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WavePacket demo example: Tully's curve crossing problems

In his famous 1990 paper, J. C. Tully proposed three standard problems for nonadiabatic quantum dynamics. They consist of two coupled Schrödinger equations where the diagonals of the (diabatic) potential energy matrices undergo a single or double crossing, or exhibit an extended coupling region. These prototypical cases have since evolved into standard benchmark systems for any approximate methods, in particular for mixed quantum-classical dynamics.

• The three examples are presented in Tully's original work where also the stochastic "fewest switches" surface hopping algorithm is introduced.
• Anther recommended source of information is D. F. Coker's article "Computer Simulation Methods for Nonadiabatic Dynamics in Condensed Systems" which apppeared in "Computer Simulation in Chemical Physics", edited by M. P. Allen and D. J. Tildesley, Kluwer (Netherlands), pp. 315-377 (1993).
• See also M. Ben-Nun's and T. J. Martinez's review article "Ab initio quantum molecular dynamics" which appeared in Advances in Chemical Physics, vol. 121, pp. 439-512 (2002) where also the issue of momentum/position jumps is discusssed in great detail.

Single crossing

The simplest example consists of a single crossing of two potential energy curves which are designed such that an incoming wavepacket passes the crossing once without returning to the crossing zone. Hence, interference effects are limited to the Stückelberg oscillations during the passage. Full quantum dynamics can often be well approximated by the semiclassical Landau-Zener formula. Learn more ...

Double crossing

A situation where a wavepacket passes two successive crossings of a pair of potential energy curves is more demanding because wavepacket intereference has to be taken into account. Tully's model problem is, however, constructed such that the two potentials in the adiabatic representation are roughly parallel. Hence, simple quantum-classical approximations may appear correct although they might fail for more general double crossing problems. Learn more ...

Extended coupling

The most demanding of the three standard examples is a situation with an extended region of strong nonadiabatic coupling. This example poses a severe challenge to quantum-classical algorithms. Learn more ...