^{1,a)}, Garnet Kin-Lic Chan

^{1,b)}, Frederick R. Manby

^{2,c)}, Martin Schütz

^{3,d)}and Hans-Joachim Werner

^{4,e)}

### Abstract

We extend the orbital-specific-virtual tensor factorization, introduced for local Møller-Plesset perturbation theory in Ref. [J. Yang, Y. Kurashige, F. R. Manby and G. K. L. Chan, J. Chem. Phys.134, 044123 (2011)10.1063/1.3528935], to local coupled cluster singles and doubles theory (OSV-LCCSD). The method is implemented by modifying an efficient projected-atomic-orbital local coupled cluster program (PAO-LCCSD) described recently, [H.-J. Werner and M. Schütz, J. Chem. Phys.135, 144116 (2011)10.1063/1.3641642]. By comparison of both methods we find that the compact representation of the amplitudes in the OSV approach affords various advantages, including smaller computational time requirements (for comparable accuracy), as well as a more systematic control of the error through a single energy threshold. Overall, the OSV-LCCSD approach together with an MP2 correction yields small domain errors in practical calculations. The applicability of the OSV-LCCSD is demonstrated for molecules with up to 73 atoms and realistic basis sets (up to 2334 basis functions).

G.K.C. acknowledges support from the Department of Energy (DOE), Office of Science Award DE-FG02-07ER46432. M.G.S. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG). H.J.W. acknowledges support from the DFG within the SimTech Cluster of Excellence at the University of Stuttgart.

I. INTRODUCTION

II. THEORY

A. Definition of the CCSD wavefunction

B. General transformations for the virtual orbitals

C. Projected atomic orbitals (PAOs)

D. Pair natural orbitals (PNOs)

E. Orbital specific virtuals (OSVs)

F. The OSV-LCCSD residuals

G. OSV-LCCSD integrals

H. Local pair approximations

III. BENCHMARK CALCULATIONS

A. Dependence of the correlation energy and computational cost on the domain sizes

B. Dependence of the correlation energy and computational cost on pair approximations

C. Reaction energies

IV. ILLUSTRATIVE APPLICATIONS

A. Barrier height of PHBH

B. Intermolecular interactions

V. CONCLUSIONS

### Key Topics

- Tensor methods
- 12.0
- Rotational correlation time
- 9.0
- Coupled cluster
- 8.0
- Integral transforms
- 8.0
- Integral equations
- 7.0

## Figures

Fraction of the OSV-LCCSD and OSV-LCCSD + ΔMP2 correlation energies relative to the CCSD value as a function of the average domain size for pyrazole (basis aVTZ, see text). No pair approximations are applied.

Fraction of the OSV-LCCSD and OSV-LCCSD + ΔMP2 correlation energies relative to the CCSD value as a function of the average domain size for pyrazole (basis aVTZ, see text). No pair approximations are applied.

The PHBH reaction.

The PHBH reaction.

The G-C dimer structures: Watson-Crick (left) and Stacked (right).

The G-C dimer structures: Watson-Crick (left) and Stacked (right).

## Tables

Average pair domain sizes AVD (including redundant functions), correlation energies (in *E* _{ h }), computation times (in min), and file sizes (in GB) for various molecules and domain selection thresholds. The percentage of correlation energy relative to the canonical CCSD value is given in parenthesis. All pairs are included in the LCCSD. Basis set: hydrogen atoms cc-pVTZ, other atoms aug-cc-pVTZ. Calculations were carried out on a single core Xeon X5690 @ 3.47 GHz. Timings for the complete LCCSD calculations (including integral evaluation and transformations, 10 iterations).

Average pair domain sizes AVD (including redundant functions), correlation energies (in *E* _{ h }), computation times (in min), and file sizes (in GB) for various molecules and domain selection thresholds. The percentage of correlation energy relative to the canonical CCSD value is given in parenthesis. All pairs are included in the LCCSD. Basis set: hydrogen atoms cc-pVTZ, other atoms aug-cc-pVTZ. Calculations were carried out on a single core Xeon X5690 @ 3.47 GHz. Timings for the complete LCCSD calculations (including integral evaluation and transformations, 10 iterations).

Average pair domain sizes AVD (including redundant functions), correlation energies (in *E* _{ h }), computation times (in min), and file sizes (in GB) for various molecules. The percentage of correlation energy relative to the canonical CCSD value is given in parenthesis. Pairs are classified using wck=321 (see text). Basis set: hydrogen atoms cc-pVTZ, other atoms aug-cc-pVTZ. Calculations were carried out on a single core Xeon X5690 @ 3.47 GHz. Timings for the complete LCCSD calculations (including integral evaluation and transformations, 10 iterations).

Average pair domain sizes AVD (including redundant functions), correlation energies (in *E* _{ h }), computation times (in min), and file sizes (in GB) for various molecules. The percentage of correlation energy relative to the canonical CCSD value is given in parenthesis. Pairs are classified using wck=321 (see text). Basis set: hydrogen atoms cc-pVTZ, other atoms aug-cc-pVTZ. Calculations were carried out on a single core Xeon X5690 @ 3.47 GHz. Timings for the complete LCCSD calculations (including integral evaluation and transformations, 10 iterations).

Comparison of OSV and PAO correlation energies relative to the canonical CCSD correlation energy for linear polyglycine chains (in percent). The pair selection thresholds *w* *ck* (see text) are varied. The aVTZ basis set has been used. Average domain size (AVD) includes redundant functions.

Comparison of OSV and PAO correlation energies relative to the canonical CCSD correlation energy for linear polyglycine chains (in percent). The pair selection thresholds *w* *ck* (see text) are varied. The aVTZ basis set has been used. Average domain size (AVD) includes redundant functions.

Comparison of OSV and PAO computational resources for linear polyglycine chains. CPU times are user times, disk space is the total disk space used by the calculations. The pair selection thresholds *w* *ck* (see text) are varied. The aVTZ basis set has been used. *N* _{AO} is the number of basis functions (CGTOs), *N* _{ el } is the number of correlated electrons.

Comparison of OSV and PAO computational resources for linear polyglycine chains. CPU times are user times, disk space is the total disk space used by the calculations. The pair selection thresholds *w* *ck* (see text) are varied. The aVTZ basis set has been used. *N* _{AO} is the number of basis functions (CGTOs), *N* _{ el } is the number of correlated electrons.

Average pair domain sizes AVD,^{a} correlation energies (in *E* _{ h }), computation times (in min), and file sizes (in GB) for various domain selection thresholds for penicillin, using the TZVPP basis set and *wck* = 321. The relative contribution of the MP2 correction (in percent) is given in parenthesis. Calculations were carried out on a single core Xeon X5690 @ 3.47 GHz. Timings for the complete LCCSD calculations (including integral evaluation and transformations, 11 iterations)

Average pair domain sizes AVD,^{a} correlation energies (in *E* _{ h }), computation times (in min), and file sizes (in GB) for various domain selection thresholds for penicillin, using the TZVPP basis set and *wck* = 321. The relative contribution of the MP2 correction (in percent) is given in parenthesis. Calculations were carried out on a single core Xeon X5690 @ 3.47 GHz. Timings for the complete LCCSD calculations (including integral evaluation and transformations, 11 iterations)

Maximum (MAX) and root mean square (RMS) errors of PAO-LCCSD and OSV-LCCSD calculations (in kJ mol^{−1}) relative to canonical CCSD values for the 52 reactions of Ref. 46 for different values of the domain selection threshold THR. The VTZ-F12 basis set has been used. ΔMP2=MP2−LMP2 is a correction for domain errors. The LMP2 values are computed with the same domains as the corresponding LCCSD values.

Maximum (MAX) and root mean square (RMS) errors of PAO-LCCSD and OSV-LCCSD calculations (in kJ mol^{−1}) relative to canonical CCSD values for the 52 reactions of Ref. 46 for different values of the domain selection threshold THR. The VTZ-F12 basis set has been used. ΔMP2=MP2−LMP2 is a correction for domain errors. The LMP2 values are computed with the same domains as the corresponding LCCSD values.

Maximum (MAX) and root mean square (RMS) errors of PAO-LCCSD and OSV-LCCSD calculations relative to canonical CCSD/CBS values^{a} for the 52 reactions of Ref. 46 for different values of the domain selection threshold THR. The VTZ-F12 basis set has been used. ΔMP2=MP2/CBS−LMP2 is a correction for both basis set and domain errors. The LMP2 values are computed with the same domains as the corresponding LCCSD values.

Maximum (MAX) and root mean square (RMS) errors of PAO-LCCSD and OSV-LCCSD calculations relative to canonical CCSD/CBS values^{a} for the 52 reactions of Ref. 46 for different values of the domain selection threshold THR. The VTZ-F12 basis set has been used. ΔMP2=MP2/CBS−LMP2 is a correction for both basis set and domain errors. The LMP2 values are computed with the same domains as the corresponding LCCSD values.

OSV-LMP2 and OSV-LCCSD barrier heights for PHBH, using the aVTZ basis set and two different localization methods. The energy difference contains the MM contribution of −2.8 kcal mol^{−1} (see text). The Hartree-Fock value is 38.74 kcal mol^{−1}. The structures correspond to snapshopt 3 in Ref. 71.

OSV-LMP2 and OSV-LCCSD barrier heights for PHBH, using the aVTZ basis set and two different localization methods. The energy difference contains the MM contribution of −2.8 kcal mol^{−1} (see text). The Hartree-Fock value is 38.74 kcal mol^{−1}. The structures correspond to snapshopt 3 in Ref. 71.

Counterpoise-corrected LCCSD/aVTZ interaction energies in (kcal mol^{−1}) for G-C base pairs. The BSSE estimate according to the CP correction is given in parenthesis. The appended letters (W), (C), and (S) specify the treatment of the intermolecular pairs (see text). Note that the CP correction is identical for (W), (C), and (S).

Counterpoise-corrected LCCSD/aVTZ interaction energies in (kcal mol^{−1}) for G-C base pairs. The BSSE estimate according to the CP correction is given in parenthesis. The appended letters (W), (C), and (S) specify the treatment of the intermolecular pairs (see text). Note that the CP correction is identical for (W), (C), and (S).

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