1. Coupled Cluster Calculations using Density Matrix Renormalization Group "like" idea Osamu Hino 1, Tomoko Kinoshita 2 and Rodney J. Bartlett 1 Quantum Theory Project University of Florida 1 Graduate University for Advanced Studies and Institute for Molecular Science, Japan 2

2. 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.

3. 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.

4. Purpose of this study To develop a coupled cluster method which has following 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.

5. Basic idea (1) Pay attention to the “direct product” structure of the cluster operators: (2) Define a pseudo density matrix for the cluster amplitude: (3) Define a small cluster operator and perform CCSDT calculation:

6. Actual implementation (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: (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.

7. CCSDT and CCSDT-1,2,3 methods Approximate treatment for the T3 amplitude CCSDT-n are easier to implement and faster. Operation count scales as CCSD=V 4 O 2, CCSDT-n= V 4 O 3, CCSDT= V 5 O 3 Compressed CCSDT, CCSDT-n= k 2 V 2 O, k 2 V 3 O.

8. Distribution of the normalized probabilities (N 2, cc-pVTZ case) VO=371, k=31. The size of the triple- amplitude becomes less than 1/1000 of the original size.

9. Energies (mE h =0.027eV) at experimental equilibrium geometry (CCSDT-1), k 3 =0.2*O 2 V 2 H2OH2ON2N2 F2F2 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

10. Energies (mE h =0.027eV) at experimental equilibrium geometry (CCSDT-3), k 3 =0.2*O 2 V 2 H2OH2ON2N2 F2F2 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

11. Clock timings per iteration (CCSDT-1/CCSD), k 3 =0.2*O 2 V 2 H2OH2ON2N2 F2F2 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 CCSD Comp. 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 -

12. Clock timings per iteration (CCSDT-3/CCSD), k 3 =0.2*O 2 V 2 H2OH2ON2N2 F2F2 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 -

13. Potenrial Energy Curve (1) (HF, aug-cc-pVDZ, HF-bond stretcing) r(eq)=0.917 angstroms k 3 =0.2*O 2 V 2

14. Potenrial Energy Curve (2) (H 2 O, aug-cc-pVDZ, OH-bonds stretcing) r(eq)=0.957 angstroms k 3 =0.2*O 2 V 2

15. 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.

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