Zhu-Nakamura theory

Stokes phenomenon plays a key role in the theory of Zhu and Naka-mma [510 512, 514], which is the only complete solution of the linear curve 

The second [510] and the third [511] of their series of papers, Zhu and Nakamura discuss approximate expression of 17i in a parameter-dependent manner, while in the fourth paper [512], they also propose more handy (empirical) formulae for LZ-like probability pLZ = A. To quote one of the latter type of formulae, suggested approximation to pLZ in case F2 0 is given as 

4The definition of f/i and U2 here follows those of Ref. [514], which is by a factor different from those in Ref. [512] 

The Zhu Nakamma theory also covers other cases such as low-energy collision dynamics, where the incident energy is much lower than the barrier of the lower adiabatic potential (see Ref. [512] for the explicit form appropriate for this parameter range). That is, it can handle correctly and conveniently a system in which nonadiabatic dynamics couples with tunnelling phenomenon. 

One great advantage of the Zhu-Nakamura theory is that its transition amplitude can be represented in terms of the parameters related to the adiabatic potentials and incident energy alone. Although the derivation started from the diabatic representation of the system, transformation to the diabatic parameters from the adiabatic counterparts is in fact not needed, once the effective parameters as a and b in Ekj. (4.28) are obtained. Thus it is applicable solely within the adiabatic representation [291]. 

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VIII. Appendix Zhu-Nakamura Theory and Summary of the Formulas A. Landau—Zener Type of Transition... 

VIII. APPENDIX ZHU-NAKAMURA THEORY AND SUMMARY OF THE FORMULAS... 

The author would like to thank all the group members in the past and present who carried out all the researches discussed in this chapter Drs. C. Zhu, G. V. Mil nikov, Y. Teranishi, K. Nagaya, A. Kondorskiy, H. Fujisaki, S. Zou, H. Tamura, and P. Oloyede. He is indebted to Professors S. Nanbu and T. Ishida for their contributions, especially on molecular functions and electronic structure calculations. He also thanks Professor Y. Zhao for his work on the nonadiabatic transition state theory and electron transfer. The work was supported by a Grant-in-Aid for Specially Promoted Research on Studies of Nonadiabatic Chemical Dynamics based on the Zhu-Nakamura Theory from MEXT of Japan. 

The Zhu-Nakamura theory provides analytical expressions separately for the following three energy regions < E[ (top of the lower adiabatic potential), iif < E < Eb (bottom of the upper adiabatic potential), and Et < E. The formulas given here contain some empirical corrections so that the formulas can cover even those small regions of parameters in which the original formulas are not necessarily very accurate. Thus the formulas given below can be directly applied to practical problems. 

Summary. An effective scheme for the laser control of wavepacket dynamics applicable to systems with many degrees of freedom is discussed. It is demonstrated that specially designed quadratically chirped pulses can be used to achieve fast and near-complete excitation of the wavepacket without significantly distorting its shape. The parameters of the laser pulse can be estimated analytically from the Zhu-Nakamura (ZN) theory of nonadiabatic transitions. The scheme is applicable to various processes, such as simple electronic excitations, pump-dumps, and selective bond-breaking, and, taking diatomic and triatomic molecules as examples, it is actually shown to work well. 

C. Zhu and H. Nakamura, Theory of nonadiabatic transition for general two-state curve crossing problems. I Nonadiabatic tunneling case, J. Chem. Phys. 101 10630 (1994). 

Surface Hopping Model (SHM) first proposed by Tully and Preston [444] is a practical method to cope with nonadiabatic transition. It is actually not a theory but an intuitive prescription to take account of quantum coherent jump by replacing with a classical hop from one potential energy surface to another with a transition probability that is borrowed from other theories of semiclassical (or full quantum mechanical) nonadiabatic transitions state theory such as Zhu-Nakamura method. The fewest switch surface hopping method [445] and the theory of natural decay of mixing [197, 452, 509, 515] are among the most advanced methodologies so far proposed to practically resolve the critical difficulty of SET and the primitive version of SHM. 

To deal with the ET rate in such a case, our strategy is to combine the generalized nonadiabatic transition state theory (NA-TST) and the Zhu-Nakamura (ZN) nonadiabatic transition probability.The generalized NA-TST is formulated based on Miller s reactive flux-flux correlation function approach. The ZN theory, on the other hand, is practically free from the drawbacks of the LZ theory mentioned above. Numerical tests have also confirmed that it is essential for accurate evaluation of the thermal rate constant to take into account the multi-dimensional topography of the seam surface and to treat the nonadiabatic electronic transition and nuclear tunneling effects properly. 

Multidimensional theory is not yet available, unformnately, not only for the nonadiabatic tunneling problem but also for general nonadiabatic transition problems. For practical applications, the Zhu-Nakamura formulas for transition amplitude including phases can be incorporated into classical or semiclassical propagation... 

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965], For further refinements of the theory and the ways to go beyond this approximation, see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], and Zhu and Nakamura [1992], The nonadiabatic transition probability for a more general case of dissipative tunneling is found in Appendix C at the end of Chapter 5. We cite here only the result of the dissipationless case, which is commensurate with the results of the papers cited above. When Etransition probability is the product of the adiabatic tunneling rate, calculated in previous sections, and a factor resembling the Landau-Zener-Stueckelberg expression ... 

C. Zhu, G. Mil nikov and H. Nakamura, Semiclassical Theory of Nonadia-batic Transition and Tunneling. In Modem Trends in Chemical Reactions Dynamics—Part I., edited by K. P. Liu and X. M. Yang (World Scientific, Singapore, 2003)... 

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