  We consider non-adiabatic transitions in multiple dimensions, which occur
when the Born-Oppenheimer approximation breaks down. We present a general,
multi-dimensional algorithm which can be used to accurately and efficiently
compute the transmitted wavepacket at an avoided crossing. The algorithm
requires only one-level Born-Oppenheimer dynamics and local knowledge of the
potential surfaces. Crucially, in contrast to standard methods in the
literature, we compute the whole wavepacket, including its phase, rather than
simply the transition probability. We demonstrate the excellent agreement with
full quantum dynamics for a range of examples in two dimensions. We also
demonstrate surprisingly good agreement for a system with a full conical
intersection.
