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Project P9: Energies and forces for materials with strong correlations

Project P9:Energies and forces for materials with strong

correlations

Robert Schade1, Ebad Kamil2

1Clausthal University of Technology, Institute for Theoretical Physics2Georg-August-University Göttingen, Institute for Theoretical Physics

11.02.2015-12.02.2015

Robert Schade1, Ebad Kamil2 Project P9: Energies and forces for materials with strong correlations

Project P9: Energies and forces for materials with strong correlationsMotivation and main goal

Motivation and main goal1 ground-state total energy calculations beyond

existing XC-functionals2 structure relaxations3 ab-initio molecular dynamics

→ approach: construct corrections to xc-functionalsbased on reduced density matrix functional theory(rDMFT):

F Wβ (ρ(1)) := max

hα,β

[ΩWβ (h)− Tr

(hρ(1)

)]i.e. a Legendre-Fenchel transform of the grandpotential ΩW

β,µ(h) with respect to the one-particleHamiltonian hα,β

FOR 1346


Project P9: Energies and forces for materials with strong correlationsOutline

1 DFT+rDMFT

2 Ways to evaluate/approximate the rDMF in practiceVia many-particle wave functionsVia Green functions

Greens functions


Project P9: Energies and forces for materials with strong correlationsDFT+rDMFT

DFT+rDMFT:

E0(N) ≈ min|φn〉,fn∈[0,1]

statµ,Λnm

(EDFT [|φn〉, fn]

+(F W

HF [ρ(1)]− F WDFT ,DC [ρ(1)]

)︸︷︷︸

hybrid functional

+(F

ˆW [ρ(1)]− FˆW

HF [ρ(1)])

︸︷︷︸high−level correction

+

− µ

(∑n

fn − N

)︸︷︷︸

particle−number constraint

−∑i,j

Λi,j (〈φi |φj〉 − δi,j)︸︷︷︸orthonormality constraints

)

with ρ(1)α,β =

∑n

〈χα|φn〉fn〈φn|χβ〉

implemented in PAW-formalism (CP-PAW, plane waves)natural tight binding orbitals as local orbital basisdouble counting via an approximate density-matrix functionalof DFT-xc-functional (Blöchl et al. [2011])



Simple example system: stretched hydrogen molecule H2

local orbitals: one s-orbital per atomdistance of atoms d = 10 Å

⇒ Hubbard-dimer with large Ut at half filling

ground-state can not be described by a single-determinantmethod

|ψ0,Ut →∞〉 =

1√2

(| ↑↓〉 − | ↓↑〉)

⇒ huge static correlation error in non-spin-polarized DFT




method E − E0 f1,2 f3,4 commentFCI (Orca,QZVPP) 0 mH E0 = −1 Hnon-spin-polarized 79 mH 0.000 0.915 static correlation errorspin-polarized 3 mH 0.000 1.000 broken spin-symmetryDFT+Müller-rDMF 2 mH 0.464 0.464DFT+exact rDMF 8 mH 0.494 0.506 correct occupations




method E − E0 f1,2 f3,4 commentFCI (Orca,QZVPP) 0 mH E0 = −1 Hnon-spin-polarized 79 mH 0.000 0.915 static correlation errorspin-polarized 3 mH 0.000 1.000 broken spin-symmetryDFT+Müller-rDMF 2 mH 0.464 0.464DFT+exact rDMF 8 mH 0.494 0.506 correct occupations

Occupations for a Hubbard dimer:

0

0.5

1

occupatio

nsf i

Ut


Project P9: Energies and forces for materials with strong correlationsWays to evaluate/approximate the rDMF in practice

Ways to evaluate/approximate the rDMF in practice:

1 Levy’s constrained search formalism (Levy [1979])2 Relation between density-matrix functional, Luttinger-Ward

functional and Greens functions (Blöchl, Pruschke,Potthoff [2013])

3 Parametrized functional approximations (e.g. Mueller,Sharma, Buijse/Baerends)



Via many-particle wave functions

Levy’s constrained search formalism (Levy [1979])

F Wβ (ρ(1)) = min

Pi ,|Ψi 〉stat

hα,β ,Λi,j ,λ

(∑i

Pi 〈Ψi |W |Ψi 〉︸︷︷︸interaction energy

+ kBT∑

i

Pi ln(Pi )︸︷︷︸−T ·entropy

−∑α,β

hα,β

(∑i

Pi 〈Ψi |c†αcβ|Ψi 〉 − ρ(1)β,α

)︸︷︷︸

density−matrix constraint

−∑i ,j

Λj ,i (〈Ψi |Ψj〉 − δi ,j)︸︷︷︸orthonormality constraint

− λ

(∑i

Pi − 1

)︸︷︷︸

probability constraint

)

⇒ nonlinear constrained optimization problem withNψ · (1 + 2Nχ+1) real variational parameters1 + N2

χ + N2ψ real equality constraints




Algorithms for solution:Car-Parrinello-method (Car and Parrinello [1985])Augmented Lagrangian-approach (Hestenes [1969], Powell[1969])

Complexity:number of orbitals: Nχ (→ 10-100)number of real variational parameters: 2Nχ+1 (→ 103 − 1030)

reduce complexity:1 restrict the space of Slater determinants:→ natural orbitals and CI-approximation

2 reduce number Nχ of local orbitals:→ local approximation and adaptive cluster approximation




1. CI-approximation and adaptive search spaceMain idea: transform the one-particle basis

natural orbitals improve convergence of CI-expansion

|ψ〉 =2Nχ∑i=1

ci |si 〉 = c1|000..〉+ c2|100..〉+ c3|010..〉+ c4|110..〉+ ...

≈M2Nχ∑

j=1

ci(j)|si(j)〉

do a basis transformation to natural orbitals

ρ(1) = U−1ρ(1)U ∈ diag(f1, ..., fNχ)




rDMF of the ground-state density-matrix of a 14-siteHubbard-chain with interaction on every site (U/t = 4)

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

100 1000 10000 100000 1e+06 1e+07 1e+08

1 UF

W(ρ

(1)

0)

number of Slater determinants

full space

HF-approx.CI (iterative construction of space)CI (singles and doubles of HF-det.)

exact result

good approximation with NCI ≈ 104 determinants(Nfull space = 108.4), but sublinear (logarithmic) convergence




2. Local approximation + adaptive cluster approximationMain idea: transform density-matrix with a unitary transformationbefore doing a cluster approximation

neglect coupling between effective bath and remaining system→ decoupling of impurity+effective bath and remaining system→ scheme with sub-exponential scaling in total system size




2. Local approximation + adaptive cluster approximationrDMF for ground-state density-matrix of the 14-site Hubbard chainat half filling and U/t = 4 with interaction only on the first site

0.045

0.05

0.055

0.06

0.065

0.07

0.075

1 2 3 4 5 6 7 8

1 UF

W1(ρ

(1)

0)

number of effective bath sites

full system

exact resultconventional cluster approximation

adaptive cluster approximation

⇒ monotonic convergence towards the exact result




3. Additional approaches under development/considerationwave function-decomposition via Schmidt-like decompositionuse variational wave functions with less variational parameters(e.g. Gutzwiller, Jastrow)



Via Green functions

1 DFT+rDMFT

2 Ways to evaluate/approximate the rDMF in practiceVia many-particle wave functionsVia Green functions

Greens functions



Via Green functions

Kadanoff-Baym Functional

The grand potential Ωβ,µ(h + W ) can also be expressed as afunctional of the Green’s function and the self energy,

Ωβ,µ(h + W ) = statG ,Σ

ΨKBβ,µ[G ,Σ,h, W ]

ΨKBβ,µ[G ,Σ, h, W ] = ΦLW

β [G , W ]−1β

∑ν

Tr

ln(

1−1

(i~ων + µ)1− hΣ(iων)

)+ Σ(iων)G(iων)

−

1βTr

ln (1 + −β(h − 1µ))

βδΦLWβ [G , W ]

δGb,a(iων)= Σa,b(iων), and G(iων) =

((i~ων + µ)1− h −Σ(iων)

)−1



Via Green functions

Density Matrix Functional (Blöchl P. E. Pruschke T. PothoffM.(2013) )

F Wβ [ρ] =

1βTr[ρ ln(ρ) + (1− ρ) ln(1− ρ)

]+ stat

h′statG ,Σ

ΦLWβ [G , W ]−

1β

∑ν

Tr

ln[1−

(G)−1(

h′ + Σ(iων)− h)]

+(h′ + Σ(iων)− h

)G(iων)−

[G(iων)−

(G)−1](

h′ − h)

.

where

ρ→ h[ρ] = µ1 +1βln[1− ρ

ρ

]→ G (iων) =

((i~ων + µ)1− h

)−1

ρ[G ] =1β

∑ν

e iβ~ων0+G (iων)



Via Green functions

Direct Density Matrix functional from Greens function

New framework for approximate scheme.

Steps

Define mapping. ρ→ G [ρ]

Error∆F W

β [ρ] ∼ 1n

(G [ρ]− G true)n n ≥ 2

Most simple attemptNon interacting Greens function.

G [ρ] = G (iωn) =(

(i~ων + µ)1− h)−1

NDE2 density matrix functional

F Wβ [ρ] ' 1

βTr[ρ ln(ρ) + (1− ρ) ln(1− ρ)

]+ ΦLW

β [G [ρ], W ]



Via Green functions

Bubble Series

+ + +

~ ~ 2~ 3

FBubble =1β

∑iνn

Trln [I − Y (iνn)] + Y (iνn)

χi1o1,i3o3(iνn) =

∑n,m

〈πi1 |Φm〉〈Φm|πo3〉〈πi3 |Φn〉〈Φn|πo1〉fn − fm

iνn + εn − εm

Yi1o1,i2o2 =∑i3,o3

χi1o1,i3o3(iνn)Wi2i3,o2o3



Via Green functions

Application of NDE2 approximation to 1D HubbardModel

0 0,5 1 1,5 2 2,5 3

k

0

0,2

0,4

0,6

0,8

1

nk

Bubble

HF

Figure : 1D Hubbard model, with two orbitals (spin up and spindown) per unit cell for U = 0.5, β = 20 and 80 k points



Via Green functions

Understanding the breakdown

For for Uβ 1 energy calculation with Bubble approximationbreaks down.

Poles of NDE2 Greens function fall at the Fermi level in thelow temperature regime.

Gαβ(iων) =∑

n

〈πα|φn〉〈φn|πβ〉

iωn − 1β ln

(1−fnfn

)Band width proportional to kBT . Perturbation theorybreaksdown.

kBT

W



Via Green functions


For for Uβ 1 energy calculation with Bubble approximationbreaks down.Poles of NDE2 Greens function fall at the Fermi level in thelow temperature regime.

Gαβ(iων) =∑

n


iωn − 1β ln

(1−fnfn

)

Band width proportional to kBT . Perturbation theorybreaksdown.

kBT

W



Via Green functions


For for Uβ 1 energy calculation with Bubble approximationbreaks down.Poles of NDE2 Greens function fall at the Fermi level in thelow temperature regime.

Gαβ(iων) =∑

n


iωn − 1β ln

(1−fnfn

)Band width proportional to kBT . Perturbation theorybreaksdown.

kBT

W



Via Green functions

Two Pole Model

Construct a Greens function which avoids collapse of thespectrum to Fermi energy.

Two pole model for the Greens function.

Gαβ(iων) =∑n

〈πα|ψj〉〈ψj |πβ〉[

P1

iων − A1+

P2

iων − A2

]



Via Green functions

Two Pole Model

Construct a Greens function which avoids collapse of thespectrum to Fermi energy.Two pole model for the Greens function.

Gαβ(iων) =∑n

〈πα|ψj〉〈ψj |πβ〉[

P1

iων − A1+

P2

iων − A2

]



Via Green functions

Parametrized Density Matrix Functionals

Explore standard density matrix functionals for benchmarkingrDMFT drivers.

Look for Hartree-Fock like functionals i.e.

F Wβ = FH

β +∑αβγδ

Wαβ,γδ

∑m,n

c(fmfn)

[〈πδ|φm〉〈φm|πα〉〈πγ |φn〉〈φn|πβ〉

]

Standard functionals available,

c(fm, fn) =

−fmfn, Hartree Fock

− f12

m f12

n , Müller (1984), BBCn corrections by Baerands et.al (2005)− f αm f αn ,

12 < α ≤ 1, Sharma et.al. (2008)



Via Green functions


Explore standard density matrix functionals for benchmarkingrDMFT drivers.Look for Hartree-Fock like functionals i.e.

F Wβ = FH

β +∑αβγδ

Wαβ,γδ

∑m,n

c(fmfn)


]


c(fm, fn) =


− f12

m f12


12 < α ≤ 1, Sharma et.al. (2008)



Via Green functions



F Wβ = FH

β +∑αβγδ

Wαβ,γδ

∑m,n

c(fmfn)


]


c(fm, fn) =


− f12

m f12

n , Müller (1984), BBCn corrections by Baerands et.al (2005)

− f αm f αn ,12 < α ≤ 1, Sharma et.al. (2008)



Via Green functions



F Wβ = FH

β +∑αβγδ

Wαβ,γδ

∑m,n

c(fmfn)


]


c(fm, fn) =


− f12

m f12


12 < α ≤ 1, Sharma et.al. (2008)



Via Green functions

Results

0 1 2 3 4 5 6 7 8 9 10U

0

0.2

0.4

0.6

0.8

1

Occ

upat

ion

f1 MÜLLER

f3 MÜLLER

f1 α=0.525

f3

α=0.525

f1 α=0.600

f3

α=0.600



Via Green functions

0

0

2

4

6

8

10

0.6 0.7 0.8 0.9 1.00.5

U

0.10

0.15

0.20

0.25

0.05

NM

AF-

-

-

-

-



Via Green functions

Semi Analytical Results

At half filling Müller gives the exact occupation numbers. Naturalorbitals at the minima are always bonding and anti-bonding orbitals.

Besides non magnetic solution (symmetric) for Müller, there exist adegenerate set of magnetic solutions. These solutions lies on a line

U


Project P9: Energies and forces for materials with strong correlationsSummary

Summary:rDMF via Greens functions:NDE2 approximation. Breakdown for Uβ 1. Testingtwo-pole model of the Greens function.rDMF Parametrized Density matrix functionals:Exploring stadard DMF for benchmarking rDMFT drivers.DFT+rDMFT:working scheme with initial resultsrDMF via many-particle wave functions:CI-approximation, non-static search spaces and adaptivecluster approximation to reduce the complexity

We would like to thank Peter Blöchl, Thomas Pruschke and theDFG Research Unit FOR 1346 for their support.



References I

T. Baldsiefen and E. K. U. Gross. Reduced density matrix functional theory atfinite temperature. I. Theoretical foundations. ArXiv e-prints, Aug. 2012.

P. E. Blöchl, C. F. J. Walther, and T. Pruschke. Method to include explicitcorrelations into density-functional calculations based on density-matrixfunctional theory. Phys. Rev. B, 84:205101, Nov 2011. doi:10.1103/PhysRevB.84.205101. URLhttp://link.aps.org/doi/10.1103/PhysRevB.84.205101.

P. E. Blöchl, T. Pruschke, and M. Potthoff. Density-matrix functionals fromgreen’s functions. Phys. Rev. B, 88:205139, Nov 2013. doi:10.1103/PhysRevB.88.205139. URLhttp://link.aps.org/doi/10.1103/PhysRevB.88.205139.

R. Car and M. Parrinello. Unified approach for molecular dynamics anddensity-functional theory. Phys. Rev. Lett., 55:2471–2474, Nov 1985. doi:10.1103/PhysRevLett.55.2471. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.55.2471.


http://link.aps.org/doi/10.1103/PhysRevB.84.205101

http://link.aps.org/doi/10.1103/PhysRevB.88.205139

http://link.aps.org/doi/10.1103/PhysRevLett.55.2471


References II

M. Hestenes. Multiplier and gradient methods. Journal of Optimization Theoryand Applications, 4(5):303–320, 1969. ISSN 0022-3239. doi:10.1007/BF00927673.

M. Levy. Universal Variational Functionals of Electron Densities, First-OrderDensity Matrices, and Natural Spin-Orbitals and Solution of thev-Representability Problem. Proceedings of the National Academy ofScience, 76:6062–6065, Dec. 1979. doi: 10.1073/pnas.76.12.6062.

M. J. D. Powell. A method for nonlinear constraints in minimization problems.In R. Fletcher, editor, Optimization, pages 283–298. Academic Press, NewYork, 1969.

L. Sorber, M. Barel, and L. Lathauwer. Unconstrained optimization of realfunctions in complex variables. SIAM Journal on Optimization, 22(3):879–898, 2012. doi: 10.1137/110832124. URLhttp://dx.doi.org/10.1137/110832124.


http://dx.doi.org/10.1137/110832124


Algorithms: some requirements for constrained optimizationalgorithm:

globally convergent, robustno constraint feasibility problem in restricted CI-space

the derivatives ∂

∂ρ(1)a,b

F W (ρ(1)) must be accessible

only matrix-vector operations, no matrix-decompositionsgood warm start capability

Candidates:Car-Parrinello (Car and Parrinello [1985])Augmented Lagrangian (Hestenes [1969], Powell [1969])sequential linearized constraints (e.g. LCL, BCL)sequential quadratic programming (e.g. SQP, SLQP)



Algorithms: basicsbasis: CI-expansion of wave function (i.e. full CI resp. ED)

|ψ〉 =2Nχ∑i=1

ai |si 〉 = a1|000..〉+ a2|100..〉+ a3|010..〉+ a4|110..〉+ ...

problem to be solved:

min~x∈D

f (~x) with ~c(~x) = 0

Lagrange function:

min~x∈D

stat~λ

L(~x , ~λ) with L(~x , ~λ) = f (~x) +∑

i

λici (~x)



Car-Parrinello approach

Algorithms: Car-Parrinello (Car and Parrinello [1985])1 derive equations of motion of the system from a fictitious

Lagrangian

L(~x , ~λ) = Tfictitious(~x) + L(~x , ~λ)

2 integrate equations of motion with additional friction termuntil stationary state is reached→ local minimum

use Verlet algorithm for integrationconstraint enforcing: the Lagrange multipliers ~λ are modifiedso that the variational parameters ~x satisfy the constraints inthe next time step




Algorithms: behaviour of Car-Parrinello approach1 calculation in full space (FCI):

√

instabilities cased by constraint enforcing curedtime complexity: T ∈ O(number of determinants)memory complexity: M ∈ O(number of determinants)

2 calculation in subspace (CI):Xconstraint enforcing subproblem (system of multivariatepolynomial equations) causes major instabilities:

can get unsolvable (solved with graph search)can get ill-conditioned (not solved yet)constraint enforcing can yield solutions that are notadiabatically connected to the ∆t = 0 solution (not solved yet)→ no proper dynamics




Algorithms: properties of the Car-Parrinello approachglobally convergent

√

small memory requirements, only matrix-vector operations√

dynamic properties can be understood nicely√

variational (i.e. f (~xk) ≥ f (~x∗))√

good warm start properties√

efficient (with a good choice of the friction parameters)√

the constraints have to be fulfilled in every iteration X→ Maratos-effect X→ constraint feasibility problem X



Augmented Lagrangian approach

Algorithms: Augmented Lagrangian [Hestenes, Powell,Flecher (1969)]

1 add penalty term to the Lagrangian to enforce constraints

L(~x , ~λ, µ) = f (~x) +∑

i

λici (~x) +µ

2

∑i

(ci (~x))2

2 solve unconstrained subproblems for increasing values of thepenalty parameter µk

min~xk

L(~xk , ~λk , µk)

and update Lagrange multipliers λk+1 = λk − µkci (xk)

use CG or L-BFGS (generalized complex variant, Sorber et al.[2012]) for unconstrained subproblemsuse exact polynomial-based line-searches




Algorithms: behaviour of Augmented Lagrangian approachtime complexity:

T ∈ O(number of determinants)

memory complexity:

M ∈ O(number of determinants · (4 ·mLBFGS + const.))

1 calculation in full space (FCI):√

2 calculation in subspace (CI):√

〈i |∂L(|ψ〉,~λ,µ)∂〈ψ| is a very good measure for adding new Slater

determinants to the search space




Algorithms: properties of the augmented Lagrangianapproach

globally convergent√

∃µ <∞ so that ∀µ ≥ µ: the local minimizer of theconstrained problem ~x∗ is a local minimizer of L(~x , ~λ∗, µ)→ no serious ill-conditioning of the subproblems

√

subproblem solver:CG: linear convergenceLBFGS: locally superlinear convergence

no Maratos effect√

no constraint feasibility problem√

not variational Xin practice not as fast as CP, because of the iterated solutionsof the subproblems X




N = 10, Nχ=20, 2Nχ = 1.048.576, NN=10 = 184.756


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