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  • 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