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