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

It is shown that the cross product t ^{I J} (R _x )≡g ^{I J} (R _x )×h ^{I J} (R _x ), where g _τ ^{I J} (R)=(c ^I (R _x )−c ^J (c ^I (R _x )+c ^J (R _x )), h _τ ^{I J} (R)=c ^I (R _x )^† c ^J (R _x ), τ is an internal nuclear coordinate, the c ^I (R) satisfy H(R)−E _I (R)]c ^I (R)=0 and H(R) is the electronic Hamiltonian matrix, is a unique property of a conical intersection at R _x . t ^{I J} (R _x )=0 when R _x 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 C_2v ^2S+1 A−^2S+1 B seam of conical intersection, analogous to those recently found in AlH₂ and CH₂. Received: 31 July 1997 / Accepted: 27 August 1997