Instanton theory of ground-state tunneling splittings with general paths

Abstract

We derive a multidimensional instanton theory for calculating ground-state tunneling splittings in Cartesian coordinates for general paths. It is an extension of the method by Mil'nikov and Nakamura [J. Chem. Phys. 115, 6881 (2001)] to include asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. The approach avoids multiple expensive matrix diagonalizations to converge the fluctuation prefactor in the ring-polymer instanton (RPI) method, and instead replaces them by an integration of a Riccati differential equation. When combined with the string method for locating instantons, we avoid the need to converge the calculation with respect to the imaginary-time period of the semiclassical orbit, thereby reducing the number of convergence parameters of the optimized object to just one: the number of equally-spaced system replicas used to represent the instanton path. The entirety of the numerical effort is thus concentrated in optimizing the shape of the path and evaluating hessians along the path, which is a dramatic improvement over RPI. In addition to the standard instanton approximations, we neglect the coupling of vibrational modes to external rotations. The method is tested on the model potential of malonaldehyde and on the water dimer and trimer, giving close agreement with RPI at a much-reduced cost.

Tunneling splittings of vibrationally excited states using general instanton paths

“…The wavefunctions that start from the "left" and from the "right" symmetry-connected minima along the MAP are constructed and used in the Herring formula at the dividing surface to obtain the ground-state tunneling splitting, which is identical in form to the JFI instanton expression from our previous work. 67 The derivation follows Ref. 52 but does not assume the mirror symmetry of the potential along the MAP.…”

“…In our previous work, 67 we derived the tunneling matrix element h, or equivalently the tunneling splitting Δ, using the JFI theory. The splitting is dominated by the Euclidean action of the MAP, while the contributions from all other paths in the harmonic neighborhood of the MAP are collected into the fluctuation prefactor.…”

“…53,61,62 Tunneling splittings of vibrationally excited states have been obtained using the related methods in symmetric systems such as malonaldehyde, 62 tropolone, 63 9-hydroxyphenalenone, 64 HO 2 , 53 formic acid dimer, 65 and vinyl radical. 66 In our recent work, 67 we generalized the JFI approach of Mil'nikov and Nakamura 52 to obtain the ground-state tunneling splittings for asymmetric paths in Cartesian coordinates. We obtained an almost perfect agreement between the JFI and RPI splittings 67 for systems in which rotations do not couple strongly to the internal degrees of freedom, such as water trimer or malonaldehyde.…”

“…66 In our recent work, 67 we generalized the JFI approach of Mil'nikov and Nakamura 52 to obtain the ground-state tunneling splittings for asymmetric paths in Cartesian coordinates. We obtained an almost perfect agreement between the JFI and RPI splittings 67 for systems in which rotations do not couple strongly to the internal degrees of freedom, such as water trimer or malonaldehyde. The development enabled us to treat large asymmetric systems that exhibit slow motion of a heavy-atom skeleton, such as the water pentamer, 17 in full dimensionality.…”

Observation of the Low‐Frequency Spectrum of the Water Trimer as a Sensitive Test of the Water‐Trimer Potential and the Dipole‐Moment Surface

“…To conclude this section, we describe calculations of the ground‐vibrational‐state tunneling splitting in the water trimer using the ring‐polymer instanton (RPI) method . These used the WHBB potential and MB‐Pol potential . The calculated splitting of about 50 cm −1 , using either potential, is more than a factor of two larger than experiment.…”