PNOF5-PT2 - UPV/EHU€¦ · PNOF5-PT2 A useful method for dealing with strongly correlated systems Mario Piris Kimika Fakultatea, UPV/EHU, and DIPC, .K.P 1072, 20080 Donostia IKERBASQUE,

PNOF5-PT2A useful method for dealing with strongly correlated systems

Mario Piris

Kimika Fakultatea, UPV/EHU, and DIPC, P.K. 1072, 20080 DonostiaIKERBASQUE, Basque Foundation for Science, 48011 Bilbao

August 29, 2013

Mario Piris PNOF5-PT2

The Natural Orbital Functional (PNOF5)Size-consistent second-order Multicon�gurational PT

Outline

1 The Natural Orbital Functional PNOF5 (JCP 134, 164102, 2011)

- generating APSG wavefunction → top-down method

2 PNOF5-PT2 (JCP 139, 064111, 2013)

- Introduction to the SC2-MCPT (JCP 122, 114104, 2005)

- Some examples to illustrate the potentiality of the method

3 Closing Remarks



The Energy in Natural Orbital Functional TheoryA generating wavefunction of PNOF5Results: Diradicals, Dissociations, TM Dimers, ...

The electronic energy E for N-electron systems

In NOFT, the energy is expressed in terms of the diagonal 1-RDM:

E [N, {ni , φi}] =∑i

niHii +∑ijkl

D[ni , nj , nk , nl ] < kl |ij >

Hii : core-Hamiltonian

< kl |ij >: 2e- integrals{ni}: occupation numbers

{φi (x)}: natural orbitals

1-RDM: Γki = niδki , Γ (x′1|x1) =∑

i niφi (x′1)φ∗i (x1)

2-RDM: D[ni , nj , nk , nl ] ⇐ reconstruction functional




The bottom-up and top-down methods (J. Mod. Phys. 4, 391, 2013)

Ψ (x1, x2, . . . , xN) =∑

CI ({ni}) ΦI (x1, x2, . . . , xN)

N-particle energy functional E [ΨN ]

⇓

E [N, {ni , φi}]Prof. J. Ugalde

⇑ ← Tuesday 27

D [Γ] = D [{ni}]N-representability conditions (D,Q,G,...)




Antisymmetrized Product of Strongly orthogonal Geminals

- PNOF5 ⇔ E(ΨAPSG , 2 conf, �xed signs for {cp}) ≡ E(ΨGVB,PP)

E exact ≤ EAPSG [{cp} , {ϕp}] ≤ EPNOF5 [{np} , {ϕp}]

(Pernal, Comput. Theor. Chem. 1003, 127, 2013)

- A generating wavefunction of PNOF5:

|0〉 =

N/2∏p=1

g †p |vac〉 =

N/2∏p=1

(√np a†p a†p −√np′ a

†p′ a†p′

)|vac〉

|0〉 = d0 |ΨHF 〉+

N/2∑p=1

dp

∣∣∣Ψp′p′

p p

⟩+

N/2∑p<q

dpq

∣∣∣Ψp′p′q′q′

p p q q

⟩+ · · ·

(JCP 139, 064111, 2013)




APSG generating wavefunction of PNOF5

|0〉 = |Φ0〉+ |Φd 〉+ |Φq〉+ ...+∣∣ΦN/2

⟩where

|Φ0〉 = d0 |ΨHF 〉

|Φd 〉 =

N/2∑p=1

dp

∣∣∣Ψp′p′

p p

⟩|Φq〉 =

N/2∑p<q

dpq

∣∣∣Ψp′p′q′q′

p p q q

⟩· · ·∣∣ΦN/2

⟩= d12..N/2

∣∣∣Ψ1′1′2′2′...N/2′ ¯N/2′

1 1 2 2 ... N/2 N/2

⟩d0 =

√n1n2...nN/2, dp = −d0

√np′np, dpq = d0

√np′nq′np nq

, · · ·




spin-parallel and spin-opposite components of the 2-RDM

Dpq,rt =npnq

2(δprδqt − δptδqr ) (1− δqp)

(1− δqp′

)Dpq,r t =

npnq

2δprδqt (1− δqp)

(1− δqp′

)+

(np

2δrp −

√npnp′

2δrp′

)δpqδrt

PNOF5:

E =N∑p=1

[np (2Hpp + Jpp)−√np′npKpp′

]+

N∑p,q=1

′′ nqnp (2Jpq −Kpq)




Examples of strongly correlated systems

1 diradicals and diradicaloids

J. Chem. Phys. 134, 164102, 2011Chem Phys Chem 12, 1673, 2011

2 homolytic dissociation

Phys. Chem. Chem. Phys. 13, 20129, 2011J. Chem. Theory Comp. 8, 2646, 2012TMs (Cr2), Phys. Chem. Chem. Phys. 15, 2055, 2013

3 IPs by the extended Koopmans' therorem

J. Chem. Phys. 136, 174116, 2012

4 PNOF5 natural and canonical orbitals

Chem. Phys. Lett. 531, 272, 2012Chem Phys Chem 13, 2297, 2012Theor. Chem. Acc. 132, 1298, 2013J. Chem. Phys. 138, 151102, 2013



Zero-order Hamiltonian and DK partitioningFirst- and second-order energy correctionsPNOF5-PT2: He2, HF, CO and N2

The reciprocal space in the MCPT

MCPT:

|0〉 : the zero-order ground state

one seeks perturbative corrections to |0〉overlapping basis in the full M-dimensional space

|K 〉 =

{|0〉 ,|ΨK 〉 ,

K = 0

K 6= 0

|ΨK 〉 : excited determinants with respect to |ΨHF 〉

reciprocal (biorthogonal) vectors

˜〈K | =

{d−10 〈ΨHF | ,〈ΨK | − dK ˜〈0|,

K = 0

K 6= 0

˜〈K | L〉 = δKL

Z. Rolik, et. al., J. Chem. Phys. 119, 1922 (2003).




Non-Hermitian zero-order Hamiltonian

Spectral resolution:

H0 =M∑

K=0

EK |K 〉 ˜〈K |

- EK -s are free parameters, de�ne the partitioning

- size-consistency is ensured by omitting projectors

E0 is conveniently taken as the zero-order ground state energy:

E0 = E (0) = ˜〈0|H |0〉 = EHF + Ecorr

EHF = 〈ΨHF | H |ΨHF 〉 , Ecorr = −N/2∑p=1

√np′npKpp′




The Davidson-Kapuy Partitioning: SC2-MCPT.

The zero-order excited energies: EK = E0 + ∆K

∆K =

εs − εa

εs + εu − εa − εb· · ·

a ≤ N ; s > N

a, b ≤ N; s, u > N

∆K : di�erences between diagonal elements of Fock operator

εi = hii +

N/2∑q=1

[2 〈iq| iq〉 − 〈iq| qi〉]

size-consistency is ensured by considering energy denominators as

di�erences of one-particle energies

A. Szabados, et. al., J. Chem. Phys. 122, 114104 (2005).




Lowest order energy corrections

E (1) = ˜〈0|V |0〉 = 0 , V = H − H0

E (2) = −M∑

K=1

˜〈0|V |K 〉 ˜〈K |V |0〉EK − E0

= −M∑

K=1

˜〈0|H |K 〉 ˜〈K |H |0〉∆K

=M∑

K=1

〈ΨHF | H |K 〉[dKE

(0) − 〈K | H |0〉]

d0∆K

non-invariance with respect to the choice of the Fermi vacuum

one needs to consider only {|Ψsa〉} and

{∣∣Ψsuab

⟩}excited det.




Second-order energy correction

E (2) = E(2)0 + E

(2)d + E

(2)q

where

E(2)0 = 2

N/2∑pr

|fpr |2

εp − εr+

N/2∑pqrt

〈pq| rt〉 [2 〈rt| pq〉 − 〈rt| qp〉]εp + εq − εr − εt

E(2)q =

N/2∑p

(np′

np

) ∣∣Kpp′∣∣2(εp′ − εp

)+

N/2∑pq

√np′nq′

npnq

[2 〈pq| p′q′〉 − 〈pq| q′p′〉]2

εp + εq − εp′ − εq′




Second-order energy correction

E (2)d = 2

N/2∑p

√np′np

|fpp′ |2

(εp′ − εp)+ 2

N/2∑pqr

√nrnr ′〈pq| rr〉〈r ′r ′| pq〉2εr − εp − εq

+2

N/2∑pr

frpεr − εp

[√np′np〈pr | p′p′

⟩−√

nrnr ′

⟨r ′r ′∣∣ pr〉]

+2

N/2∑p

√np′np

N/2∑r

frp′ 〈pp| p′r〉εr + εp′ − 2εp

+

N/2∑q

fqp 〈pq| p′p′〉εp + εq − 2εp′

+1

2

N/2∑rs

〈pp| rs〉〈rs| p′p′〉εr + εs − 2εp

+

N/2∑qr

〈pq| rp′〉〈pr | qp′〉εp + εq − εr − εp′

+

N/2∑qr

[〈pq| |rp′〉+ 〈pr | |qp′〉] 〈pq| p′r〉εp + εq − εr − εp′




Dissociation of the helium dimer (aug-cc-pV5Z)

typical phenomenon of dispersion interaction (Eb = 0.021 kcal/mol)

electron correlation e�ect is almost entirely atomic intrapair

the interpair correlation should be mostly dispersion type

3.0 3.5 4.0 4.5 5.0 5.5 6.0R (Å)

-5.79618

-5.79612

-5.79606

-5.79600

-5.79594

-5.79588

Ene

rgy

(Har

tree

s)

PNOF5-SC2-MCPT

full CI calculations (JCP 137, 204117, 2012)

contribution of∣∣Ψsu

ab

⟩with a 6= b :

2/3 ≈ 100% of vdW for same-spin e−

1/3→ 50% of vdW for opposite-spin e−

contribution of∣∣∣Ψsu

pp

⟩:

50% of vdW for opposite-spin e−

Note: excitations∣∣Ψsu

pp

⟩are already in |0〉




PNOF5-PT2:∣∣Ψsu

pp

⟩are excluded from E(2)

E (2) = 2

N/2∑pr

|fpr |2

εp − εr+

N/2∑pqrt(q 6=p)

〈pq| rt〉 [2 〈rt| pq〉 − 〈rt| qp〉]εp + εq − εr − εt

+2

N/2∑p

√np′np

|fpp′ |2

(εp′ − εp)+

N/2∑q(q 6=p)

fqp 〈pq| p′p′〉εp + εq − 2εp′

+2

N/2∑pr

frpεr − εp

[√np′np〈pr | p′p′

⟩−√

nrnr ′

⟨r ′r ′∣∣ pr〉]

+2

N/2∑pqr(q 6=p)

{√np′np

〈pq| rp′〉〈pr | qp′〉εp + εq − εr − εp′

+

√nrnr ′〈pq| rr〉〈r ′r ′| pq〉2εr − εp − εq

+

√np′np

[〈pq| |rp′〉+ 〈pr | |qp′〉] 〈pq| p′r〉εp + εq − εr − εp′

}

+

N/2∑pq(q 6=p)

√np′nq′npnq

[2 〈pq| p′q′〉 − 〈pq| q′p′〉]2

εp + εq − εp′ − εq′




Dissociation of the helium dimer (aug-cc-pV5Z)

PNOF5 curve is repulsive (no correlation between pairs)

PNOF5-PT2:∣∣Ψsu

pp

⟩are excluded from E (2)

dissociation limit = 2× E(He)→ size-consistency of the method

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0R (Å)

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Ene

rgy

(kca

l/mol

)

PNOF5PNOF5-PT2MP2

Eb = 0.009 kcal/mol

Eb + 1/3Eb = 0.012 kcal/mol ≈ EMP2b

→ Include more orbitals in each geminal:

- better intrapair correlation

- better reference for the PT




Dissociation curves of the hydrogen �uoride (cc-pVTZ)

1 2 3 4R (Å)

-100.3

-100.2

-100.1

-100.0

-99.9

Ene

rgy

(Har

tree

s)

PNOF5PNOF5-PT2PNOF5-SC2-MCPTMP2

De(kcal/mol)

PNOF5 114.5

PNOF5-PT2 134.2

Experiment 141.1

the excitations between the lowest-weakly and highest-strongly

occupied orbitals, which in the limit correspond to the degenerate

orbitals, are removed from the whole dissociation process.




Dissociation of the CO and N2 molecules (cc-pVTZ)

1 2 3 4R (Å)

-250

-200

-150

-100

-50

0

50

Ene

rgy

( kc

al /

mol

)

PNOF5 (CO)PNOF5 (N2)

PNOF5-PT2 (CO)PNOF5-PT2 (N2)

De(N2) De(CO)

PNOF5 239.3 225.6

PNOF5-PT2 221.8 238.0

Experiment 225.1 256.2

PNOF5 shows a wrong order : De(CO) < De(N2).

PNOF5-PT2 recovers the correct order De(N2) < De(CO).

(zero-energy has been set at their corresponding energy at 6 Å)




Selected molecular properties: Re , De , ωe

PNOF5 PNOF5-PT2 Experimental

Mol Re ωe De Re ωe De Rexp ωe De

HF 0.915 4149.3 114.5 0.924 4047.0 134.2 0.917 4138.4 141.1

N2 1.090 2468.7 239.3 1.103 2326.3 221.8 1.098 2358.6 225.1

CO 1.116 2313.8 225.6 1.129 2199.0 238.0 1.128 2169.8 256.2

PNOF5 underestimates Re and overestimates ωe .

PNOF5-PT2 increases the predicted equilibrium bond lengths.

For frequencies, the PNOF5-PT2 values decrease toward the experimentaldata, particularly, in case of multiple bonds.

PNOF5-PT2 dissociation energies have improved substantially over thePNOF5 ones, getting closer to the experimental marks.




Closing Remarks

the generating PNOF5-wavefunction is a two-con�gurationAPSG (GVB or PP).

a computationally tractable second-order perturbation theoryPNOF5-PT2 has been derived from the SC2-MCPT. Thisansatz involves only double excitations from di�erent spatialorbitals to account for the interpair correlation.

future work: better description of the intrapair electron

correlation with an extended version of PNOF5




Acknowledgements

involved in this work:

Jon M. MatxainXabier LopezFernando RuiperezTxema MerceroJesus UgaldeEduard Matito (Girona)

Financial support comes from the Basque Government (S-PC12UN005)

The SGI/IZO-SGIker UPV/EHU is greatfully acknowledged for generousallocation of computational resources.

Thank you for your attention !!!


Documents

PNOF5-PT2 - UPV/EHU€¦ · PNOF5-PT2 A useful method for dealing with strongly correlated systems Mario Piris Kimika Fakultatea, UPV/EHU, and DIPC, .K.P 1072, 20080 Donostia IKERBASQUE,