1. Theoretical Chemistry for Electronic Excited States
Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Theoretical Chemistry for Electronic Excited States
Supplementary File: Chapter 4

2. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.1 Schematic representation of a trajectory on a potential surface undergoing a nonadiabatic transition at the surface crossing. In the region of the surface crossing, the electronic states φ1(t), φ2(t) mix under the influence of the derivative coupling. The probability of changing from one surface
to another depends on the weight of the coefficient C2; when C2 decreases the states mix at the
crossing. For a review see the perspective article by Tully.1

3. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.2 Quantum description of the process shown in Figure 4.1. The nuclear motion is now described by a Gaussian wavepacket gj(Qj,t) (represented as a pictogram in the figure), which has a different weight Aj(s)(t) on each of the two potential surfaces. The decay through the surface crossing is now described by a mixing under the influence of quantum mechanics.
Adapted from C. S. M. Allan, B. Lasorne, G. A. Worth and M. A. Robb, J. Phys. Chem. A,
2010, 114, 8713– Copyright 2010 American Chemical Society.

4. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.3 Multiple trajectories Qj on a single potential surface sheet as a function of time. A Gaussian wavepacket gj(Q,t) is ‘‘floated’’ (i.e. the centre of the Gaussian lies on the trajectory) on each trajectory. These wavepackets mix to form the nuclear wavefunction |Ψ(Q,t)〉.
Adapted from C. S. M. Allan, B. Lasorne, G. A. Worth and M. A. Robb, J. Phys. Chem. A,
2010, 114, 8713– Copyright 2010 American Chemical Society.

5. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.4 A cross-section at a given time t in Figure 4.3 showing the probability (or |ψ(Q,t)|2 the effective weight) of each gj(Q,t) in ψ(Q,t)|2.
Adapted from C. S. M. Allan, B. Lasorne, G. A. Worth and M. A. Robb, J. Phys. Chem. A,
2010, 114, 8713– Copyright 2010 American Chemical Society.

6. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.5 Adiabatic (a) to diabatic (b) transformation. The apex of the cone goes away and is replaced by a ‘‘seam’’ of crossing.

7. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.6 Toluene electron dynamics (fixed nuclei): (a) toluene cation potential surfaces; the vertical stick indicates the equilibriumgeometry of the neutral species. (b) The spin density as a function of time from an electron dynamics simulation starting from a superposition of the two lowest energy radical cation states
at the equilibriumgeometry of the neutral species. The numbering of the C atoms begins with the atom
bonded to the methyl group. Note that the radical pair oscillates between atoms 3–6 and 2–5.
Adapted with permission from M. Vacher, J. Meisner, D. Mendive-Tapia, M. J. Bearpark and
M. A. Robb, J. Phys. Chem. A, 2015, 119, 5165–5172. Copyright 2015 American
Chemical Society.

8. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.7 (a) VB structures in the ‘‘moat’’ of the conical intersection shown in the plane of the branching space
(GD: gradient difference; DC: derivative coupling). (b) The electron dynamics in Figure 4.6 oscillates along the
direction labelled DC. The geometry is at the point shown by the dot on the GD axis.
Adapted with permission from M. Vacher, J. Meisner, D. Mendive-Tapia, M. J. Bearpark and M. A. Robb,
J. Phys. Chem. A, 2015, 119, 5165–5172. Copyright 2015 American Chemical Society.

9. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 4.8 Nuclear trajectories in the branching space initiated with |Ψ0〉, θ = 90 green; 1/√2|ψ0〉 + 1/√2|ψ1〉, θ = 45 blue; |Ψ1〉, θ = 0 red. The dashed lines show the average evolution for the ensemble of trajectories. The solid lines show the evolution of the unsampled trajectory starting at the equilibrium geometry of the neutral system.