The wave functions for an atom with a single electron have three index numbers called quantum numbers. These indices allow a set of solutions to share the same form and yet allow each solution to be unique. A set of quantum numbers is sort of like a finger print: even though you have the same basic form as other human beings, you are unique and you have a unique finger print. Your finger print isn’t what makes you an individual, but it is a way to differentiate you from others. A set of quantum numbers doesn’t make an orbital, but it allows us to distinguish among them. But a set of quantum numbers is more than just an identifier. Their values give us information about the shape and orientation of the probability density.

- The principle quantum number,
*n*, indicates the energy of the electron in that orbital. The higher the value of*n*is, the greater the energy the electron has. The principle quantum number has values that range from 1 to infinity, theoretically, but the principle quantum number also indicates the distance to the nucleus. High energy electrons far from the nucleus tend to get lost. Which brings us to an important digression: if an atom has got only one electron, doesn’t it only have one orbital? NO! Every atom has an infinite number of orbitals because every wave equation has an infinite number of solutions. At any given instant an electron is occupying only one of these orbitals, but all the other possible energy states still exist. - The azimuthal quantum number,
*l*, indicates the shape of the probability density that the electron occupies (or the shape of the orbital, if you prefer). The possible values for*l*depend on the value for*n*of that orbital, and*l*has values from zero up to*n*– 1. For example, if*n*= 3,*l*= 0, 1, 2. Many students balk at this point and demand to know which value is the right one. They are all equally right. Remember that we are building a large set of orbitals. - The magnetic quantum number,
*m*, indicates the orientation of the probability density, or the direction the orbital is pointing. The value the magnetic quantum number can have depends on the value of the azimuthal quantum number with values ranging from negative_{l}*l*up to positive*l*. For example, if*l*= 2,*m*= -2, -1, 0, 1, 2._{l }

## Shells, Subshells, and Orbitals in the Many Electron Atom

For atoms with more than one electron, we can use the hydrogen atom orbitals as a starting point. For an atom with more than one electron, the value of *n* is only a rough guideline to the electron’s energy. The value for the *l* quantum number gives orbitals that vary in energy, even when they have the same *n* value. In general, for equal values of n, the lower the value of* l *is, the lower the energy is.

The set of orbitals with the same *n* value is called a shell. The set of orbitals with the same values for both *n* and *l* are called a subshell. The energy of an electron in any of these orbitals is exactly the same as long as the value for both *n *and *l* are the same. We use the word **degenerate **to signify that different orbitals have the same energy. The subshell orbitals have names, though they don’t seem much like names. They are labeled with a number representing the shell (or * n* value) and a letter that was originally used to describe experimental results in spectroscopic experiments.

The orbitals also have characteristic shapes depending on their* l* values. The shapes of these orbitals are important because they are used in various bond theories to predict and explain molecular shapes.

If an atom has more than one electron, we need an additional index that the hydrogen atom wave functions don’t need. The spin quantum number, *m _{s}*, describes the magnetic direction of the electron in that energy state. No matter what the values of the other quantum numbers might be, the two possible values for the spin quantum number are positive and negative one half. We can think of the spin quantum number as designating whether an electron is a magnetic north or a magnetic south. Diagrams of electrons use arrows pointing either up or down to illustrate this property.

In any given atom, each electron has a unique energy state at any given instant. The first three quantum numbers describe the total energy, probable distance, the most likely volume of space, and the spin quantum number designates magnetic direction. That means that a unique set of four quantum numbers is all we need to identify a unique energy state. **Pauli Exclusion Principle** states that, in any given atom, none of the electrons can have the same values for all four quantum numbers. Because three of the numbers identify the orbital and each electron can have a spin of +/- ½, each orbital can hold at most two electrons.

Subshell Summary

Value of n |
Subshell Name | Number of orbitals in the subshell | Orbital Shape(Centered at the Nucleus) | Maximum # Electrons |
---|---|---|---|---|

1 | 1s | 1 | Sphere | 2 |

2 | 2s2p | 13 | SphereDouble teardrop | 26 |

3 | 3s3p
3d |
13
5 |
SphereDouble teardrop
Four leafed clover |
26
10 |

4 | 4s4p
4d 4f |
13
5 7 |
Sphere
Double teardrop Four leafed clover Sea urchin? Or some other really lumpy thing. |
26
10 14 |