Noncrossing Rule and Conical Intersections

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- 

to have degenerate solutions (i.e., an unavoided crossing), the discriminant must vanish, and it is necessary to satisfy two independent conditions  

This requires the existence of at least two independently variable nuclear coordinates. Since in a diatomic molecule there is only one variable coordinate—the interatomic distance—so the noncrossing rule can be stated as follows  

For states of different (spatial or spin) symmetry, H12 is always zero, and the two surfaces cross when Htl = H21. This is possible for a suitable 

However in a system of three or more atoms (N is the number of atoms), there are enough degrees of freedom for the rule to break down the two conditions (Eq. [5]) can be simultaneously satisfied by choosing suitable values for two independent variables, while the other n - 2 degrees of freedom (n = 3N-6) are free to be varied without leaving the crossing region. 

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