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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