^{1}, Michael V. Pak

^{1}and Sharon Hammes-Schiffer

^{1,a)}

### Abstract

Fundamental issues associated with the application of the nuclear-electronic orbital (NEO) approach to hydrogen transfer systems are addressed. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wavefunctions are calculated with molecular orbital methods. The positions of the nuclear basis function centers are optimized variationally. In the application of the NEO approach to hydrogen transfer systems, the hydrogen nuclei and all electrons are treated quantum mechanically. Within the NEO framework, the transferring hydrogen atom can be represented by two basis function centers to allow delocalization of the proton vibrational wavefunction. In this paper, the NEO approach is applied to the and model systems. Analyses of technical issues pertaining to flexibility of the basis set to describe both single and double well protonpotential energy surfaces, linear dependency of the hydrogen basis functions, multiple minima in the basis function center optimization, convergence of the number of hydrogen basis function centers, and basis set superposition error are presented. The accuracy of the NEO approach is tested by comparison to grid calculations for these model systems.

We gratefully acknowledge the support of AFOSR Grant No. F49620-01-1-0046 and NIH Grant No. GM56207. We wish to thank Dr. Alexander Soudackov for providing the FGH routines and Dr. Simon Webb for his helpful suggestions and many stimulating discussions.

I. INTRODUCTION

II. METHODS

III. RESULTS

IV. CONCLUSIONS AND FUTURE DIRECTIONS

### Key Topics

- Wave functions
- 34.0
- Protons
- 20.0
- Basis sets
- 11.0
- Potential energy surfaces
- 7.0
- Ground states
- 6.0

## Figures

One-dimensional proton potential energy curves and the corresponding ground state nuclear wavefunctions for the system with HeHe distances of (a) 1.8, (b) 2.2, and (c) 2.6 Å. These curves were calculated on a one-dimensional grid along the HeHe axis at the full CI/6-31G level, and the proton vibrational wavefunctions were calculated with the FGH method.

One-dimensional proton potential energy curves and the corresponding ground state nuclear wavefunctions for the system with HeHe distances of (a) 1.8, (b) 2.2, and (c) 2.6 Å. These curves were calculated on a one-dimensional grid along the HeHe axis at the full CI/6-31G level, and the proton vibrational wavefunctions were calculated with the FGH method.

NEO-full CI ground state energies as functions of the HeHe distance for (a) and (b) . All energies are relative to the minimum of the system.

NEO-full CI ground state energies as functions of the HeHe distance for (a) and (b) . All energies are relative to the minimum of the system.

NEO-full CI ground state energy as a function of the hydrogen basis function separation for two (solid) and three (dashed) hydrogen basis functions, where the third basis function center is placed at the midpoint. The results are given for the (a) system with HeHe distance of 2.6 Å and (b) system with HeHe distance of 1.8 Å.

NEO-full CI ground state energy as a function of the hydrogen basis function separation for two (solid) and three (dashed) hydrogen basis functions, where the third basis function center is placed at the midpoint. The results are given for the (a) system with HeHe distance of 2.6 Å and (b) system with HeHe distance of 1.8 Å.

Full CI ground state energy as a function of the distance between the He nucleus and a “ghost” hydrogen electronic basis function center for the He atom (solid) and the cation (dashed). Note that the intramolecular BSSE is much larger for the He atom. The energies are relative to the energy of the He and atoms in the absence of the “ghost” hydrogen electronic basis function center.

Full CI ground state energy as a function of the distance between the He nucleus and a “ghost” hydrogen electronic basis function center for the He atom (solid) and the cation (dashed). Note that the intramolecular BSSE is much larger for the He atom. The energies are relative to the energy of the He and atoms in the absence of the “ghost” hydrogen electronic basis function center.

A schematic representation of the system used to estimate the intramolecular BSSE for NEO-HF calculations on the system with a HeHe distance of 2.6 Å. The X’s represent the nuclear basis function centers, and the open circles represent the electronic basis function centers. In (a), the proton and electronic basis function centers are identical, as in typical NEO calculations. The separation between basis function centers is 1.60 Å on the left and 0.928 Å on the right. In (b), two additional electronic basis function centers are included, so each calculation includes four electronic and two nuclear basis function centers. The additional electronic basis function centers ensure a consistent electronic basis set. The intramolecular BSSE is estimated to be the difference between the relative energies of the configurations in (a) and (b).

A schematic representation of the system used to estimate the intramolecular BSSE for NEO-HF calculations on the system with a HeHe distance of 2.6 Å. The X’s represent the nuclear basis function centers, and the open circles represent the electronic basis function centers. In (a), the proton and electronic basis function centers are identical, as in typical NEO calculations. The separation between basis function centers is 1.60 Å on the left and 0.928 Å on the right. In (b), two additional electronic basis function centers are included, so each calculation includes four electronic and two nuclear basis function centers. The additional electronic basis function centers ensure a consistent electronic basis set. The intramolecular BSSE is estimated to be the difference between the relative energies of the configurations in (a) and (b).

## Tables

Optimized nuclear basis function center separations for a range of relatively short HeHe distances in the system. The two hydrogen basis function center positions were optimized variationally at the NEO-full CI level. The smallest eigenvalue from diagonalization of the electronic overlap matrix is also given for each HeHe distance.

Optimized nuclear basis function center separations for a range of relatively short HeHe distances in the system. The two hydrogen basis function center positions were optimized variationally at the NEO-full CI level. The smallest eigenvalue from diagonalization of the electronic overlap matrix is also given for each HeHe distance.

Energy differences between the lowest two vibronic states corresponding to the hydrogen stretch for the system with a HeHe distance of 1.9 Å. The numbers in parentheses indicate the number of hydrogen basis function centers. The DZSPDN nuclear basis set includes two each of -, and -type Gaussians, resulting in a total of 20 nuclear basis functions per hydrogen center. The DZSNB (or DZSPNB) include only - (or - and -) type Gaussians, and the nuclear basis set includes two each of - and -type Gaussians and one set of -type Gaussians. The grid calculations were performed with the FGH method^{3,20} for a one-dimensional grid potential obtained at the full CI level. The 6-31G electronic basis set was used for all calculations.

Energy differences between the lowest two vibronic states corresponding to the hydrogen stretch for the system with a HeHe distance of 1.9 Å. The numbers in parentheses indicate the number of hydrogen basis function centers. The DZSPDN nuclear basis set includes two each of -, and -type Gaussians, resulting in a total of 20 nuclear basis functions per hydrogen center. The DZSNB (or DZSPNB) include only - (or - and -) type Gaussians, and the nuclear basis set includes two each of - and -type Gaussians and one set of -type Gaussians. The grid calculations were performed with the FGH method^{3,20} for a one-dimensional grid potential obtained at the full CI level. The 6-31G electronic basis set was used for all calculations.

Energy differences between the lowest two vibronic states for the system with a HeHe distance of 1.6 Å. Two hydrogen basis function centers were used for the NEO calculations. The DZSPDN nuclear basis set includes two each of -, and -type Gaussians, resulting in a total of 20 nuclear basis functions per hydrogen center. The DZSNB (or DZSPNB) include only - (or - and -) type Gaussians. The grid calculations were performed with the FGH method^{3,20} for a one-dimensional grid potential obtained at the full CI level. The 6-31G electronic basis set was used for all calculations.

Energy differences between the lowest two vibronic states for the system with a HeHe distance of 1.6 Å. Two hydrogen basis function centers were used for the NEO calculations. The DZSPDN nuclear basis set includes two each of -, and -type Gaussians, resulting in a total of 20 nuclear basis functions per hydrogen center. The DZSNB (or DZSPNB) include only - (or - and -) type Gaussians. The grid calculations were performed with the FGH method^{3,20} for a one-dimensional grid potential obtained at the full CI level. The 6-31G electronic basis set was used for all calculations.

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