[PPT] - Introduction and Overview of the Reduced Density Matrix Functional PowerPoint Presentation

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications

Introduction and Overview of the Reduced Density Matrix Functional Theory

N. N. Lathiotakis

Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens

April 13, 2016

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications

Outline

1

Connection to Hartree Fock

2

Density matrices and N-representability

3

Reduced density matrix functional theory (RDMFT)

4

Functionals minimization/performance

5

Comparison with DFT

6

Application to prototype systems: Correlation energies, Molecular dissociation, Homogeneous electron gas, Spectra, IPs, Electronic Gap

7

Conclusion

Oxford, 13 April 2016

SLIDE 3

Connection to Hartree Fock RDMFT Functionals and Minimization Applications

Hartree Fock

Wave function is one Slater Determinant: Φ(x1, x2, · · · xN) = ϕ1(x1) ϕ1(x2) · · · ϕ1(xN) ϕ2(x1) ϕ2(x2) · · · ϕ2(xN) . . . . . . ... . . . ϕN(x1) ϕN(x2) · · · ϕN(xN) We need to minimize: Etot = Φ| ˆ H|Φ Φ|Φ Minimization chooses N orbitals out an infinite dimension space (or of dimension M > N for practical applications).

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications

Energy in Hartree-Fock

Spin-orbitals ϕ(x) = ϕ(r)α(ω). For spin compensated systems: Etot = 2

N/2

j=1

h(1)

jj + 2 N/2

j,k=1

Jjk −

N/2

j,k=1

Kjk h(1)

jj =

d3r ϕ∗

j(r)

−1

2∇2

r + V (r)

ϕj(r)

Jjk =

d3r
d3r′ |ϕj(r)|2 |ϕk(r′)|2

|r − r′| Kjk =

d3r
d3 r′ ϕj(r) ϕ∗

j(r′) ϕk(r′) ϕ∗ k(r)

|r − r′|

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications

Energy in Hartree-Fock

Spin-orbitals ϕ(x) = ϕ(r)α(ω). For spin compensated systems: Etot = 2

∞

j=1

njh(1)

jj + 2 ∞

j,k=1

njnk Jjk −

∞

j,k=1

njnk Kjk h(1)

jj =

d3r ϕ∗

j(r)

−1

2∇2

r + V (r)

ϕj(r)

Jjk =

d3r
d3r′ |ϕj(r)|2 |ϕk(r′)|2

|r − r′| Kjk =

d3r
d3r′ ϕj(r) ϕ∗

j(r′) ϕk(r′) ϕ∗ k(r)

|r − r′| Where nj and nk occupation numbers

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications

Hartree Fock Functional in RDMFT

Etot = 2

∞

j=1

nj h(1)

jj + 2 ∞

j,k=1

njnkJjk −

∞

j,k=1

njnk Kjk Assume that this functional is minimized w.r.t. nj, ϕj. It is not bound! nj should satisfy extra conditions. Ensemble N-representability conditions of Coleman: 0 ≤ nj ≤ 1, and 2

∞

j=1

nj = N The first reflects the Pauli principle and the second fixes the number of particles. No extrema between 0 and 1: collapses to HF Theory

Oxford, 13 April 2016

SLIDE 7

Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

Density matrices

N-body density matrix (NRDM) Γ(N)(r1, r2..rN; r′

1, r′ 2..r′ N) = Ψ∗(r′ 1, r′ 2..r′ N) Ψ(r1, r2..rN)

Reduce the order of the density matrix (pRDM) Γ(p)(r1, ..rp; r′

1, ..r′ p) =

N p d3rp+1..d3rNΨ∗(r′

1, ..r′ p, rp+1..rN) Ψ(r1, ..rN)

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

Density matrices

N-body density matrix (NRDM) Γ(N)(r1, r2..rN; r′

1, r′ 2..r′ N) = Ψ∗(r′ 1, r′ 2..r′ N) Ψ(r1, r2..rN)

Reduce the order of the density matrix (pRDM) Γ(p)(r1, ..rp; r′

1, ..r′ p) =

N p d3rp+1..d3rNΨ∗(r′

1, ..r′ p, rp+1..rN) Ψ(r1, ..rN)

Recurrence relation Γ(p−1)(r1, ..rp−1; r′

1, ..r′ p−1) =

p N − p + 1

d3rp Γ(p)(r1, ..rp; r′

1, ..r′ p−1, rp)

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

Density matrices

One-body reduced density matrix (1RDM) Γ(1)(r, r′) = 2 N − 1

d3r2 Γ(2)(r, r2; r′, r2) =: γ(r; r′)

Expectation value of p-body operator: ˆ O = Tr

Γ(p) ˆ

O

Total energy: expectation value of the Hamiltonian (2-body)

The e-e interaction energy simple functional of 2RDM: Eee = d3r1 d3r2 ρ(2)(r1, r2) |r1 − r2) ρ(2)(r1, r2) = Γ(2)(r1, r2; r1, r2) (second reduced density) Why don’t we minimize the total energy with respect to Γ(2)?

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

N-representability of the 2RDM

Remember Γ(2)(r1, r2; r′

1, r′ 2) =

N(N − 1) 2

d3r3..d3rNΨ∗(r′

1, r′ 2, r3..rN)Ψ(r1..rN)

with Ψ: antisymmetric, normalized wave function For Γ(2) several necessary N-representability conditions are known1. These conditions are not sufficient. If they were sufficient they would provide the solution of the many electron problem.

1work and talk of D. A. Mazziotti. Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

N-representability of the 1RDM

For γ ensemble N-representability, necessary and sufficient conditions were proven by Coleman2: 0 ≤ nj ≤ 1,

j

nj = N

ccupation numbers nj and the natural orbitals ϕj:

γ(r, r′) =

∞

j=1

nj ϕ∗

j(r′) ϕj(r)

Plus orthonormality of ϕj

2Rev. Mod. Phys. 35, 668 (1963)

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

N-representability of the 1RDM

For γ ensemble N-representability, necessary and sufficient conditions were proven by Coleman2: 0 ≤ nj ≤ 1,

j

nj = N

ccupation numbers nj and the natural orbitals ϕj:

γ(r, r′) =

∞

j=1

nj ϕ∗

j(r′) ϕj(r)

Plus orthonormality of ϕj 2RDM vs 1RDM functional theory: (simple functional but complicated N-rep) vs (totally unknown functional but simple N-rep.)

2Rev. Mod. Phys. 35, 668 (1963)

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

N-representability of the 1RDM

Coleman’s conditions are also sufficient for pure state N-rep for even number of electrons and spin compensated systems. Given the exact functional of the 1RDM the ensemble N-rep conditions are enough. Are there pure state N-rep conditions? Generalized Pauli constrains3 for (N, M); N: number of electrons, M: size of the Hilbert space. Recently: a method to generate them for all pairs (M, N). Unfortunately, their number explodes as N, M increase. Their incorporation in RDMFT calculations for 3-electron systems improves the results for approximate 1RDM functionals.

3Talks on Thursday Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

RDMFT Foundations

Gilbert’s Theorem (T. Gilbert Phys. Rev. B 12, 2111 (1975)): γgs(r; r′) 1−1 ← → Ψgs(r1, r2...rN) Every ground-state observable is a functional of the ground-state 1RDM. The exact total energy functional Etot = Ekin + Eext + Eee Eee[γ] = min

Ψ→γΨ|Vee|Ψ

(Domain of γ: Pure state N-representable (Lieb 1979)) Eee[γ] = min

Γ(N)→γΨ|Vee|Ψ

(Domain of γ: Ensemble state N-representable (Valone 1980))

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Density Matrices N-representability Foundations

RDMFT Foundations

Total energy Etot = Ekin + Eext + Eee Ekin = d3r d3r′ δ(r − r′)

−∇2

2

γ(r; r′)

Eext =

d3r vext(r) γ(r; r)

Eee = EH + Exc EH = d3r d3r′ γ(r; r) γ(r′; r′) |r − r′| Exchange-correlation energy does not contain any kinetic energy contributions

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

M¨ uller type functionals

Exc = −1 2

∞

j,k=1

f(nj, nk)

d3rd3r′ ϕj(r)ϕ∗

j(r′)ϕk(r′)ϕ∗ k(r)

| r − r′ | Hartree-Fock: f(nj, nk) = njnk M¨ uller functional4: f(nj, nk) = √njnk Goedecker-Umrigar5: f(nj, nk) = √njnk(1 − δjk) + n2

jδjk

Power functional6 f(nj, nk) = (njnk)α, α ∼ 0.6. ML: Pade approximation for f, fit for a set of molecules7

4A. M¨

uller, Phys. Lett. 105A, 446 (1984); M. A. Buijse, E. J. Baerends, Mol. Phys. 100, 401 (2002)

5S. Goedecker, C. J. Umrigar, Phys. Rev. Lett. 81, 866 (1998).

6Sharma et al, PRB 78, 201103R (2008) 7Marques, et al, PRA 77, 032509 (2008). Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

BBC functionals

A hierarchy of 3 corrections8 BBC3: f(nj, nk) =            −√njnk j = k both weakly occupied njnk j = k both strongly occupied j(k) anti−bonding , k(j) not bonding n2

j

j = k √njnk

therwise.

AC3: Similar to BBC3 with C2,C3 corrections analytic9

8O. Gritsenko, et al, JCP 122, 204102 (2005)

9Rohr, et al, JCP 129, 164105 (2008). Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

PNOF

PNOFn, n=1,6: Γ(2)

pqrs = npnq(δprδqs − δpsδqr) + λpqrs[γ]

Approximations for the cumulant part λpqrs[γ]. PNOF5: Satisfies sum rule, positivity, particle-hole symmetry10 Pairs (p, ¯ p): np + n¯

p = 1,

EPNOF5 =

N

p=1

[np(2Hpp + Jpp) − √n¯

pnpKp¯ p]

+

N

p,q=1

q=p,q=¯ p

npnq(2Jpq − Kpq) Jpq,Kpq: Coulomb and Exchange integrals; Hpp single particle term

10Piris et al, JCP 134, 164102 (2011). Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

Phase dependent functionals

L¨

wdin-Shull11 (exact for 2 electrons ):

Eee = 1 2

j,k

fjfk√njnkKjk where fj = ±1. Usually f1fj = −1. Antisymmetrized product of strongly orthogonal geminals (APSG)12. Phase dependent functionals are useful in TD-RDMFT:

ccupation numbers from BBGKY TD equations are time

dependent for phase dependent functionals.

11LS, JCP 25, 1035 (1956). 12Surjan, Topics in Current Chem. 203 63, (1999) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

Minimization

We need to minimize the quantity F = Etot−µ  

∞

j=1

nj − N  −

∞

j,k=1

ǫjk

d3rϕ∗

j(r)ϕk(r) − δjk

Minimize with respect to nj and ϕj

Minimization with respect to nj can have border minima (pinned states)

1

E nj

At the solution, µ = dE/dnj, ∀ fractional nj

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

Minimization

We need to minimize the quantity F = Etot−µ  

∞

j=1

nj − N  −

∞

j,k=1

ǫjk

d3rϕ∗

j(r)ϕk(r) − δjk

Minimize with respect to nj and ϕj

Minimization with respect to nj can have border minima (pinned states) At the solution, µ = dE/dnj, ∀ fractional nj Minimization with respect to ϕj is complicated; not a diagonalization problem. The Lagrangian Matrix ǫjk is Hermitian at the extremum. Orbital optimization remains the bottleneck of RDMFT.

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

Scaling of RDMFT

Scaling depends only on the Hilbert space size M (all orbitals are occupied) and not the number of electrons. Nominal scaling M5. For comparison, DFT, HF: M4; CI, CCSD(T), MP4: M7 Orbital minimization can be extremely expensive. Several iterative schemes have been developed (effective Hamiltonian schemes), however the problem still remains.

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

Comparison with DFT

Similarly to DFT, RDMFT is not a variational method. The kinetic energy is a known functional of the 1RDM. On the contrary, it is NOT a functional of the density. Without the exact kinetic energy functional,DFT resorts on the fictitious Konh-Sham (KS) system to restore the quantum mechanics. In RDMFT no KS systems is required and does not exist. RDMFT the quasiparticle picture (like KS) is not built in the theory. In KS-DFT, kinetic energy parts “pollute” the xc energy term. In RDMFT, no such terms exist. Easier to create functionals. The KS system with a single Slater determinant is difficult to describe non-dynamic correlations. Due to the non-idempotency of the 1RDM, RDMFT can describe more naturally non-dynamic correlation.

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Functionals Minimization Comparison with DFT

The price to pay!

RDMFT is much less efficient than DFT due to:

Nominal scaling M 5 vs M 4. In principle, all orbitals are occupied. Extra occupation number optimization. Orbital optimization does not reduce to eigenvalue iterative problem.

For routine calculations that DFT works well, there is no need to use RDMFT. Where can RDMFT be useful? In systems with strong non-dynamic correlations where single Slater description is poor: Molecular dissociation, diradicals, band gaps of semicunductors, insulators and highly correlated periodic systems.

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

H2 dissociation

1 2 3 4

d [Å]

1.1
1
0.9
0.8
0.7

Etot [a.u.]

RHF CI Müller GU BBC3

Energy of the H2 molecule

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Molecular dissociation with PNOF5

N2 Molecule

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Molecular dissociation with APSG

K. Pernal, J. Chem. Theory and Comput. 10, 4332 (2014).

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Homogeneous electron gas

0.01 0.1 1 10 100 rs [a.u.]

0.25
0.2
0.15
0.1
0.05

Correlation Energy [Ha]

Monte Carlo (Ortiz-Ballone) Müller Functional (Csányi-Arias) Müller Functional (Cioslowski-Pernal) Csányi-Arias CHF Csányi-Goedecker-Arias (CGA) BBC1 BBC2

Correlation energy of the HEG as a function of rs13

13N.N.L. et al, Phys. Rev. B 75, 195120 (2007) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Power-functional for HEG

0.25 0.5 1 2 4 rs

0.2
0.15
0.1
0.05

Correlation Energy [a.u.]

Exact Müller (α=0.50) α=0.55 α=0.60 α=0.70

Correlation energy for different α

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Benchmark for finite systems

Benchmark for 150 molecules and radicals (G2/97 test set)14 6-31G* basis set, Comparison with CCSD(T)

Method ¯ ∆ ∆max ¯ δ δmax ¯ δe M¨ uller 0.55 1.23 (C2Cl4) 135.7% 438.3% (Na2) 0.0193 GU 0.26 0.79 (C2Cl4) 51.63% 114.2% (Si2) 0.0072 BBC1 0.29 0.75 (C2Cl4) 69.91% 159.1% (Na2) 0.0098 BBC2 0.18 0.50 (C2Cl4) 45.02% 125.0% (Na2) 0.0058 BBC3 0.068 0.27 (SiCl4) 18.37% 50.8% (SiH2) 0.0017 PNOF 0.102 0.42 (SiCl4) 20.84% 59.1% (SiCl4) 0.0021 PNOF0 0.072 0.32 (SiCl4) 17.11% 46.0% (Cl2) 0.0015 ML(cl. shell) 0.059 0.18 (pyridine) 11.02% 35.7% (Na2) 0.0015 MP2 0.040 0.074 (C2Cl4) 11.86% 35.7% (Li2) 0.0015 B3LYP 0.75 2.72 (SiCl4) 305.0% 2803.7% (Li2) 0.022

14N.L. et al, J. Chem. Phys. 128, 184103 (2008) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Ec for finite systems

1.0
0.8
0.6
0.4
0.2

Ec

(ref) (a.u.)

3.0
2.5
2.0
1.5
1.0
0.5

0.0 Ec (a.u)

(a)

Muller GU BBC1 BBC3

Correlation energy for finite systems

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Ec for finite systems

1.0
0.8
0.6
0.4
0.2

0.0 Ec

(ref) (a.u.)

3.0
2.5
2.0
1.5
1.0
0.5

0.0 Ec (a.u)

(b)

Muller GU PNOF PNOF0

Correlation energy for finite systems

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Ec for finite systems

1.0
0.8
0.6
0.4
0.2

0.0 Ec

(ref) (a.u.)

4.0
3.0
2.0
1.0

0.0 Ec (a.u)

(c)

MP2 B3LYP BBC3 PNOF0

Correlation energy for finite systems

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Atomization energies

set 1 set 2 ¯ δ (%) δmax (%) ¯ δ (%) δmax (%) R(O)HF 42.4 195 (F2) 53.8 233(F2) Mueller 32.7 138 (Na2) 40.6 130(Na2) GU 43.7 239 (ClF3) 50.4 180(F2) BBC1 31.0 107 (ClF3) 34.8 75(O2) BBC2 26.9 142 (ClO) 40.1 142(F2) BBC3 18.0 117 (Li2) 25.6 103(Li2) PNOF 25.5 161 (ClF3) 30.4 127(F2) PNOF0 17.5 76 (Li2) 23.9 73(Cl2) MP2 6.24 34 (Na2) 7.94 35(Na2) B3LYP 11.7 40 (BeH) 12.1 38(F2)

set 1: G2/97 test set, 6-31G*-basis set 2: subset of 50 molecules, cc-pVDZ-basis

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

One-electron spectrum

In RDMFT there is no Kohn-Sham scheme which offers

ne-electron (quasiparticle) spectrum.

Exact KS-DFT ǫHOMO = IP; Koopmans’ theorem in HF. Two proposals and an approximate framework:

Extended Koopmans’ theorem (EKT). IP is an eigenvalue of λij = ǫjk √njnk Energy derivative at half occupancy ǫi = ∂E ∂ni

{nj}=optimal,ni=1/2

Local RDMFT: Minimize RDMFT functionals under the additional constraint that the orbitals satisfy a KS-like equation with a local potential.

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Local RDMFT

Minimize RDMFT functionals under the additional constraint that the orbitals are eigenfunctions of a single particle Hamiltonian with a local potential.

−∇2

2 + vext(r) + vrep(r)

ϕj(r) = ǫjϕj(r).

Local potential in RDMFT is an approximation: true natural

rbitals do not come from a local potential

Total energy will be higher than full RDMFT. Optimization of a local potential: smaller scale problem than

rbital optimization.

Are the obtained single-electron properties reasonable?

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Ionization potential

All methods give errors of a few % in the calculation of IP’s for a set of atoms/molecules (He, H2, LiH, H2O, HF, CH4, CO2, NH3, Ne, C2H4, C2H2). Local RDMFT for two aromatic molecules15:

15NNL et al, JCP 141, 164120 (2014) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Fundamental gap

Extend the whole theory to fractional particle number Lagrange multiplier µ is the chemical potential µ(N) is a step function with step at integer N The fundamental gap is given as the discontinuity of µ at integer particle number16 ∆ = I − A = lim

η→0 (µ(N + η) − µ(N − η))

16Helbig et al, Europhys. Lett. 77, 67003 (2007) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Results for LiH

3.6 3.8 4 4.2 4.4 4.6

M

0.4
0.3
0.2
0.1

µ (Ha)

GU Müller BBC1 BBC2 BBC3 BBC3 without SI PNOF

The discontinuity of µ at N = 4 electrons for LiH17

17Helbig et al, Phys. Rev. A 79, 022504 (2009) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Fundamental gap

System RDMFT RDMFT Other Experiment µ(M) step I − A theoretical Li 0.177 0.202 0.175 0.175 Na 0.175 0.198 0.169 0.169 F 0.538 0.549 0.514 LiH 0.269, 0.293 0.271 0.286 0.271

Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Band gaps for solids

Ge Si GaAs BN C LiF LiH MnO NiO FeO CoO

100
80
60
40
20

20 40

δ ( % )

DFT-LDA GW RDMFT | γ

α | 2 (α = 0.7)

RDMFT | γ

α | 2 (α = 0.65)

Sharma et al, PRB 78, 201103R (2008) Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Equilibrium lattice parameter for solids

Solid Expt. DFT-LDA M¨ uller α = 0.7 Diamond 6.74 6.68 6.78 6.75 Si 10.26 10.188 10.49 10.55 BN 6.83 6.758 6.86 6.82 ∆ 0.0 1.03 1.07 1 ∆: Absolute percentage deviation

RDMFT: Sharma et al, PRB 78, 201103R (2008). Oxford, 13 April 2016

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Connection to Hartree Fock RDMFT Functionals and Minimization Applications Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap

Conclusion

RDMFT: a framework for electronic correlations where the 1RDM is the fundamental variable. Kinetic energy is an explicit functional of the 1RDM. No kinetic energy contributions in the xc energy term. No need for a KS auxiliary system. RDMFT is not computationally efficient (M5 scaling, Orbital

ptimization bottleneck).

RDMFT is a promising alternative to DFT Target: not to replace DFT but to give answers for problems the DFT results are not satisfactory RDMFT is promising in systems with static correlations (molecular dissociation) Present results show that potentially it can be successful in strongly correlated periodic systems

Oxford, 13 April 2016