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