Atomic Orbitals

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(2) radial density versus r.

Density versus radius r: In this case, the square of the wave function is plotted against r. These plots are sometimes misleading. For example, the 1s orbital plot looks like
```
probability
  |
  |.
  |
  | .
  |
  |   .
  |
  |      .
  |          .
  |               .
  |                     .
  |                             .
  |________________________________________._______ r
```
You may feel the probability of finding the electron is the highest in the nucleus, yet you have learned that the electron is most likely at a distance r = 53 pm from the center of the atom.
Radial density (RD) versus r: To really represent the probability of finding the electron at r at a given time, the radial distribution against r is often plotted. In this plot, instead of plotting square-of-the-wave-function, we modify square-of-the-wave-function by the volume associated with r, (4*pi*r²). This modification converts electron density to radial electron density.
The radial density plot of 1s orbital has a shape as shown below:
```
probability
  |
  |
  |
  |
  |
  |        .  '   .
  |     .           .
  |   .                 .
  | .                            .
  |________________________________________._______ r
```
At the center of the atom, the value of the wavefunction is large, but when r = 0, the volume element (4*pi*r²) is almost zero when r -> 0. Thus, the radial distribution rises as r increases, reaching a maximum at some value of r. For the H atom, the maximum of the radial distribution is at r = 53 pm.

Contributors and Attributions

Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo)