On the multidimensional surface intersection problem and classical trajectory ’’surface hopping’’

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Abstract
It is found that within the two (electronic) ‐state approximation, the multidimensional surface intersection problem may be reduced to an equivalent local one‐dimensional curve crossing problem. The unique direction along which the adiabatic surfaces avoid a crossing is found to be normal to the ’’surface of avoided intersection.’’ One may apply this result to the surface hopping trajectory method of Tully and Preston without having to define the surfaces of avoided intersection beforehand. Furthermore, the Landau–Zener transition probability at such an avoided intersection may be calculated from information obtained along the trajectory. These results are applied to trajectory calculations of H+2+H2 collisions.