By Hiroki Nakamura
Written in view that the full analytical options to the fundamental difficulties in nonadiabatic transitions have lately been formulated through the writer. For graduate scholars and researchers in molecular physics, atomic physics, actual chemistry and theoretical physics.
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25) should coincide with the first one of Eq. 25) for abritrary A and B, we can easily obtain Tt = T2 = T3 = i. 26) Now, we have the single-valued general solution valid in the whole asymptotic region. For instance, we can obtain the following connection formula for the Chapter 4- 48 Background Mathematics physical solution on the real axis: ( • , * ) • tix, •) (x < 0) <—> -i(x, •) (x > 0) . 27) The ordinary Airy function Ai(z) corresponds to this solution with A = 0. 27) represents the famous connection formula of the WKB solutions acrossing the turning point (see Eq.
5. Electronic energy diagram of the 1CTU, l7r u , and 2CT„ states of the Ne+—Ne system. : variable screening model; : model potential used. (Taken from Ref. ) the previous section. R-plane (see Eq. 22)). In order to carry out the quantum mechanical numerical calculations, however, we always stay on the real ii-axis and need the explicit information of the nonadiabatic couplings. Even in the diabatic representation, which is often employed because of its convenience, the nonadiabatic couplings are necessary to obtain the diabatic couplings , unless the diabatic potential matrix is known from the beginning.
8) The next naive question is how the connection formulas, Eqs. 6), can be derived mathematically. And also, what happens when the energy is close to the potential barrier top which cannot be approximated by a linear curve but is quadratic? These questions are related to the so called Stokes phenomenon of asymptotic solutions of ordinary differential equations of the second order. This will be explained in the next subsection. Another important concept is the comparison equation method. We can use the connection formulas, Eqs.