Term symbol

In quantum mechanics, the term symbol is an abbreviated description of the angular momentum quantum numbers in a multi-electron atom. It is related with the energy level of a given electron configuration. LS coupling is assumed. The ground state term symbol is predicted by Hund's rules.

The term symbol has the form::$\{\}^\{2S+1\}!L\_\{J\}$:where : "S" is the total spin quantum number. 2"S"+1 is the spin multiplicity: the maximum number of different possible states of "J" for a given ("L","S") combination.: "L" is the total orbital quantum number in spectroscopic notation. The symbols for L = 0,1,2,3,4,5 are S,P,D,F,G,H respectively.:"J" is the total angular momentum quantum number.

When used to describe electron states in an atom, the term symbol usually follows the electron configuration, e.g., in the case of carbon, the ground state is 1s²2s²2p^{2 3}P₀. The 3 indicates that 2S+1=3 and so S=1, the P is spectroscopic notation for L=1, and 0 is the value of J.

The term symbol is also used to describe compound systems such as mesons or atomic nuclei, or even molecules (see molecular term symbol). In that last case, Greek letters are used to designate the (molecular) orbital angular momenta.

For a given electron configuration
* The combination of an "S" value and an "L" value is called a term, and has a statistical weight (i.e., number of possible microstates) of (2"S"+1)(2"L"+1);
* A combination of "S", "L" and "J" is called a level. A given level has a statistical weight of (2"J"+1), which is the number of possible microstates associated with this level in the corresponding term;
* A combination of "L", "S", "J" and "M_J" determines a single state. As an example, for "S" = 1, "L" = 2, there are (2×1+1)(2×2+1) = 15 different microstates corresponding to the ³D term, of which (2×3+1) = 7 belong to the ³D₃ (J=3) level. The sum of (2"J"+1) for all levels in the same term equals (2"S"+1)(2"L"+1). In this case, "J" can be 1, 2, or 3, so 3 + 5 + 7 = 15.

Term symbol parity

The parity of a term symbol is calculated as :$P=(-1)^\{sum\_i\; l\_i\},!$, where "l_i" is the orbital quantum number for each electron. In fact, only electrons in odd orbitals contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd "l" such as in p, f,...) will make an odd term symbol, while an even number of electrons in odd orbitals will make an even term symbol, irrespective of the number of electrons in even orbitals.

When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted: :$\{\}^2!mathrm\; P\_\{1/2\}^o$ has odd parity, but $\{\}^3!mathrm\; P\_0$ has even parity.

Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for "gerade" (German for 'even') or "ungerade" ('odd')::$\{\}^2!mathrm\; P\_\{1/2,u\}$ for odd parity and $\{\}^3!mathrm\; P\_\{0,g\}$ for even.

Ground state term symbol

It is relatively easy to calculate the term symbol for the ground state of an atom. It corresponds with a state with maximal "S" and "L".
# Start with the most stable electron configuration. Full shells and subshells do not contribute to the overall angular momentum, so they are discarded.
#* If all shells and subshells are full then the term symbol is $\{\}^1!S\_0$.
# Distribute the electrons in the available orbitals, following the Pauli exclusion principle. First, we fill the orbitals with highest "m_l" value with one electron each, and assign a maximal "m_s" to them (i.e. +1/2). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning "m_s" = −1/2 to them.
# The overall "S" is calculated by adding the "m_s" values for each electron. That is the same as multiplying ½ times the number of unpaired electrons. The overall "L" is calculated by adding the "m_l" values for each electron (so if there are two electrons in the same orbital, then we add twice that orbital's "m_l").
# Calculate "J" as:
#* if less than half of the subshell is occupied, take the minimum value $J\; =\; |L\; -\; S|$;
#* if more than half-filled, take the maximum value $J\; =\; L\; +\; S$;
#* if the subshell is half-filled, then "L" will be 0, so $J\; =\; S$.

As an example, in the case of fluorine, the electronic configuration is: 1s²2s²2p⁵.

1. Discard the full subshells and keep the 2p⁵ part. So we have five electrons to place in subshell p ("l" = 1).

2. There are three orbitals ("m_l" = 1, 0, −1) that can hold up to 2(2"l"+1) = 6 electrons. The first three electrons can take "m_s" = 1/2 (↑) but the Pauli exclusion principle forces the next two to have "m_s" = −1/2 (↓) because they go to already occupied orbitals.

* Fourth, extract smaller tables representing each possible term. Each table will be (2"L"+1)(2"S"+1), and will contain "1"s as entries. The first table extracted corresponds to "M_L" ranging from −2 to +2 (so "L" = 2), with a single value for "M_S" (implying "S" = 0). This corresponds to a ¹D term. The remaining table is 3×3. Then we extract a second table, removing the entries for "M_L" and "M_S" both ranging from −1 to +1 (and so "S" = "L" = 1, a ³P term). The remaining table is a 1×1 table, with "L" = "S" = 0, i.e., a ¹S term. :
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