If two diabatic potential curves cross with opposite signs of slopes, then the lower and upper adiabatic potential curves have an attractive well and a potential barrier, respectively (see Figure 1.2). At energies lower than the top of the lower adiabatic potential, quantum mechanical tunneling can occur. This is called nonadiabatic tunneling and this type of potential curve crossing is called nonadiabatic tunneling type curve crossing (see Figure 1.2) [48]. This type of curve crossing plays important roles in chemical physics, since the so-called conical intersection of potential energy surfaces is ubiquitous in molecular systems [88]. A two-dimensional schematic view of conical intersection is drawn in Figure 5.1. The famous Jahn-Teller effect is related to this conical intersection [89-91]. When the potential energy surfaces are described by N independent coordinates, they can have a real potential surface crossing of (N − 2) dimensions, if the electronic symmetry of the two states is the same. This is called the Neumann-Wigner noncrossing rule. This can be simply proved as follows. Let V1(x), V2(x), and V12(x), be two diabatic potentials and the diabatic coupling between the two, where x collectively represents the relative nuclear coordinates. Then the corresponding adiabatic potentials W1,2(x) with W1(x) ≥ W2(x) are given by

W1,2(x) = 12 [ V1(x) + V2(x) ±

√ (V1(x) − V2(x))2 + 4V12(x)2

] . (5.1)

In order for the two adiabatic potentials W1,2(x) to cross, two terms in the square root should be zero at the same time that gives two conditions. Thus, the potential surface crossing is of (N − 2) dimensions or lower. In the case of a one-dimensional system, no crossing is possible. The two adiabatic curves can come close together but never cross on the real axis. This is called avoided crossing. In the case of N = 2, real crossing is possible at one point, namely, at the apex of two cones, at energy EX as shown in Figure 5.1 (conical intersection). If the electronic symmetry of the two states is different, then there is no diabatic coupling, i.e., V12(x) = 0 and the crossing surface can be (N − 1) dimension. If we cut the two cones in Figure 5.1 by a vertical plane near the apex, then we can easily see that the two potential curves of the cross section are of the nonadiabatic tunneling type. The crossing of two diabatic potential curves with the same sign of slopes is called Landau-Zener type [48]. These two types of curve crossings present mathematically as well as physically quite different problems. The basic differential equations for the two cases show very different characteristics [48]. Physically, as is clear from the scheme of potential energy curves shown in Figure 1.2, the transmission-namely, the nonadiabatic tunneling-occurs in the case of nonadiabatic tunneling type. For both nonadiabatic tunneling type and the Landau-Zener type, the transition between the two adiabatic states is induced by

the nonadiabatic coupling, which is given by the following expression:

Nonadiabatic coupling ∝ [∇x V12][V1 − V2] − V [∇x (V1 − V2)](W1 − W2)2 ∇x , (5.2)

which is a quantum mechanical operator. The transition is called nonadiabatic transition. Since the adiabatic states are defined as the eigenstates of the electronic Hamiltonian at fixed nuclear coordinates, this coupling comes from the nuclear motion, namely, the nuclear Laplacian operator, as is seen from the above expression.