Transcript Slide 1
CC and CI in terms that even a Physicist can understand
Karol Kowalski
William R Wiley Environmental Molecular Sciences Laboratory and Chemical Sciences Division, Pacific Northwest National Laboratory
2
How it started
Coester & Kummel (1958,1960) Čižek (1966) Paldus & Čižek (1971) Bartlett Monkhorst Mukherjee Lindgren Kutzelnigg … and many others
3
CC reviews
J. Paldus, X. Li, “A critical assessment of coupled cluster methods in quantum chemistry,” Advances in Chemical Physics 110, 1 (1999).
R.J. Bartlett, M. Musial, “Coupled-cluster theory in quantum chemistry,” Reviews of Modern Physics 79, 291 (2007).
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What we want to solve
H
E
Molecular/Atomic Physics, Quantum Chemistry (electronic Schrödinger equations) Many Particle Systems Solid State Physics Nuclear Physics
Exact solution of Schrödinger equation Weyl formula (dimensionality of full configuration interaction space) – exact solution of Schrödinger equation
f
(
n
,
N
,
S
) 2
n S
1 1
N n
/ 1 2
S
N
/
n
2 1
S
1 n – total number of orbitals N – total number of correlated electrons S – spin of a given electronic state C 2 molecule : 12 electrons, 100 orbitals : # FCI config.
10 17 !
!
!
Efficient approximations are needed
5
Approximate wavefunction (WF) methods
Hartree-Fock method (single determinant) E HF is used to define the correlation energy E In molecules E HF E=E-E HF but without E making any reliable predictions is impossible accounts for 99% of total energy Correlated methods (going beyond single determinant description) Configuration interaction method (linear parametrizaton of WF) Perturbative methods (MBPT-n) Coupled Cluster methods and many other approaches
7
Many-Fermion Systems
Creation/annihilation operators {
a
,
a
} 0 {
a
,
a
} 0 {
a
,
a
} Indices & designate the one particle states: in chemistry spinorbitals Second quantized form of the Hamiltonian (welcome to the Fock space)
H
E
0 ,
h
a
a
1 4 , , ,
v
a
a
a
a
F=
n
1
H n
n
1 (
H
1 ...
H
1 )
A n times
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Wick Theorem
The basic tool in deriving CC equations
M
M
1
M
2 ...
M k
N
[
M
1
M
2 ...
M k
]
N
[
M
1
M
2 ...
M k
]
a
a
In normal product of the operator string M (N[M]) all the creations operator are permuted to the left of all annihilation operators, attaching (+/-) phase depending on the parity of the required permutation.
Commutator of two operators A & B represented by connected diagrams only
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Particle-hole formalism
Special form of the Bogoliubov-Valatin transformation (choosing a new Fermi Vacuum)
b
a a
if if
i a b
a a
if if
i a b
0 Slater determinant i,j,k,… occupied single particle states a,b,c, …. unoccupied single particle states
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CC and CI methods
CI formalism (
D
0
D
1
D
2 ( 1
C
1
C
2 ...
...
C N D
)
N
) N stands for the number of electrons
C n
1 (
n
!
) 2
i
1 ...
a
1 ...
i n a n c a
1
i
1 ...
...
i n a n
a a
1 ...
a a
n a i n
...
a i
1
a a
1 ...
a a
n a i n
...
a i
1
a
1 ...
a n i
1 ...
i n
reference function (HF determinant) Intermediate normalization 1
C
2
i a
j b ij c ab
CC and CI methods
CC method
e T
Intermediate normalization
T
T
1
T
2
T
3 ...
T N
1
T n
1 (
n
!
) 2
i
1
a
1 ...
...
i n a n t a
1 ...
i
1 ...
i n a n
a a
1 ...
a a n a i n
...
a i
1 cluster amplitudes ( 1
T
1 2 !
T
2 1 3 !
T
3 ...
1
N
!
T N
) For fermions the expansion for e T terminates (Pauli principle) 11
CI and CC methods
Full CI and full CC expansions are equivalent (and this is the only case when CI=CC) ( 1
C
)
e T
C C
2 1
T
1
T
2 1 2
T
1 2
C C
3 4
T
3
T
1
T
2
T
4
T
1
T
3 1 6
T
1 3 1 2
T
2 2 1 2
T
1 2
T
2 1 24
T
1 4 ...
CI amplitudes are calculated from the variational principle while the cluster amplitudes are obtained from projective methods 12
CC formalism
Working equations:
e
T
|
He T
Ee T
e
T He T
E
From Campbell-Hausdorff formula
e
B Ae B
A
[
A
,
B
] 1 2 [[
A
,
B
],
B
] 1 3 !
[[[
A
,
B
],
B
],
B
] ...
We get
e
T He T
{
H
[
H
,
T
] 1 2 [[
H
,
T
],
T
] 1 3 !
[[[
H
,
T
],
T
],
T
] 1 4 !
[[[[
H
,
T
],
T
],
T
],
T
]} (
He T
)
C
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CC formalism
Separating the equations for cluster amplitudes from the equation for energy 1
P
Q P
Q
n
1 ,...,
N i
1
a
1 ...
...
i n a n
a
1 ...
i
1 ...
i n a n
a
1 ...
a n i
1 ...
i n Q
(
He T
)
C
0
E
(
He T
)
C
Step1: we solve energy independent equations for cluster amplitudes Step 2 :having cluster amplitudes we Can calculate the energy 14
Approximations: CCD
CC with doubles (CCD):
T
T
2
ij ab
(
He T
2 )
C
0
i
j
;
a
b
ij ab
(
H
HT
2 1 2
HT
2 2 )
C
0
i
j
;
a
b E CCD
(
He T
2 )
C
E CCD
(
H
( 1
T
2 ))
C
E HF
(
HT
2 )
C
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Approximations: CCD
H N
H
H
F N
V N F N
,
f
N
[
a
a
]
V N
1 4 , , ,
v
N
[
a
a
a
a
]
v
|| 16
Approximations: CCSD
CC with singles and doubles (CCSD):
T
T
1
T
2
i a
(
He T
1
T
2 )
C
0
i
;
a
ij ab
(
He T
1
T
2 )
C
0
i
j
;
a
b E CCSD
(
He T
1
T
2
E HF
)
C
(
HT
2 1 2
HT
1 2 )
C
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CCSD and Thouless Theorem
Thouless theorem & two Slater determinants
e T
1 CCSD wavefunction
CCSD
e T
1
T
2
e T
2
e T
1 1
e T
2 0 CCSD provides better description of the static correlation effects (than the CCD approach) 18
CC approximations: CCSDT
T
CC with singles, doubles, and triples (CCSDT):
T
1
T
2
T
3
i a
(
He T
1
T
2
T
3 )
C
ab ij
(
He T
1
T
2
T
3 )
C
0
i
;
a
0
i
j
;
a
b
abc ijk
(
He T
1
T
2
T
3 )
C
0
i
j
k
;
a
b
c E CCSDT
E
(
He T
1
T
2 )
C HF
(
HT
2 1 2
HT
1 2 )
C
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CC and Perturbation Theory (Linked Cluster Theorem)
Linked Cluster Theorem states: Perturbative expansion for the energy is expressed in terms of closed (having no external lines) connected diagrams only Perturbative expansion for the wavefunction is epxressed in terms of linked diagrams (having no disconnected closed part) only
E
....
....
....
....
T
2 ( 1 )
T
2 ( 1 )
T
2 ( 1 )
T
2 ( 3 ) Cluster operator T is represented by connected diagrams only 20
CC and Perturbation Theory
Enable us to categorize the importance of particular cluster amplitudes
T
1
T
2
T
1 ( 2 ) ...
T
2 ( 1 ) ...
T
3
T
3 ( 2 ) ...
T
4
T
4 ( 3 ) ...
Enable us to express higher-order contributions through lower-order contribution (CCSD(T))
T
3
R
3 ( 0 )
V N T
2
E CCSD
(
T
)
E CCSD
E
[ 4 ]
E
[ 5 ] 21
CCSD(T) method
Driving force of modern computational chemistry (ground-state problems) Belongs to the class of non-iterative methods Enable to reduce the cost of the inclusion of triple excitations to n
o 3 n u 4 (N 7
) : required triply excited amplitudes can be generated on-the-fly.
Storage requirements as in the CCSD approach 22
Size-consistency of the CC energies A
R H
H A
H B
A
B
B
Cluster operator is represented by the connected diagrams only:
T
T A
T B
e T A e T B
A
B E
E A
E B
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Numerical cost Method
CCSD CCSD(T) CCSDT CCSDTQ
Numerical Complexity
N 6 N 7 N 8 N 10
Global Memory Requirements
N 4 N 4 N 6 N 8
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Equation-of-Motion Coupled Cluster Methods: Excited-State CC extension
0
e T
cluster operator reference function (HF determinant)
K
R K e T
“excitation” operator
H R K
E K R K
H R K
E K R K
H
e
T He T
similarity transformed Hamiltonian
Equation-of-Motion Coupled Cluster Methods: Excited-State CC extension
EOMCCSD: singly-excited states
K EOMCCSD
(
R K
,
o
R K
, 1
R K
, 2 )
e T
1
T
2 EOMCCSDT: singly and doubly excited states
K EOMCCSDT
(
R K
,
o
R K
, 1
R K
, 2
R K
, 3 )
e T
1
T
2
T
3 Perturbative methods: EOMCCSD(T) formulations 26
CC methods: across the energy and spatial scales CC methods can be universally applied across energy and spatial scales!
Bartlett, Musial Rev. Mod. Phys. (2007) Dean, Hjorth-Jensen, Phys. Rev. B (2004)
Performance of the CC methods
K. Kowalski,D.J. Dean, M. Hjorth-Jensen, T. Papenbrock, P. Piecuch, PRL 92, 132501 (2004) 28
Performance of the CC method
29 R.J. Bartlett Mol. Phys. 108, 2905 (2010).
Performance of the CC methods
Bartlett & Musial, Rev. Mod. Phys.
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Illustrative examples of large-scale excited-state calculations – components of light harvesting systems 1 L a
5,5
state POL1 basis set
5 4,5 4 3,5 3 2,5 2 1,5 1 2
Expt.
EOMCCSD CR-EOMCCSD(T)
3 4 5
Number of rings
6 7
Functionalization of porphyrines System Leading excitations
H L, H-1 L+1 H-1 L, H L+1 H L, H-1 L+1, H-2 L+2, H-3 L+3
CR-EOMCCSD(T) (eV)
2.32 (Expt. 2.27 eV) 1.86 (Expt. 1.91 eV) 1.91 (Expt. 1.84 eV) H L 1.78
H L 1.36
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K. Kowalski, S. Krishnamoorthy, O. Villa, J.R. Hammond, N. Govind, J. Chem. Phys. 132, 154103 (2010); K. Kowalski, R.M. Olson, S. Krishnamoorthy, V. Tipparaju, E. Apra, J. Chem. Theory Comput. 7, 2200 (2011)
Multiscale Approaches: localized excited states in extended systems
Localized excited-states in materials catalysis photocatalytic decomposition of organic pollutants photolysis of water solar energy conversion
Visible Light Photoresponse of pure and N doped TiO 2 (active-space EOMCCSD calculations, 400 correlated electrons): TiO 2 N-doped TiO 2 EOMCCSd
EOMCCSd
3.84 eV 2.79 eV
33 N. Govind, K. Lopata, R. Rousseau, A. Andersen, K. Kowalski, J. Phys. Chem. Lett.
“Visible Light Absorption of N-Doped TiO 2 Rutile Using (LR/RT)-TDDFT and Active Space EOMCCSD Calculations,” J. Phys. Chem. Lett. 2, 2696 (2011).
Why CC method is so popular in computational chemistry (and less popular in physics) ???
Simpler form of the interactions (1/r) CC functionalities are available in many quantum chemistry packages ACES III (parallel) CFOUR (some pieces in parallel) DALTON (serial) GAMESS (CCSD/CCSD(T) – parallel) Gaussian (serial) MOLPRO (parallel) NWCHEM (parallel) PQS (CCSD/CCSD(T) – parallel)
Tensor Contraction Engine (TCE)
Highly parallel codes are needed in order to apply the CC theories to larger molecular systems Symbolic algebra systems for coding complicated tensor expressions: Tensor Contraction Engine (TCE)
Parallel performance Parallel structure of the TCE CC codes
Tile structure: S 1 S 2 … S 1 S 2
Occupied spinorbitals
… Tensor structure: S 1 S 2 ……….
S 1
unccupied spinorbitals
S 2 ……….
T a i
T
[ [
h m p n
] ]
Parallel performance
An example of the scalability of the triples part of the CR EOMCCSD(T) approach for GFPC described by the cc-pVTZ basis set (648 basis set functions). Timings were determined from calculations on the Franklin Cray-XT4 computer at NERSC using 1024, 16384, 20000, 24572, and 34008 cores).
Scalability of the non-iterative EOMCC code 94 %parallel efficiency using 210,000 cores
Scalability of the triples part of the CR- EOMCCSD(T) approach for the FBP-f-coronene system in the AVTZ basis set. Timings were determined from calculations on the Jaguar Cray XT5 computer system at NCCS.
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Scalability of the iterative EOMCC methods
Alternative task schedulers use “global task pool” improve load balancing reduce the number of synchronization steps to absolute minimum larger tiles can be effectively used 39
Towards future computer architectures
The CCSD(T)/Reg-CCSD(T) accelerators codes have been rewritten in order to take advantage of GPGPU Preliminary tests show very good scalability of the most expensive N
7
part of the CCSD(T) approach
Concluding remark
If you know the nature of the interactions in your system there is a good chance that the CC methods will give you the right results for the right reasons (assuming you have an access to a large computer)
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