FCIQMC, (Un)linked Stochastic Coupled Cluster Theory and Other

FCIQMC,(Un)linked Stochastic Coupled Cluster Theory

and Other Animals

Alex Thom

Department of Chemistry, University of Cambridge

ˆ FCIQMC Coupled Cluster Coupled Cluster Monte Carlo $

Alex ThomStochastic Coupled Cluster Theory

Electronic Structure Theory: Scaling

100 101 102 103

# Water molecules

10-6

10-4

10-2

100

102

104

106

108

1010

1012

1014Calculation Time / s

minute

hour

day

week

yeardecade

centurymillennium

239Pu half life

aeon

age of universe

force-field

poor SCF

good SCF

MP2

CCSD

CCSD(T)FCI



Electronic Structure Theory: Accuracy

HF DFT:GGADFT:Hybrid MP2 CCSD CCSD(T) CCSDT CCSDT(Q)Method

−80

−60

−40

−20

0

20

40

60

80

Devia

tion f

rom

experi

ment

/ kJ

/mol

Helgaker T. et al., Molecular Electronic Structure TheoryCurtiss L.A., Raghavachari K., Redfern P.C., Pople, J.A. J. Chem. Phys. 106, 1063 (1997)Klopper W., Helgaker T. J. Phys. B 32, R103 (1999)Klopper W., Ruscic B., Tew D., Bischoff F., Wolfsegger S. Chem. Phys. 356, 14 (2009)Klopper W., Bachorz R., Hattig C., Tew D. Theor. Chem. Acc. 126. 289 (2010)



Determinant Space

Hartree-Fock Triple excitation

I For an N -electron system, N spin-orbitals are chosen out of2M spin-orbitals {φ1, φ2, ..., φ2M} to form a SlaterDeterminant,|Di〉.

I Complete space of determinants is finite, but exponentiallygrowing in N and M as

(2MN

)



Determinant Space

Hartree-Fock Triple excitation

I For an N -electron system, N spin-orbitals are chosen out of2M spin-orbitals {φ1, φ2, ..., φ2M} to form a SlaterDeterminant,|Di〉.

I Complete space of determinants is finite, but exponentiallygrowing in N and M as

(2MN

)ˆ FCIQMC Coupled Cluster Coupled Cluster Monte Carlo $


Full Configuration Interaction

I We can project the many-electron Hamiltonian into the spaceof Slater Determinants and solve in this space.Hij = 〈Di|H|Dj〉.

I Solve H|ΨCI〉 = E|ΨCI〉 as |ΨCI〉 =∑

i ci|Di〉.I Iterative diagonalization of the sparse Hamiltonian in this

space gives the “Full Configuration Interation” (FCI) solution.

I As H contains at most 2-electron operators, matrix elementsbetween determinants are a simple combination of one- andtwo-electron Hamiltonian integrals.

I Energy is variationally minimized — all of the basis setcorrelation energy captured.

I Exponentially scaling with system size (N or M)



Dynamics

I At time τ , given a psip at j, at time τ + δτ spawn a new oneat randomly chosen i based on −δτHij. This samples

∑j.

I Psips at i die based on .

I Start S = EHF . Once enough psips created change S inresponse to population. e.g. for growth, make S morenegative, so more death.

I Opposite-signed psips at the same determinant annihilate.



Dynamics

∂ci∂τ

= −Hiici −∑j→i

Hijcj.


∑j.

I Psips at i die based on .





Dynamics

∂ci∂τ


Hijcj.


∑j.

I Psips at i die based on δτHii.





Dynamics

∂ci∂τ


Hijcj. X


∑j.






Dynamics

∂ci∂τ


Hijcj.


∑j.






Dynamics

∂ci∂τ

= − [Hii − S]ci −∑j→i

Hijcj.


∑j.

I Psips at i die based on δτ(Hii − s). Usually S < EHF .





Dynamics

∂ci∂τ

= − [Hii − S]ci −∑j→i

Hijcj.XXXX


∑j.






Dynamics

∂ci∂τ

= − [Hii − S]ci −∑j→i

Hijcj.


∑j.






Example

I Initialconfiguration

I Spawning

I Death

I Annihilation

I Spawning

I Death

I Annihilation

I Many stepslater



Example


I Spawning

I Death

I Annihilation

I Spawning

I Death

I Annihilation

I Many stepslater

X

X



Example


I Spawning

I Death

I Annihilation

I Spawning

I Death

I Annihilation

I Many stepslater



Example


I Spawning

I Death

I Annihilation

I Spawning

I Death

I Annihilation

I Many stepslater

X

X

X



Example


I Spawning

I Death

I Annihilation

I Spawning

I Death

I Annihilation

I Many stepslater

XX



Example


I Spawning

I Death

I Annihilation

I Spawning

I Death

I Annihilation

I Many stepslater



Performance

I With a system-dependent critical number of psips, FCIQMCreproduces CI energy essentially exactly.

I Critical psip number, Nc, needed to allow the correct signstructure of the wavefunction to develop.

I Below this, energy is wrong. Plateau shows what Nc is.

0 10000 20000 30000 40000 50000iteration

100

101

102

103

104

105

106

107

# psips



The Initiator Approximation

I Only allow psips on determinants with a significant population(e.g. 3+) to spawn to unoccupied determinants.

I Vastly reduces Nc. Introduces systematic, but controllableerror.

I Scaling exponential with increasing system size — even withinitiators.

I Able to calculate energies for previously insoluble systems(larger by many orders of magnitude).

Deidre Cleland, George H. Booth, and Ali Alavi, J. Chem. Phys. 132, 041103 (2010)

George H. Booth, and Ali Alavi, J. Chem. Phys. 132, 174104 (2010)

Deidre Cleland, George Booth, and Ali Alavi, J. Chem. Phys. 134, 024112 (2011)



The exponential Ansatz

I Coupled Cluster Theory uses a cluster operator T =∑

i tiai

I which can be split into excitation levels (and truncated)T = T 1 + T 2 + T 3 + . . .

T 1 =∑ia

tai aai T 2 =

∑i<ja<b

tabij aabij · · ·

I Use an exponential to generate the wavefunction

|ΨCC〉 = eT |D0〉I Exponential generates products of excitors

eT = 1 + T 1 + T 2 + · · ·+ 12 T

21 + 1

2 T 1T 2 + · · ·



The exponential Ansatz

I Coupled Cluster Theory uses a cluster operator T =∑

i tiaiI which can be split into excitation levels (and truncated)T = T 1 + T 2 + T 3 + . . .

T 1 =∑ia

tai aai T 2 =

∑i<ja<b

tabij aabij · · ·

I Use an exponential to generate the wavefunction

|ΨCC〉 = eT |D0〉I Exponential generates products of excitors

eT = 1 + T 1 + T 2 + · · ·+ 12 T

21 + 1

2 T 1T 2 + · · ·



Size Consistency

I Consider a single atom.

I Excitation aabij is present in both CISD and CCSD.

I Consider two separated non-interacting atoms: X and Y.

I Simultaneous excitation aaXbXaY bYiXjX iY jYneeded for same level of

description of each atom.

I Present in CCSD in 12 T

22 product aaXbXiXjX

aaY bYiY jY.

I Only present in CISDTQ level and beyond.

I As number of electrons increases, truncated CI rapidlyworsens.



Size Consistency

1 2 3 4 5 6Truncation Level

10-3

10-2

10-1

100

101

102

103

Error in Energy per atom / kJ/mol

CC1 kcal/mol

1 kcal/mol = 10 meVˆ FCIQMC Coupled Cluster Coupled Cluster Monte Carlo $


Size Consistency


10-3

10-2

10-1

100

101

102

103


CC1 kcal/molNe1 CI



Size Consistency


10-3

10-2

10-1

100

101

102

103


CC1 kcal/molNe1 CI

Ne2 CI



Size Consistency


10-3

10-2

10-1

100

101

102

103


CC1 kcal/molNe1 CI

Ne2 CI

Ne3 CI



CI vs. CC Theory



〈Di|(H − E)|ΨCI〉 = 0



ci − δτ∑j

(Hij − Eδij)cj = ci





CI vs. CC Theory



〈Di|(H − E)|ΨCI〉 = 0



ci − δτ∑j


ci(τ)− δτ∑j

(Hij − Eδij)cj(τ) = ci(τ + δτ)





CI vs. CC Theory



〈Di|(H − E)|ΨCC〉 = 0



ci − δτ∑j


ci(τ)− δτ∑j






CI vs. CC Theory



〈Di|(H − E)|ΨCC〉 = 0




(Hij − Eδij)〈Dj|ΨCC〉 = 〈Di|ΨCC〉

ci(τ)− δτ∑j






Sampling the CC equations

ti(τ)− δτ∑j


.I ∀i, ∀j, calculate 〈Dj|ΨCC(τ)〉 and update ti.

I ∀j, calculate 〈Dj|ΨCC(τ)〉. Now randomly pick i, and updateti.

eT = 1 +∑i

tiai +1

2!

∑i,j

titjaiaj +1

3!

∑i,j,k

titjtkaiajak + . . .

.I Pick a random term in eT and work out j and 〈Dj|ΨCC(τ)〉

from just this term. Now randomly pick i, and update ti.




ti(τ)− δτ∑j


.I ∀i, ∀j, calculate 〈Dj|ΨCC(τ)〉 and update ti.I ∀j, calculate 〈Dj|ΨCC(τ)〉. Now randomly pick i, and updateti.

eT = 1 +∑i

tiai +1

2!

∑i,j

titjaiaj +1

3!

∑i,j,k







ti(τ)− δτ∑j



eT = 1 +∑i

tiai +1

2!

∑i,j

titjaiaj +1

3!

∑i,j,k


.

I Pick a random term in eT and work out j and 〈Dj|ΨCC(τ)〉from just this term. Now randomly pick i, and update ti.




ti(τ)− δτ∑j



eT = 1 +∑i

tiai +1

2!

∑i,j

titjaiaj +1

3!

∑i,j,k






Discretization

I Represent T =∑

i tiai as a population of excips for eachexcitation.

I Need to sample

eT = 1 +∑i

tiai +1

2!

∑i,j

titjaiaj +1

3!

∑i,j,k


I From list of excips randomly select a number of them(remembering signs). Need normalized probabilities.

+ai +b

i +bi −aj −bj +c

j +cj +c

j −abij +acik −abcijk

AJWT Phys. Rev. Lett. 105, 263004 (2010)



Discretization

+acik −bj

→ −aabcijk

I Collapse this into a single excitor and apply to the reference,resulting in a determinant.

−aabcijk |D0〉 = −|Dabcijk 〉

I Spawning: Pick random excitation from |Dabcijk 〉 (e.g. c

i ) and

spawn according to −δτ〈Dci |H|Dabc

ijk 〉. . . . Create new +ci

excip.I Death: Die according to δτ(〈Dabc

ijk |H|Dabcijk 〉 − S). . . . Create

new +abcijk excip.

I Annihilate as in FCIQMC.



Discretization

+acik −bj → −aabcijk




i ) and





new +abcijk excip.




Discretization





i ) and


ijk 〉.

. . . Create new +ci



new +abcijk excip.




Discretization





i ) and



excip.

I Death: Die according to δτ(〈Dabcijk |H|Dabc

ijk 〉 − S). . . . Create

new +abcijk excip.




Discretization





i ) and




ijk |H|Dabcijk 〉 − S).

. . . Create

new +abcijk excip.




Discretization





i ) and





new +abcijk excip.




Discretization





i ) and





new +abcijk excip.

I Annihilate as in FCIQMC.ˆ FCIQMC Coupled Cluster Coupled Cluster Monte Carlo $


Coupled Cluster Monte Carlo



Performance

I Initial excip growth much faster than FCIQMC.

I Critical number of excips required otherwise growth unstable.

I Critical number of excips seems to scale with size of space.e.g. CCSD Nc ∼ N4.

I Able to reproduce large coupled cluster calculations in lesstime.

I Initiator approximation also applicable.

I Cluster is initiator iff all component excips have sufficientamplitude.

I Much fewer excips, so much faster.



H2O cc-pVDZ

−0.24

−0.22

−0.20

−0.18

−0.16

−0.14

−0.12

Ene

rgy

/Har

trees ECCSDTQ

〈E(τ)〉〈E(τ)〉ma

0 1000 2000 3000 4000 5000Iterations

10−6

10−5

10−4

10−3

10−2

10−1

Ene

rgy

Err

or/H

artre

es

|〈E(τ)〉ma − ECCSDTQ||〈E(τ)〉equil − ECCSDTQ|



Ne cc-pVQZ initiator

0 20000 40000 60000 80000 100000 120000 140000 160000# excips

−0.3365

−0.3360

−0.3355

−0.3350

−0.3345

−0.3340

−0.3335

−0.3330C

orr

ela

tion E

nerg

y/H

ar



Scaling

TZ QZ 5Z 6ZBasis

104

105

106

107

108

109

CPU Time / s

minute

day

week

year

decade

millennium

239Pu half life

aeon

age of universe

Full CCSDTQ timeMC CCSDTQ time 105

106

107

108

109

1010

1011

Storage / bytes

Full CCSDTQ memMC CCSDTQ mem



Parallel Scaling

10 20 30 40 50 60 70 80 90 100Cores

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Speedup rel. to 24

Ideal ScalingCCMC



The Uniform Electron Gas

rs = 1

186−1342−1514−11030−14218−1

1/# basis functions

−1.10

−1.05

−1.00

−0.95

−0.90

Corr

ela

tion E

nerg

y/e

lect

ron /

eV n14 iFCIQMC

n = 14 Cutoff 64Ry, M = 4218, triples: 3× 108, full 1034

n = 54 Cutoff 64Ry, M = 4218, triples: 2× 1010, full 10117J. J. Shepherd, G. H. Booth and A. Alavi J. Chem. Phys. 136, 244101 (2012)

J. J. Shepherd, A. Gruneis, G. H. Booth, G. Kresse, and A. Alavi Phys. Rev. B 86, 035111 (2012)




rs = 1

186−1342−1514−11030−14218−1

1/# basis functions

−1.10

−1.05

−1.00

−0.95

−0.90

Corr

ela

tion E

nerg

y/e

lect

ron /

eV n14 iFCIQMC

n14 CCSD







rs = 1

186−1342−1514−11030−14218−1

1/# basis functions

−1.10

−1.05

−1.00

−0.95

−0.90

Corr

ela

tion E

nerg

y/e

lect

ron /

eV n14 iFCIQMC

n14 CCSDn14 iCCSDT







rs = 1

186−1342−1514−11030−14218−1

1/# basis functions

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6C

orr

ela

tion E

nerg

y/e

lect

ron /

eV DMC

n54 iFCIQMC







rs = 1

186−1342−1514−11030−14218−1

1/# basis functions

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6C

orr

ela

tion E

nerg

y/e

lect

ron /

eV DMC

n54 iFCIQMCn54 CCSD MC







rs = 1

186−1342−1514−11030−14218−1

1/# basis functions

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6C

orr

ela

tion E

nerg

y/e

lect

ron /

eV DMC

n54 iFCIQMCn54 CCSD MCn54 iCCSD MC







10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n14 FCI

J. J. Shepherd, G. H. Booth and A. Alavi J. Chem. Phys. 136, 244101 (2012)





10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n14 FCI

n14 MP2

J. J. Shepherd, G. H. Booth and A. Alavi J. Chem. Phys. 136, 244101 (2012)J. J. Shepherd, A. Gruneis, G. H. Booth, G. Kresse, and A. Alavi Phys. Rev. B 86, 035111 (2012)




10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n14 FCI

n14 MP2n14 CCSD





10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n14 FCI

n14 MP2n14 CCSDn14 CCSDT





10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n14 RPA+SOSEX

n14 FCIn14 MP2n14 CCSDn14 CCSDT





10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n54 iCCSD

P. Lopez Rıos, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs Phys. Rev. E 74, 066701 (2006)J. J. Shepherd, A. Gruneis, G. H. Booth and A. Alavi Phys. Rev. B 85, 081103(R) (2012)




10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n54 iCCSD

n54 BF DMC

P. Lopez Rıos, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs Phys. Rev. E 74, 066701 (2006)

J. J. Shepherd, A. Gruneis, G. H. Booth and A. Alavi Phys. Rev. B 85, 081103(R) (2012)




10-1 100 101

rs / a0

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2C

orr

ela

tion E

nerg

y/e

lect

ron /

eV n54 iCCSD

n54 BF DMCn54 FCIQMC

P. Lopez Rıos, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs Phys. Rev. E 74, 066701 (2006)J. J. Shepherd, A. Gruneis, G. H. Booth and A. Alavi Phys. Rev. B 85, 081103(R) (2012)



Linked Coupled Cluster Thoery

〈Di|(H − E)eT |D0〉 = 0. Unlinked

〈Di|e−T (H − E)eT |D0〉 = 0. Linked





〈Di|e−T (H − E)eT |D0〉 = 0. LinkedBaker–Campbell–Hausdorff expansion simplifies this significantly.

e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ] + · · ·






e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ]






e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ]

[[H, T ], T ] = HT T − T HT − T HT + T T H






e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ]

[[H, T ], T ] = HT T − T HT − T HT + T T H[[H, tαaα], tβ aβ] = tαtβ(Haαaβ − aαHaβ − aβHaα + aαaβH






e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ]


〈Di|

tαtβ(hγ aγ aαaβ − aαHaβ − aβHaα + aαaβH|D0〉Pick a single spawning, sampling H.






e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ]


〈Di|tαtβ(hγ aγ aαaβ − aαHaβ − aβHaα + aαaβH|D0〉Pick a single spawning, sampling H. Determines Di.






e−T HeT = H + [H, T ] +1

2![[H, T ], T ] +

1

3![[[H, T ], T ], T ]

+1

4![[[[H, T ], T ], T ], T ]


〈Di|tαtβ(hγ aγ aαaβ − aαhγ aγ aβ − aβhγ aγ aα + aαaβhγ aγ |D0〉Pick a single spawning, sampling H. Determines Di. Use the sameexcitation for H in all terms.




Pros:

I Exact cancellation should reduce plateaux and thus difficulty.

I Energy (=Shift) is decoupled from excitor population.

I Only need clusters of up to four excitors — should scale muchbetter

Cons:

I Far less simple: Lots of terms to code up.



Conclusions

I Stochastic Coupled Cluster (CCMC) behaves very similar toFCIQMC

I Truncated CC allows polynomial scaling spaces to be used.

I Much simpler to implement than deterministic CC.

I Arbitrary truncation (e.g. CCSDTQ5) uses identical code.

I Required number of excips appears to scale with size of space.

I Initiator approximation very successful at reducing criticalnumber of excips.

I Feasible on workstations, and very parallelizable.



Directions

I Solids. Complex excips. (Volunteer applications welcomed!)

I Large-scale parallelization.

I f12-CCMC.

I CASCC.

I Excitation Energies.

I Available in MOLPRO (Alavi group’s NECI code).

I New open-source code, HANDE, now available from ImperialCollege.

I Applications...



Acknowledgements

MAGDALENE COLLEGECAMBRIDGE

I Ali AlaviI George Booth — FCIQMCI Deirdre Cleland — InitiatorsI James Shepherd — FCIQMC UEGI James Spencer — Open Source implementationI Alberto Zoccante — Linked CC



FCIQMC, (Un)linked Stochastic Coupled Cluster Theory and Other

Documents

Transcript of FCIQMC, (Un)linked Stochastic Coupled Cluster Theory and Other