2-STATE LINEAR CURVE CROSSING PROBLEMS REVISITED .4. THE BEST ANALYTICAL FORMULAS FOR SCATTERING MATRICES

Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau-Zener (LZ) and the nonadiabatic tunneling (NT) case: two f ormulas valid at b(2) greater than or equal to 0 and b(2) less than or equal to 0 in the LZ case, and three formulas valid at b(2) less than or equal to-1, -1 less than or equal to b(2) less than or equal to 1 and b(2) greater than or equal to 1 in the NT case, where b(2) represe nts the effective energy. Simple and compact formulas which work far b etter than the LZ formula are proposed for nonadiabatic transition pro bability by one passage of crossing point for both the LZ and NT cases . Furthermore, compact expressions are derived, for the first time, fo r the nonadiabatic tunneling probability at b(2) less than or equal to 1, i.e., at energies lower than the bottom of the upper adiabatic pot ential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expan ded compared to the previous formulas by employing certain empirical c orrections. Besides, these formulas have convenience to enable an exte nsion to general curved potentials.