^{1}, Christopher F. Williams

^{1}and John M. Herbert

^{1,a)}

### Abstract

Previously, Turi and Borgis [J. Chem. Phys.117, 6186 (2002)] parametrized an electron-water interaction potential, intended for use in simulations of hydrated electrons, by considering in the “static exchange” (essentially, frozen-core Hartree–Fock) approximation, then applying an approximate Phillips–Kleinman procedure to construct a one-electron pseudopotential representing the electron-water interaction. To date, this pseudopotential has been used exclusively in conjunction with a simple point charge water model that is parametrized for bulk water and yields poor results for small, neutral water clusters. Here, we extend upon the work of Turi and Borgis by reparametrizing the electron-water pseudopotential for use with the AMOEBA water model, which performs well for neutral clusters. The result is a one-electron model Hamiltonian for , in which the one-electron wave function polarizes the water molecules, and vice versa, in a fully self-consistent fashion. The new model is fully variational and analytic energy gradients are available. We have implemented the new model using a modified Davidson algorithm to compute eigenstates, with the unpaired electron represented on a real-space grid. Comparison to *ab initio* electronic structure calculations for cluster isomers ranging from to reveals that the new model is significantly more accurate than the Turi–Borgis model, for both relative isomer energies and for vertical electron detachment energies. Electron-water polarizationinteractions are found to be much more significant for cavity states of the unpaired electron than for surface states.

This work was supported by the American Chemical Society’s Petroleum Research Fund, by a National Science Foundation CAREER award (CHE-0748448), and by an allocation from the Ohio Supercomputer Center. The Visual Molecular Dynamics program^{89} was used for visualization.

I. INTRODUCTION

II. BACKGROUND

A. Pseudopotential theory

B. TURI-BORGIS MODEL

III. NEW MODEL

A. AMOEBA water model

B. Electron-water potential

C. Polarization potential

D. Energy gradients

IV. SIMULATION ALGORITHM

V. COMPUTATIONAL DETAILS

VI. ANALYSIS OF THE NEW MODEL

A. VEBEs

B. Conformational energies

C. Pseudopotentials

D. Energy decompositions

VII. CONCLUSIONS

### Key Topics

- Polarization
- 78.0
- Water energy interactions
- 23.0
- Wave functions
- 23.0
- Electrostatics
- 20.0
- Ab initio calculations
- 19.0

## Figures

Comparison of VEBEs obtained with various *ab initio* methods to those obtained at the CCSD(T) level, for a set of 24 cluster isomers ranging from to , each of which is a stationary point at the level. The MP2 and CCSD(T) calculations employ the basis set, while the density-functional calculations use the basis. LRC-BOP calculations employ a range-separation parameter of , as recommended in Ref. 76. The diagonal line indicates where the predicted VEBE is the same as the CCSD(T) value.

Comparison of VEBEs obtained with various *ab initio* methods to those obtained at the CCSD(T) level, for a set of 24 cluster isomers ranging from to , each of which is a stationary point at the level. The MP2 and CCSD(T) calculations employ the basis set, while the density-functional calculations use the basis. LRC-BOP calculations employ a range-separation parameter of , as recommended in Ref. 76. The diagonal line indicates where the predicted VEBE is the same as the CCSD(T) value.

Comparison of VEBEs computed at the level of theory to those predicted using model Hamiltonians, for (a) 32 isomers ranging from to , and (b) 59 isomers ranging from to . The diagonal line indicates where the model Hamiltonian and MP2 predications are identical.

Comparison of VEBEs computed at the level of theory to those predicted using model Hamiltonians, for (a) 32 isomers ranging from to , and (b) 59 isomers ranging from to . The diagonal line indicates where the model Hamiltonian and MP2 predications are identical.

Comparison of VEBEs computed at the level of theory to those predicted using model Hamiltonians, for isomers (with ) that were (a) extracted from a MD simulation, as described in the text; and (b) subsequently optimized, using the TB model.

Comparison of VEBEs computed at the level of theory to those predicted using model Hamiltonians, for isomers (with ) that were (a) extracted from a MD simulation, as described in the text; and (b) subsequently optimized, using the TB model.

Structures of the , , and isomers used to benchmark relative conformational energies. Each geometry shown here is a stationary point at the level.

Structures of the , , and isomers used to benchmark relative conformational energies. Each geometry shown here is a stationary point at the level.

Energies of tetrameric clusters on (a) the potential surface and (b) the potential surface. Note that the two panels use different energy scales. *Ab initio* geometries for each cluster are depicted in Fig. 4.

Energies of tetrameric clusters on (a) the potential surface and (b) the potential surface. Note that the two panels use different energy scales. *Ab initio* geometries for each cluster are depicted in Fig. 4.

Energies of pentameric clusters on (a) the potential surface and (b) the potential surface. Note that the two panels use different energy scales. *Ab initio* geometries for each cluster are depicted in Fig. 4.

Energies of pentameric clusters on (a) the potential surface and (b) the potential surface. Note that the two panels use different energy scales. *Ab initio* geometries for each cluster are depicted in Fig. 4.

Stationary points of isomers on the TB potential surface compared to stationary points at the level.

Stationary points of isomers on the TB potential surface compared to stationary points at the level.

Energies of hexameric clusters on (a) the potential surface and (b) the potential surface. Note that the two panels use different energy scales. *Ab initio* geometries for each cluster are depicted in Fig. 4. Isomer hex2 is omitted, for reasons discussed in the text.

Energies of hexameric clusters on (a) the potential surface and (b) the potential surface. Note that the two panels use different energy scales. *Ab initio* geometries for each cluster are depicted in Fig. 4. Isomer hex2 is omitted, for reasons discussed in the text.

Comparison of the TB model (dashed curves) and the new potential (solid curves) for , in four one-dimensional slices. To facilitate the comparison, both potentials are shown without polarization. In (a)–(c), represents the center of mass, whereas in (d), at the oxygen atom.

Comparison of the TB model (dashed curves) and the new potential (solid curves) for , in four one-dimensional slices. To facilitate the comparison, both potentials are shown without polarization. In (a)–(c), represents the center of mass, whereas in (d), at the oxygen atom.

Plots of the pseudopotential for using various -type exchange potentials of the form , where is defined in Eq. (38). Values of the scaling parameter range from (the top curve, in red, which is least attractive) to (the bottom curve, in black, which is most attractive), in increments of 0.2. In (a)–(c), represents the center of mass, whereas in (d), at the oxygen atom.

Plots of the pseudopotential for using various -type exchange potentials of the form , where is defined in Eq. (38). Values of the scaling parameter range from (the top curve, in red, which is least attractive) to (the bottom curve, in black, which is most attractive), in increments of 0.2. In (a)–(c), represents the center of mass, whereas in (d), at the oxygen atom.

Histograms of the average repulsion energy, for a library of isomers.

Histograms of the average repulsion energy, for a library of isomers.

Histograms illustrating the change in the polarization energy of the water molecules when an electron is added to a water cluster assembled in its anion geometry, as defined in Eq. (39). Expectation values are binned over a library of isomers. Results are shown only for the polarizable model.

Histograms illustrating the change in the polarization energy of the water molecules when an electron is added to a water cluster assembled in its anion geometry, as defined in Eq. (39). Expectation values are binned over a library of isomers. Results are shown only for the polarizable model.

Histogram of the total polarization energy, as defined in Eq. (41), binned over a library of isomers. Results are shown only for the polarizable model developed here.

Histogram of the total polarization energy, as defined in Eq. (41), binned over a library of isomers. Results are shown only for the polarizable model developed here.

Histogram of the electron-water polarization energy binned over a library of isomers. For the TB model, , whereas for the polarizable model, is defined in Eq. (40).

Histogram of the electron-water polarization energy binned over a library of isomers. For the TB model, , whereas for the polarizable model, is defined in Eq. (40).

Histogram of the components of the electron-water polarization energy for the polarizable model developed here, binned over a library of isomers. The quantities and are the total polarization energy and the polarization energy in the field of the neutral dipoles, respectively.

Histogram of the components of the electron-water polarization energy for the polarizable model developed here, binned over a library of isomers. The quantities and are the total polarization energy and the polarization energy in the field of the neutral dipoles, respectively.

## Tables

Parameters that determine the electron-water potential developed in this work.

Parameters that determine the electron-water potential developed in this work.

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