Reduced Density Matrix Functional Theory forMany Electron Systems

S. Sharma1,2, J. K. Dewhurst1,2 and E. K. U. Gross2

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

12 June 2009

S. Sharma Reduced Density Matrix Functional Theory

Schrodinger equation for N particles:

H Ψi(x1,x2 . . . ,xN ) = Ei Ψi(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|

S. Sharma Reduced Density Matrix Functional Theory

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]

S. Sharma Reduced Density Matrix Functional Theory

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]

S. Sharma Reduced Density Matrix Functional Theory

Reduced density matrix functional theory

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!Proof by A. J. Coleman [RMP 35, 668 (1963)]

S. Sharma Reduced Density Matrix Functional Theory

Reduced density matrix functional theory

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)

S. Sharma Reduced Density Matrix Functional Theory

Set of all ensemble N -representable 1-RDM

Gives the necessary and sufficient N -representability conditions

γ is Hermitian

Tr γ = N

0 ≤ ni ≤ 1, γφi = niφi

S. Sharma Reduced Density Matrix Functional Theory

Set of all ensemble N -representable 1-RDM

Gives the necessary and sufficient N -representability conditions

γ is Hermitian

Tr γ = N

0 ≤ ni ≤ 1, γφi = niφi

S. Sharma Reduced Density Matrix Functional Theory

Set of all ensemble N -representable 1-RDM

Gives the necessary and sufficient N -representability conditions

γ is Hermitian

Tr γ = N

0 ≤ ni ≤ 1, γφi = niφi

S. Sharma Reduced Density Matrix Functional Theory

Set of all ensemble N -representable 1-RDM

Gives the necessary and sufficient N -representability conditions

γ is Hermitian

Tr γ = N

0 ≤ ni ≤ 1, γφi = niφi

S. Sharma Reduced Density Matrix Functional Theory

Reduced density matrix functional theory

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

S. Sharma Reduced Density Matrix Functional Theory

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

S. Sharma Reduced Density Matrix Functional Theory

Exchange-correlation functionals (T → 0)

DefineF [γ] ≡ inf

Γ(2)N →γ

Tr {Γ(2)N (T + Vee)}

Exc[γ] ≡ F [γ]− T [γ]− EH[γ]

Then

E[γ] = T [γ] +∫vext(r) γ(r, r) d3r + EH[γ] + Exc[γ]

Given the exact γ 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

S. Sharma Reduced Density Matrix Functional Theory

Muller functional, Phys. Rev. Lett. 105A, 446 (1984)

Exc[γ] = −12

∫γp(r, r′)∗γ1−p(r, r′)

|r− r′|d3r d3r′ (p = 1/2)

γp(r, r′) =∑i

npiφ∗i (r′)φi(r)

“Power functional”, Phys. Rev. B 78, R201103 (2008)

Exc[γ] = −12

∫|γα(r, r′)|2

|r− r′|d3r d3r′ (1/2 ≤ α ≤ 1)

Hartree-Fock functional

Exc[γ] = −12

∫|γ(r, r′)|2

|r− r′|d3r d3r′

S. Sharma Reduced Density Matrix Functional Theory

Muller functional, Phys. Rev. Lett. 105A, 446 (1984)

Exc[γ] = −12

∫γp(r, r′)∗γ1−p(r, r′)

|r− r′|d3r d3r′ (p = 1/2)

Γ(r1, r2, r1′, r2

′) =∫

Ψ(r1, r2...rN)Ψ∗(r1′, r2

′...rN)dr3...drN

“Power functional”, Phys. Rev. B 78, R201103 (2008)

Exc[γ] = −12

∫|γα(r, r′)|2

|r− r′|d3r d3r′ (1/2 ≤ α ≤ 1)

Hartree-Fock functional

Exc[γ] = −12

∫|γ(r, r′)|2

|r− r′|d3r d3r′

S. Sharma Reduced Density Matrix Functional Theory

Muller functional, Phys. Rev. Lett. 105A, 446 (1984)

Exc[γ] = −12

∫γp(r, r′)∗γ1−p(r, r′)

|r− r′|d3r d3r′ (p = 1/2)

“Power functional”, Phys. Rev. B 78, R201103 (2008)

Exc[γ] = −12

∫|γα(r, r′)|2

|r− r′|d3r d3r′ (1/2 ≤ α ≤ 1)

Hartree-Fock functional

Exc[γ] = −12

∫|γ(r, r′)|2

|r− r′|d3r d3r′

S. Sharma Reduced Density Matrix Functional Theory

Correlation energy for atoms and molecules (G2)

(LDA: ∼ 600%, B3LYP: ∼ 300%)S. Sharma Reduced Density Matrix Functional Theory

Lieb’s conjecture for LiH

S. Sharma Reduced Density Matrix Functional Theory

Band gap using chemical potential

∆ ≡ I(N)−A(N)= µ(N + η)− µ(N − η)

S. Sharma Reduced Density Matrix Functional Theory

Band gap for solids

Sharma et al. Phys. Rev. B 78, R201103 (2008)

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′|d

3rd3r′ and cr = 2∫ n+(r)|r−r′|d

3rd3r′

S. Sharma Reduced Density Matrix Functional Theory

Band gap using chemical potential

∆ ≡ I(N)−A(N)= µ(N + η)− µ(N − η)

S. Sharma Reduced Density Matrix Functional Theory

Band gaps for solids

S. Sharma Reduced Density Matrix Functional Theory

Natural-orbital minimisation

E[γ] = E[ni,Φi]

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

S. Sharma Reduced Density Matrix Functional Theory

Occupation number minimization

Constraints:∑

i ni = Nand 0 ≤ ni ≤ 1

Definegi(κ) ≡ dE/dni − κ and

gi(κ) ≡{gi(1− ni) gi > 0gini gi ≤ 0

1 Compute dE/dni2 Find κ such that

∑i gi(κ) = 0

3 Make change in occupationnumber: ni → ni + λgi(κ), forlargest λ which keepsoccupancies in [0, 1]

4 Goto step 1, or exit onceconvergence is achieved

S. Sharma Reduced Density Matrix Functional Theory

Code used: ELK

J. K. Dewhurst, S. Sharma and E. K. U. Gross

Code available at: http://elk.sourceforge.net/

S. Sharma Reduced Density Matrix Functional Theory

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.

S. Sharma Reduced Density Matrix Functional Theory

Atomisation energy for molecules

S. Sharma Reduced Density Matrix Functional Theory

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)

S. Sharma Reduced Density Matrix Functional Theory

Band gap for finite systems with exact functional

S. Sharma Reduced Density Matrix Functional Theory

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

S. Sharma Reduced Density Matrix Functional Theory

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

S. Sharma Reduced Density Matrix Functional Theory

Band gap for finite systems with exact functional

S. Sharma Reduced Density Matrix Functional Theory

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

S. Sharma Reduced Density Matrix Functional Theory

Chemical potential for solids

S. Sharma Reduced Density Matrix Functional Theory