introduction and overview of the reduced density matrix ... › confs › pauli2016 › files ›...

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



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




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).




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′|




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




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




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 )





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)





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)?





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.






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)






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)






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





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))





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




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).





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).





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).





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)





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





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.





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.





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.





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.




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





Molecular dissociation with PNOF5

N2 Molecule





Molecular dissociation with APSG

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





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)





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





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)





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






-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







-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






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





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.





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?





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




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)





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)





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





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)





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).





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


introduction and overview of the reduced density matrix ... › confs › pauli2016 › files ›...

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