Landau-Zener transitions in a linear chain

Landau-Zener ~LZ! theory treats a quantum system placed in a slowly varying external field. If such a system was prepared in a state of its discrete spectrum, it adiabatically follows this state until its time-dependent energy level crosses another one. Near the crossing point the adiabaticity can be violated and the system can escape from the state it occupied initially to another one. Landau and Zener found the transition probability for two-level crossing. The crossing of more than two levels at the same time is generally an unlikely coincidence. However, in some systems such a multilevel crossing may occur systematically, due to the high symmetry of the underlying Hamiltonian. The transition matrix for special cases of multilevel crossing was studied in Refs. 3–8. Presently only a few exact results for multilevel crossing are known. One of them relates to a multiplet of atomic electronic states with a total spin S or total rotational moment J larger than 1/2 in a varying external magnetic field. The Zeeman splitting between 2S11 or 2J11 levels regularly vanishes at nodes of the magnetic field. Another exactly solvable model displaying multilevel crossing is the so-called bow-tie model, whose physical interpretation is not obvious. Since its creation in 1932, LZ theory has had numerous applications. They include molecular predissociation, slow atomic and molecular collisions, and electron transfer in biomolecules. Recently Wernsdorfer et al. employed the LZ theory to describe consistently the steplike shape of the hysteresis loop in special molecules with large magnetic moments called nanomagnets. Using the LZ probability formula these authors were able to find the extremely small tunnel splitting of the classic degenerate ground states and even to reveal oscillations of this value in an external magnetic field. This beautiful experiment, together with its clever treatment, is a triumph of quantum mechanics and, in particular, LZ theory. The problem considered in this article is closely related to another application of LZ theory: electronic transfer in donor-acceptor complexes. In this process of biological and chemical importance, an electron tunnels between initial and final positions through a long chain of identical sites. There are two limiting cases for such a process. In the first case there is no coherence between two sequential tunneling processes connecting nearest-neighbor sites. In this case the probability of tunneling through several sites is very small in comparison to that for one-site tunneling. This limiting case