Reduced Density Matrix Functional Theory for Many Electron Systems

RDMFTNumerics

Reduced Density Matrix Functional Theory forMany Electron Systems

S. Sharma1,2, J. K. Dewhurst3, N. N. Lathiotakis2,4

and E. K. U. Gross2

1 Fritz Haber Institute of the Max Planck Society, Berlin, Germany2 Institut fur Theoretische Physik, Freie Universitat Berlin, Germany

3 School of Chemistry, The University of Edinburgh, Scotland4 Theoretical and Physical Chemistry Institute, The National Hellenic Research

Foundation, Greece

24 July 2008

S. Sharma Reduced Density Matrix Functional Theory

RDMFTNumerics

Many-body quantum theory

Begin with the Schrodinger equation for N particles:

HΨi(x1,x2 . . . ,xN ) = EiΨi(x1,x2, . . . ,xN )

x ≡ {r, σ}

H = −12

N∑i

∇2i +

N∑i

N∑j 6=i

1|ri − rj |

+N∑i

vext(ri)

vext(ri) = −M∑ν

Zν

|Rν − ri|


RDMFTNumerics

Many-body quantum theory

Many-body theory involves solving for functions in 3N coordinates:Ψ(r1, r2, . . . , rN )

Element of a anti-symmetrised N -body Hilbert spaceH ⊗H ⊗ · · · ⊗H

Neon atom (10 electrons) wavefunction stored with 100 points ineach dimension would require ∼ 106 tons of DVDs for storage


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Reduced density matrix functional theory

Density

ρ(r) = N

∫Ψ(r, r2, . . . , rN )Ψ∗(r, r2, . . . , rN ) d3r2 . . . d

3rN .

E[ρ] = T [ρ] + Eext[ρ] + Eee[ρ]

One-reduced density matrix (1-RDM)

γ(r, r′) = N

∫Ψ(r, r2, . . . , rN )Ψ∗(r′, r2, . . . , rN ) d3r2 . . . d

3rN .

E[γ] =∫d3r′d3rδ(r− r′)

[−∇

2

2

]γ(r, r′) + Eext[γ] + Eee[γ]


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Density

ρ(r) = N

∫Ψ(r, r2, . . . , rN )Ψ∗(r, r2, . . . , rN ) d3r2 . . . d

3rN .

E[ρ] = T [ρ] + Eext[ρ] + Eee[ρ]

One-reduced density matrix (1-RDM)

γ(r, r′) = N

∫Ψ(r, r2, . . . , rN )Ψ∗(r′, r2, . . . , rN ) d3r2 . . . d

3rN .

E[γ] =∫d3r′d3rδ(r− r′)

[−∇

2

2

]γ(r, r′) + Eext[γ] + Eee[γ]


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One-Reduced density matrix functional theory (1-RDMFT)requires solving for functions in 6 coordinates: γ(r, r′)

Diagonalising the density matrix gives the natural orbitals andoccupation numbers

γ(r, r′) =∑

i

niφi(r)φ∗i (r′)

∫γ(r, r′)φi(r′)d3r′ = niφi(r)


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Gilbert’s Theorem [PRB 12, 2111 (1975)] (HK for 1-RDM)

Total energy is a unique functional E[γ] of the 1-RDM

Ground-state energy can be calculated by minimizing

F [γ] ≡ E[γ]− µ

[∫γ(r, r) d3r −N

]Must ensure that γ is N -representable!Hence an additional constraint that 0 ≤ ni ≤ 1Proof by A. J. Coleman [RMP 35, 668 (1963)]


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Three major differences from DFT

Exact kinetic-energy functional is known explicitly

T [γ] = −12

∫δ(r− r′)∇2γ(r, r′) d3r d3r′

so no kinetic energy in Exc (≡ Eee − EH)

There exists no Kohn-Sham system reproducing the exact γ(because γKS is idempotent)

There exists no variational equation

δF [γ]γ(r, r′)

= 0


RDMFTNumerics

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No simple variational equation

N -representability condition 0 ≤ ni ≤ 1 leads to border minimum

One can still minimise but δF [γ]δγ(r,r′) 6= 0 at minimum


RDMFTNumerics

functionalsresults

Functionals

Given the exact 1-RDM we can compute the exact kinetic energy,Hartree energy and external potential energy explicitly

BUT the exchange-correlation interaction energy is an implicitfunctional of γ

Bother.


RDMFTNumerics

functionalsresults

Functionals

Given the exact 1-RDM we can compute the exact kinetic energy,Hartree energy and external potential energy explicitly

BUT the exchange-correlation interaction energy is an implicitfunctional of γ

Bother.


RDMFTNumerics

functionalsresults

Functionals of γ

Conditions to be satisfied by approximate functionals

xc hole integrates to -1

Lieb’s Conjecture: for Coulomb systems E(N) is upwardlyconvex

Lieb Oxford bound Exc ≥ −1.68∫ρ4/3(r)d3r


RDMFTNumerics

functionalsresults

Hartree-Fock

Exc[γ] = −12

∫|γ(r, r′)|2

|r− r′|d3r d3r′

xc hole integrates to -1 but E(N) is not upwardly convex.


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Muller functional [PRL 105A, 446 (1984)]

Exc[γ] = −12

∫(γp(r, r′))∗γ1−p(r, r′)

|r− r′|d3r d3r′ (p = 1/2)

xc hole integrates to -1 and energy is upward convex. Lieb-Oxfordbound violated for the homogeneous electron gas.


RDMFTNumerics

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Our functional (Sharma et al. cond-mat/0801.3787)

Exc[γ] = −12

∫|γα(r, r′)|2

|r− r′|d3r d3r′ (0.5 ≤ α ≤ 1)

Energy is upward convex. Lieb-Oxford bound satisfied for thehomogeneous electron gas. But xc hole does not integrate to -1.


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Results

Homogeneous electron gas

Finite systems

Solids


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functionalsresults

Results for the homogeneous electron gas


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functionalsresults

Results for the homogeneous electron gas


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Correlation energy for atoms and molecules


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Mott insulators

Solid TN Band gap

MnO 118 3.7FeO 198 2.5CoO 292 2.4NiO 523 4.1

Table: Neel temperature (in K) and band gap (in eV) above Neeltemperature for transition metal oxides.


RDMFTNumerics

functionalsresults

Band gaps for solids


RDMFTNumerics

Numerical issues

Full-potential linearised augmented planewaves (FP-LAPW)

potential is fully described without any shape approximation

core is treated as Dirac spinors and valence as Pauli spinors

space divided into interstitial and muffin-tin regions

this is one of the most precise methods available

MT

I

II

I


RDMFTNumerics

Numerical issues






MT

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RDMFTNumerics

Numerical issues






MT

I

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RDMFTNumerics

Numerical issues






MT

I

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RDMFTNumerics

Numerical issues






MT

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RDMFTNumerics

Natural-orbital minimisation

E[γ] = E[ni,Φi]

Self-consistent Kohn-Sham calculation is performed (anyfunctional can be used) : ψKS

j (r)1 Natural orbitals are expanded in Kohn-Sham wave functions

ΦRDMi (r) =

∑j

cij ψKSj (r)

2 Compute gradients of the total energy w.r.t. cij3 Use steepest-descent along the gradient cij → cij + λdE/dcij4 Use Gramm-Schmidt to orthogonalise the natural-orbitals

5 Goto step 1, or exit once convergence is achieved


RDMFTNumerics

Occupation number minimisation

Constraints:∑

i ni = N and 0 ≤ ni ≤ 1

Define gi(κ) ≡ dE/dni − κ and

gi(κ) ≡{gi(1− ni) gi > 0gini gi ≤ 0

1 Compute dE/dni

2 Find κ such that∑

i gi(κ) = 03 Make change in occupation number: ni → ni + λgi(κ), for

largest λ which keeps occupancies in [0, 1]4 Goto step 1, or exit once convergence is achieved


RDMFTNumerics

(q −→ 0 term)

Exchange correlation energyinvolves integrals:∫ Φ∗

ik(r)Φjk′(r)Φ∗jk′(r

′)Φik(r′)

|r− r′|d3rd3r′

Require integral over k-pointdifferences (q = k− k′):

I =∫

BZ

f(q)q2

d3q

Integrable but very slowconvergence w.r.t. k-points.

Precalculate weights usingmany q-points (millions)

Wq =∫

Bq

1q2

d3q,

then

I '∑q

Wqf(q).


RDMFTNumerics

Code used: EXCITING

J. K. Dewhurst, S. Sharma and E. K. U. GrossThe code is released under GPL and is freely available at:http://exciting.sourceforge.net/


RDMFTNumerics

Summary

RDMFT for periodic solids is implemented within a FP-LAPWcode.

Produces very good results for wide range of systems.

New algorithm for minimisation of energy with respect tooccupation numbers.


RDMFTNumerics

Atomisation energy for molecules


RDMFTNumerics

Band gap using chemical potential

Perdew et al. PRL 49 1691 (82), Helbig et al. EPL 77 67003 (07)

µ(M) =δE(M)δM

= −I(N) N − 1 < M < N

µ(M) =δE(M)δM

= −A(N) N < M ≤ N + 1

∆ = −A(N) + I(N) = µ(N + η)− µ(N − η)


RDMFTNumerics

Band gap for finite systems with exact functional


RDMFTNumerics

Band ap for LiH [EPL 77, 67003 (2007)]


RDMFTNumerics

Band ap for LiH [EPL 77, 67003 (2007)]


RDMFTNumerics

Band gap for solids

Sharma et al. cond-mat/0801.3787

µ(η) has discontinuity at η = 0 withEg = limη→0+(µ(η)− µ(−η)) being identical to exactfundamental gap.

In the vicinity of η = 0 one finds a linear behavior

µ(η) = µ(η = 0−) +{

clη for η < 0Eg + crη for η > 0

with cl = 2∫ n−(r)

|r−r′|d3rd3r′ and cr = 2

∫ n+(r)|r−r′|d

3rd3r′


RDMFTNumerics

Band gap for finite systems with exact functional


RDMFTNumerics

Chemical potential for solids


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Reduced Density Matrix Functional Theory for Many Electron Systems