Reduced Density Matrix Functional Theory for Many Electron Systems

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

1 Fritz Haber Institute of the Max Planck Society, Berlin, Germany 2 Institut für Theoretische Physik, Freie Universität Berlin, Germany

12 June 2009

S. Sharma Reduced Density Matrix Functional Theory

Schrödinger equation for N particles:

Ĥ Ψi(x1,x2 . . . ,xN ) = Ei Ψi(x1,x2, . . . ,xN )

x ≡ {r, σ}

Ĥ = −1 2

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 . . . d3rN .

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 . . . d3rN .

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 . . . d3rN .

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 . . . d3rN .

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 and occupation 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] = −1 2

∫ 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)

Define F [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 implicit functional of γ

S. Sharma Reduced Density Matrix Functional Theory

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

Exc[γ] = − 1 2

∫ 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] = − 1 2

∫ |γα(r, r′)|2

|r− r′| d3r d3r′ (1/2 ≤ α ≤ 1)

Hartree-Fock functional

Exc[γ] = − 1 2

∫ |γ(r, r′)|2

|r− r′| d3r d3r′

S. Sharma Reduced Density Matrix Functional Theory

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

Exc[γ] = − 1 2

∫ 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] = − 1 2

∫ |γα(r, r′)|2

|r− r′| d3r d3r′ (1/2 ≤ α ≤ 1)

Hartree-Fock functional

Exc[γ] = − 1 2

∫ |γ(r, r′)|2

|r− r′| d3r d3r′

S. Sharma Reduced Density Matrix Functional Theory

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

Exc[γ] = − 1 2

∫ γp(r, r′)∗γ1−p(r, r′)

|r− r′| d3r d3r′ (p = 1/2)

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

Exc[γ] = − 1 2

∫ |γα(r, r′)|2

|r− r′| d3r d3r′ (1/2 ≤ α ≤ 1)

Hartree-Fock functional

Exc[γ] = − 1 2

∫ |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)

µ̃(η) has discontinuity at η = 0 with Eg = limη→0+(µ̃(η)− µ̃(−η)) being identical to exact fundamental gap.

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

µ̃(η) = µ(η = 0−) + {

clη for η < 0 Eg + 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 (any functional can be used) : ψKSj (r)

1 Natural orbitals are expanded in Kohn-Sham wave functions

ΦRDMi (r) = ∑ j

cij ψ KS j (r)

2 Compute gradients of the total energy w.r.t. cij 3 Use steepest-descent along the gradient cij → cij + λdE/dcij 4 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 = N and 0 ≤ ni ≤ 1

Define gi(κ) ≡ dE/dni − κ and

g̃i(κ) ≡ { gi(1− ni) gi > 0 gini gi ≤ 0

1 Compute dE/dni 2 Find κ such that

∑ i g̃i(κ) = 0

3 Make change in occupation number: ni → ni + λg̃i(κ), for largest λ which keeps occupancies in [0, 1]

4 Goto step 1, or exit once convergence 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-LAPW code.

Produces very good results for wide range of systems.

New algorithm for minimisation of energy with respect to occupation numbers.

S. Sharma Reduced Density Matrix Functional Theory

Atomisation energy for molecules