NEW COUPLED-CLUSTER METHODS
FOR MOLECULAR POTENTIAL ENERGY SURFACES:
I. GROUND-STATE APPROACHES
Piotr Piecuch
Department of Chemistry, Michigan State University,East Lansing, Michigan 48824
P. Piecuch and K. Kowalski, in: Computational Chemistry: Reviews of Current Trends, edited by
J. Leszczynski (World Scientific, Singapore, 2000), Vol. 5, pp. 1-104; K. Kowalski and P. Piecuch,
J. Chem. Phys. 113, 18-35 (2000); 113, 5644-5652 (2000); J. Molec. Struct.: THEOCHEM 547,
191-208 (2001); Chem. Phys. Lett. 344, 165-175 (2001); P. Piecuch, S.A. Kucharski, and K.
Kowalski, Chem. Phys. Lett. 344, 176-184 (2001); P. Piecuch, S.A. Kucharski, V. Spirko, and K.
Kowalski, J. Chem. Phys. 115, 5796-5804 (2001); P. Piecuch, K. Kowalski, and I.S.O. Pimienta,
Int. J. Mol. Sci., submitted (2001); P. Piecuch and K. Kowalski, Int. J. Mol. Sci., submitted
(2001); K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966-2978 (2001); P. Piecuch, K.
Kowalski, I.S.O. Pimienta, and S.A. Kucharski, in: Accurate Description of Low-Lying Electronic
States and Potential Energy Surfaces, ACS Symposium Series, Vol. XXX, edited by M.R. Hoffmann
and K.G. Dyall (ACS, Washington, D.C., 2002), pp. XXX-XXXX; K. Kowalski and P. Piecuch,
J. Chem. Phys., submitted (2001); P. Piecuch, S.A. Kucharski, and V. Spirko, J. Chem. Phys.
111, 6679-6692 (1999); K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 8490-8502 (2000); 115,
643-651 (2001); K. Kowalski and P. Piecuch, Chem. Phys. Lett. 347, 237-246 (2001).
1
The “holy grail” of the ab initio electronicstructure theory:
The development of simple, “black-box,” and affordable methods that can pro-
vide highly accurate (∼ spectroscopic) description of ENTIRE GROUND- AND
EXCITED-STATE POTENTIAL ENERGY SURFACES
Na...FHNa+HF
NaF+H
V/cm-1
14000
12000
10000
8000
6000
4000
2000
0
R(Na-F)/Ao 4.543.532.52
R(F-H)/Ao1.4 1.2 1 0.8 0.6
4 6 8 10 12 14
1.52.5
3.54.5
10
8
6
4
2
0
R(Na-F)/bohr
R(F-H)/bohr
Energy/eV
Na F + H+ -
Na FH..Na + FH
Na + FH*
Examples of applications:
• dynamics of reactive collisions
• highly excited and metastable ro-vibrational states of molecules
• rate constant calculations
• collisional quenching of electronically excited molecular species
Motivation:
• elementary processes that occur in combustion (e.g., reactions involving
OH and NxOy)
• collisional quenching of the OH and other radical species
IN THIS PRESENTATION, WE FOCUS ON NEW
“BLACK-BOX” COUPLED-CLUSTER METHODS FOR
GROUND-STATE POTENTIAL ENERGY SURFACES
2
SINGLE-REFERENCE COUPLED-CLUSTER (CC) THEORY
(J. Cızek, 1966, 1969; J. Cızek and J. Paldus, 1971)
|Ψ〉 = eT(A)|Φ〉, T (A) =
mA∑
k=1
Tk
T1|Φ〉 =∑
ia
tia|Φai 〉, T2Φ =
∑
i > ja > b
tijab|Φabij 〉, Tk|Φ〉 =
∑
i1 > i2 > · · · > ika1 > a2 > · · · > ak
ti1i2...ika1a2...ak|Φa1a2...ak
i1i2...ik〉
mA = N – exact theorymA < N – approximate methods
mA = 2 T = T1 + T2 CCSD n2on
4u (n2
on2u)
mA = 3 T = T1 + T2 + T3 CCSDT n3on
5u (n3
on3u)
mA = 4 T = T1 + T2 + T3 + T4 CCSDTQ n4on
6u (n4
on4u)
PROBLEMS WITH THE STANDARD CC
APPROXIMATIONS
(T (A) =∑mA
k=1 Tk, mA < N)
Example: N2
[K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 5644 (2000); K. Kowalski and P. Piecuch, Chem.
Phys. Lett. 344, 165 (2001); P. Piecuch, S.A. Kucharski, and K. Kowalski, ibid. 344, 176 (2001)]
2.0 3.0 4.0 5.0 6.0R F−F (bohr)
−199.12
−199.10
−199.08
−199.06
−199.04
−199.02
−199.00
Ene
rgy
(har
tree)
CCSDCCSD(T)CCSD(TQf)CCSDT (practically exact)
1.5 2.0 2.5 3.0 3.5 4.0 4.5R N−N (bohr)
−109.3
−109.2
−109.1
−109.0
−108.9
−108.8
−108.7
−108.6
Ene
rgy
(har
tree)
CCSDCCSD(T)CCSD(TQf)CCSDT(Qf)Full CICCSDT
3
Existing solutions
• Multi-Reference CC Methods (Jeziorski, Monkhorst, Paldus, Piecuch,Bartlett, Mukherjee, Lindgren, Kaldor et al.)[also, Multi-Reference CI and MBPT Approaches; cf. the presentationby Professor Mark S. Gordon]
• State-Selective, Active-Space CC Methods (e.g., the CCSDt and CCSDtqapproaches of Piecuch et al.)
• Externally-Corrected CC Methods (e.g., the RMRCCSD approach of Pal-dus and Li)
THE PROPOSED RESEARCH FOCUSES ON METHODSTHAT COMBINE THE SIMPLICITY OF THE STANDARDSINGLE-REFERENCE CC APPROACHES, SUCH ASCCSD(T), WITH THE EFFICIENCY WITH WHICH THEMULTI-REFERENCE METHODS DESCRIBE GROUND-STATE POTENTIAL ENERGY SURFACES
SPECIFIC GOALS
• New CC methods for ground-state potential energy surfaces:
– method of moments of CC equations
– renormalized CC approaches
4
METHOD OF MOMENTS OF COUPLED-CLUSTER
EQUATIONS: A NEW APPROACH TO THE
MANY-ELECTRON CORRELATION PROBLEM
A new relationship in quantum-mechanical theory of
many-fermion (many-electron) systems
δ = E − E(A) = Λ[Ψ;M(j)J (mA), j > mA]
E(A) – the electronic energy obtained using the approximate coupled-
cluster calculations (e.g., CCSD)
E – the exact energy (the exact eigenvalue of the electronic Hamil-
tonian)
Ψ – the exact wave function
M(j)J (mA) - the generalized moments of coupled-cluster equations
EMMCC = E(A) + δMMCC
MMCC - the method of moments of coupled-cluster equations - provides us
with fundamentally new ways of performing electronic structure calculations
and renormalizing the failing standard tools of quantum chemistry.
5
The MMCC Energy Formula (the Ground-State Problem)
δ = E − E(A) =N∑
n=mA+1
n∑
j=mA+1
〈Ψ|QnCn−j(mA)Mj(mA)|Φ〉/〈Ψ|eT (A)|Φ〉
Mj(mA) =(HNe
T (A))C,j, Cn−j(mA) =
(eT
(A))n−j
Mj(mA)|Φ〉 = Qj
(HNe
T (A))C|Φ〉 =
∑
J
M(j)J (mA) |Φ(j)
J 〉
M(j)J (mA) = 〈Φ(j)
J |(HNe
T (A))C|Φ〉 − generalized moments of the CC equations
Approximate MMCC Methods: The MMCC(mA,mB)
Approaches
E(mA,mB) = E(A) +
mB∑
n=mA+1
n∑
j=mA+1
〈Ψ|QnCn−j(mA)Mj(mA)|Φ〉/〈Ψ|eT (A)|Φ〉
Various approximate forms of |Ψ〉 lead to different classes of MMCC(mA,mB) schemes.
6
The MMCC(2,3), MMCC(2,4), and MMCC(3,4) Methods
E(2, 3) = ECCSD + 〈Ψ|Q3M3(2)|Φ〉/〈Ψ|eT1+T2|Φ〉
E(2, 4) = ECCSD + 〈Ψ|Q3M3(2) +Q4 [M4(2) + T1M3(2)]|Φ〉/〈Ψ|eT1+T2|Φ〉
E(3, 4) = ECCSDT + 〈Ψ|Q4M4(3)|Φ〉/〈Ψ|eT1+T2+T3|Φ〉
M3(2)|Φ〉 = Q3
(HNe
T1+T2)C|Φ〉 =
∑
i<j<k
a<b<c
Mabcijk(2)|Φabc
ijk〉
M4(2)|Φ〉 = Q4
(HNe
T1+T2)C|Φ〉 =
∑
i<j<k<l
a<b<c<d
Mabcdijkl (2)|Φabcd
ijkl 〉
M4(3)|Φ〉 = Q4
(HNe
T1+T2+T3)C|Φ〉 =
∑
i<j<k<l
a<b<c<d
Mabcdijkl (3)|Φabcd
ijkl 〉
Mabcijk(2) = 〈Φabc
ijk|(HNe
T1+T2)C|Φ〉
Mabcdijkl (2) = 〈Φabcd
ijkl |(HNe
T1+T2)C|Φ〉
Mabcdijkl (3) = 〈Φabcd
ijkl |(HNe
T1+T2+T3)C|Φ〉
7
The MMCC(2,3)/CISDT, MMCC(2,3)/CISDt,
MMCC(2,4)/CISDTQ, and MMCC(2,4)/CISDtq Methods
MMCC(2,3)/CISDT
|ΨCISDT〉 = (1 + C1 + C2 + C3) |Φ〉
MMCC(2,3)/CISDt (CISDt = active-space CISDT)
|ΨCISDt〉 =
[1 + C1 + C2 + C3
(abCIjk
)]|Φ〉
MMCC(2,4)/CISDTQ
|ΨCISDTQ〉 = (1 + C1 + C2 + C3 + C4) |Φ〉
MMCC(2,4)/CISDtq (CISDtq = active-space CISDTQ)
|ΨCISDtq〉 =
[1 + C1 + C2 + C3
(abCIjk
)+ C4
(abCDIJkl
)]|Φ〉
8
The Renormalized and Completely Renormalized CCSD[T],
CCSD(T), CCSD(TQ), and CCSDT(Q) Methods
MBPT(2)-like choices of Ψ lead to the renormalized (R) and completely renormalized
(CR) CCSD[T], CCSD(T), CCSD(TQ), CCSDT(Q), etc. approaches. Here are some
examples of these methods:
CR-CCSD(T) ← completely renormalized CCSD(T)
|ΨCCSD(T)〉 =(
1 + T1 + T2 +R(3)0 (VNT2)C +R
(3)0 VNT1
)|Φ〉
ECR−CCSD(T) = ECCSD + 〈ΨCCSD(T)|Q3M3(2)|Φ〉/〈ΨCCSD(T)|eT1+T2|Φ〉
R-CCSD(T) ← renormalized CCSD(T)
M3(2) ≈ (VNT2)C,3
ER−CCSD(T) = ECCSD + 〈ΨCCSD(T)|Q3 (VNT2)C |Φ〉/〈ΨCCSD(T)|eT1+T2|Φ〉↓
1.0 → CCSD(T)
CR-CCSD(TQ)
|ΨCCSD(TQ)〉 = |ΨCCSD(T)〉+ 12T
22 |Φ〉
ECR−CCSD(TQ) = ECCSD + 〈ΨCCSD(TQ)|{Q3M3(2)
+ Q4 [M4(2) + T1M3(2)]}|Φ〉/〈ΨCCSD(TQ)|eT1+T2|Φ〉
9
CR-CCSDT(Q)
|ΨCCSDT(Q)〉 =(1 + T1 + T2 + T3 + T1T2 + 1
2T22
)|Φ〉
ECR−CCSDT(Q) = ECCSDT + 〈ΨCCSDT(Q)|Q4M4(3)|Φ〉/〈ΨCCSDT(Q)|eT1+T2+T3|Φ〉
The (C)R-CCSD[T], (C)R-CCSD(T), (C)R-CCSD(TQ), and (C)R-CCSDT(Q)
methods can be viewed as the extensions of the standard CCSD[T], CCSD(T),
CCSD(TQf), and CCSDT(Qf) methods, respectively.
The computer costs of the (C)R-CCSD[T], (C)R-CCSD(T), (C)R-CCSD(TQ), and
(C)R-CCSDT(Q) calculations are the same as the costs of the corresponding stan-
dard CCSD[T], CCSD(T), (C)R-CCSD(TQ), and (C)R-CCSDT(Q) calculations.
For example, the cost of calculating the noniterative (T) correction of the standard
CCSD(T) approximation is kn3on
4u. The costs of calculating the noniterative (T)
corrections of the R-CCSD(T) and CR-CCSD(T) methods are kn3on
4u and 2kn3
on4u,
respectively.
10
Representative Results
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0R H−F (bohr)
−99.95
−100.00
−100.05
−100.10
−100.15
Ene
rgy
(har
tree
)
CR−CCSD[T]CR−CCSD(T)CCSDCCSD[T]CCSD(T)Full CI
HF
2.0 3.0 4.0 5.0 6.0R F−F (bohr)
−199.12
−199.10
−199.08
−199.06
−199.04
−199.02
−199.00 CCSDCCSD(T)CCSD(TQf)CR−CCSD(T)CR−CCSD(TQ)CCSDT (practically exact)
F2
1.5 2.0 2.5 3.0 3.5 4.0 4.5R N−N (bohr)
−109.3
−109.2
−109.1
−109.0
−108.9
−108.8
−108.7
−108.6CCSDCCSD(T)CCSD(TQf)CR−CCSD(TQ)Full CICCSDT
N2
Selected vibrational energies G(v) (v is the vibrational quantum number) and dissociationenergies De (in cm−1) of the HF molecule as described by the aug-cc-pVTZ basis set.The RKR represent total energies and all theoretically computed energies represent errorsrelative to RKR.
v RKR CCSD CCSDT CCSD(T) CCSD(TQf) R-CCSD(T) CR-CCSD(T) CR-CCSD(TQ)
0 2051 15 -7 -7 -7 -4 -3 -61 6012 52 -19 -18 -17 -9 -4 -162 9802 96 -28 -25 -23 -9 -2 -223 13424 144 -36 -32 -29 -9 2 -275 20182 252 -54 -47 -43 -6 12 -37
10 34363 623 -116 -136 -124 1 49 -7011 36738 728 -131 -175 -159 1 60 -7712 38955 850 -148 -232 -211 -2 72 -8413 41007 993 -166 -9 87 -9115 44576 1370 -207 -55 123 -10919 49027 2881 -325 227 -159
De 49362 5847 -453 207 -156
11
The PES for the Be + FH→ BeF + H Reaction
2 3 4 5 6 7 81.2
2
3
4
5
6
7
8
Full CI Results
R (bohr)Be−F
R
(
bohr
)F
−H
−1.05
−1.02
−0.99
−0.96
−0.931
−0.92
−0.905
−0.89
−0.8
−0.5
2 3 4 5 6 7 81.2
2
3
4
5
6
7
8
2 3 4 5 6 7 81.2
2
3
4
5
6
7
8
CCSD(T) Results
R (bohr)Be−F
−1.05
−1.02
−0.99
−0.96
−0.931
−0.92
−0.905
−0.89
−0.8
−0.5
2 3 4 5 6 7 81.2
2
3
4
5
6
7
8
2 3 4 5 6 7 81.2
2
3
4
5
6
7
8
CR−CCSD(T) Results
R (bohr)Be−F
R
(
bohr
)F
−H
−1.05
−1.02
−0.99
−0.96
−0.931
−0.92
−0.905
−0.89
−0.8
−0.5
2 3 4 5 6 7 81.2
2
3
4
5
6
7
8
Differences with Full CI for CCSD(T)
−0.027
−0.024
−0.021
−0.018
−0.015
−0.012
−0.009
−0.006
−0.003
0
0.003
Ene
rgy
(har
tree
)
−0.03
−0.02
−0.01
0
0.01
R (bohr)F−H
23
45
67
8 R (bohr)Be−F
2345678
Differences with Full CI for CR−CCSD(T)
−0.027
−0.024
−0.021
−0.018
−0.015
−0.012
−0.009
−0.006
−0.003
0
0.003
Ene
rgy
(har
tree
)
−0.03
−0.02
−0.01
0
0.01
R (bohr)F−H
23
45
67
8 R (bohr)Be−F
2345678
12
Future Work (Methods and Algorithms, Ground-State
Problem)
• Incorporation of the standard and renormalized CCSD(T),
CCSD(TQ), and CCSDT(Q) methods in GAMESS (years 1
and 2)
• Development of the ground-state MMCC schemes with the
non-perturbative choices of Ψ (years 1–3)
• Extensions of the MMCC and renormalized CC methods to
open-shell states and reference configurations of the ROHF
type (years 2 and 3)
•Work with Professor Mark S. Gordon and coworkers on par-
allelizing the MMCC and renormalized CC methods within
GAMESS (years 2 and 3)
• Personnel: 3 (PI, 1 postdoc, 1 student)
• Present computer resources: 2- and 32-CPU Origin systems
at MSU
• Collaborations: Professor Mark S. Gordon and coworkers at
Iowa State University and Ames Laboratory; also, Professor
Stanis law A. Kucharski (Silesian University)
• Expected computer needs: 55,000 MPP hours at NERSC
13