A criterion for the confluence of two seams of conical intersection in triatomic molecules

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It is shown that the cross product tIJ(Rx)≡gIJ(Rx)×hIJ(Rx), where gτIJ(R)=(cI(Rx)−cJ\(\)(cI(Rx)+cJ(Rx)), hτIJ(R)=cI (Rx)†\(\)cJ(Rx), τ is an internal nuclear coordinate, the cI(R) satisfy [H(R)−EI(R)]cI(R)=0 and H(R) is the electronic Hamiltonian matrix, is a unique property of a conical intersection at Rx. tIJ(Rx)=0 when Rx is located at the intersection of two (or more) seams of conical intersection. This criterion for an intersection of two seams of conical intersection has important implications for algorithms that seek to locate such points. Here it␣is␣used to analyze the trifurcation of a generic C2v2S+1A−2S+1B seam of conical intersection, analogous to those recently found in AlH2 and CH2.