The Landau-Zener theory of curve crossing model

1 The Landau-Zener theory of curve crossing model 

We assume that the electronic state is in the state 2 at the beginning. For simplicity, we also assume that Fi — F2 to be positive. We assume that the nuclear coordinate X has initially large positive value and propagates leftward (uphill) in constant velocity v, that is, X = —vt. The wavefunctions of the two electronic states can be represented by i(r X) and respectively. The total wavefunction is then represented as 

Considering X as a time-dependent external parameter, the Schrodinger equation is written in a matrix equation 

Equation (4.2) is to be solved as an initial value problem under the conditions Cl(—00) = 0 and C2(—oo) = 1, so that the nonadiabatic transition probability (or, equivalently, the probability of the system remains as the diabatic state 2) is given by P = C2(- -oo)p. Eliminating C2, the problem is reduced to a second order differential equation 

This results in Ci 1 — as t oo, which is the probability of adiabatic passage. Cases with different signs of F and F2 are also obtained in the same manner. With a slight generalization, the transition probability is, 

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