(a) Schematic representation of the H3 system and the trajectory (along the vector r) of the third H atom towards a fixed H2 molecule. This trajectory is defined by a small angle, α = 1°, with respect to the X-axis. (b) PES for the ground (E 0, green) and first excited (E 1, blue) state of the H3 system as a function of the distance r along this trajectory. (c) Evolution of the X-component of the non-adiabatic coupling vector between the HOMO and LUMO KS states along the trajectory.
(a) PES for the H3 system along the path defined by the hyperspherical coordinates ρ = 2.5 a 0 and ϕ = 120° as a function of the angle θ, see text. The ground (blue) and first excited (red) PES are shown. (b) HOMO and LUMO KS levels as a function of the angle θ for the ground state (blue), excited state (red), and Slater transition-state (green) densities. (c) NACVs (x-component on the second H atom) as a function of θ: (circles/blue) reference ab initio results by Abrol et al. 48 (taken from Ref. 46 ); (triangles/green) real-time TDDFT results by Baer; 46 (crosses/black) linear-response TDDFT results by Tavernelli et al.; 9 (triangles/violet) linear-response TDDFT results by Hu et al.; 49 (squares/red) our results, using a LDA Slater transition-state calculation and Eq. (18) .
The atomic structure of formaldimine corresponding to the two symmetric ground state configurations (a,b) and transition configuration (c), where β = 90° and α ∼ 15°. α is the angle between the XY-plane and the vector that goes from the N atom to the H atom; β is the angle between the projection of that vector in the XY-plane and the negative X-axis.
Potential energy surfaces (PES) for the ground state (E 0) and first excited state (E 1) for the formaldimine molecule as a function of the angles α and β, see Fig. 3 .
as a function of the angles α and β of formaldimine molecule for the HOMO and LUMO KS states corresponding to (a) the ground state E 0 calculation; (b) the first excited state E 1 calculation; and (c) the Slater Transition State calculation.
NACV between HOMO and LUMO KS states for the formaldimine molecule in the transition region (α ∼ 15°, β = 90°). The arrows show the direction and magnitude of the components on the different atoms α of the formaldimine molecule; (a) ground state, (b) excited state, and (c) Slater transition-state calculations.
The atomic structure of cis-azobenzene. The angle α between the planes of the two phenyl rings defines the path that have been analyzed between the cis- (α = 0°, shown in this figure) and trans- (α = 180°) configurations.
(Top) Potential energy surfaces (PES) E 0 (blue) and E 1 (green) as a function of the angle α (see Fig. 7 ). (Bottom) Modulus of the projection on some atoms (N1, N2, C8, and C14, see top figure) of the non-adiabatic coupling vector between the HOMO and LUMO KS states along the isomerization pathway as a function of the angle α.
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