Coupled-cluster and perturbation methods for … and perturbation methods for macromolecules So Hirata Quantum Theory Project and MacroCenter Departments of Chemistry & Physics, University

Coupled-cluster and perturbation methods for

macromolecules

So HirataQuantum Theory Project

and MacroCenterDepartments of Chemistry & Physics, University of Florida

ContentsAccurate electronic structure methods

for small moleculesAccurate vibrational structure methods

for small molecules

Electronic and vibrational methodsfor polymers

Accurate electronic methodsfor clusters

and molecular crystals



for small molecules




Automated symbolic algebra Hirata JPCA (2003); Hirata TCA (2006); Hirata JP Conf. Ser. (2006)

Definition of a many-electron theory[ ] [ ] [ ] 0000

212121212121 0 ;0 ; ΦΦ=ΦΦ=ΦΦ= +−−+−−+−−C

TTTTabijC

TTTTaiC

TTTT HeeHeeHeeE

Mathematical expressions

A parallel computer program

Implemented methods Hirata JPCA (2003); Hirata TCA (2006); Hirata JP Conf Ser (2006)

CI

PTCC Combined CC+PTCCSD(T)

CCSD(2)T

, CCSD(3)TCCSD(2)TQ

, CCSD(3)TQCCSDT(2)Q

, CR-CCSD(T)Hirata et al. JCP (2004)Shiozaki et al.

(2007)

Linear ExpansionCIS, CISD, CISDT, CISDTQ

Hirata JPCA (2003)

CIS+perturbationCIS(D), CIS(3), CIS(4)

Hirata JCP (2005)

PerturbationMP2,

MP3, MP4Hirata JPCA (2003)Excited State Theories

EOM-CCSDEOM-CCSDT

EOM-CCSDTQHirata JCP (2004)

Ionization TheoryIP-EOM-CCSD

IP-EOM-CCSDTIP-EOM-CCSDTQKamiya & Hirata JCP (2006)

Cluster ExpansionCCD, CCSD, CCSDT,

CCSDTQ, LCCD,LCCSD, QCISD

Hirata JPCA (2003)

Electron Attachment TheoryEA-EOM-CCSD

EA-EOM-CCSDTEA-EOM-CCSDTQ

Kamiya & Hirata (2007)

EOM-CC+perturbationEOM-CCSD(2)T

, EOM-CCSD(2)TQEOM-CCSD(3)T

Shiozaki et al.

(2007)

Implemented methods Hirata JPCA (2003); Hirata TCA (2006); Hirata JP Conf Ser (2006)

CC PT

CI

Active-SpaceCCSDt, CCSDTq, CCSDtq, CISDt, CISDTq, CISDtq

Fan & Hirata JCP

(2006)

EOM-CCSDt, EOM-

CCSDTq, EOM-

CCSDtq, IP-EOM-

CCSDt, IP-EOM-

CCSDTq, IP-EOM-

CCSDtq, EA-EOM-

CCSDt, EA-EOM-

CCSDTq, EA-EOM-

CCSDtq

Fan, Kamiya & Hirata (2007)

Scalar Relativistic ECP+ Spin-Orbit ECP

CC, CI, MBPTEOM-CC, IP-EOM-CCEA-EOM-CC, CIS+PT

CC+PTHirata et al. JCP (2007)

1-Electron PropertiesTransition Moments

CC, CI, MBPTEOM-CC

Hirata

JCP (2004)

Geometries and binding energies Hirata, Yanai, de Jong, Nakajima & Hirao JCP (2004)

Photoelectron and relativity Hirata, Yanai, Harrison, Kamiya & Fan JCP (2007)

Explicitly-correlated CC

e1e21

12r−

( ) ( ), j kij i kf ϕΨ ≈ ∏∑ r r r

( )k kϕΨ ≈ ∏∑ r

Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008)

ˆˆ0

STe +Ψ = Φ

Toru ShiozakiUniversity of Florida

Muneaki KamiyaUniversity of Florida

ˆ0

TeΨ = Φ

{ }† †ˆ klkl ijS F t jiαβ α β= ( )12expklF r klαβ αβ γ= −

{ }† †ˆijS t jiαβ α β={ }† †ˆ ab

ijT t a b ji=

Occupiedi, j, k, ...

Virtuala, b, c, ...

Complementary Complete Virtual α′, β ′, γ ′, ...

Orbital basisp, q, r, ...

Complete Basisκ, λ, μ, ...

Complete Virtual α, β, γ, ...

0abijF ≡




{ }† †ˆ abijT t a b ji=

{ }( )ˆ† †0 0

Cˆ 0T ab

iji j ba He tΦ Φ = a

ˆ0

TeΨ = Φ


Virtuala, b, c, ...








{ } ( )ˆ† † ˆ*0 0

Cˆ 0k kll T S

iji j F He tαβαβ +Φ Φ = a

ˆˆ0

STe +Ψ = Φ

{ }† †ˆ klkl ijS F t jiαβ α β=


Virtuala, b, c, ...








{ } ( )† † *0 0

C

ˆˆˆ 0kl kli

T SjHei j F tαβαβ +Φ Φ = a

{ } { }† † †1ˆ4vH f κλμν

κλ νκ μλ κ λ= +

{ }† †ˆ klkl ijS F t jiαβ α β=

pqijv Fαβ

αβ


Virtuala, b, c, ...







( )112 12

pq pq rs pq k pq kij rs ij k ij k ij

pq pq rs pq k pq krs ij k ij k ijij

a

pq

a

i

a

j

a

v F v F v F v F

r f v F v F v F

v Fαβαβ

κλ α ακλ α α

′

′′

′

′−′

′ ′= − − −

= − − −


Virtuala, b, c, ...





ab

Edward F. ValeevVirginia Tech

Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008); Same team (in preparation)



Diagrammatic evaluation of expectation values

Identification of special intermediates such as f12 /r12

The resolution of the identity insertions

CABS substitutions

Special common subexpression

eliminations

Factorization

Strength reduction

Common subexpression

eliminations

Code synthesis


Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev (in preparation)




Ne H2

O F2

The first complete CCSD-R12 results!!



for small molecules




Anharmonicity in polyethylene Keçeli & Hirata (in preparation)

HF/6-31G* MP2/6-31G*

Murat KeçeliUniversity of Florida

Anharmonicity in polyacetylene Keçeli & Hirata (in preparation)

HF/6-31G* MP2/6-31G*

Murat KeçeliUniversity of Florida

CCSD for polymers Hirata, Bartlett, et al. CPL (2001); Hirata, Bartlett, et al. JCP (2004)

Fast MP2 for polymers Shimazaki & Hirata (in preparation)

2[ ][ ] [ ] [ ]

cell [ ][ ] [ ] [ ], , ,

BZ

, ,

14

ji a b

i ji b

aj a

kk k k

kk k ki j a b i jk bk k a

i j a bE

K e e e e=

+ − −∑∑

i j a bk k k k+ = +112r−

ik jk

ak bk

( )3cost 2K∝

momentum conservation


i j a bk k k k+ = +Normal MP2 Accuracy

100%Speed

1

( )120, , , , ,pK Kk

Ka Ka Ka Kaππ π π−

= ± ± ±K

( )3cost 2K∝


( ) ( )3 314cost 2K∝

i j a bk k k k+ = +(mod 4) MP2 Accuracy

100.2%Speed

11

4 80, , , ,pKk

Ka Ka Kaπ π π

= ± ± < +K


cost 1∝

0pk =

0i j a bk k k k= = = =Γ

MP2 Accuracy

94%Speed

80



for small molecules




Fast methods for water clusters Hirata et al. MP (2005)

( )n n

ij i j ii j i

E E E E E<

′ ′ ′ ′= − − +∑ ∑

Pair energy in the presence of dipole field

Self-consistent dipolesMonomer in dipole field

N-body (N > 2) Coulomb in dipole-dipole approximation

1 and 2-bodyCoulombExchangeCorrelation

Fast methods: ESP+BSSE Kamiya, Hirata & Valiev JCP (2007)

( )n n

ij i j ii j i

E E E E E<

′ ′ ′ ′= − − +∑ ∑

Partial charges thatreproduce the electrostaticpotential around the molecule

N-body (N > 2) Coulomb

with more accurate short-range potentials


Counterpoise BSSE correction

Fast methods for water clusters Hirata et al. MP (2005); Kamiya, Hirata, Valiev JCP (2008)

One-body Two-body Higher

Coulomb Exact Exact >dipole

Exchange Exact Exact Neglected

Correlation Exact Exact Neglected

Total energy 99.999 –

100.001%

Binding energy A few kcal/mol

Excitation energy A few hundredths eV

Accurate

Fast

Systematic

Easy to implement

BSSE correction

Analytical derivatives


0.00.0

1.41.0

2.21.7

12.214.4

Binary interaction MP2

Conventional MP2

kcal/mol


A record

equation-of-motion coupled-

cluster singles and doubles (EOM-CCSD)

calculation with aug-cc-pVDZ

of a 247-

atom cluster

(740 cm–1)

Molecular crystals

Maddox (Nature, 1988): “One of the continuing scandals in the physical sciences is that it remains

in general impossible to predict the structure of even the simplest crystalline solids from a

knowledge of their chemical composition. …

Solids such as crystalline water (ice) are still thought to lie

beyond mortals’

ken.”

Forces and force constants Hirata et al. MP (2005); Hirata (in preparation)

LRn n

ij ji i

i j i

E EE E Ex x x x xx

E

<

′ ′∂ ∂′ ′⎛ ⎞∂ ∂ ∂= − − +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝

∂+

⎠ ∂∑ ∑

2 22 2 2n nij ji i

i j i

E EE E Ex y x y x y x y x y<

⎛ ⎞′ ′∂ ∂′ ′∂ ∂ ∂= − − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠∑ ∑

( )n n

ij i j ii j i

E E E E E<

′ ′ ′ ′= − − +∑ ∑

Crystalline hydrogen fluoride Sode, Keçeli & Hirata (in preparation)

Olaseni

SodeUniversity of Florida

Solid formic acid: energetics Hirata (in preparation)

MP2 aug-cc-pVTZ

BSSE CCSD aug-cc-pVDZ

+0.7 kcal/mol

±0.0 kcal/mol ±0.0 kcal/mol

+2.6 kcal/mol

+0.5 kcal/mol

+2.9 kcal/mol

Solid formic acid: k

= 0

frequencies Hirata (in preparation)

β1

Solid formic acid: phonon dispersion Hirata (in preparation)

β1 β1

Solid formic acid: inelastic neutron Hirata (in preparation)

α β1 β2

Acknowledgements

US Department of EnergyUS National Science FoundationUniversity of Florida Division of Sponsored ResearchHewlett–Packard Company & ACSAmerican Chemical Society PRFJapan Society for the Promotion of Science

Documents

Coupled-cluster and perturbation methods for … and perturbation methods for macromolecules So Hirata Quantum Theory Project and MacroCenter Departments of Chemistry & Physics, University