^{1}and Benjamin J. Schwartz

^{1,a)}

### Abstract

Mixed quantum/classical (MQC) simulations treat the majority of a system classically and reserve quantum mechanics only for a few degrees of freedom that actively participate in the chemical process(es) of interest. In MQC calculations, the quantum and classical degrees of freedom are coupled together using pseudopotentials. Although most pseudopotentials are developed empirically, there are methods for deriving pseudopotentials using the results of quantum chemistry calculations, which guarantee that the explicitly-treated valence electron wave functions remain orthogonal to the implicitly-treated core electron orbitals. Whether empirical or analytically derived in nature, to date all such pseudopotentials have been subject to the frozen core approximation (FCA) that ignores how changes in the nuclear coordinates alter the core orbitals, which in turn affects the wave function of the valence electrons. In this paper, we present a way to go beyond the FCA by developing pseudopotentials that respond to these changes. In other words, we show how to derive an analytic expression for a pseudopotential that is an explicit function of nuclear coordinates, thus accounting for the polarization effects experienced by atomic cores in different chemical environments. We then use this formalism to develop a coordinate-dependent pseudopotential for the bonding electron of the sodium dimer cation molecule and we show how the analytic representation of this potential can be used in one-electron MQC simulations that provide the accuracy of a fully quantum mechanical Hartree-Fock (HF) calculation at all internuclear separations. We also show that one-electron MQC simulations of using our coordinate-dependent pseudopotential provide a significant advantage in accuracy compared to frozen core potentials with no additional computational expense. This is because use of a frozen core potential produces a charge density for the bonding electron of that is too localized on the molecule, leading to significant overbinding of the valence electron. This means that FCA calculations are subject to inaccuracies of order ∼10% in the calculated bond length and vibrational frequency of the molecule relative to a full HF calculation; these errors are fully corrected by using our coordinate-dependent pseudopotential. Overall, our findings indicate that even for molecules like , which have a simple electronic structure that might be expected to be well-treated within the FCA, the importance of including the effects of the changing core molecular orbitals on the bonding electrons cannot be overlooked.

_{2}

^{+}” [J. Chem. Phys.138, 054110 (2013)]

This work was supported by the National Science Foundation (NSF) under Grant No. CHE-0908548. We gratefully acknowledge the Institute for Digital Research and Education (IDRE) at UCLA for use of the hoffman2 computing cluster and William Glover and Jennifer Casey for helpful discussion.

I. INTRODUCTION

II. BACKGROUND: MOLECULAR PSEUDOPOTENTIAL THEORY

A. Philips-Kleinman pseudopotential theory

B. A reformulation of the PK pseudopotential theory

III. COORDINATE-DEPENDENT PSEUDOPOTENTIALS

A. A coordinate-dependent pseudopotential for

B. Analytic representation of coordinate-dependent pseudopotentials

IV. DEMONSTRATION OF GOING BEYOND THE FROZEN CORE APPROXIMATION FOR THE SODIUM DIMER CATION

V. CONCLUSIONS

## Figures

Cross-sections of (a) , (b) , and (c) (right panel) taken through the bonding axis at an internuclear distance of 3.7 Å. The white dots show the location of the Na^{+} nuclei.

Cross-sections of (a) , (b) , and (c) (right panel) taken through the bonding axis at an internuclear distance of 3.7 Å. The white dots show the location of the Na^{+} nuclei.

Slices of the numerically calculated for (left panels) and fits of these slices (right panels) to Eq. (12) . Slices were taken through the bonding axis with internuclear spacings of (a) 2.5 Å, (b) 3.7 Å, and (c) 5.0 Å, and all axis labels are in atomic units. The white dots show the location of the Na^{+} nuclei.

Slices of the numerically calculated for (left panels) and fits of these slices (right panels) to Eq. (12) . Slices were taken through the bonding axis with internuclear spacings of (a) 2.5 Å, (b) 3.7 Å, and (c) 5.0 Å, and all axis labels are in atomic units. The white dots show the location of the Na^{+} nuclei.

Bond length (**R**) coordinate dependence (points) of the fitting parameters (a) and (b) from Eq. (12) , which themselves are fit to (a) twelfth-order and (b) thirteenth-order polynomials, respectively (solid curves), plotted in atomic units.

Bond length (**R**) coordinate dependence (points) of the fitting parameters (a) and (b) from Eq. (12) , which themselves are fit to (a) twelfth-order and (b) thirteenth-order polynomials, respectively (solid curves), plotted in atomic units.

Gas-phase potential energy surfaces of the system calculated from MQC MD simulations with a frozen core pseudopotential (blue hexagons), our coordinate-dependent pseudopotential (orange squares) and from fixed-point RHF calculations of the LUMO of (green circles) and UHF calculations of the HOMO of (black stars) using GAUSSIAN 03.

Gas-phase potential energy surfaces of the system calculated from MQC MD simulations with a frozen core pseudopotential (blue hexagons), our coordinate-dependent pseudopotential (orange squares) and from fixed-point RHF calculations of the LUMO of (green circles) and UHF calculations of the HOMO of (black stars) using GAUSSIAN 03.

Charge densities for calculated from MQC simulations employing (a) a frozen core approximation pseudopotential, (b) our coordinate-dependent pseudopotential (CDP), and (c) one generated from an unrestricted Hartree-Fock calculation using GAUSSIAN 03. The electron density increases from the blue to the red contours, and the purple contour marks zero effective charge density. Calculations were performed at a bond distance of 3.50 Å and slices were taken through the bonding axis. The white dots show the location of the Na^{+} nuclei.

Charge densities for calculated from MQC simulations employing (a) a frozen core approximation pseudopotential, (b) our coordinate-dependent pseudopotential (CDP), and (c) one generated from an unrestricted Hartree-Fock calculation using GAUSSIAN 03. The electron density increases from the blue to the red contours, and the purple contour marks zero effective charge density. Calculations were performed at a bond distance of 3.50 Å and slices were taken through the bonding axis. The white dots show the location of the Na^{+} nuclei.

## Tables

Fitting function ξ_{fit} and corresponding parameter functions *x* (where *x* represents the general functional form for each of the eight fitting parameters for ξ_{fit}) for the coordinate-dependent pseudopotential for the bonding electron of . **R** is the internuclear distance in atomic units.

Fitting function ξ_{fit} and corresponding parameter functions *x* (where *x* represents the general functional form for each of the eight fitting parameters for ξ_{fit}) for the coordinate-dependent pseudopotential for the bonding electron of . **R** is the internuclear distance in atomic units.

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