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# Atomic Orbitals

### Skills to develop

• Describe the shapes of ns, np, and nd atomic orbitals.
• Explain the variation of wavefunctions as the radius increases.

### Atomic Orbitals

Atomic orbitals are (energy) states or wave forms of electrons in the atom. If we insist on the particle nature of electrons, then the probability of finding an electron in an atomic orbital is proportional to the square of the wavefunction. The values of the wavefunction can be either positive or negative, but the probability is always a positive value.

Most of us are not used to thinking of electrons as waves, and we still refer to the density as the probability of finding the electron (a particle). The electron density diagrams given in many text books can be plotted using the appropriate wave functions. They are not the results of artists' imagination. In this demonstration, the computer will plot the densities according to an algorithm, which is based on the wavefunctions of various atomic orbitals.

The representation of atomic orbitals and their visualization has fascinated young and old scientists for ages. This link gives interesting pictures and molves. Copngratulation to Dean Dauger - dauger@physics.ucla.edu, who has been a student winner twice in a row in Computer In Physics's Software Contest. His real-time visualization of the quantum mechanical atomic orbitals is a real treat.

Some more links regarding atomic orbitals:

### Dot-density as Electron-Density Plots

The probabilities of finding electrons anywhere in a three dimensional space around the hydrogen nucleus are proportional to the squares of the values of the wavefunction in the volume element corresponding to that space. Thus, the plot of probability should be done in a 3-dimensional space. However, such a plot not only requires a high degree of programming and viewing skills, it also takes a lot of computer time and a better monitor to do the job right. The next best way to represent them are looking at cross sections. This is the way we treat this subject here.

Lately, many diagrams to represent atomic orbitals have been made available via the web. The picture shown here is a 3pz atomic orbital. The learning matter of chemistry is linked to a few useful places that provide various forms.

### Two Forms of Electron Probability Density Plots

Each dot-density plot in this simulation is accompanied by two plots,
(1) density versus radius r, and
1. 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.

2. 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.

### The Simulation of Electron Density

The simulation of electron density and showing orbitals is very well done in

The following program is no longer available, but the description is left here to you to used when you look at the above simulation

The computer plots of cross section of electron clouds for 1s, 2s, 2p, 3s, 3p, and one 3d orbital are available. When the orbital is plotted the radial density is plotted versus r in the right lower corner of the screen. The function used for the plot is shown on the bottom of the screen. You do not need to copy the wave function, as you will get that later in a chemistry course.

The purposes of the computer simulation are given below:

1. To show you a simulated plot of the wavefunction by a desk top computer.

2. To show you the shapes of 1s, 2s, 2p, 3s, 3p, and one 3d orbitals.

3. From the dot-density plots, please construct 3-dimensional pictures in your own mind.

4. From the plots, construct the nodal surfaces or planes in your mind.

### Problems to Solve

1. Find the maximum radial distribution for 2s, and 3s orbitals in Bohr's radius from the graph.

2. Find the maximum radial distribution for 3s, 3p, and 3d orbitals in Bohr's radius from the graph. Compare these three values and detect a trend as the azimuthal quantum number l increases.

3. Which orbital among 4s, 4p, 4d, and 4f will have the smallest radius, at which the maximum radial distribution is found? This is an extension of 2.

The Orbitron is a link that gives wonderful views of the atomic orbitals.

### Confidence Building Questions

• Which one of the following orbitals has one and only one nodal (spherical) shell or surface?
1s, 2s, 3s, or 4s.

Consider...
An ns orbital has (n-1) nodal shell(s). When n = 2, there is 1 nodal shell.

Excellent...
How does a electron get through a nodal shell?

• Which one of the following orbitals has one and only one nodal plane?
1s, 2s, 2pz, 3s, 3dxy.

Consider...
The 2s orbital has a nodal shell, whereas the 2pz.qn orbital or 2p orbitals have a nodal plane.

Excellent...
A nodal plane is a plane in which the probability of finding a electron is zero. The 2s orbital has a nodal shell, whereas the 2pz orbital or 2p orbitals have a nodal plane.

• What shape do all s orbitals have?
a. spherical
b. dumbbell
c. donut
d. disk

Consider...
Not all d orbitals have the same shape.

Excellent...
All p orbitals have a dumbbell shape, but from the plot, you should recognize the difference between 2p and 3p orbitals.

• Although the maximum electron density of a 1s orbital is at the nucleus, why the radial density for this orbital is zero at the nucleus?
a. The volume factor (4*pi*r2) is zero at r = 0.
b. Electron can not get into a nucleus.
c. No two particles can occupy the same space.
d. The angular momentum keeps the electron away from the nucleus.

Consider...
Only a is accepted as the correct answer, although reasons given in other items sound all right.

Excellent...
Electron density means finding the electron per unit volume. Radial electron density means finding the electron at r.

• What is the distance (in pm) for the maximum radial density of the 1s orbital in the hydrogen, H, atom.