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Fig. 10. Contours of the energy difference [E(1'B,)-E(1'A,)] in the C», restricted coordinated space near point A. The interna coordinates are the Cartesian coordinates of the end atom (se¢ Fig. 3 of ref. [1]). Energy increment = 10 mh. Solid lines, > 0; dashec lines, <0; bold contour = 0 = intersection seam in C),. The dot indicated as A is the point where the intersection seam given by Figs. £ and 6 penetrates this x-y plane. Asterisk: see Sect. 4.5 We call the confluence A of the two seams a node. Here the standard dimensionality rules [3] regarding intersec- tions require a modification which is readily understood in terms of the derivation of these rules (see, for example, our discussion in Ref. [3b]): In the present case we have, in the region of interest, two surfaces on which Hj, = 0 (one of them being the C),-conserving

Figure 10 Contours of the energy difference [E(1'B,)-E(1'A,)] in the C», restricted coordinated space near point A. The interna coordinates are the Cartesian coordinates of the end atom (se¢ Fig. 3 of ref. [1]). Energy increment = 10 mh. Solid lines, > 0; dashec lines, <0; bold contour = 0 = intersection seam in C),. The dot indicated as A is the point where the intersection seam given by Figs. £ and 6 penetrates this x-y plane. Asterisk: see Sect. 4.5 We call the confluence A of the two seams a node. Here the standard dimensionality rules [3] regarding intersec- tions require a modification which is readily understood in terms of the derivation of these rules (see, for example, our discussion in Ref. [3b]): In the present case we have, in the region of interest, two surfaces on which Hj, = 0 (one of them being the C),-conserving

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Fig. 1. Parameter space of the scale-independent shape coordinates. The labelled points are defined in Section 3.1
Fig. 1. Parameter space of the scale-independent shape coordinates. The labelled points are defined in Section 3.1
Fig. 3. Illustration of adiabatic wavefunction sign change about intersection The case at hand is a particularly favorable one since all dominant configurations in the adiabatic states ‘P, and ‘¥, contain doubly occupied orbitals only and, hence, also for the diabatic states ¢,, ¢,. Consequently,
Fig. 3. Illustration of adiabatic wavefunction sign change about intersection The case at hand is a particularly favorable one since all dominant configurations in the adiabatic states ‘P, and ‘¥, contain doubly occupied orbitals only and, hence, also for the diabatic states ¢,, ¢,. Consequently,
Fig. 2. The four regions generated by the surfaces AH = 0 and Hj. = 0 around an intersection. The latter is indicated by the heavy dot
Fig. 2. The four regions generated by the surfaces AH = 0 and Hj. = 0 around an intersection. The latter is indicated by the heavy dot
Fig. 4. Corralling an intersection. See text The method is illustrated in Fig. 4 for the determi- nation of the intersection point in Co, symmetry. The
Fig. 4. Corralling an intersection. See text The method is illustrated in Fig. 4 for the determi- nation of the intersection point in Co, symmetry. The
Fig. 5. Projection of the 1!4’-2’4’ intersection seam of ozone onto the scale-independent shape coordinate space Because of the very short bond length of 723 at point A and the correspondingly small bond angle at atom 1
Fig. 5. Projection of the 1!4’-2’4’ intersection seam of ozone onto the scale-independent shape coordinate space Because of the very short bond length of 723 at point A and the correspondingly small bond angle at atom 1
Table 1 Bond lengths of ozone at various geometries (about 33.5°), the molecule is obviously very unstable at this geometry. The energy along the seam is displayed in Fig. 8, which confirms that the C2, intersections A, E, G lie in a highly repulsive region of the PES. The repulsive character in the region of point A is also illustrated by Fig. 9, which shows a projection of the intersection seam on the PES contour map for the circumference r = 4.6 A, which is about midway on the seam. Even for this con- stant value of r, the energy difference between the two different C>, points A and C is about 80 mh. Because of the variation in r, shown in Fig. 6, the difference between points A and C in Fig. 8 is in fact about 370 mh.
Table 1 Bond lengths of ozone at various geometries (about 33.5°), the molecule is obviously very unstable at this geometry. The energy along the seam is displayed in Fig. 8, which confirms that the C2, intersections A, E, G lie in a highly repulsive region of the PES. The repulsive character in the region of point A is also illustrated by Fig. 9, which shows a projection of the intersection seam on the PES contour map for the circumference r = 4.6 A, which is about midway on the seam. Even for this con- stant value of r, the energy difference between the two different C>, points A and C is about 80 mh. Because of the variation in r, shown in Fig. 6, the difference between points A and C in Fig. 8 is in fact about 370 mh.
Fig. 6. Variation of the molecular circumference along the intersec- tion seam of Fig. 5
Fig. 6. Variation of the molecular circumference along the intersec- tion seam of Fig. 5
Fig. 7. Variation of the molecular shape along the intersection seam described by Figs. 5 and 6
Fig. 7. Variation of the molecular shape along the intersection seam described by Figs. 5 and 6
Fig. 8. Variation of the molecular energy along the intersection seam described by Figs. 5 and 6
Fig. 8. Variation of the molecular energy along the intersection seam described by Figs. 5 and 6
Fig. 9. Projection of the intersection seam, given by Figs. 5 and 6, onto the contours of the ground state in the plane r = 4.6. Energy increment between contours = 10 mh
Fig. 9. Projection of the intersection seam, given by Figs. 5 and 6, onto the contours of the ground state in the plane r = 4.6. Energy increment between contours = 10 mh
Fig. 11. Intersection seams near the confluence point A. Hj;, Hy, H)2 are matrix elements between diabatic states
Fig. 11. Intersection seams near the confluence point A. Hj;, Hy, H)2 are matrix elements between diabatic states
Fig. 12. Two possibilities for the four branches of the 1'4’-2'4’ intersection seam in the shape-scale coordinate space. Shaded planes, Cy,-restricted coordinate regions
Fig. 12. Two possibilities for the four branches of the 1'4’-2'4’ intersection seam in the shape-scale coordinate space. Shaded planes, Cy,-restricted coordinate regions
Fig. 13. Natural orbitals of ozone at three points along the seam from point C to point A. Contours of o orbitals, in a molecular plane contours of x, in a plane above and parallel to molecular plane
Fig. 13. Natural orbitals of ozone at three points along the seam from point C to point A. Contours of o orbitals, in a molecular plane contours of x, in a plane above and parallel to molecular plane
Table 2. Dominant configurations of the 1'A’ and 2'A’ states
Table 2. Dominant configurations of the 1'A’ and 2'A’ states
Fig. 14. Occupation numbers of the natural orbitals o, 21, m2 along the 1'4’-2'4’ intersection seam from point C to point A. Solid dots, occupancies for state 1; hollow dots, occupancies for state 2
Fig. 14. Occupation numbers of the natural orbitals o, 21, m2 along the 1'4’-2'4’ intersection seam from point C to point A. Solid dots, occupancies for state 1; hollow dots, occupancies for state 2
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