(a) Chemical structure of the dendritic molecule consisting of two two-ring, one three-ring, and one four-ring linear poly(phenylene ehynylene) (PPE) units linked by meta-substitution. (b) Fragments of the dendrite (2,2-PPE and 4-PPE molecules) used to test the numerical procedure to identify unavoided crossings.
Variation of the adiabatic energies obtained during a short-time NA-ESMD dynamics of the 2,2-PPE and 4-PPE molecules separated by 100 Å. Blue and red lines correspond to the excited states fully localized on 2,2-PPE and 4-PPE fragments, respectively, as determined by the transition density matrix analysis.
Typical trivial unavoided crossings between non-interacting adiabatic states obtained from a NA-ESMD simulation of the 2,2-PPE and 4-PPE fragments separated by 500 Å. (a) Time evolution of the adiabatic state energies. The molecular fragments where the transition densities are localized for each state, are also shown. The arrows indicate the dynamics of the system that remains on the same diabatic state. (b) Variation of the respective time-dependent nonadiabatic coupling (NACT) between S2 and S3 states.
Variation of the time-dependent fraction of the transition densities localized at the 2,2-PPE (blue lines) and 4-PPE (red lines) fragments when they are separated by different distances. (a), (b), and (c) are obtained from the NA-ESMD simulations without considering any specific treatment of unavoiding crossings. (d), (e), and (f) are obtained from the NA-ESMD simulations including the Min-Cost algorithm dealing with trivial unavoiding crossings.
Comparison of the variation of the time-dependent fraction of the transition densities localized at the 2,2-PPE (blue lines) and 4-PPE (red lines) fragments with r f ∼ 100 Å obtained using different procedures: the method proposed in this article (solid lines), a simple projection scheme to detect the trivial unavoided crossings (dashed lines), and an alternative algorithm to deal with cases of sβα(t; t + Δt) < Slim (dotted lines).
Time evolution of (a) the energy gap , and (b) the overlap with (n = 0,…, N c(N q − 1)) (see Eq. (7)) throughout the time-interval defined by a classical time step Δt at which a potential unavoided crossing between non-interacting states takes place. The value of the overlap sαβ(t; t + Δt) ≡ ϕ α( r ; R (t)) · ϕ β( r ; R (t − Δt)) is shown as a dashed line.
Distribution of (a) the energy gaps ΔEαβ and (b)NACTαβ at the moment of Sβ → Sα hops, and distribution of (c) energy gaps ΔEαβ and (d)NACTαβat the moment of Sβ → Sα unavoided crossings.
Histogram of the overlaps sαβ(t; t + Δt) taken at Sβ → Sα hops, and at trivial Sβ → Sα unavoided crossings.
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