# Guide Hydrogen-ion wave function Contents:

## Wave function for the hydrogen molecular ion - IOPscience

Fingerprint hydrogen ions. Dirac equation. Wave functions. Ground state. Excited states.

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Journal of Chemical Physics , 12 , []. We note, first, that most nuclei are orders of magnitude heaver than an electron.

### Solving The Hydrogen Molecule Cation Exactly

For the lightest nucleus, the proton,. This mass difference is large enough to have important physical consequences. Let us think classically about this mass difference first. That is, it will simply follow the heavy particle wherever it goes, and , it will move rapidly in response to the heavy particle motion. As an illustration of this phenomenon, consider the simple mechanical system pictured below:.

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Considering this as a classical system, we expect that the motion will be dominated by the large heavy particle, which is attached to a fixed wall by a spring. The small, light particle, which is attached to the heavy particle by a spring will simply follow the heavy particle and execute rapid oscillations around it. The figure below bottom panel illustrates this:. In this illustration, when the heavy particle moves even a tiny amount, the light particle executes many oscillations around the heavy particle. What we can conclude, therefore, is that we can approximately fix the position of the heavy particle and just allow the light particle to move around before we advance the heavy particle to its next position.

### Supplementary files

Light particles, such as electrons, tend to have very diffuse, delocalized wave functions, while heavy particles, such as the nuclei, tend to have wave functions that are very localized about the classical positions. This is illustrated in the figure below:. Figure 1: Crude picture of matter.

Nuclear wave functions are shown above the axis, electronic wave functions are shown below. Thus, taking the idea of solving the electronic problem for fixed nuclear positions seriously, we can write the molecular wave function as.

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This is not an exact wave function for the molecule but an approximate one. Hence, we call this the Born-Oppenheimer approximation to the wave function.

The total classical energy of the molecule is. Therefore, the total Hamiltonian of the molecule is. The details of these operators are not that important for this discussion. Let us consider how the electronic and nuclear kinetic energy operators act on the wave function. Fingerprint hydrogen ions. Dirac equation. Wave functions. Ground state. Excited states. Journal of Chemical Physics , 12 , [].

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