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Propagation through conical crossings: An asymptotic semigroup
Author(s) -
Lasser Caroline,
Teufel Stefan
Publication year - 2005
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20087
Subject(s) - semigroup , semiclassical physics , surface hopping , conical intersection , mathematics , conical surface , trajectory , markov process , intersection (aeronautics) , order (exchange) , scale (ratio) , statistical physics , quantum , mathematical analysis , quantum mechanics , geometry , physics , statistics , engineering , aerospace engineering , finance , economics
We consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics. Our main result is the construction of a semi‐group in order to approximate the Wigner function associated with the solution of the Schrödinger equation at leading order in the semiclassical parameter. The semigroup stems from an underlying Markov process that combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing. Our semigroup can be viewed as a rigorous mathematical counterpart of so‐called trajectory surface hopping algorithms, which are of major importance in molecular simulations in chemical physics. The key point of our analysis, the incorporation of the nonadiabatic transitions, is based on the Landau‐Zener type formula of Fermanian‐Kammerer and Gérard10 for the propagation of two‐scale Wigner measures through conical crossings. © 2005 Wiley Periodicals, Inc.