Coupled Cluster Calculations using Density Matrix Renormalization Group

"like" idea

Osamu Hino1, Tomoko Kinoshita2

and Rodney J. Bartlett1

Quantum Theory ProjectUniversity of Florida1

Graduate University for Advanced Studiesand Institute for Molecular Science, Japan2

Background(1)

The CCSD is one of the most successful methods in the field of quantum chemisry. It is size-extensive (so as the many body perturbation theory), numerically accurate and efficient enough to manipulate medium size (about 50 electrons) systems. However, chemists often needs so called chemical accuracy (about 0.05eV error in energy) and CCSD sometimes fails in giving this accuracy. It is well-known that we have to incorporate higher order cluster operators to attain the chemical accuracy.

† † †exact 1 2 HF 1 2

1exp , ,

4a abi ij

ai abij

T T T t a i T t a b ji

CCSD (Coupled Cluster Singles and Doubles)

Background(2)

The CCSDT yields practically almost the same results as the full CI as long as the Hartee-Fock wavefunction is a “good” approximation. But, the computational costs of CCSD and CCSDT are roughly proportional to the 6th and 8th order of the system size. If we want to calculate on a molecule including 50 electrons, CCSDT calculation will take 2500 times as much cpu-time as the CCSD calculation. The CCSDT is impractical unless the system size is modest.

3exact 1 2 HF

† † †1 2

† † †3

,

exp ,

1, ,

4

1

36a abi ij

ai abi

abcijk

kj abcij

T T

T t a i T t a

T

T t a kji jib b c

CCSDT (Coupled Cluster Singles Doubles and Triples)

Purpose of this study

To develop a coupled cluster method which hasfollowing features:

(1) including the triple cluster operator(2) as efficient as CCSD (3) as accurate as CCSDT

These are contradictory to one another. However,we found that there is a way to avoid that

difficulty.The key idea is closely related to that of DMRG.

Basic idea

(1) Pay attention to the “direct product” structure of the cluster operators:

† † † † † †3

1 1( ) ( )

36 36abc abcijk ijk

abcijk abcijk

T t a b c kji t a i b c kj

(2) Define a pseudo density matrix for the cluster amplitude:

ab acd bcd ai bjij ikl jkl X X X

cdkl X

P t t Q Q (3) Define a small cluster operator and perform CCSDT calculation:

†3

1 ˆ ˆ ˆ ˆ, , 36

ka

XYZ iXYZ ai

T t XYZ X Q a i k VO

Actual implementation 2 abc

ijkt(1) Calculate the second order triples:

(2) Calculate the pseudo density matrix (using the sparcity of the amplitude):

(3) Calculate the eigenvalues and eigenvectors within the accuracy required:

2 2 2ab acd bcdij ikl jkl

cdkl

P t t

21 2

1

, , ,...k

ab ai bjij X X X k k

X

P Q Q threshold

(4) Define a small cluster operator and perform CCSDT calculation. If k is significantly less than OV, we can save a lot of computational costs. We call this method the compressed CCSDT.

CCSDT and CCSDT-1,2,3 methodsApproximate treatment for the T3 amplitude

1 2 3

1 2

2

HF

HF

HF

1 2 HF

e CCSDT

e CCSDT-3

e CCSDT-2

1 CCSDT-1

, F

T T Tabc abc abcijk ijk ijk

c

T Tabc abc abcijk ijk ijk

c

Tabc abc abcijk ijk ijk

c

abc abc abcijk ijk ijk c

abcijk i j k a b c

D t H

D t H

D t H

D t H T T

D H H F

ock operator

CCSDT-n are easier to implement and faster.Operation count scales asCCSD=V4O2 , CCSDT-n= V4O3 , CCSDT= V5O3

Compressed CCSDT, CCSDT-n= k2V2O, k2V3O.

Distribution of the normalized probabilities(N2, cc-pVTZ case)

0

0.02

0.04

0.06

0.08

0.1

0 10 20 30 40 50

PROBABILITIES

PR

OB

AB

ILIT

IES

k

VO=371, k=31.The size of the triple- amplitude becomes less than 1/1000 of the original size.

1 1

0.9431k VO

X XX X

Energies (mEh=0.027eV) at experimental equilibrium geometry (CCSDT-1), k3=0.2*O2V2

H2O N2 F2

cc-pVDZ

HF

CCSD

Comp. CCSDT-1

CCSDT-1

CCSDT

-76026.795

-213.290

-216.488

-216.416

-216.505

-108954.153

-313.041

-326.501

-325.488

-325.039

-198685.670

-406.392

-415.275

-415.552

-415.747

cc-pVTZ

HF

CCSD

Comp. CCSDT-1

CCSDT-1

CCSDT

-76057.163

-280.834

-289.293

-288.887

-288.674

-108983.507

-397.508

-418.178

-417.355

-417.302

-198752.043

-550.165

-568.579

-568.979

-569.521

Energies (mEh=0.027eV) at experimental equilibrium geometry (CCSDT-3), k3=0.2*O2V2

H2O N2 F2

cc-pVDZ

HF

CCSD

CCSDT-1

Comp.CCSDT-3

CCSDT-3

CCSDT

-76026.795

-213.290

-216.416

-216.357

-216.192

-216.505

-108954.153

-313.041

-325.488

-325.363

-324.139

-325.039

-198685.670

-406.392

-415.552

-414.687

-414.795

-415.747

cc-pVTZ

HF

CCSD

CCSDT-1

Comp.CCSDT-3

CCSDT-3

CCSDT

-76057.163

-280.834

-288.887

-288.777

-288.262

-288.674

-108983.507

-397.508

-417.355

-416.593

-415.430

-417.302

-198752.043

-550.165

-568.979

-567.335

-567.421

-569.521

Clock timings per iteration (CCSDT-1/CCSD) , k3=0.2*O2V2

H2O N2 F2

cc-pVDZ

CCSD

Comp. CCSDT-1

CCSDT-1

CCSDT

0.484(s)

1.000

1.107

4.168

57.470

2.082(s)

1.000

1.197

4.723

114.094

2.037(s)

1.000

1.274

6.462

155.666

cc-pVTZ

CCSDComp. CCSDT-1 CCSDT-1

CCSDT

22.667(s)

1.000

0.959

4.033

77.372

28.671(s)

1.000

1.143

12.670

295.034

49.750(s)

1.000

1.147

15.066

-

Clock timings per iteration (CCSDT-3/CCSD) , k3=0.2*O2V2

H2O N2 F2

cc-pVDZ

CCSD

CCSDT-1

Comp. CCSDT-3

CCSDT-3

CCSDT

0.484(s)

1.000

4.168

2.736

6.630

57.470

2.082(s)

1.000

4.723

3.483

7.635

114.094

2.037(s)

1.000

6.462

3.710

8.056

155.666

cc-pVTZ

CCSD

CCSDT-1

Comp. CCSDT-3

CCSDT-3

CCSDT

22.667(s)

1.000

4.033

2.737

10.447

77.372

28.671(s)

1.000

12.670

3.252

16.917

295.034

49.750(s)

1.000

15.066

3.376

18.854

-

Potenrial Energy Curve (1) (HF, aug-cc-pVDZ, HF-bond stretcing)

r(eq)=0.917 angstromsk3=0.2*O2V2

-300

-250

-200

-150

-100

-50

0

0.5 1 1.5 2 2.5 3 3.5 4

MRCICCSDCCSDT-1CCSDT-3COMP.SDT-1COMP.SDT-3

Ele

ctro

nic

En

ergy

+10

0000

(mE

h)

bond length

Potenrial Energy Curve (2) (H2O, aug-cc-pVDZ, OH-bonds stretcing)

r(eq)=0.957 angstromsk3=0.2*O2V2

-300

-200

-100

0

100

0.5 1 1.5 2 2.5 3 3.5

MRCICCSDCCSDT-1CCSDT-3COMP.SDT-1COMP.SDT-3

Ele

ctro

nic

En

ergy

+76

000(

mE

h)

bond length

Summary

Inclusion of the compressed triples into the CC method is very effective. Computational costs are much reduced without losing accuracy.

In some cases, the compressed CCSDT-n methods are more stable under the deformed the geometry even when the parent CCSDT-n methods become unstable.

Now, implementing compressed CCSDT, CCSDTQ-1.

Acknowledgement

This work was supported in part by U. S. Army Research Office under MURI (Contract No. AA-5-72732-B1).