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).