Sketch of the symmetric and antisymmetric allylic stretch coordinates, and , and of the torsion angle of fulvene.
Two-dimensional cuts of PESs of the and electronic states of fulvene for . The figure shows the PESs as a function of the symmetric allylic stretch and the torsion angle . The torsion angle is chosen such that and correspond to the planar geometry of fulvene and correspond to a twisted geometry.
One-dimensional cuts of the adiabatic [(a) and (c)] and diabatic [(b) and (d)] PESs for a torsion angle (twisted structure). The value of the asymmetric stretch is for (a) and (b) and for (c) and (d).
Same as Fig. 3, but for a torsion angle and (planar structure).
Diabatic potential couplings as a function of the coupling coordinate for (a) and (b) and for (solid line) and (dashed line).
Initial symmetric and antisymmetric wave functions and of para- and ortho-fulvene, respectively, as a function of the torsion angle and the antisymmetric stretch .
Population of the (adiabatic) excited electronic state (upper panels) and absolute value of the autocorrelation function (lower panels) for the antisymmetric initial state (ortho-fulvene, solid gray lines) and for the symmetric initial state (para-fulvene, dashed black lines). Panels (a) and (b) show the results for the couplings and fitted to the ab initio PESs. For comparison, panels (c) and (d) show the results without diabatic coupling, .
Expectation values for the antisymmetric (solid gray lines) and symmetric (dashed black lines) initial wave functions, representing para- and ortho-fulvene, respectively, cf. Fig. 6. The top, middle, and lower panels show the expectation values for the wave functions on the excited and ground adiabatic electronic state and for the sum of states, respectively. The diabatic coupling strength is for panels (a)–(c) and for panels (d)-(f).
Expectation values for the antisymmetric (solid gray lines) and symmetric (dashed black lines) initial wave functions, representing para- and ortho-fulvene, respectively. The notations are as in Fig. 8.
Fulvene oriented along the -axis of laboratory fixed coordinates with origin at the center of mass. The nuclei of the carbon and hydrogen atoms are labeled and 7–12, respectively. The Cartesian and cylindrical coordinates of the nuclei are denoted and , see examples for and , respectively. Cylindrical coordinates of the centers of masses of the CH-fragments labeled are denoted , see the example . Alternatively one may employ molecule-fixed coordinates , , where is parallel to the bond, is in the plane of the cyclopentadienyl ring, and is perpendicular to and , see text.
The coordinates (“symmetric allylic stretch”), (“antisymmetric allylic stretch”), (“wagging-type allylic stretch”), and (“translational allylic coordinate”) and corresponding displacements of the CH fragments, parallel to the -axis, for the cyclopentadienyl ring of the model fulvene (schematic).
Equivalent periodic domains [panel (a)] and [panel (c)], cf. Eq. (A20). The construction of panel (c) from (a) via (b) is carried out in two steps, using the periodicities of the angles: First the triangular domain ABC [panel (a)] is transferred into the equivalent domain [panel (b)]. Second, the triangular domain [panel (b)] is transferred into the equivalent domain [panel (c)]. For example, the equivalent coordinates of D are , or , , cf. Eq. (A20).
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