Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections

Abstract

Dynamical consideration that goes beyond the common Born-Oppenheimer approximation (BOA) becomes necessary when energy differences between electronic potential energy surfaces become small or vanish. One of the typical scenarios of the BOA breakdown in molecules beyond diatomics is a conical intersection (CI) of electronic potential energy surfaces. CIs provide an efficient mechanism for radiationless electronic transitions: acting as "funnels" for the nuclear wave function, they enable rapid conversion of the excessive electronic energy into the nuclear motion. In addition, CIs introduce nontrivial geometric phases (GPs) for both electronic and nuclear wave functions. These phases manifest themselves in change of the wave function signs if one considers an evolution of the system around the CI. This sign change is independent of the shape of the encircling contour and thus has a topological character. How these extra phases affect nonadiabatic dynamics is the main question that is addressed in this Account. We start by considering the simplest model providing the CI topology: two-dimensional two-state linear vibronic coupling model. Selecting this model instead of a real molecule has the advantage that various dynamical regimes can be easily modeled in the model by varying parameters, whereas any fixed molecule provides the system specific behavior that may not be very illustrative. After demonstrating when GP effects are important and how they modify the dynamics for two sets of initial conditions (starting from the ground and excited electronic states), we give examples of molecular systems where the described GP effects are crucial for adequate description of nonadiabatic dynamics. Interestingly, although the GP has a topological character, the extent to which accounting for GPs affect nuclear dynamics profoundly depends on topography of potential energy surfaces. Understanding an extent of changes introduced by the GP in chemical dynamics poses a problem of capturing GP effects by approximate methods of simulating nonadiabatic dynamics that can go beyond simple models. We assess the performance of both fully quantum (wave packet dynamics) and quantum-classical (surface-hopping, Ehrenfest, and quantum-classical Liouville equation) approaches in various cases where GP effects are important. It has been identified that the key to success in approximate methods is a method organization that prevents the quantum nuclear kinetic energy operator to act directly on adiabatic electronic wave functions.

Vibronic interaction in CO3− photo-detachment: Jahn–Teller effects beyond structural distortion and general formalisms for vibronic Hamiltonians in trigonal symmetries

“…in simulating the A band of the CO − 3 NIPE spectrum. The almost barrier-less trough allows hinder-less pseudo-rotation around the conical intersection, resulting in significant geometric phase effects [68][69][70][71][72]. One consequence is that the vibronic ground state of this JT problem is an E-type doubly degenerate state (vide infra),73,74 instead of an A-type non-degenerate state, which is anticipated based on the BO approximation.…”

Variational nonadiabatic dynamics in the moving crude adiabatic representation: Further merging of nuclear dynamics and electronic structure

“…One of the main features of this dynamics is radiationless population transfer between adiabatic states, which takes place in the vicinity of the CI. Another feature is related to a nontrivial geometric (or Berry) phase induced by CIs between D 1 and D 2 , the nuclear density corresponding to an adiabatic state exhibits a nodal line upon skirting one CI, 17,20,51 this node disappears after encircling a second CI. 52,53 Since the wavepacket starts on D 2 it must display a nodal line between the two CIs on D 2 and the absence of the nodal line between the CIs on D 1 .…”

Combining SchNet and SHARC: The SchNarc Machine Learning Approach for Excited-State Dynamics

“…The virtual property that SchNarc is generating is then the multi-dimensional antiderivative of the latter expression in equation 19, Ψ i | ∂H el R | Ψ j . Noticeably, due to the Berry phase [12,104,105] the NAC vector field is not conservative [11] and a line integral remains path dependent. Hence this approach does not include the effects of the Berry phase, which is also neglected in approaches such as the Zhu-Nakamura approximation [51,52] that does not contain a phase at all, or the phase correction algorithm [32,82].…”