Efficient algorithms for semiclassical instanton calculations based on discretized path integrals

Abstract

A calculation of the rovibronic energies and spectrum of the B A 1 1 electronic state of Si H 2 J. Chem. Phys. 123, 244312 (2005) Path integral instanton method is a promising way to calculate the tunneling splitting of energies for degenerated two state systems. In order to calculate the tunneling splitting, we need to take the zero temperature limit, or the limit of infinite imaginary time duration. In the method developed by Richardson and Althorpe [J. Chem. Phys. 134, 054109 (2011)], the limit is simply replaced by the sufficiently long imaginary time. In the present study, we have developed a new formula of the tunneling splitting based on the discretized path integrals to take the limit analytically. We have applied our new formula to model systems, and found that this approach can significantly reduce the computational cost and gain the numerical accuracy. We then developed the method combined with the electronic structure calculations to obtain the accurate interatomic potential on the fly. We present an application of our ab initio instanton method to the ammonia umbrella flip motion. © 2014 AIP Publishing LLC.[http://dx

The isotope effects on a hydrogen transfer using path integral instanton method

“…[34] for details. The factor with L and L r in Equation (6) is a ratio of six zero modes between J 00 kink and J 00 0 in b !…”

“…The tilde of g det 0 indicates excluding six zero modes for translations and rotations of the effective isomorphic polymer. The zero factors can be cancelled out between Hessians for the single kink and no kink effective isomorphic polymers, leaving the factor with L and L r in Equation (6), which is a ratio of these six zero modes in J 00 kink and J 00 0 [34] in b ! 1 limit.…”

“…; u kink L6 Þ, where u kink kr is a vector for rth vector of translations and rotations for kth time slice, see section III-B of Ref. [34] for details. Indices k and r run from 0 to L and from 1 to 6, respectively.…”

“…[34] The following formulations are written using the mass-weighted atomic coordinates q ðsÞ È É . The formula for the tunnelling splitting in the dilute kink gas approximation is written as…”

“…[34] In this paper, we have applied our method for calculating tunnelling splittings using the path integral instanton method with discretised imaginary time slices. We have applied the method to various isotopomers, HO 2 , DO 2 , TO 2 , H 15 O 2 , D 15 O 2 and T 15 O 2 that have two equal geometrical states for the hydrogen.…”