- Reading
- Brown: Quantum Mechanics and Atomic Orbitals
- Brown: Representation of Orbitals
- OpenStax: Development of Quantum Theory

For beginning chemistry students, it is easy to get confused when we say that electrons exist in **orbitals** not **orbits**. What do we mean by this distinction?

In the Bohr model of the hydrogen atom we imagined the electron orbiting the proton the way a planet moves in a definite predictable path around the sun. We call such a path an orbit. If electrons moved around a nucleus the way a planet moved around the sun we would call their path orbits as well. Electrons do not however move in such predictable paths.There is a principle called the Heisenberg Uncertainty Principle, which states that you cannot know the position and momentum of a particle like an electron at the same time. This ends up meaning that if you know where an electron is at some point in time, you cannot know with certainty where it will go next. Of course this means that the electron does not orbit the nucleus the way a planet orbits the sun.

How then do we describe its motion around the nucleus? Although we do not claim to know how an electron moves from one moment to the next, if we map its locations over time we could plot a 3D map which would show where the electron is likely to be found. This map is derived mathematically. This region is called an orbital. For example, for an electron in a hydrogen atom the orbital is roughly in the shape of a sphere. Over 90% of the time the electron will be found in that spherical region.

Here is a representation of an electron in the ground state of a hydrogen atom. The darker the shading the more likely it is to find the electron in that area.

What happens when the electron absorbs enough energy to go to what we previously called a higher Bohr orbit? That is, how should we imagine orbitals with higher energy? If we label the lower energy orbital 1s (corresponding to a n = 1 Bohr orbit) we could label the higher energy orbitals (corresponding to a n = 2, and n =3 Bohr orbits) 2s or 3s. A diagram is shown below.

As the electron absorbs energy the map of its probability density, where it is most likely to be found, changes. The probability distribution can be thought of like the layers of an onion. At higher energies the electron can be found in a greater number of layers – with a zero probability of being found at nodes – or between layers.

Go to the Dynamic Periodic Table . Click on the orbitals tab and then hover over hydrogen. Then hover over helium. Note the spherical shape of the orbital is shown. Hover over the various elements in the first or second group, and notice the diagram of the orbital shown in the middle of the page. Every time you go to a higher period on the table we add an onion layer. Although the periodic table is showing the outer electron of different elements, when a single electron of hydrogen absorbs energy it takes on the a similar probability density.

We designate orbitals using quantum numbers that give us information about the spatial distribution of an electron.
We will initially use three quantum numbers, *n, l, and ml*.
For a ground state electron in hydrogen the three numbers are
*n = 1, l = 0 , ml = 0*

The letter n designates the principal quantum number. This number can be a positive integer like 1,2,3 etc. and indicates the energy of the orbital. Higher energy means the electron is further from the nucleus.

The letter *l* indicates the shape of the orbital.
This number is called the angular momentum quantum number and can have values from 0 to n – 1.
So in the first energy level, where n = 1, there is 1 shape available corresponding to *l* = 0.
We also call this an “s” shape.
In the 2nd energy level, where n = 2, there are 2 shapes available corresponding to *l* = 0, and *l* = 1.
We also call this an “s” and a “p” shape.

Different shapes mean different electron density distribution.

In these digarms the nucleus is located at the intersection of the axis lines.

Here are the shapes corresponding to *l* = 2.

The letter ml designates the orientation of the orbital in space.
It can have values between *-l* and *l*.
When *l* = 0 (“s” shape) only 1 orientation is possible.
When *l* = 0 (“s” shape) *ml* = 0.
When *l* = 1 (“p” shape) 3 orientations are possible.
When *l* = 1 (“p” shape) *ml* = -1, *ml* = 0, or *ml* = 1.

Different *ml* values mean different orientations in space. For an orbital with *l* = 0 only 1 orientation is possible.

For orbitals with *l* = 1 only 3 orientations are possible.

Here is a summary of shapes and their orientation for *l* = 0,1,2, and 3.

The angular momentum quantum number, *l*, specifies the shape of an orbital with a particular principal quantum number.
This quantum number divides the shells into smaller groups of orbitals called subshells (sublevels).
Usually, a letter code is used to identify *l* to avoid confusion with n:

l |
0 | 1 | 2 | 3 | ... |

letter | s | p | d | f | ... |

The subshell with n = 2 and *l* = 1 is the 2p subshell
if n = 3 and *l* = 0, it is the 3s subshell, and so on.
For multi-electron atoms the value of *l* also has a slight effect on the energy of the subshell.
The energy of the subshell increases with *l*.

Here is a way of showing the subshells available for the first threee energy levels in a hydrogen atom.

Here is a summary table.

n | l |
Subshell | ml |
Number of orbitals | Number of electrons |
---|---|---|---|---|---|

1 | 0 | 1s | 0 | 1 | 2 |

2 | 0 | 2s | 0 | 1 | 2 |

2 | 1 | 2p | -1,0,1 | 3 | 6 |

3 | 0 | 3s | 0 | 1 | 2 |

3 | 1 | 3p | -1,0,1 | 3 | 6 |

3 | 2 | 3d | -2,-1,0,1,2 | 5 | 10 |

4 | 0 | 4s | 0 | 1 | 2 |

4 | 1 | 4p | -1,0,1 | 3 | 6 |

4 | 2 | 4d | -2,-1,0,1,2 | 5 | 10 |

4 | 3 | 4f | -3,-2,-1,0,1,2,3 | 7 | 14 |