Overlap of the Nuclear Wave Functions
The nuclear wave functions will be analyzed in the center of mass coordinates. In the center of mass, the two nuclei approach each other with the same momentum, thus, their wave functions will have the same dimensions in the axis aligned with their relative velocity vector. Before interacting, the particle’s wave functions, from a stationary perspective, will look like the top in figure 4-10a. The wave function in the transverse (orthogonal) direction to travel, will be determined by the particles independent transverse wave function. As the particles approach (middle figure 4-10a) and loose velocity due to the coulomb repulsion, their relative wave functions will broaden (bottom Figure 4-10a) as their momentum decreases in the axis of the velocity vector.
figure 4-10a
We saw above that the transition to the “He-d” linear momentum state bond only takes place at 0.19 and 0.33 angstroms. If the kinetic energy is too high, the relative nuclear velocity will be high at the given separation. A high velocity does two bad things for the transition state;
1) the relative wave length is short so that the overlap of the initial and final states is poor and,
2) a high velocity decreases the time for reaction at the transition separation.
Because of this, the transition will only take place if the energy is high enough for the particles to reach the transition separation, but low enough to slow the particles and stretch the wave functions for good transition overlap. Figure 4-10b shows the competing reaction channels. There is the conventional collision, with the particles deflecting, or the transition to the linear momentum bonding state. The activation energy is equal to the negative value of the linear momentum state energy. The linear momentum state "work" done by the electron to drop to a lower energy level is manifest in the repulsion energy of the two particles.
The time for a reaction to take place can be estimated from the time required for the nuclear wave functions to pass the critical separations determined by the angular momentum. As the particles slow, due to repulsion, their wave functions broden significantly. The wider wave function coupled with the lower velocity, creats a realitively narrow energy band for a high probability of transition to the linear momentum state. Figure 4-10c shows how the "reaction time" is increased as the relative kinetic energy of the two particles comes closer to the transition energy (slows down). The increase in the reaction time is simply the result of the widening wave function, traveling slower through the critical separation dimension determined by the angular momentum. This "reaction time" coupled with the exponetial probability of transition, due to the difference in the transition energies, determines the overall transition probability. If the transition to the linear momentum state does not occur, the helium will collide as depicted in the center of figure 4-10b. The collision will de-channel the helium. This de-channeling of the helium lowers the overall reaction probability. De-channeling can also occur due to flaws in the crystal. Both of these de-channeling affects are encorporated in the model introduced at the end of this chapter.
FIgure 4-10c
Construction of the Activated Complex “He-d” Nuclear Wave Function
The nuclear wave function at the transition to the “He-d” molecule will be constructed of the high order (high “n”) wave functions. The actual calculations to find the pseudo potential “felt” by the nuclei are beyond our capabilities. These calculations would involve the detailed
construction of nuclear wave functions and the very detailed calculation of the electron energy associated with each location in the nuclear wave function. This would be an iterative process as the electron wave function, as well as the nuclear wave function is constructed of many (4 in the electron case) states. The time evolution of the activated complex would then have to be analyzed, a very complex undertaking!
For an intuitive look at the construction of the nuclear wave function, we will work briefly through an example. If we just assume a pseudo potential for the nuclei (with the general shape of the He-d pseudo potential), we can then generate high order wave functions to construct a nuclear wave function.
Figure 10-11 shows the potential generated for this exercise. 
Figure 10-12 shows the n=8 wave function generated using the center of mass of “He-d” and the given potential. As part of the exercise we “constructed” all of the wave functions for n=1 to 8 from the Fourier transform of the wave forms calculated with the Schrodinger equation. The “sine” function was used as he transform function. Figure 10-12 also shows how the n=8 transformed function looks, compared to the original, using 22 frequency terms to construct the wave function. This is similar to the idea mentioned above about constructing electron wave functions from the “almost free” solid state electron wave functions. Since the solid state electron wave functions where just “cosine” functions, these functions could be used directly to construct the desired wave function of the “He-d” binding electron.
Figure 10-13 shows the Fourier reconstruction of all 8 nuclear wave functions. Also shown is an arbitrarily chosen nuclear wave function constructed from these wave functions. Note that since the wave functions are composed of a Fourier fitted function, there are some aberrations caused by too few frequencies used to construct them. 
Figure 10-14 shows a bar graph with the coefficients of each of the 8 nuclear energy levels used to construct the wave function. The constructed wave function was arbitrarily chosen and its shape has no meaning beyond the fact that any shape may be constructed. This is pertinent to the overlap of the “activation complex” from the starting nuclear energy levels using Fourier transformation. The infinite complexity comes when trying to calculate electron wave functions to go with the nuclear wave functions, all having properly aligned energies. This is far beyond our capabilities.