^{1}, Jonathan H. Skone

^{1}and Sharon Hammes-Schiffer

^{1,a)}

### Abstract

The nuclear-electronic orbital (NEO) method is combined with vibronic coupling theory to calculate hydrogen tunneling splittings in polyatomic molecules. In this NEO-vibronic coupling approach, the transferring proton and all electrons are treated quantum mechanically at the NEO level, and the other nuclei are treated quantum mechanically using vibronic coupling theory. The dynamics of the molecule are described by a vibronic Hamiltonian in a diabatic basis of two localized nuclear-electronic states for the electrons and transferring proton. This *ab initio* approach is computationally practical and efficient for relatively large molecules, and the accuracy can be improved systematically. The NEO-vibronic coupling approach is used to calculate the hydrogen tunneling splitting for malonaldehyde. The calculated tunneling splitting of is in excellent agreement with the experimental value of . This approach also enables the identification of the dominant modes coupled to the transferring hydrogen motion and provides insight into their roles in the hydrogen tunneling process.

We thank Michael V. Pak, Arindam Chakraborty, Alexander V. Soudackov, and Professor Marcel Nooijen for helpful discussions. We gratefully acknowledge the support of AFOSR Grant No. FA9550-07-1-0143 and NSF Grant No. CHE-07-49646.

I. INTRODUCTION

II. THEORY AND METHODS

A. General theoretical formulation

B. Calculation of diabatic nuclear-electronic wave functions

C. Summary of procedure

III. APPLICATION TO MALONALDEHYDE

A. Computational methods

1. Calculation of normal mode coordinates

2. Selection of the reference configuration

3. Calculation of localized proton orbitals

4. Calculation of potential energy matrix elements and derivatives

5. Solution of vibronic Hamiltonian matrix equation

B. Results

IV. CONCLUSIONS

### Key Topics

- Protons
- 82.0
- Tunneling
- 58.0
- Wave functions
- 44.0
- Normal modes
- 42.0
- Potential energy surfaces
- 15.0

## Figures

The intramolecular hydrogen transfer process in malonaldehyde.

The intramolecular hydrogen transfer process in malonaldehyde.

(a) The determination of the localized proton orbitals for malonaldehyde by fitting to the one-dimensional proton vibrational wave function. The solid line represents the one-dimensional proton vibrational wave function generated with the Fourier grid Hamiltonian method. The dashed and dotted lines represent the two sets of Gaussians comprising the localized proton orbitals. The sum of all four Gaussians leads to the solid curve. The sum of each pair of dotted or dashed Gaussians leads to each localized proton orbital. (b) Schematic depiction of the two localized proton orbitals corresponding to the transferring proton for malonaldehyde. The localized proton orbitals are depicted in blue as three-dimensional contour plots on the left and the right.

(a) The determination of the localized proton orbitals for malonaldehyde by fitting to the one-dimensional proton vibrational wave function. The solid line represents the one-dimensional proton vibrational wave function generated with the Fourier grid Hamiltonian method. The dashed and dotted lines represent the two sets of Gaussians comprising the localized proton orbitals. The sum of all four Gaussians leads to the solid curve. The sum of each pair of dotted or dashed Gaussians leads to each localized proton orbital. (b) Schematic depiction of the two localized proton orbitals corresponding to the transferring proton for malonaldehyde. The localized proton orbitals are depicted in blue as three-dimensional contour plots on the left and the right.

The physical description of normal mode coordinates for the following modes: (a) 4, (b) 11, and (c) 13.

The physical description of normal mode coordinates for the following modes: (a) 4, (b) 11, and (c) 13.

The potential energy matrix element as a function of a specified normal mode coordinate with all other normal mode coordinates set to zero for (a) and (b) . The diagonal matrix elements and are shown with solid lines. The dashed and dotted lines correspond to the eigenvalues of the matrix, where the dashed lines include both linear and quadratic coupling constants and the dotted lines include only linear coupling constants. The two diagonal matrix elements (solid) are identical for the symmetric mode in (a) but are qualitatively different for the asymmetric mode in (b). The eigenvalues (dashed) are virtually identical to the diagonal matrix elements (solid) for the asymmetric mode in (b) everywhere except near zero.

The potential energy matrix element as a function of a specified normal mode coordinate with all other normal mode coordinates set to zero for (a) and (b) . The diagonal matrix elements and are shown with solid lines. The dashed and dotted lines correspond to the eigenvalues of the matrix, where the dashed lines include both linear and quadratic coupling constants and the dotted lines include only linear coupling constants. The two diagonal matrix elements (solid) are identical for the symmetric mode in (a) but are qualitatively different for the asymmetric mode in (b). The eigenvalues (dashed) are virtually identical to the diagonal matrix elements (solid) for the asymmetric mode in (b) everywhere except near zero.

## Tables

Frequencies and descriptions of the vibrational modes of malonaldehyde for all nuclei except the transferring hydrogen. The electronic basis set is , and the nuclear basis set for NEO-HF is DZSNB.

Frequencies and descriptions of the vibrational modes of malonaldehyde for all nuclei except the transferring hydrogen. The electronic basis set is , and the nuclear basis set for NEO-HF is DZSNB.

Parameters of the Gaussians for one of the localized proton orbitals. The centers are relative to the symmetry axis, and the other localized proton orbital has identical parameters except the centers have the opposite signs.

Parameters of the Gaussians for one of the localized proton orbitals. The centers are relative to the symmetry axis, and the other localized proton orbital has identical parameters except the centers have the opposite signs.

Linear coupling constants in eV. Only the normal modes with nonzero linear coupling constants are given.

Linear coupling constants in eV. Only the normal modes with nonzero linear coupling constants are given.

Tunneling splittings calculated as the number of modes with excited vibrational states included is converged. Case 1 neglects the quadratic coupling constants, while cases 2 and 3 include both linear and quadratic coupling constants. Cases 1 and 2 use the same two localized proton orbitals for all modes, and case 3 uses different localized proton orbitals for the displacements along mode 4. For all cases, the modes are added in order of decreasing impact on the calculated tunneling splitting, starting with the greatest impact at the top with modes 4 and 11. Note that the order of the modes added is different for the three cases, leading to blank entries in the table.

Tunneling splittings calculated as the number of modes with excited vibrational states included is converged. Case 1 neglects the quadratic coupling constants, while cases 2 and 3 include both linear and quadratic coupling constants. Cases 1 and 2 use the same two localized proton orbitals for all modes, and case 3 uses different localized proton orbitals for the displacements along mode 4. For all cases, the modes are added in order of decreasing impact on the calculated tunneling splitting, starting with the greatest impact at the top with modes 4 and 11. Note that the order of the modes added is different for the three cases, leading to blank entries in the table.

Number of excited vibrational states included for each mode when convergence is achieved.

Number of excited vibrational states included for each mode when convergence is achieved.

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