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Section 14.3: The H_{2}^{+} Ion

Animation 1: Energy Eigenfunction  Probability Density
Animation 2: Energy Eigenfunction  Probability Density
Animation 3: Energy Eigenfunction  Probability Density
Animation 4: Wave Function  Probability Density
Please wait for the animation to completely load.
One of the simplest extensions from atoms to molecules is that of the H_{2}^{+} ion. The idealized H_{2}^{+} ion is described by two fixed protons a distance r_{s} apart, and one electron shared by the two protons. The electron is a distance r_{1} and r_{2} away from each proton, respectively. This situation is modeled in one dimension in the animation.^{1}
Animation 1, Animation 2, and Animation 3 show the onedimensional model with the protons becoming progressively closer. In each animation you can view the energy eigenfunction or the probability density associated with an electron bound to the H_{2}^{+} ion. To see the other bound states, simply clickdrag in the energy level diagram on the left to select a level. The selected level will turn red. You should notice that there are two states with almost the same energy. The lowest state is symmetric about the origin and the next lowestlying state is antisymmetric about the origin. When the protons are close together, there is a noticeable difference in the probability density associated with the symmetric and the antisymmetric state. For the symmetric state, it is relatively likely to find the electron between the two protons, while for the antisymmetric state, it is relatively unlikely to find the electron between the two protons.
We can extend this analysis to describe what happens in a neutral H_{2} molecule (two electrons). In this case, shown in Animation 4, the two energy eigenfunctions in the animation represent the symmetric and antisymmetric combinations of the individual electron wave functions,
Ψ_{S} = (1/2^{1/2})[ ψ(x_{1}) + ψ(x_{2}) ] (14.6)
and
Ψ_{A} = (1/2^{1/2})[ ψ(x_{1}) − ψ(x_{2}) ], (14.7)
respectively. In the animation, the symmetric and antisymmetric spatial wave functions are shown along with ψ(x_{1}), ψ(x_{2}), and −ψ(x_{2}). The total wave function (including spin) describing the two electrons must be antisymmetric upon exchange of the electrons. Therefore a symmetric spatial wave function requires an antisymmetric spin state (yielding an overall antisymmetric state) and an antisymmetric spatial wave function requires a symmetric spin state (yielding an overall antisymmetric state).
^{1}This scenario is often also modeled with two finite wells that are separated. In Section 11.4, we explored this situation.
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