# 8.8: Term Symbols Gives a Detailed Description of an Electron Configuration

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Conveniently for chemists, an atom’s electronic state depends entirely on its unfilled sub shells. Because electrons distribute themselves in a symmetric manner, the inner shell electrons end up canceling out each other’s momenta. For an atom in the configuration \(1s^2 2s^2 p^2\), only the two p‐electrons matter. For an atom in the configuration \(1s^22s^12p^1\), we have to examine the 2s and 2p electrons. Atoms have quantum numbers that are directly analogous to the electronic quantum numbers.

## The Total Orbital Angular Momentum Quantum Number: L

One might naively think that you could get the total angular momentum of an atom by simply adding up the \(l\) values of the individual electrons. The problem with this idea is that the angular momenta of the various electrons are not necessarily pointing in the same direction. If two electrons are revolving in the same direction as each other, you would add their \(l\) values.

\[ L = \sum_i^n l_i \label{8.8.4A}\]

If they were revolving opposite to each other, you would subtract them. If they are revolving at some off ‐ angle relative to each other, you would partially subtract them. To figure out all of the possible combinations of l for a pair of electrons, simply add them together to get the co‐aligned case, subtract them to get the opposing case, and then fill in all the numbers in between to get the off‐angle cases. If you prefer to have a formula, you can use this:

\[L = |l_1 + l_2 | , |l_1 + l_2 ‐ 1| , \ldots, |l_1 ‐ l_2 | \label{8.8.5A}\]

## The Total Magnetic Quantum Number: \(M_l\)

\(M_l\) is the total z‐component of all of the relevant electrons’ orbital momentum. Where \(L\) told you how much angular momentum there was, \(M_l\) tells you which direction it is pointing. Like \(L\), a given configuration can have several possible values of \(M_l\), depending on the electrons’ relative orientation. Unlike \(L\), \(M_l\) is allowed to have negative values. To list the possible \(M_l\) values for a two electron system, take the case where both \(m_l\) are positive, then take the case where they are both negative, and then fill in the numbers in between.

\[M_l = m_{l1} + m_{l2}, m_{l1} + m_{l2} ‐ 1, ... , ‐m_{l1} ‐ m_{l2} \label{8.8.6}\]

### The Total Spin Magnetic Quantum Number: \(M_s\)

\(M_s\) is the sum total of the z‐components of the electrons’ inherent spin. Do not confuse it with \(M_l\), which is the sum total of the z‐component of the **orbital angular momentum**. It is easily computed by finding all of the possible combinations of \(m_s\). Since \(m_s\) for each individual electron can only be +1/2 or ‐ 1/2, this isn’t too complicated.

\[M_s = m_{s1} + m_{s2}, m_{s1} +m_{s2} ‐ 1 , ... , m_{s1} ‐ m_{s2} \label{8.8.7}\]

## The Total Intrinsic Spin Quantum Number: \(S\)

The sum total of the spin vectors of all of the electrons is called \(S\). The difference between \(S\) and \(M_s\) is subtle, but vital for understanding multiplicity. \(M_s\) measures the total z‐component of the electrons’ spins, while \(S\) measures the entire resultant vector. The values of \(S\) are computing in a manner very similar to \(M_s\). Because \(S\) measures the magnitude of a vector, it cannot ever be negative.

\[S = |s_1 + s_2 |, |s_1 +s_2 ‐ 1| , ... ,| s_1 ‐ s_2 | \label{8.8.8}\]

## The Total Angular Momentum Quantum Number J

The total orbital angular momentum of an atom (measured in terms of \(L\)), and the total spin angular momentum of an atom (measure in \(S\)) combine to form total angular momentum, a number that is quantized by the number \(J\). \(L\) and do not necessarily have to be pointing in the same direction (Figure \(\PageIndex{1}\)), so \(J\) can range from \(L + S\) to \(|L – S|\).

Symbol | Name | Allowed Range |
---|---|---|

\(L\) | Total orbital angular momentum | \(|l_1 + l_2 |, ..., |l_1 ‐ l_2 |\) |

\(M_l\) | Magnet Quantum number | \( [m_{l1} + m_{l2}, ..., ‐ m_{l1} ‐ m_{l2} ]\) |

\(M_s\) | Spin Magnetic Quantum Number | \( | m_{s1} + m_{s2} |,..., |m_{s1} ‐ m_{s2} | \) |

\(S\) | Inherent Spin Number | \(|s_1 + s_2 |,..., |s_1 ‐ s_2 |\) |

\(M\) | Multiplicity | 2S+1 |

\(J\) | Total Angular Momentum | \(L+S,..., | L-S |\) |

## Multiplicity

Multiplicity is a simple ‐ sounding concept that defies simple explanations. You know from your first ‐ year education that a singlet is when the net spin (S) is equal to zero (e.g. all the electrons are spin paired), and a triplet happens when the net spin is equal to 1 (e.g. two electrons are pointing in the same direction). They are called “singlet” and “triplet” because there are 3 ways to combine a pair of electron spins to get S=1, but only one way to get \(S=0\). If you draw a picture of the possible ways that two electrons can arrange their spins, you get something like this:

While this picture is an improvement over the simple up‐down model, it is still misleading. The three spin axes of an electron share a Heisenberg Uncertainty Principle. The more you know about \(S_x\), the less you can know about either \(S_y\) or \(S_z\). The same is true for all other combinations of x, y and z. Since we have defined \(S_z\) as a known and fixed value, the values of \(S_x\) and \(S_y\) must be **completely unknown**. This causes the x and y orientations of the electrons to become smeared out across all possible values:

## Constructing Term Symbols

Atomic term symbols contain two pieces of information. They tell you the total orbital angular momentum of the atom (\(L\)), and they tell you the multiplicity (\(M\)). \(L\) is denoted by a simple code, similar to the code used to delineate the types of atomic orbitals:

- \(L=0 \rightarrow S \)
- \(L=1 \rightarrow P\)
- \(L=2 \rightarrow D\)
- \(L=3 \rightarrow F\)

Note that while the notation is similar, L does NOT say anything about what types of orbitals the electrons are in. A state that has the term symbol P does NOT necessarily have an open p‐shell. The multiplicity is indicated by appending a number to the upper left of the symbol. A \(L=2\), \(M=3\) state would be represented by \(^3D\). The secret to writing the term symbols for an atom is to discover what combinations of \(L\) and \(M\) are possible for that atom with that specific electronic configuration. An atom that only has closed shells will always be \(1S\).

Each term symbol represents a discrete energy level. We can place these levels in the correct order by using these simple rules:

- 1: High multiplicity values mean low energy
- 2: If there is a tie, high \(L\) values mean low energy
- 3a: If there is still a tie and the shell is less than half full, then low J means low energy
- 3b: If the shell is more than half full, then high J means low energy

These rules reliably predict the ground state. They have only erratic agreement with experiment when ordering the other levels.

## Shortcuts

There is a deep symmetry that connects different electronic configurations. It turns out that a \(p^1\) configuration has the same term symbols as a \(p^5\). Similarly, \(p^2 = p^4\). A similar relationship can be used to figure out high electron number term symbols for the \(d\) and \(f\) orbitals.

## Contributors

Mattanjah de Vries (Chemistry, University of California, Santa Barbara)