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Fig. 11. Preliminary valence description of the molecule.

Figure 11 Preliminary valence description of the molecule.

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Fig. 1. A contour plot of the S; excited electronic state of NOCI as function of the two lengths of the Jacobi vectors: ra, the distance between the center of mass of NO and Cl, r, the distance NO is shown for @ = 127°, @ being the angle between the two Jacobi vectors. Superimposed are contours of the wave packet (averaged over all angles) for t= 0, 12, 24, and 36 fs describing the photodissociation into NO + Cl. 4. Examples Let us return to MCTDH. For NOCI, Fig. 1 displays the propaga- tion of the wave packet with MCTDH. Since the wave packet re- mains rather localized and the form of the potential along r, slowly varies with rg, the correlation effects coming from terms of
Fig. 1. A contour plot of the S; excited electronic state of NOCI as function of the two lengths of the Jacobi vectors: ra, the distance between the center of mass of NO and Cl, r, the distance NO is shown for @ = 127°, @ being the angle between the two Jacobi vectors. Superimposed are contours of the wave packet (averaged over all angles) for t= 0, 12, 24, and 36 fs describing the photodissociation into NO + Cl. 4. Examples Let us return to MCTDH. For NOCI, Fig. 1 displays the propaga- tion of the wave packet with MCTDH. Since the wave packet re- mains rather localized and the form of the potential along r, slowly varies with rg, the correlation effects coming from terms of
Fig. 2. A contour plot of the electronic ground state of H + H2 as function of the two lengths of the Jacobi vectors: rg, the distance between the center of mass of Hz and H, r, the distance Hz is shown for 0 =0° (collinear geometry). Superimposed are contours of the wave packet (averaged over all angles) for t= 20 fs describing the reactive H + H2 collision.
Fig. 2. A contour plot of the electronic ground state of H + H2 as function of the two lengths of the Jacobi vectors: rg, the distance between the center of mass of Hz and H, r, the distance Hz is shown for 0 =0° (collinear geometry). Superimposed are contours of the wave packet (averaged over all angles) for t= 20 fs describing the reactive H + H2 collision.
Fig. 3. Jacobi description of H2 + Hz collision. The ahedtral angle, ~1, (not shown on the Figure) is the angle between the (Ri,R 3) and (R. R3) planes (see Ref. [17] for precise conventions). The resulting KEO is erirveed ae in Ref. [38].
Fig. 3. Jacobi description of H2 + Hz collision. The ahedtral angle, ~1, (not shown on the Figure) is the angle between the (Ri,R 3) and (R. R3) planes (see Ref. [17] for precise conventions). The resulting KEO is erirveed ae in Ref. [38].
Fig. 4. Definition of the three valence vectors for the HONO rs The dihedral angle, #1, (not shown on the Figure) is the angle between the (R 1, R3) and (Ro, R3) planes (see Ref. [17] for precise conventions). Similarly, the cis-trans photoisomerisation following x > 2™ excitation of ethene was studied in Ref. [52,53]. The set of poly- spherical coordinates depicted in Fig. 5 and, in particular, a valence vector, Rs, pointing from one C-atom to the c other one was used. We have also introduced two Jacobi vectors Ri and R. 2 since they provide angles that can easily parameterize the pyramidalizations of the two CH sub-groups. Note, however, that the choice of a va- lence vector, Rs, is essential for a good description of the torsion of the C-C double bond.
Fig. 4. Definition of the three valence vectors for the HONO rs The dihedral angle, #1, (not shown on the Figure) is the angle between the (R 1, R3) and (Ro, R3) planes (see Ref. [17] for precise conventions). Similarly, the cis-trans photoisomerisation following x > 2™ excitation of ethene was studied in Ref. [52,53]. The set of poly- spherical coordinates depicted in Fig. 5 and, in particular, a valence vector, Rs, pointing from one C-atom to the c other one was used. We have also introduced two Jacobi vectors Ri and R. 2 since they provide angles that can easily parameterize the pyramidalizations of the two CH sub-groups. Note, however, that the choice of a va- lence vector, Rs, is essential for a good description of the torsion of the C-C double bond.
Fig. 5. Mixed Jacobi/valence description of the C,H, system.
Fig. 5. Mixed Jacobi/valence description of the C,H, system.
Fig. 7. Jacobi description of the Hs0; system. G, is the center of mass of the first monomer, G3 is the center of mass of the second monomer. G42 is the center of the dimer, H,03.
Fig. 7. Jacobi description of the Hs0; system. G, is the center of mass of the first monomer, G3 is the center of mass of the second monomer. G42 is the center of the dimer, H,03.
Fig. 8. Mixed Jacobi/valence description of the H;03 system, Go2 is the center of mass of the O2 subsystem, or, in other words, the middle of Ry, the valence vector joining one O atom to the other one. where q is a coordinate vector and 7(q) denotes the hierarchical approximation to “q). A similar approach has been adopted for the Zundel cation [61], where, however, the cluster expansion Eq. Now, regarding the potential, the exact PES is a function of 15 internal coordinates. A direct use of the potential is thus impossi- ble. In addition, it is impossible to convert the potential to MCTDH product form by using the potfit algorithm [28,29] since the full product grid is employed for such a transformation. The problem of representing a high-dimensional PES for quantum-dynamical computation has already been considered in the context of MULTI- MODE simulations [9], where the method is called n-mode repre- sentation [9,56,27], as well as in the more general context of the high dimensional model representation (HDMR) [57-60]. In such hierarchical representations a multidimensional function depend- ing on f variables is approximated as:
Fig. 8. Mixed Jacobi/valence description of the H;03 system, Go2 is the center of mass of the O2 subsystem, or, in other words, the middle of Ry, the valence vector joining one O atom to the other one. where q is a coordinate vector and 7(q) denotes the hierarchical approximation to “q). A similar approach has been adopted for the Zundel cation [61], where, however, the cluster expansion Eq. Now, regarding the potential, the exact PES is a function of 15 internal coordinates. A direct use of the potential is thus impossi- ble. In addition, it is impossible to convert the potential to MCTDH product form by using the potfit algorithm [28,29] since the full product grid is employed for such a transformation. The problem of representing a high-dimensional PES for quantum-dynamical computation has already been considered in the context of MULTI- MODE simulations [9], where the method is called n-mode repre- sentation [9,56,27], as well as in the more general context of the high dimensional model representation (HDMR) [57-60]. In such hierarchical representations a multidimensional function depend- ing on f variables is approximated as:
Fig. 9. The set of eight relative position vectors for malonaldehyde. A set of 21 polyspherical coordinates [17] is used to parameter- ize the molecule of malonaldehyde. After separation of the center of mass motion, the system can be parameterized by eight relative position vectors, which are depicted in Fig. 9. R3 and Ro describe the “frame” of the molecule and are probably well adapted to de- scribe the collective motions leading to the ring deformation. They define the Body-Fixed (BF) frame. Here, we use the technique of the ‘separation into subsystems’, which is described in Section 2.4 of Ref. [17]. In other words, some angles will be defined with respect
Fig. 9. The set of eight relative position vectors for malonaldehyde. A set of 21 polyspherical coordinates [17] is used to parameter- ize the molecule of malonaldehyde. After separation of the center of mass motion, the system can be parameterized by eight relative position vectors, which are depicted in Fig. 9. R3 and Ro describe the “frame” of the molecule and are probably well adapted to de- scribe the collective motions leading to the ring deformation. They define the Body-Fixed (BF) frame. Here, we use the technique of the ‘separation into subsystems’, which is described in Section 2.4 of Ref. [17]. In other words, some angles will be defined with respect
Mode combinations. For a definition of the transition state normal mode coordinates qi see Ref. [72]. The coordinates Q; in polyspherical coordinates are different from the coordinates Q; in normal coordinates.
Mode combinations. For a definition of the transition state normal mode coordinates qi see Ref. [72]. The coordinates Q; in polyspherical coordinates are different from the coordinates Q; in normal coordinates.
Mean values and variances of the error of an mth order cluster expansion in logical coordinates Q,.. Results in cm~!.
Mean values and variances of the error of an mth order cluster expansion in logical coordinates Q,.. Results in cm~!.
Fig. 10. The Benzopyran molecule: the chemical bond that breaks during the ring opening is the chemical bond between the Oxygen atom and the Carbon atom linked to two Hydrogen atoms.
Fig. 10. The Benzopyran molecule: the chemical bond that breaks during the ring opening is the chemical bond between the Oxygen atom and the Carbon atom linked to two Hydrogen atoms.
Fig. 13. The set of vectors used to parametrize the system. G; denotes the center-of- mass of the ith subsystem. In the polyspherical approach, for each subgroup one defines a body-fixed frame using two vectors belonging to the subgroup: one vector defines the z body-fixed axis, and the second one the (x, Z; x > O half-plane). The spherical coordinates of the other vec- tors within the subgroup are defined with respect to this body- fixed frame. The rotation of each subsystem with respect to the body-fixed frame of the whole molecule is parametrized by the three Euler of the body-fixed frame of the subsystem. The same oc- curs for the subsubsystems of the gr; subsystems: the rotation of each subsubsystem with respect to the body-fixed frame of grz is described by the three Euler angles of the body-fixed frame of the subsubsystem, and so on... There are many ways of choosing the body-fixed frame, since all the vectors in the subgroup can
Fig. 13. The set of vectors used to parametrize the system. G; denotes the center-of- mass of the ith subsystem. In the polyspherical approach, for each subgroup one defines a body-fixed frame using two vectors belonging to the subgroup: one vector defines the z body-fixed axis, and the second one the (x, Z; x > O half-plane). The spherical coordinates of the other vec- tors within the subgroup are defined with respect to this body- fixed frame. The rotation of each subsystem with respect to the body-fixed frame of the whole molecule is parametrized by the three Euler of the body-fixed frame of the subsystem. The same oc- curs for the subsubsystems of the gr; subsystems: the rotation of each subsubsystem with respect to the body-fixed frame of grz is described by the three Euler angles of the body-fixed frame of the subsubsystem, and so on... There are many ways of choosing the body-fixed frame, since all the vectors in the subgroup can
Fig. 12. The different subsystems.
Fig. 12. The different subsystems.
Table 3
Table 3
dynamics simulations. The correlation in both the wave function and the PES, as well as the complexity of the KEO are directly af- fected by the coordinate system. In the present paper, we show a clear connection between the correlation and the convergence of the MCTDH algorithm, and we stress the importance of curvilinear coordinates when dealing with motions of large amplitude. We ex- hibit some general trends in the choice of curvilinear coordinates, which we hope will help researchers wanting to study molecular systems with a quantum mechanical approach when motions of large amplitude are present.
dynamics simulations. The correlation in both the wave function and the PES, as well as the complexity of the KEO are directly af- fected by the coordinate system. In the present paper, we show a clear connection between the correlation and the convergence of the MCTDH algorithm, and we stress the importance of curvilinear coordinates when dealing with motions of large amplitude. We ex- hibit some general trends in the choice of curvilinear coordinates, which we hope will help researchers wanting to study molecular systems with a quantum mechanical approach when motions of large amplitude are present.
Variation of the, different coordinates depicted in Fig. 13. For instance, R17 is the length of the vector R 17, 91617 is the angle between R 1g and Ri, 151617 is the dihedral angle between Ris, Rig and R17. The angles «, and y, are defined as the Pi angles, and the f; angles as the 6; ; angles but the , £, and 1, are also the three Euler angles of the Ith subsystem. FC denotes the Franck-Condon geometry, M, the local minimum in the first electronic state between FC and the transition state, TS; denotes the transition state in the first electronic state before the conical intersection, CI denotes the minimum of the seam of the conical intersection. The columns 2-1, 3-2, 4-3, 4-1 denote the differences between the values of the coordinates for the previous geometries: 1 is for FC, 2 for Mj, 3 for TS;, and 4 for Cl. VE is the gradient at the FC geometry, D<o is the transition vector at the TS; geometry. VAE and VHj;2 the unscaled gradient difference and the derivative coupling, respectively, at the minimum of the seam of the conical intersection.
Variation of the, different coordinates depicted in Fig. 13. For instance, R17 is the length of the vector R 17, 91617 is the angle between R 1g and Ri, 151617 is the dihedral angle between Ris, Rig and R17. The angles «, and y, are defined as the Pi angles, and the f; angles as the 6; ; angles but the , £, and 1, are also the three Euler angles of the Ith subsystem. FC denotes the Franck-Condon geometry, M, the local minimum in the first electronic state between FC and the transition state, TS; denotes the transition state in the first electronic state before the conical intersection, CI denotes the minimum of the seam of the conical intersection. The columns 2-1, 3-2, 4-3, 4-1 denote the differences between the values of the coordinates for the previous geometries: 1 is for FC, 2 for Mj, 3 for TS;, and 4 for Cl. VE is the gradient at the FC geometry, D<o is the transition vector at the TS; geometry. VAE and VHj;2 the unscaled gradient difference and the derivative coupling, respectively, at the minimum of the seam of the conical intersection.
Now, in order to clearly separate the coupling between the dif- ferent subsystems from the rest, we write
Now, in order to clearly separate the coupling between the dif- ferent subsystems from the rest, we write
and T® is the KEO of monomer "i{=1, 2). yo is the total angular momentum of monomer it= 1, 2). Ti is the angular momentum asso- ciated \ with Ri, i(=1, 2). T®?) is the angular momentum associated with R°) . The exact KEO can be recast as follows
and T® is the KEO of monomer "i{=1, 2). yo is the total angular momentum of monomer it= 1, 2). Ti is the angular momentum asso- ciated \ with Ri, i(=1, 2). T®?) is the angular momentum associated with R°) . The exact KEO can be recast as follows
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