Non-crossing rule

A conical intersection needs at least two nuclear degrees of freedom to form. In a ID system states of different symmetry will cross as Wy = 0 for i j and so when Wu = 0 the surfaces are degenerate. There is, however, no coupling between the states. States of the same symmetry in contrast cannot cross, as both Wij and Wu are nonzero and so the square root in Eq. (68) is always nonzero. This is the basis of the well-known non-crossing rule. 

Katriel J, Davidson ER (1980) The non-crossing rule triply degenerate ground-state geometries of CH. Chem Phys Lett 76 259... 

There may be special difficulties in reactions where the ordering of orbitals centred on the metal changes along the actual reaction path, because of configuration interaction and the non-crossing rule for states. 

As an example we can take the excited states of NO. It has been shown that there are two excited states of the same symmetry ( 11) whose vibrational levels are best interpreted on the basis of diabatic curves which cross as in Fig. 1 (75-7 7). One of these states (B) arises from the electron excitation to an antibonding valence molecular orbital and the other (C) from excitation to a Rydberg orbital. The Born-Oppenheimer adiabatic curves cannot cross (by virtue of the non-crossing rule which is to be discussed in a later section) and must fullow the dashed curves shown in the figure. 

Historically the first application of symmetry to potential energy surfaces was to prove the so-called non-crossing rule. In its simplest form this may be stated as potential energy curves for states of diatomic molecules of the same symmetry do not cross . We have already seen in section 2 that this should be qualified to apply to adiabatic curves, as in some situations it may be convenient to define diabatic curves wdiich do cross. 

The established methods of proving the non-crossing rule which are in the literature (32,33) have recently been criticised by Naqvi and Byers-Brown (34), but their proof in turn has been criticized by Longuet-Higgins (35). All proofs attempt to show that crossing will only occur if conditions are satisfied for two functions of the internuclear distanced which are assumed to be independent. If we write this in the general form... 

Once the two sides of a correlation diagram have been established, the states of the same symmetry and multiplicity are connected by straight lines in such a way as to observe the non-crossing rule identical states cannot cross as the strength of the interaction is changed. When this is done we have completed the correlation diagram. 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

Molecular Crystals as a Consequence of the Non-Crossing Rule (Level Anti-Crossing). 

Enhancement via Albrecht s 5-term derives from the non-Condon dependence of the electronic transition moment upon the vibrational coordinate. Unlike the A-term, the 6-term arises from the vibronic mixing of two excited states and it is non-zero for scattering due to both totally symmetric and non-totally symmetric fundamentals, provided that they are responsible for vibronic coupling of the states. The latter only takes place for a vibrational fundamental whose irreducible representation is contained in the direct product of the irreducible representations of the two states. Thus, 6-term activity for a totally symmetric mode requires that the latter must vibronically couple two states of the same symmetry. As a consequence of the non-crossing rule this holds only for few excited states which are lying very close together. 

Each reactant state correlates with some state of the products along the potential. Vibrations and rotations that are similar in the reactant and product (conserved modes), remain in the same quantum state throughout the channel, in the sense that their quantum numbers remain the same throughout. Other modes that change between reactants and products (transitional modes), are subject to correlation rules. Channels with the same angular momentum are not permitted to cross, similar to the non-crossing rule in diatomic molecules. 

Figure 9. Schematic adiabatic channel potentials for a dissociation reaction, illustrating the development of barriers due to convergence of vibrational levels and the non-crossing rule. At the energy indicated, two of the channels are open and two are closed.

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