A study on topological properties of nuclear wavefunctions induced by conical intersection
Date Issued
2014
Date
2014
Author(s)
Liu, Yi-Hsien
Abstract
The geometric phase induced by conical intersection can be understood as structural change of adiabatic state line bundle. However, we proved that for real Hamiltonian, this structural change only occur for real line bundle and the corresponding complex line bundle is still trivial. Since nuclear wavefunction is a complex-valued function (or section), only complex line bundle is relevant. Hence we conclude that the existence of conical intersection cannot be determined by topological invariants for real Hamiltonian. We also showed that due to reduction of seam dimension, conical intersection disappear from the nearby region for most conical intersection points of original real Hamiltonian under complex perturbation.
As an example, we consider the A3 molecule with A being an alkali atom. We showed that the conical intersection disappears under arbitrarily weak magnetic field except at some specific nuclear position. The half-integer pseudo-rotational quantum number, which is believed to be an evidence of existence of conical intersection, is also explained
using a perturbed complex Hamiltonian without conical intersection. Hence half-integer quantization of pseudo-rotational degrees of freedom does not prove the existence of conical intersection.
For systems with complex Hamiltonian, a quantity called conical number is defined for 2-surface in nuclear manifold Mnu, denoted by CN(S). This quantity can be used to determine whether structural change occurred around the given surface or not, thus CN(S) gives a lower bound of conical intersections inside S. We also showed that the relation between this number and conical intersection is an analog of integral Gauss’ law and charged particles, and an example of how this structural change affects the pseudorotational spectrum is given. Another result is that the calculation method proposed by S. C. Althorpe to include geometric phase[38][39] is generalized to arbitrary system with real Hamiltonian.
Subjects
錐形交叉
幾何位相
拓樸不變量
Type
thesis
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