937)I LAPW (linearized augumented plane waves), O. Andersen (1979)I LMTO (local muffin tin orbitals)I Wien2k, Excite, Elk, FLEUR (Julich), (most of them are commercial)

LAPW - H. Chen & S.

Domain decomposition: Ω is divided into intra-atomic spheresand an interstitial regions

&%'$

Ωout

Ωin &%'$

&%'$

Ωout

C1 C2

The institial region – plane waves,the atomic regions inside the spheres – spherical coordinates.For sake of simplicity, we explain it for a single atom and k = 0.

We do not follow the description in physics!

Denote by PK (Ωout ) the space of functions in Ωout expanded by plane waves

PK (Ωout ) =

u ∈ H1#(Ωout ), u(r) =

∑|k|≤ 2π

L K

ckek(r)

BNL the space of functions in Ωin expanded by radial basis functions

BNL(Ωin) =

u ∈ H1(Ωin), u(x) = w(r , θ, φ) =∑

n≤N,l≤L,|m|<l

cnlmχn(r)Ylm(θ, φ)

,

where χnNn=0 is the basis functions on [0,R],

In practise one chooses selected eigenfunctions of radial SE. For theory we choosesimply polynomials of degree no greater than N!

SKNL(Ω) = PK (Ωout )⊕ BNL(Ωin) =

u ∈ L2

#(Ω), u|Ωin ∈ BNL and u|Ωout ∈ PK

.

We introduce a (number) parameter ρ

maxK ,N, L ≤ % ≤ C minK ,N, L.

Discontinuous Galerkin - (less efficient)

Find λDG ∈ R and uDG ∈ SKNL, such that ‖uDG‖L2

#(Ω) = 1 and

aDG(uDG, v) = λDG(uDG, v) ∀ v ∈ SKNL,

where the DG bilinear form aDG : SKNL × S

KNL → R is defined by

aDG(u, v) =

∫Ωin

(12∇u∇v + Vext uv) +

∫Ωout

(12∇u∇v + Vext uv)

−12

∫Γ∇u · [v ]ds −

12

∫Γ∇v · [u]ds +

∫Γσ[u] · [v ]ds, (1)

where σ = Cσ% is the discontinuity-penalization parameter.with the jumps and averages, e.g.

[u] = u+n+ + u−n− , u =12

(u+ + u−)

Mortar formulationIntroducing Lagrange multiplier space

ML = spanYlm, 0 ≤ l ≤ L, |m| ≤ l

b : SKNL × L2(S2) , b(v , ψ) =

∫Γψ(v+ − v−)dΓ

with v± the traces of v from inside and outside the sphere.

a(uLAPW , v) + b(Ψ,uLAPW )− λ(uLAPW ,w) = 0 , ∀w ∈ SKNL ,

subordinated to the constraints b(Ψ,u) = 0, ∀Ψ ∈ ML, where

a(u, v) =

∫Ωin

(12∇u∇v + Vextuv) +

∫Ωout

(12∇u∇v + Vextuv)

b(Ψ,u) = −12

∫Γ

Ψ · (u+ − u−)ds ,

Modified LAPW basis functionsUsing scattering expansion eik·r = 4π

∑lm

i l jl(kr)Y ∗lm(k)Ylm(r).

ωk(r) =

|Ω|−

12 e−ik·r in Ωout ,

L∑lm

βklmχ0(r)Ylm(r) in Ωin,

where the coefficients

βklm = 4πi l jl (kR)Y∗lm(k)/χ0(R) (2)

Figure: Radial basis function. Left: χi (r) (1 ≤ i ≤ N). Right: χ0(r).

Broken Sobolev norms

Therefore, constraint condition could be eliminated explicitely,obtaining the nonconforming method

a(uLAPW , v)− λ(uLAPW ,w) = 0 , ∀w ∈ Vδ ,

Vδ =

vδ ∈ SKNL : b(vδ, ψ) = 0, ∀ ψ ∈ ML

,

Analysis in broken Sobolew norm

‖u‖2H1DG

= ‖u‖2H1(Ωin) + ‖u‖2H1(Ωout )+ σ‖[u]‖2L2(Γ),

LAPW : ‖u‖2H1d

= ‖u‖2H1(Ωin) + ‖u‖2H1(Ωout ).

Matrices

Matrix coefficients can be computed precisley also on theexterior domain. e.g. mass (overlap) matrix p,q ∈ PK (Ωout ),

Mapq =

1|Ω|

∫Ωout

e−i(kq−kp)·r = U(kp − kq),

where U(k) is the Fourier transform of char. fct. of the unit ball

U(k) =

|Ωout |/|Ω| if k = 0,

−4πR2j1(kR)/(k |Ω|) if k 6= 0.

Mass (overlap) matrices are often ill conditioned. All arisingmatrices are full.

Regularity

For first analysis, we consider an effective Schrodingereqn.,(−1

2∆ + Veff )u = λu, with an effective potential

Veff =−Z|x|

+ vs(x) with vs ∈ C(Ω).

Weighted Sobolev space with index γ on Ωin

Ks,γ(Ωin) =

u : |r |α−γ∂αu ∈ L2(Ωin) ∀ |α| ≤ s.

Asymptotics of u for r → 0, (Mellin calculus (Kontratiev .... ) )

u(x) = w(r , θ, φ) ∼∑

j

mj∑k=0

cjk (θ, φ)r−pj lnk r , (3)

where cjk belong to finite dimensional subspaces Lj ⊂ C∞(S2)

Regularity

Lemma (Flad, S.,Schultze (2008) (Hartree-Fock))The eigenfunctions ui are asymptotically well behaved, i.e.k = 0, pj ∈ N, Let ui(x) = w(r , θ, φ), then and u admits theasymptotic smoothness property

|∂βx u(x)| . |x |1−|β| for x 6= 0 and |β| ≥ 1.

w ∈ Hs([0,R]× S2) , ∀s ∈ R .

In Ω ⊂ R3 there holds

‖ui‖H1(Ω) ≤ ‖w‖H1(Ω×S2) .

i.e. w ∈ C∞([0,R]× S2), is w analytic? Fournais, Soerensen,

Thomas-Ostenhoff (2009)

Approximation properties in Ωin

Inside the sphere we use spherical coordinates

PropositionLet u(x) = w(r , θ, φ)(spherial coordinates), ifw ∈ Hs([0,R]× S2), then for any s ∈ R, there exists a constantC such that

infU in

NL∈BNL(Ωin)‖u − U in

NL‖H1(Ωin) ≤ C(L1−s + N1−s)‖w‖Hs([0,R]×S2). (4)

If uNL ∈ BNL(Ωin), then for 0 ≤ a < b ≤ R and r ≤ s, we have

‖wNL‖Hs([a,b]×S2) ≤ C(Ns−r + Ls−r )‖wNL‖H r ([a,b]×S2]). (5)

Approximation (and inverse properties) in Ωout

In the interstitial region we use plane waves.

PropositionIf u ∈ Hs(Ωout ), then for any s ∈ Z+ and r < s, there exists aconstant C such that

infUout

K ∈PK (Ωout )‖u − Uout

K ‖H r (Ωout ) ≤ CK r−s‖u‖Hs(Ωout ). (6)

If uK ∈ PK (Ωout ), then for r < s the following inverse estimateholds

‖uK‖Hs(Ωout ) ≤ CK s−r‖uK‖H r (Ωout ). (7)

Spectral convergence - effective Schrodinger equation

Theorem (Convergence rate )Let (λi ,ui) be an eigenpair of the effective Schrodinger eqn.. IfCσ and K ,N,L are sufficiently large, then there exists aneigenpair (λDG

i ,uDGi ) of the DG scheme, s. t. for any s > 1,

|λDGi − λi |+ ‖uDG

i − ui‖DG ≤ Cs%1−s‖ui‖Hs . (8)

Under the above assumptions the error of mod. LAPW solution(λi,δ,ui,δ) can be bounded by

|λi − λi,δ|+ ‖ui − ui,δ‖H1d≤ Cs,i%

1−s+1/2‖ui‖Hs .

% = %(KNL) ∼ maxK ,L,N, for any s > 1. (not optimal! improvable)

Spectral convergence

Scetch of proof:

Standard numerical analysis (theory) implies:Error ‖u − uδ‖H1

d∼

∼ error of best approximation infvδ∈SK N,L

‖u − vδE‖H1d

++ consistency errors

Consistency: Although uδ is not continuous, discrete operatorsare consistent as ρ→ E∞.Remark: Optimal bounds would inlcude |λ− λδ|2 and %1−s.

Spectral convergence - Kohn Sham equations

Assumptions: LDA: Exc(n) =∫

Ω E(n(x)

)dx

1. There ex. α ∈ (0,1] s.t. |E(2)(t)|+ |tE(3)| ≤ 1 + tα−1

2. The bilinear formaδ(Λ,Φ)(Ψ, Γ) := J (2)(Ψ)(Ψ, Γ)−

∑i,j

〈λi,jψj , γi〉 sat.

aδ(Λ,Φ)(Ψ,Ψ) ≥ C‖Ψ‖2(H1

d )N ∀Ψ .

Theorem (Convergence rate )Under the above assumptions the error of mod. LAPW solution(Λδ,Φδ) can be bounded by

|Λ− Λδ|+ ‖Φ− Φδ‖H1d≤ Cs%

1−s+1/2‖Φ‖Hs .

% = %(KNL) ∼ maxK ,L,N, for any s > 1. (not optimal! improvable)

Numerical results

100 10110−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

truncation of planewaves K2

num

eric

al e

rrors

planewavesR=1.0R=3.0

0 2 4 6 8 10 12 14 16 18 2010−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

truncation of planewaves K2

num

eric

al e

rrors

R=0.5R=1.0R=2.0R=3.0R=4.0

y=0.5e−0.2x

Figure: Comparing numerical errors of planewave methods andnonconforming methods.

Numerical results

100.3 100.4 100.5 100.6 100.7 100.8 100.910−4

10−3

10−2

10−1

100

|k|<Nc

eige

nval

ue e

rror

planewave methodslope=−3.0DG method (R=2.0)DG method (R=3.0)

Figure: Comparing numerical errors of plane waves and DG methods.

Numerical results -LAPW

0 2 4 6 8 10 1210−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

degree of polynomials

num

eric

al e

rrors

R=1.0R=4.0

Figure: Numerical errors of the nonconforming approximations withrespect to the order of polynomials radial basis with fixed K = 6.

Numerical results

1 2 3 4 5 6 7 8 9 1010−7

10−6

10−5

10−4

10−3

10−2

10−1

100

degree of radial basis

eige

nval

ue e

rrors

polynomialsy=exp(−1.5x)slater orbitals

Figure: Comparison of polynomials radial basis pi and Slater typebasis e−

Z2 |Ex|pi

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

radial distance (a.u.)

wave

func

tion

(a.u

.)

k=1

k=2

k=3

k=4

planewave k=8

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

x

eige

nfun

ctio

ns

R=2.0, N=5, K=4planewave K=5

Figure: Radial wavefunctions obtained by plane waves and DGdiscretizations.