Polyatomic molecules and conical intersection

The non-crossing rule for a diatomic molecule has been based on eq. (6.43). lb achieve the crossing we had to make two independent terms vanish with only one parameter (the intemuclear distance R) able to vary. It is important to note that in the case of a polyatomic molecule the formula would be the same, but the number of parameters would be laiger 3Af — 6 in a molecule with N nuclei. For N = 3 one has already, therefore, three such parameters. No doubt even for a three-atomic molecule we would be able to make the two terms equal to zero and, therefore, achieve E+ = E-, i.e. the crossing of the two diabatic hypersurfaces. 

Let us investigate this possibility, which, for reasons that will become clear later, is called conical intersection. We will approach this concept by a few steps. 

All the quantities in eq. (6.43) depend on n = iN — 6 coordinates of the nuclei. These coordinates may be chosen in many different ways, the only thing we should bother about is that they have to determine the positions of N point objects. To begin, let us construct a Cartesian system of 3A coordinates O n)- Let us locate (Fig. 6.13) nucleus I at the origin (in this way we eliminate three degrees of freedom connected with the translation of the system), nucleus 2 will occupy the point X2 on the x axis, i.e. 2 = 2 2 = 0. In this way we have eliminated two rotations of the system. The total system may still be rotated about the x axis. This last possibility 

Thus six degrees of freedom have been eliminated Irom the 3N coordinates. The other nuclei may be indicated by vectors (jc,-, y,-, z,) for i = 4,5.N. Asv/e can see there has been a lot of arbitrariness in these choices. By the way, if the molecule was diatomic, the third rotation need not be determined and the number of variables would be equal ton = 3x2 — 5=1. 

This choice of coordinate qfstem may be viewed a little differently. We may construct a Cartesian coordinate system with the origin at atom 1 and the axes X2, JC3, 3 and Xi, yi, z, for i = 4,5. IV. Thus, we have a Cartesian coordinate system with 3 + 3 N - 3) = 3N — 6 = n axes, which may be labelled (in 

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A very useful starting point for the study of non-adiabatic processes, which are common in photochemistry and photophysics, is the vibronic coupling model Hamiltonian. The model is based on a Taylor expansion of the potential surfaces in a diabatic electronic basis, and it is able to correctly describe the dominant feature resulting from vibronic coupling in polyatomic molecules a conical intersection. The importance of such intersections is that they provide efficient non-radiative pathways for electronic transitions. Not only is the position and shape of the intersection described by the model, but it also predicts which nuclear modes of motion are coupled to the electronic transition which takes place as the system evolves through the intersection. 

In photochemistry one must deal with a new type of potential surface feature (surface crossings and conical intersections), and we now introduce this subject. In diatomic molecules the PES of two states (e.g., the ground state and the first excited state) will intersect only if the states have a different (spatial or spin) symmetry. However, this statement is not true in polyatomic sys-... 

Simulating the harpooning effect in the NaCl molecule Polyatomic molecules and the conical intersection ((DO)... 

In recent years, computational testimonies for the existence of conical intersections in many polyatomic systems became abundant and compelling [6-11]. The current consensus concerning the ubiquitous presence of conical intersections in polyatomic molecules is due in large part to computational experiments. ... 

Reactions in which structural change is simultaneously occurring in more than one structural parameter can be depicted as interaction between surfaces with coordinates described by the structural parameters. For many photochemical reactions it has been found that transfer from an excited to a ground state involves a conical intersection (Cl), which can be thought of as a funnel that permits transition from one energy surface (state) to another. The efficiency of the transformation depends on the structural similarity between the excited state and the corresponding ground state molecular ensemble. There can be a number of CIs for the excited states of a typical polyatomic molecule. The transition occurs without luminescence. Conical... 

Conical intersections between electronically adiabatic potential energy surfaces are not only possible but actually quite frequent, if not prevalent, in polyatomic systems. Some examples are triatomic systems whose isolated atoms have 2S ground states [1,2] such as H3 and its isotopomers (DH2, HD2, HDT, etc.), LiH2 and its isotopomers, and tri-alkali systems such as Na3 and LiNaK. Many other kinds of polyatomic molecules also display such intersections. The reason is that they have three or more internal nuclear motion degrees of freedom, and only two independent relations between electronic Ham-... 

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