introduction and overview of the reduced density matrix functional theory · 2016-11-14 ·...
TRANSCRIPT
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
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|H |Φ〉〈Φ|Φ〉
Minimization chooses N orbitals out an infinite dimensionspace (or of dimension M > N for practical applications).
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
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 thenumber of particles.
No extrema between 0 and 1: collapses to HF Theory
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
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) =
(
Np
)∫
d3rp+1..d3rNΨ∗(r′1, ..r
′p, rp+1..rN ) Ψ(r1, ..rN )
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
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) =
(
Np
)∫
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
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:
〈O〉 = Tr{
Γ(p)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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
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 areknown1.
These conditions are not sufficient. If they were sufficient theywould provide the solution of the many electron problem.
1work and talk of D. A. Mazziotti.
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
N -representability of the 1RDM
For γ ensemble N -representability, necessary and sufficientconditions were proven by Coleman2:
0 ≤ nj ≤ 1,∑
j
nj = N
occupation 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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
N -representability of the 1RDM
For γ ensemble N -representability, necessary and sufficientconditions were proven by Coleman2:
0 ≤ nj ≤ 1,∑
j
nj = N
occupation 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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
N -representability of the 1RDM
Coleman’s conditions are also sufficient for pure state N-repfor even number of electrons and spin compensated systems.
Given the exact functional of the 1RDM the ensemble N-repconditions are enough.
Are there pure state N-rep conditions?
Generalized Pauli constrains3 for (N,M); N : number ofelectrons, 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-electronsystems improves the results for approximate 1RDMfunctionals.
3Talks on Thursday
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
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 theground-state 1RDM.
The exact total energy functional
Etot = Ekin + Eext +Eee
Eee[γ] = minΨ→γ〈Ψ|Vee|Ψ〉
(Domain of γ: Pure state N-representable (Lieb 1979))
Eee[γ] = minΓ(N)→γ
〈Ψ|Vee|Ψ〉
(Domain of γ: Ensemble state N-representable (Valone 1980))
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Density MatricesN-representabilityFoundations
RDMFT Foundations
Total energyEtot = 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 kineticenergy contributions
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison with DFT
Muller 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
Muller 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. Muller, 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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison with DFT
BBC functionals
A hierarchy of 3 corrections8
BBC3:
f(nj, nk) =
−√njnk j 6= k both weakly occupied
njnkj 6= k both strongly occupiedj(k) anti−bonding , k(j) not bonding
n2j j = k√
njnk otherwise.
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison 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-holesymmetry10
Pairs (p, p): np + np = 1,
EPNOF5 =
N∑
p=1
[np(2Hpp + Jpp)−√npnpKpp]
+
N∑
p,q=1
q 6=p,q 6=p
npnq(2Jpq −Kpq)
Jpq,Kpq: Coulomb and Exchange integrals; Hpp single particleterm
10Piris et al, JCP 134, 164102 (2011).
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison with DFT
Phase dependent functionals
Lowdin-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:occupation numbers from BBGKY TD equations are timedependent for phase dependent functionals.
11LS, JCP 25, 1035 (1956).
12Surjan, Topics in Current Chem. 203 63, (1999)
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison 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)
0 1
E
nj
At the solution, µ = dE/dnj , ∀ fractional nj
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison 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 adiagonalization problem.
The Lagrangian Matrix ǫjk is Hermitian at the extremum.
Orbital optimization remains the bottleneck of RDMFT.
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison with DFT
Scaling of RDMFT
Scaling depends only on the Hilbert space size M (all orbitalsare 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 (effectiveHamiltonian schemes), however the problem still remains.
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison with DFT
Comparison with DFT
Similarly to DFT, RDMFT is not a variational method.The kinetic energy is a known functional of the 1RDM. Onthe contrary, it is NOT a functional of the density.Without the exact kinetic energy functional,DFT resorts onthe fictitious Konh-Sham (KS) system to restore the quantummechanics.In RDMFT no KS systems is required and does not exist.RDMFT the quasiparticle picture (like KS) is not built in thetheory.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 todescribe non-dynamic correlations.Due to the non-idempotency of the 1RDM, RDMFT candescribe more naturally non-dynamic correlation.
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
FunctionalsMinimizationComparison with DFT
The price to pay!
RDMFT is much less efficient than DFT due to:
Nominal scaling M5 vs M4.In principle, all orbitals are occupied.Extra occupation number optimization.Orbital optimization does not reduce to eigenvalue iterativeproblem.
For routine calculations that DFT works well, there is no needto use RDMFT.
Where can RDMFT be useful? In systems with strongnon-dynamic correlations where single Slater description ispoor: Molecular dissociation, diradicals, band gaps ofsemicunductors, insulators and highly correlated periodicsystems.
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
H2 dissociation
1 2 3 4d [Å]
-1.1
-1
-0.9
-0.8
-0.7
Eto
t [a.u
.]
RHFCIMüllerGUBBC3
Energy of the H2 molecule
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Molecular dissociation with PNOF5
N2 Molecule
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Molecular dissociation with APSG
K. Pernal, J. Chem. Theory and Comput. 10, 4332 (2014).
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Homogeneous electron gas
0.01 0.1 1 10 100r
s[a.u.]
-0.25
-0.2
-0.15
-0.1
-0.05
0C
orre
latio
n E
nerg
y [H
a]
Monte Carlo (Ortiz-Ballone)Müller Functional (Csányi-Arias)Müller Functional (Cioslowski-Pernal)Csányi-Arias CHFCsányi-Goedecker-Arias (CGA)BBC1BBC2
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Power-functional for HEG
0.25 0.5 1 2 4rs
-0.2
-0.15
-0.1
-0.05
0C
orre
latio
n E
nerg
y [a
.u.]
ExactMüller (α=0.50)α=0.55α=0.60α=0.70
Correlation energy for different α
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental 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 δeMuller 0.55 1.23 (C2Cl4) 135.7% 438.3% (Na2) 0.0193GU 0.26 0.79 (C2Cl4) 51.63% 114.2% (Si2) 0.0072
BBC1 0.29 0.75 (C2Cl4) 69.91% 159.1% (Na2) 0.0098BBC2 0.18 0.50 (C2Cl4) 45.02% 125.0% (Na2) 0.0058BBC3 0.068 0.27 (SiCl4) 18.37% 50.8% (SiH2) 0.0017PNOF 0.102 0.42 (SiCl4) 20.84% 59.1% (SiCl4) 0.0021PNOF0 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) 0.0015B3LYP 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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Ec for finite systems
-1.0 -0.8 -0.6 -0.4 -0.2E
c
(ref) (a.u.)
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0E
c (a.
u)(a)
MullerGUBBC1BBC3
Correlation energy for finite systems
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Ec for finite systems
-1.0 -0.8 -0.6 -0.4 -0.2 0.0E
c
(ref) (a.u.)
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0E
c (a.
u)(b)
MullerGUPNOFPNOF0
Correlation energy for finite systems
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Ec for finite systems
-1.0 -0.8 -0.6 -0.4 -0.2 0.0E
c
(ref) (a.u.)
-4.0
-3.0
-2.0
-1.0
0.0E
c (a.
u)(c)
MP2B3LYPBBC3PNOF0
Correlation energy for finite systems
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Atomization energies
set 1 set 2δ (%) 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*-basisset 2: subset of 50 molecules, cc-pVDZ-basis
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
One-electron spectrum
In RDMFT there is no Kohn-Sham scheme which offersone-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 theadditional constraint that the orbitals satisfy a KS-likeequation with a local potential.
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Local RDMFT
Minimize RDMFT functionals under the additional constraintthat the orbitals are eigenfunctions of a single particleHamiltonian with a local potential.
[
−∇2
2+ vext(r) + vrep(r)
]
ϕj(r) = ǫjϕj(r).
Local potential in RDMFT is an approximation: true naturalorbitals do not come from a local potential
Total energy will be higher than full RDMFT.
Optimization of a local potential: smaller scale problem thanorbital optimization.
Are the obtained single-electron properties reasonable?
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Ionization potential
All methods give errors of a few % in the calculation of IP’sfor 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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental 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 atinteger particle number16
∆ = I −A
= limη→0
(µ(N + η)− µ(N − η))
16Helbig et al, Europhys. Lett. 77, 67003 (2007)
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Results for LiH
3.6 3.8 4 4.2 4.4 4.6M
-0.4
-0.3
-0.2
-0.1
0
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
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Fundamental gap
System RDMFT RDMFT Other Experimentµ(M) step I −A theoretical
Li 0.177 0.202 0.175 0.175Na 0.175 0.198 0.169 0.169F 0.538 0.549 0.514LiH 0.269, 0.293 0.271 0.286 0.271
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Band gaps for solids
Ge Si GaAs BN C LiF LiH MnO NiO FeO CoO
-100
-80
-60
-40
-20
0
20
40
δ ( %
)
DFT-LDAGWRDMFT | γ α |2 (α = 0.7)RDMFT | γ α |2 (α = 0.65)
Sharma et al, PRB 78, 201103R (2008)
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Equilibrium lattice parameter for solids
Solid Expt. DFT-LDA Muller α = 0.7
Diamond 6.74 6.68 6.78 6.75Si 10.26 10.188 10.49 10.55BN 6.83 6.758 6.86 6.82
∆ 0.0 1.03 1.07 1
∆: Absolute percentage deviationRDMFT: Sharma et al, PRB 78, 201103R (2008).
Oxford, 13 April 2016
Connection to Hartree FockRDMFT
Functionals and MinimizationApplications
Prorotype systemsCorrelation energiesQuasiparticle spectrumFundamental gap
Conclusion
RDMFT: a framework for electronic correlations where the1RDM 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, Orbitaloptimization bottleneck).
RDMFT is a promising alternative to DFT
Target: not to replace DFT but to give answers for problemsthe DFT results are not satisfactory
RDMFT is promising in systems with static correlations(molecular dissociation)
Present results show that potentially it can be successful instrongly correlated periodic systems
Oxford, 13 April 2016