Geometric phase effects on transition-state resonances and bound vibrational states of H3 via a time-dependent wavepacket method

Abstract

A time-dependent wavepacket method for including geometric phase (GP) e †ects in hyperspherical coordinates is given. It uses the hybrid basis/grid representation and prime-factor fast Fourier transformation (PFFT) technique to propagate a wavepacket while the GP e †ects are treated on the Ñy using the method of Billing and Markovic (J. Chem. Phys., 1993Phys., , 99, 2674. The calculated vibrational states of the Ðrst electronically excited molecule with and without GP e †ects are found to be signiÐ-H 3 cantly di †erent and in excellent agreement with previous theoretical investigations. In contrast, GP e †ects on transition-state resonances of ground-state are minor, although some notable di †erences can be observed in the autocorrelation functions H 3 after 90 fs wavepacket propagation.

Geometric phase effect in isotopomers of X3 systems: Use of a split basis technique for the cone states of HD2

“…Since we plan to study the geometric properties produced by this seam, we follow a suggestion by Kuppermann and co‐workers 14, 46 and employ hyperspherical coordinates (ρ,θ,ϕ) as these are best suited for such a purpose 21, 47–51. Thus, following previous work 21, we have where with X, Y, and Z standing for atoms A, B, and C. Equating now the three interatomic distances with each other, we find that the seam is a straight line for which ρ is arbitrary but ϕ and θ have the values ϕ s and θ s determined by the masses only, namely where t is given in the form Equations (5) to (7) are valid when all three masses are different. In case two atomic masses are equal, namely m B = m C , we get for θ s the simplified expression while ϕ s assumes the value π when m A > m B , and the value zero when m A < m B .…”

“…We have therefore used the following split‐basis technique. For θ<θ s , we expand the nuclear wave function in terms of a single‐valued basis as follows: where C n are expansion coefficients that satisfy C n (ρ) f (θ)=CPFFT[ψ(ρ,θ,ϕ)], f (θ) is a θ‐dependent orthonormal polynomial type basis 21, and (CPFFT −1 ) where CPFFT stands for (inverted) cosine prime‐factor fast‐Fourier transform; n =1,2,…. Note that this cosine‐type basis allows to impose the total electronuclear wave functions to be symmetric with respect to the 𝒫 2 symmetry operation which interchanges the two deuterium atoms with respect to the C 2 v axis, as it must be since D is a boson 1.…”

“…Such calculations have been carried out both without consideration and with consideration of the GP effect and shown that this effect plays a significant role for Li 3 . Being a fundamental prototype, H 3 and its isotopic variants have also been extensively studied 10–23. In particular, studies by one of the authors and Yu 20, 21 have focused on the role of the GP effect on the transition‐state resonances and vibrational states of H 3 , which has been investigated using the time‐dependent wavepacket approach.…”

Singularities in the Hamiltonian at electronic degeneracies

“…Instead, Billing and Markovic [33] adopted hyperspherical coordinates to include the GP eect in the nuclear wave function of X 3 molecules, which have a D 3h conical intersection seam. A similar method has been utilized by the authors to study the vibrational spectra and transition state resonances of H 3 [16] and Li 3 [18,19]. Such an approach has most recently been extended to isotopomers of X 3 systems [34].…”

“…Note that bR must change by p for any nuclear motion which encircles the conical intersection (i.e., a pseudo-rotation). In order to meet this boundary condition, it is commonly used [4,9,10,16,18,19,23,25,32,57±61] for X 3 systems the form…”