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
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
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 ≡
Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008)
Toru ShiozakiUniversity of Florida
Muneaki KamiyaUniversity of Florida
{ }† †ˆ abijT t a b ji=
{ }( )ˆ† †0 0
Cˆ 0T ab
iji j ba He tΦ Φ = a
ˆ0
TeΨ = Φ
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, ...
Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008)
Toru ShiozakiUniversity of Florida
Muneaki KamiyaUniversity of Florida
{ } ( )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=
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, ...
Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008)
Toru ShiozakiUniversity of Florida
Muneaki KamiyaUniversity of Florida
{ } ( )† † *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
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, ...
Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008)
Toru ShiozakiUniversity of Florida
( )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
′
′′
′
′−′
′ ′= − − −
= − − −
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, ...
ab
Edward F. ValeevVirginia Tech
Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev PCCP (2008); Same team (in preparation)
Toru ShiozakiUniversity of Florida
Muneaki KamiyaUniversity of Florida
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
Edward F. ValeevVirginia Tech
Explicitly-correlated CC Shiozaki, Kamiya, Hirata & Valeev (in preparation)
Toru ShiozakiUniversity of Florida
Muneaki KamiyaUniversity of Florida
Edward F. ValeevVirginia Tech
Ne H2
O F2
The first complete CCSD-R12 results!!
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
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
Fast MP2 for polymers Shimazaki & Hirata (in preparation)
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∝
Fast MP2 for polymers Shimazaki & Hirata (in preparation)
( ) ( )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
Fast MP2 for polymers Shimazaki & Hirata (in preparation)
cost 1∝
0pk =
0i j a bk k k k= = = =Γ
MP2 Accuracy
94%Speed
80
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
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
Muneaki KamiyaUniversity of Florida
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
Fast methods for water clusters Hirata et al. MP (2005)
0.00.0
1.41.0
2.21.7
12.214.4
Binary interaction MP2
Conventional MP2
kcal/mol
Fast methods for water clusters Hirata et al. MP (2005)
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