Resonance energy as tunneling matrix element

As shown in Fig. 7.2, at a shorter proton-proton separation (/ 16 a.u. or 8 A), the electron in the Is state in the vicinity of one proton has an appreciable probability of tunneling to the Is state in the vicinity of another proton. The tunneling matrix element can be evaluated using the perturbation theory we presented in Chapter 2. A schematic of this problem is shown in Fig. 7.3. By defining a pair of one-center potentials, Ul and Ur, we define the right-hand-.side states and the left-hand-side states. Because the potential Ul is different from the potential of a free proton, Uro, the wavefunction i ii, and the energy level Eo are different from the Is state of a free hydrogen atom. (The same is true for Ur and We will come back to the effect of such a distortion later in this section. 

In the following, we present a treatment of the hydrogen molecular ion problem using a time-dependent Schrddinger equation  

According to the Wigner theorem (see Appendix A), because of time-reversal. symmetry, the functions i At) and tl R(r) can always be made real. Also, since the ground-state wavefunction does not have a node, we can always make the 

Substituting Eq. (7.11) into Eq. (7.6), the problem is reduced to the same problem we encountered in Chapter 2. The potentials of the left-hand-side and right-hand-side problems satisfy Equations (2.21) and (2.22). Following the 

A specific solution of Eq. (7.6) now depends on the initial condition. If at t=0 the electron is in the left-hand-side state, the solution is  

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