^{1,2,3}, F. R. Manby

^{4}, M. D. Towler

^{5,6,7}and D. Alfè

^{1,2,3,5}

### Abstract

We present a detailed study of the energetics of waterclusters (H_{2}O)_{ n } with *n* ⩽ 6, comparing diffusion Monte Carlo (DMC) and approximate density functional theory(DFT) with well converged coupled-cluster benchmarks. We use the many-body decomposition of the total energy to classify the errors of DMC and DFT into 1-body, 2-body and beyond-2-body components. Using both equilibrium cluster configurations and thermal ensembles of configurations, we find DMC to be uniformly much more accurate than DFT, partly because some of the approximate functionals give poor 1-body distortion energies. Even when these are corrected, DFT remains considerably less accurate than DMC. When both 1- and 2-body errors of DFT are corrected, some functionals compete in accuracy with DMC; however, other functionals remain worse, showing that they suffer from significant beyond-2-body errors. Combining the evidence presented here with the recently demonstrated high accuracy of DMC for ice structures, we suggest how DMC can now be used to provide benchmarks for larger clusters and for bulk liquid water.

The work of M.D.T. is supported by EPSRC Grant No. EP/I0301311, and that of F.R.M. by EPSRC Grant No. EP/F0002191. DMC calculations were performed on the Oak Ridge Leadership Computing Facility, located in the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy (DOE) under contract DE-AC05-00OR22725 (USA).

I. INTRODUCTION

II. TECHNIQUES

III. DMC AND DFT COMPARED WITH BENCHMARKS

A. The monomer

B. The dimer

1. The stationary points

2. A thermal sample of dimer configurations

C. The trimer

D. Thermal samples of the tetramer, pentamer and hexamer

E. Stable isomers of the hexamer

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Density functional theory
- 129.0
- Polymers
- 41.0
- Water energy interactions
- 18.0
- Total energy calculations
- 13.0
- Basis sets
- 12.0

## Figures

Errors of DFT and DMC distortion energy of the H_{2}O monomer for a thermal sample of 100 configurations (see text). Quantities shown are deviations of calculated energies from Partridge-Schwenke benchmark values with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses), and with DMC (black squares) plotted against the PS distortion energy itself. Units: m*E* _{h}.

Errors of DFT and DMC distortion energy of the H_{2}O monomer for a thermal sample of 100 configurations (see text). Quantities shown are deviations of calculated energies from Partridge-Schwenke benchmark values with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses), and with DMC (black squares) plotted against the PS distortion energy itself. Units: m*E* _{h}.

Errors of DMC and DFT approximations relative to CCSD(T) benchmarks for total energies of thermal sample of 198 dimer configuration, plotted vs O–O distance. Symbols represent PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds), PBE0 (red crosses) and DMC (black squares). Units: m*E* _{h}.

Errors of DMC and DFT approximations relative to CCSD(T) benchmarks for total energies of thermal sample of 198 dimer configuration, plotted vs O–O distance. Symbols represent PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds), PBE0 (red crosses) and DMC (black squares). Units: m*E* _{h}.

Errors of DFT approximations for total energies of thermal sample of 198 dimer configurations when 1-body part is corrected by replacing the DFT 1-body energy by the essentially exact Partridge-Schwenke function. As in Fig. 2, errors are relative to CCSD(T) benchmarks and are plotted vs O–O distance. Symbols represent PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses). Errors of DMC (black squares) are shown for comparison. Units: m*E* _{h}.

Errors of DFT approximations for total energies of thermal sample of 198 dimer configurations when 1-body part is corrected by replacing the DFT 1-body energy by the essentially exact Partridge-Schwenke function. As in Fig. 2, errors are relative to CCSD(T) benchmarks and are plotted vs O–O distance. Symbols represent PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses). Errors of DMC (black squares) are shown for comparison. Units: m*E* _{h}.

Errors of DFT and DMC total energy of the H_{2}O trimer for a thermal sample of 50 configurations drawn from a classical simulation of liquid water (see text). Quantities shown are deviations of calculated energies from CCSD(T) benchmark energies near the basis-set limit, with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses), and with DMC (black squares) plotted against the benchmark energy itself. Units: m*E* _{h}.

Errors of DFT and DMC total energy of the H_{2}O trimer for a thermal sample of 50 configurations drawn from a classical simulation of liquid water (see text). Quantities shown are deviations of calculated energies from CCSD(T) benchmark energies near the basis-set limit, with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses), and with DMC (black squares) plotted against the benchmark energy itself. Units: m*E* _{h}.

Errors of DFT and DMC total energy of the H_{2}O pentamer for a thermal sample of 25 configurations drawn from a classical simulation of liquid water (see text). Quantities shown are deviations of calculated energies from CCSD(T) benchmark energies near the basis-set limit, with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses), and with DMC (black squares) plotted against the benchmark energy itself. Units: m*E* _{h}.

Errors of DFT and DMC total energy of the H_{2}O pentamer for a thermal sample of 25 configurations drawn from a classical simulation of liquid water (see text). Quantities shown are deviations of calculated energies from CCSD(T) benchmark energies near the basis-set limit, with PBE (black pluses), BLYP (purple triangles), B3LYP (green diamonds) and PBE0 (red crosses), and with DMC (black squares) plotted against the benchmark energy itself. Units: m*E* _{h}.

Comparison of DFT values of 2-body energies (upper panel) and 3-body energies (lower panel) of four isomers of the H_{2}O hexamer with benchmark values from CCSD(T). Numbering of isomers is prism: 1, cage: 2, book: 3, ring: 4. Units: m*E* _{h}.

Comparison of DFT values of 2-body energies (upper panel) and 3-body energies (lower panel) of four isomers of the H_{2}O hexamer with benchmark values from CCSD(T). Numbering of isomers is prism: 1, cage: 2, book: 3, ring: 4. Units: m*E* _{h}.

## Tables

Mean values and rms fluctuations of DMC and DFT errors of monomer energy for a thermal sample of 100 configurations (see text). The Partridge-Schwenke energy function is used as the “exact” energy, and the energy zero for DMC and DFT approximations is taken to be the energy in the PS equilibrium geometry. Units: m*E* _{h}.

Mean values and rms fluctuations of DMC and DFT errors of monomer energy for a thermal sample of 100 configurations (see text). The Partridge-Schwenke energy function is used as the “exact” energy, and the energy zero for DMC and DFT approximations is taken to be the energy in the PS equilibrium geometry. Units: m*E* _{h}.

Comparison of DMC energies and DFT energies given by the PBE, BLYP, B3LYP, and PBE0 functionals with CCSD(T) benchmarks for the 10 stationary points of the H_{2}O dimer. For each set of energies, the zero of energy has been taken so that the energy of the global minimum geometry is equal to zero. Numbering of stationary points follows that of previous authors (see, e.g., Ref. 31). Energy units: m*E* _{h}.

Comparison of DMC energies and DFT energies given by the PBE, BLYP, B3LYP, and PBE0 functionals with CCSD(T) benchmarks for the 10 stationary points of the H_{2}O dimer. For each set of energies, the zero of energy has been taken so that the energy of the global minimum geometry is equal to zero. Numbering of stationary points follows that of previous authors (see, e.g., Ref. 31). Energy units: m*E* _{h}.

DMC and DFT mean and rms fluctuation of errors of the total energy for thermal sample of 198 dimer configurations. In the case of DFT, the approximations denoted by PBE-Δ_{1} etc. represent the total energy after correction for errors of the 1-body energy (see text). Units: m*E* _{h}.

DMC and DFT mean and rms fluctuation of errors of the total energy for thermal sample of 198 dimer configurations. In the case of DFT, the approximations denoted by PBE-Δ_{1} etc. represent the total energy after correction for errors of the 1-body energy (see text). Units: m*E* _{h}.

Deviations of energies given by corrected DFT approximations away from CCSD(T) benchmark energies for thermal samples of (H_{2}O)_{ n } 3 ⩽ *n* ⩽ 6 configurations. Notations DFT, DFT-Δ_{1}, and DFT-Δ_{12} indicate DFT approximations uncorrected, corrected for 1-body errors, and corrected for both 1- and 2-body errors. Each entry gives the mean deviation, with rms fluctuation of the deviation in parentheses. Mean and rms deviations of DMC energies away from CCSD(T) benchmarks are shown for comparison. Energy units: m*E* _{h} per water monomer.

Deviations of energies given by corrected DFT approximations away from CCSD(T) benchmark energies for thermal samples of (H_{2}O)_{ n } 3 ⩽ *n* ⩽ 6 configurations. Notations DFT, DFT-Δ_{1}, and DFT-Δ_{12} indicate DFT approximations uncorrected, corrected for 1-body errors, and corrected for both 1- and 2-body errors. Each entry gives the mean deviation, with rms fluctuation of the deviation in parentheses. Mean and rms deviations of DMC energies away from CCSD(T) benchmarks are shown for comparison. Energy units: m*E* _{h} per water monomer.

Total energies of selected isomers of the water hexamer relative to that of the prism, calculated by different methods. In all cases, the geometry of the isomer is the relaxed geometry given by MP2 calculations with the AVTZ basis, as given in the supplementary information of Ref. 61. All energies were calculated in the present work, except for the MP2 and CCSD(T) energies marked with † from Ref. 37 and the DMC energies from Ref. 61. Entries DFT-*n* with *n* = 2 and 3 are DFT energies corrected for 1- and 2-body errors, and corrected for 1-, 2- and 3-body errors, respectively. Values in parentheses represent errors compared with the CCSD(T) energies from Ref. 37. Energy units: m*E* _{h}.

Total energies of selected isomers of the water hexamer relative to that of the prism, calculated by different methods. In all cases, the geometry of the isomer is the relaxed geometry given by MP2 calculations with the AVTZ basis, as given in the supplementary information of Ref. 61. All energies were calculated in the present work, except for the MP2 and CCSD(T) energies marked with † from Ref. 37 and the DMC energies from Ref. 61. Entries DFT-*n* with *n* = 2 and 3 are DFT energies corrected for 1- and 2-body errors, and corrected for 1-, 2- and 3-body errors, respectively. Values in parentheses represent errors compared with the CCSD(T) energies from Ref. 37. Energy units: m*E* _{h}.

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