Reduced Density Matrix Functional Theory for Many Electron ...sharma/talks/mafelap.pdf · Reduced...

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|


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]



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)]



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)


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



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


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


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


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′



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


Exc[γ] = −12

∫|γα(r, r′)|2

|r− r′|d3r d3r′ (1/2 ≤ α ≤ 1)


Exc[γ] = −12

∫|γ(r, r′)|2




Exc[γ] = −12

∫γp(r, r′)∗γ1−p(r, r′)

|r− r′|d3r d3r′ (p = 1/2)


Exc[γ] = −12

∫|γα(r, r′)|2

|r− r′|d3r d3r′ (1/2 ≤ α ≤ 1)


Exc[γ] = −12

∫|γ(r, r′)|2



Correlation energy for atoms and molecules (G2)

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

Lieb’s conjecture for LiH


Band gap using chemical potential

∆ ≡ I(N)−A(N)= µ(N + η)− µ(N − η)


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′



∆ ≡ I(N)−A(N)= µ(N + η)− µ(N − η)


Band gaps for solids


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


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


Code used: ELK

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

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


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.


Atomisation energy for molecules



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)


Band gap for finite systems with exact functional


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


Band gap for finite systems with exact functional


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


Chemical potential for solids


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