^{1,a)}

### Abstract

Interaction energies for seven weakly bound dimers involving helium, argon, water, and methane are computed using large correlation-consistent basis sets augmented with bond functions. The estimates of the coupled-cluster singles, doubles, and noniterative triples [CCSD(T)] complete basis set limit are obtained using both the conventional approach and several variants of the explicitly correlated CCSD(T)-F12 method. It is shown that both bond functions and the F12 approach significantly speed up the convergence of the CCSD(T)/aug-cc-pV*X*Z interaction energies with the basis set cardinal number *X*. However, the extent of improvement provided by each technique varies with the character of the interactions—the F12 method works best for polar, electrostatics-bound dimers, while for dispersion-dominated complexes the addition of bond functions is more efficient. The convergence rate afforded by different coupled-cluster variants is fairly consistent across the entire attractive region of the potential curve, while the improvement provided by the F12 correction increases along the repulsive wall. The use of large basis sets and the agreement between conventional and explicitly correlated approaches allow us to assess the importance of different residual approximations present in the popular CCSD(T)-F12 implementations.

This work has been supported by the Auburn University startup funding.

I. INTRODUCTION

II. DETAILS OF THE COMPUTATIONAL PROCEDURE

III. NUMERICAL RESULTS AND DISCUSSION

A. Potential energy curves

B. Sensitivity of interactionenergies to the geminal exponent β

IV. SUMMARY

### Key Topics

- Basis sets
- 34.0
- Rotational correlation time
- 10.0
- Weak interactions
- 10.0
- Water energy interactions
- 8.0
- Polymers
- 6.0

## Figures

Performance of the CCSD(T) and CCSD(T)-F12b approaches for the radial potential energy curve of the He–CH_{4} complex passing through the global minimum. The values displayed are relative to the benchmark interaction energy (estimated from the scaled-triples CCSD(T)-F12c/aQZM calculation) and normalized by the absolute deviation of the CCSD(T)/aQZ interaction energy from the benchmark at a given intermolecular distance *R*.

Performance of the CCSD(T) and CCSD(T)-F12b approaches for the radial potential energy curve of the He–CH_{4} complex passing through the global minimum. The values displayed are relative to the benchmark interaction energy (estimated from the scaled-triples CCSD(T)-F12c/aQZM calculation) and normalized by the absolute deviation of the CCSD(T)/aQZ interaction energy from the benchmark at a given intermolecular distance *R*.

Performance of the CCSD(T) and CCSD(T)-F12b approaches for the radial potential energy curve of the CH_{4}–CH_{4} complex passing through the global minimum. The values displayed are relative to the benchmark interaction energy (estimated from the scaled-triples CCSD(T)-F12c/aQZM calculation) and normalized by the absolute deviation of the CCSD(T)/aQZ interaction energy from the benchmark at a given intermolecular distance *R*.

Performance of the CCSD(T) and CCSD(T)-F12b approaches for the radial potential energy curve of the CH_{4}–CH_{4} complex passing through the global minimum. The values displayed are relative to the benchmark interaction energy (estimated from the scaled-triples CCSD(T)-F12c/aQZM calculation) and normalized by the absolute deviation of the CCSD(T)/aQZ interaction energy from the benchmark at a given intermolecular distance *R*.

Performance of the CCSD(T) and CCSD(T)-F12b approaches for the radial potential energy curve of the H_{2}O–H_{2}O complex passing through the global minimum. The values displayed are relative to the benchmark interaction energy (estimated from the scaled-triples CCSD(T)-F12c/aQZM calculation) and normalized by the absolute deviation of the CCSD(T)/aQZ interaction energy from the benchmark at a given intermolecular distance *R*.

Performance of the CCSD(T) and CCSD(T)-F12b approaches for the radial potential energy curve of the H_{2}O–H_{2}O complex passing through the global minimum. The values displayed are relative to the benchmark interaction energy (estimated from the scaled-triples CCSD(T)-F12c/aQZM calculation) and normalized by the absolute deviation of the CCSD(T)/aQZ interaction energy from the benchmark at a given intermolecular distance *R*.

Dependence of the MP2-F12 (left panels) and CCSD(T)-F12b (right panels) interaction energies on the geminal exponent β for the He–CH_{4} (top) and CH_{4}–H_{2}O (bottom) dimers. The MP2-F12 calculations use the 3C(FIX) *Ansatz* just like the CCSD(T)-F12b approach. The triples term in CCSD(T)-F12b is not scaled.

Dependence of the MP2-F12 (left panels) and CCSD(T)-F12b (right panels) interaction energies on the geminal exponent β for the He–CH_{4} (top) and CH_{4}–H_{2}O (bottom) dimers. The MP2-F12 calculations use the 3C(FIX) *Ansatz* just like the CCSD(T)-F12b approach. The triples term in CCSD(T)-F12b is not scaled.

## Tables

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the He–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −34.34 ± 0.07 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the He–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −34.34 ± 0.07 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the Ar–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −139.52 ± 0.12 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the Ar–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −139.52 ± 0.12 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the He–CH_{4} complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −29.43 ± 0.08 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the He–CH_{4} complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −29.43 ± 0.08 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the Ar–CH_{4} complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −141.16 ± 0.41 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the Ar–CH_{4} complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −141.16 ± 0.41 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the H_{2}O–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −1745.0 ± 1.2 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the H_{2}O–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −1745.0 ± 1.2 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the CH_{4}–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −354.8 ± 0.5 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the CH_{4}–H_{2}O complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −354.8 ± 0.5 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the CH_{4}–CH_{4} complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −187.30 ± 0.30 cm^{−1}.

CCSD(T)/a*X*Z and CCSD(T)-F12b/a*X*Z interaction energies (in cm^{−1}) for the minimum geometry of the CH_{4}–CH_{4} complex as functions of the basis set cardinal number *X*. The extrapolated value (rows “ext.”) in the *X* column is computed using interaction energies in bases a(*X* − 1)Z and a*X*Z. The midbond functions are chosen as hydrogenic functions from the same a*X*Z orbital basis set. The benchmark CCSD(T)/CBS interaction energy amounts to −187.30 ± 0.30 cm^{−1}.

Mean unsigned errors (MUE, in cm^{−1}) of different CCSD(T)/CCSD(T)-F12 variants and basis sets. The errors are averaged over the van der Waals minimum geometries for the seven dimers considered in this work. The benchmark CCSD(T)/CBS interaction energies have been obtained as described in Sec. III .

Mean unsigned errors (MUE, in cm^{−1}) of different CCSD(T)/CCSD(T)-F12 variants and basis sets. The errors are averaged over the van der Waals minimum geometries for the seven dimers considered in this work. The benchmark CCSD(T)/CBS interaction energies have been obtained as described in Sec. III .

Mean unsigned relative errors (MURE, in percent) of different CCSD(T)/CCSD(T)-F12 variants and basis sets. The errors are averaged over the van der Waals minimum geometries for the seven dimers considered in this work. The benchmark CCSD(T)/CBS interaction energies have been obtained as described in Sec. III .

Mean unsigned relative errors (MURE, in percent) of different CCSD(T)/CCSD(T)-F12 variants and basis sets. The errors are averaged over the van der Waals minimum geometries for the seven dimers considered in this work. The benchmark CCSD(T)/CBS interaction energies have been obtained as described in Sec. III .

Mean unsigned relative errors (MURE, in percent) of the conventional and explicitly correlated MP2, CCSD, and (T) contributions to the interaction energy. The errors are averaged over the van der Waals minimum geometries for the seven dimers considered in this work. The benchmark CBS values have been obtained as described in Sec. III . The F12a and F12b triples corrections are identical.

Mean unsigned relative errors (MURE, in percent) of the conventional and explicitly correlated MP2, CCSD, and (T) contributions to the interaction energy. The errors are averaged over the van der Waals minimum geometries for the seven dimers considered in this work. The benchmark CBS values have been obtained as described in Sec. III . The F12a and F12b triples corrections are identical.

Median unsigned relative errors (MeURE, in percent) of different CCSD(T)/CCSD(T)-F12 variants and basis sets for the radial interaction energy curves of all seven complexes. The benchmark CCSD(T)/CBS interaction energies have been computed at the scaled-triples CCSD(T)-F12c/aQZM level.

Median unsigned relative errors (MeURE, in percent) of different CCSD(T)/CCSD(T)-F12 variants and basis sets for the radial interaction energy curves of all seven complexes. The benchmark CCSD(T)/CBS interaction energies have been computed at the scaled-triples CCSD(T)-F12c/aQZM level.

MP2-F12 and CCSD(T)-F12b interaction energies (in cm^{−1}) for the near-minimum geometry of the He–CH_{4} complex, computed using several different choices of the correlation factor. The calculations in the first three columns employed a single Slater-type correlation factor fitted to six GTGs. The geminal exponent β was fixed at 1.0 (first column), taken as the recommended value from Ref. ^{ 86 } (like in most calculations in this work, second column), or chosen to minimize the MP2-F12 interaction energy (third column). The calculations in the column marked “Optimized GTGs” utilized six GTGs with even-tempered exponents and coefficients obtained by minimizing the MP2-F12 interaction energy as described in the text. The diagonal 3C(FIX) *Ansatz* and the same DF/RI bases as in the rest of this work were used. The numbers in parentheses are the values of β. The benchmark MP2/CBS and CCSD(T)/CBS interaction energies amount to −22.283 and −29.425 cm^{−1}, respectively.

MP2-F12 and CCSD(T)-F12b interaction energies (in cm^{−1}) for the near-minimum geometry of the He–CH_{4} complex, computed using several different choices of the correlation factor. The calculations in the first three columns employed a single Slater-type correlation factor fitted to six GTGs. The geminal exponent β was fixed at 1.0 (first column), taken as the recommended value from Ref. ^{ 86 } (like in most calculations in this work, second column), or chosen to minimize the MP2-F12 interaction energy (third column). The calculations in the column marked “Optimized GTGs” utilized six GTGs with even-tempered exponents and coefficients obtained by minimizing the MP2-F12 interaction energy as described in the text. The diagonal 3C(FIX) *Ansatz* and the same DF/RI bases as in the rest of this work were used. The numbers in parentheses are the values of β. The benchmark MP2/CBS and CCSD(T)/CBS interaction energies amount to −22.283 and −29.425 cm^{−1}, respectively.

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