Multidimensional Hamiltonian for tunneling with position-dependent mass

A multidimensional Hamiltonian for tunneling is formulated, based on the mode with imaginary frequency of the transition state as a reaction coordinate. To prepare it for diagonalization, it is transformed into a lower-dimension Hamiltonian by incorporating modes that move faster than the tunneling into a coordinate-dependent kinetic energy operator, for which a Hermitian form is chosen and tested for stability of the eigenvalues. After transformation to a three-dimensional form, which includes two normal modes strongly coupled to the tunneling mode, this Hamiltonian is diagonalized in terms of a basis set of harmonic oscillator functions centered at the transition state. This involves a sparse matrix which is easily partially diagonalized to yield tunneling splittings for the zero-point level and the two fundamental levels of the coupled modes. The method is tested on the well-known benchmark molecule malonaldehyde and a deuterium isotopomer, for which these splittings have been measured. Satisfactory agreement with experiment results is obtained.

Figure 1

Contour energy plots comparing the 2D Morse potential (a) with the imaginary-mode Hamiltonian potential (b) for the collinear exchange reaction $\text{B}+\text{C}\to \text{A}+\text{BC}$ analyzed in the Appendix. Panel (c) plots the minimum energy path for the Morse potential (solid line) and for the imaginary-mode Hamiltonian (dashed line).

Figure 2

Illustration of the coupling effects described by a 2D version of Hamiltonian (14) for malonaldehyde-$d_0$, which describes tunneling with renormalized mass coupled to the symmetric stretching of the hydrogen bridge (mode $y_1$). Plot of the maximum probability density $|{\Psi }_0^{(+)}(x,y_1){|}^2$ when $y_1$ is varied versus the tunneling coordinate $x$ (dimensionless). See text for details.