In atomic physics, a term symbol is an abbreviated description of the total spin and orbital angular momentum quantum numbers of the electrons in a multielectron atom. So while the word symbol suggests otherwise, it represents an actual value of a physical quantity.
For a given electron configuration of an atom, its state depends also on its total angular momentum, including spin and orbital components, which are specified by the term symbol. The usual atomic term symbols assume LS coupling (also known as Russell–Saunders coupling) in which the allelectron total quantum numbers for orbital (L), spin (S) and total (J) angular momenta are good quantum numbers.
In the terminology of atomic spectroscopy, L and S together specify a term; L, S, and J specify a level; and L, S, J and the magnetic quantum number M_{J} specify a state. The conventional term symbol has the form ^{2S+1}L_{J}, where J is written optionally in order to specify a level. L is written using spectroscopic notation: for example, it is written "S", "P", "D", or "F" to represent L = 0, 1, 2, or 3 respectively. For coupling schemes other that LS coupling, such as the jj coupling that applies to some heavy elements, other notations are used to specify the term.
Term symbols apply to both neutral and charged atoms, and to their ground and excited states. Term symbols usually specify the total for all electrons in an atom, but are sometimes used to describe electrons in a given subshell or set of subshells, for example to describe each open subshell in an atom having more than one. The ground state term symbol for neutral atoms is described, in most cases, by Hund's rules. Neutral atoms of the chemical elements have the same term symbol for each column in the sblock and pblock elements, but differ in dblock and fblock elements where the groundstate electron configuration changes within a column, where exceptions to Hund's rules occur. Ground state term symbols for the chemical elements are given below.
Term symbols are also used to describe angular momentum quantum numbers for atomic nuclei and for molecules. For molecular term symbols, Greek letters are used to designate the component of orbital angular momenta along the molecular axis.
The use of the word term for an atom's electronic state is based on the Rydberg–Ritz combination principle, an empirical observation that the wavenumbers of spectral lines can be expressed as the difference of two terms. This was later summarized by the Bohr model, which identified the terms with quantized energy levels, and the spectral wavenumbers of these levels with photon energies.
Tables of atomic energy levels identified by their term symbols are available for atoms and ions in ground and excited states from the National Institute of Standards and Technology (NIST).^{[1]}
The usual atomic term symbols assume LS coupling (also known as Russell–Saunders coupling), in which the atom's total spin quantum number S and the total orbital angular momentum quantum number L are "good quantum numbers". (Russell–Saunders coupling is named after Henry Norris Russell and Frederick Albert Saunders, who described it in 1925^{[2]}). The spinorbit interaction then couples the total spin and orbital moments to give the total electronic angular momentum quantum number J. Atomic states are then well described by term symbols of the form:
where
L =  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  ... 
S  P  D  F  G  H  I  K  L  M  N  O  Q  R  T  U  V  (continued alphabetically)^{[note 1]} 
The orbital symbols S, P, D and F are derived from the characteristics of the spectroscopic lines corresponding to s, p, d, and f orbitals: sharp, principal, diffuse, and fundamental; the rest are named in alphabetical order from G onwards (omitting J, S and P). When used to describe electronic states of an atom, the term symbol is often written following the electron configuration. For example, 1s^{2}2s^{2}2p^{2 3}P_{0} represents the ground state of a neutral carbon atom. The superscript 3 indicates that the spin multiplicity 2S + 1 is 3 (it is a triplet state), so S = 1; the letter "P" is spectroscopic notation for L = 1; and the subscript 0 is the value of J (in this case J = L − S).^{[1]}
Small letters refer to individual orbitals or oneelectron quantum numbers, whereas capital letters refer to manyelectron states or their quantum numbers.
For a given electron configuration,
The product as a number of possible states with given S and L is also a number of basis states in the uncoupled representation, where , , , ( and are zaxis components of total spin and total orbital angular momentum respectively) are good quantum numbers whose corresponding operators mutually commute. With given and , the eigenstates in this representation span function space of dimension , as and . In the coupled representation where total angular momentum (spin + orbital) is treated, the associated states (or eigenstates) are and these states span the function space with dimension of
as . Obviously, the dimension of function space in both representations must be the same.
As an example, for , there are (2×1+1)(2×2+1) = 15 different states (= eigenstates in the uncoupled representation) corresponding to the ^{3}D term, of which (2×3+1) = 7 belong to the ^{3}D_{3} (J = 3) level. The sum of for all levels in the same term equals (2S+1)(2L+1) as the dimensions of both representations must be equal as described above. In this case, J can be 1, 2, or 3, so 3 + 5 + 7 = 15.
The parity of a term symbol is calculated as
where is the orbital quantum number for each electron. means even parity while is for odd parity. In fact, only electrons in odd orbitals (with odd) contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd such as in p, f,...) correspond to an odd term symbol, while an even number of electrons in odd orbitals correspond to an even term symbol. The number of electrons in even orbitals is irrelevant as any sum of even numbers is even. For any closed subshell, the number of electrons is which is even, so the summation of in closed subshells is always an even number. The summation of quantum numbers over open (unfilled) subshells of odd orbitals ( odd) determines the parity of the term symbol. If the number of electrons in this reduced summation is odd (even) then the parity is also odd (even).
When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted:
Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for gerade (German for "even") or ungerade ("odd"):
It is relatively easy to calculate the term symbol for the ground state of an atom using Hund's rules. It corresponds with a state with maximum S and L.
As an example, in the case of fluorine, the electronic configuration is 1s^{2}2s^{2}2p^{5}.
+1  0  −1  

↑↓  ↑↓  ↑ 
In the periodic table, because atoms of elements in a column usually have the same outer electron structure, and always have the same electron structure in the "sblock" and "pblock" elements (see block (periodic table)), all elements may share the same ground state term symbol for the column. Thus, hydrogen and the alkali metals are all ^{2}S_{1⁄2}, the alkaline earth metals are ^{1}S_{0}, the boron column elements are ^{2}P_{1⁄2}, the carbon column elements are ^{3}P_{0}, the pnictogens are ^{4}S_{3⁄2}, the chalcogens are ^{3}P_{2}, the halogens are ^{2}P_{3⁄2}, and the inert gases are ^{1}S_{0}, per the rule for full shells and subshells stated above.
Term symbols for the ground states of most chemical elements^{[3]} are given in the collapsed table below.^{[4]} In the dblock and fblock, the term symbols are not always the same for elements in the same column of the periodic table, because open shells of several d or f electrons have several closely spaced terms whose energy ordering is often perturbed by the addition of an extra complete shell to form the next element in the column.
For example, the table shows that the first pair of vertically adjacent atoms with different groundstate term symbols are V and Nb. The ^{6}D_{1⁄2} ground state of Nb corresponds to an excited state of V 2112 cm^{−1} above the ^{4}F_{3⁄2} ground state of V, which in turn corresponds to an excited state of Nb 1143 cm^{−1} above the Nb ground state.^{[1]} These energy differences are small compared to the 15158 cm^{−1} difference between the ground and first excited state of Ca,^{[1]} which is the last element before V with no d electrons.
Term symbol of the chemical elements  

Group →  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  
↓ Period  
1  H ^{2}S_{1/2}

He ^{1}S_{0}
 
2  Li ^{2}S_{1/2}

Be ^{1}S_{0}

B ^{2}P_{1/2}

C ^{3}P_{0}

N ^{4}S_{3/2}

O ^{3}P_{2}

F ^{2}P_{3/2}

Ne ^{1}S_{0}
 
3  Na ^{2}S_{1/2}

Mg ^{1}S_{0}

Al ^{2}P_{1/2}

Si ^{3}P_{0}

P ^{4}S_{3/2}

S ^{3}P_{2}

Cl ^{2}P_{3/2}

Ar ^{1}S_{0}
 
4  K ^{2}S_{1/2}

Ca ^{1}S_{0}

Sc ^{2}D_{3/2}

Ti ^{3}F_{2}

V ^{4}F_{3/2}

Cr ^{7}S_{3}

Mn ^{6}S_{5/2}

Fe ^{5}D_{4}

Co ^{4}F_{9/2}

Ni ^{3}F_{4}

Cu ^{2}S_{1/2}

Zn ^{1}S_{0}

Ga ^{2}P_{1/2}

Ge ^{3}P_{0}

As ^{4}S_{3/2}

Se ^{3}P_{2}

Br ^{2}P_{3/2}

Kr ^{1}S_{0}
 
5  Rb ^{2}S_{1/2}

Sr ^{1}S_{0}

Y ^{2}D_{3/2}

Zr ^{3}F_{2}

Nb ^{6}D_{1/2}

Mo ^{7}S_{3}

Tc ^{6}S_{5/2}

Ru ^{5}F_{5}

Rh ^{4}F_{9/2}

Pd ^{1}S_{0}

Ag ^{2}S_{1/2}

Cd ^{1}S_{0}

In ^{2}P_{1/2}

Sn ^{3}P_{0}

Sb ^{4}S_{3/2}

Te ^{3}P_{2}

I ^{2}P_{3/2}

Xe ^{1}S_{0}
 
6  Cs ^{2}S_{1/2}

Ba ^{1}S_{0}

Lu ^{2}D_{3/2}

Hf ^{3}F_{2}

Ta ^{4}F_{3/2}

W ^{5}D_{0}

Re ^{6}S_{5/2}

Os ^{5}D_{4}

Ir ^{4}F_{9/2}

Pt ^{3}D_{3}

Au ^{2}S_{1/2}

Hg ^{1}S_{0}

Tl ^{2}P_{1/2}

Pb ^{3}P_{0}

Bi ^{4}S_{3/2}

Po ^{3}P_{2}

At ^{2}P_{3/2}

Rn ^{1}S_{0}
 
7  Fr ^{2}S_{1/2}

Ra ^{1}S_{0}

Lr ^{2}P_{1/2}?

Rf ^{3}F_{2}

Db ^{4}F_{3/2}?

Sg ^{5}D_{0}?

Bh ^{6}S_{5/2}?

Hs ^{5}D_{4}?

Mt ^{4}F_{9/2}?

Ds ^{3}F_{4}?

Rg ^{2}D_{5/2}?

Cn ^{1}S_{0}?

Nh ^{2}P_{1/2}?

Fl ^{3}P_{0}?

Mc ^{4}S_{3/2}?

Lv ^{3}P_{2}?

Ts ^{2}P_{3/2}?

Og ^{1}S_{0}?
 
La ^{2}D_{3/2}

Ce ^{1}G_{4}

Pr ^{4}I_{9/2}

Nd ^{5}I_{4}

Pm ^{6}H_{5/2}

Sm ^{7}F_{0}

Eu ^{8}S_{7/2}

Gd ^{9}D_{2}

Tb ^{6}H_{15/2}

Dy ^{5}I_{8}

Ho ^{4}I_{15/2}

Er ^{3}H_{6}

Tm ^{2}F_{7/2}

Yb ^{1}S_{0}
 
Ac ^{2}D_{3/2}

Th ^{3}F_{2}

Pa ^{4}K_{11/2}

U ^{5}L_{6}

Np ^{6}L_{11/2}

Pu ^{7}F_{0}

Am ^{8}S_{7/2}

Cm ^{9}D_{2}

Bk ^{6}H_{15/2}

Cf ^{5}I_{8}

Es ^{4}I_{15/2}

Fm ^{3}H_{6}

Md ^{2}F_{7/2}

No ^{1}S_{0}
 

The process to calculate all possible term symbols for a given electron configuration is somewhat longer.
As an example, consider the carbon electron structure: 1s^{2}2s^{2}2p^{2}. After removing full subshells, there are 2 electrons in a plevel (), so there are
different states.
+1  0  −1  M_{L}  M_{S}  

all up  ↑  ↑  1  1  
↑  ↑  0  1  
↑  ↑  −1  1  
all down  ↓  ↓  1  −1  
↓  ↓  0  −1  
↓  ↓  −1  −1  
one up one down 
↑↓  2  0  
↑  ↓  1  0  
↑  ↓  0  0  
↓  ↑  1  0  
↑↓  0  0  
↑  ↓  −1  0  
↓  ↑  0  0  
↓  ↑  −1  0  
↑↓  −2  0 
M_{S}  

+1  0  −1  
M_{L}  +2  1  
+1  1  2  1  
0  1  3  1  
−1  1  2  1  
−2  1 



where the floor function denotes the greatest integer not exceeding x.
The detailed proof can be found in Renjun Xu's original paper.^{[5]}For configurations with at most two electrons (or holes) per subshell, an alternative and much quicker method of arriving at the same result can be obtained from group theory. The configuration 2p^{2} has the symmetry of the following direct product in the full rotation group:
which, using the familiar labels Γ^{(0)} = S, Γ^{(1)} = P and Γ^{(2)} = D, can be written as
The square brackets enclose the antisymmetric square. Hence the 2p^{2} configuration has components with the following symmetries:
The Pauli principle and the requirement for electrons to be described by antisymmetric wavefunctions imply that only the following combinations of spatial and spin symmetry are allowed:
Then one can move to step five in the procedure above, applying Hund's rules.
The group theory method can be carried out for other such configurations, like 3d^{2}, using the general formula
The symmetric square will give rise to singlets (such as ^{1}S, ^{1}D, & ^{1}G), while the antisymmetric square gives rise to triplets (such as ^{3}P & ^{3}F).
More generally, one can use
where, since the product is not a square, it is not split into symmetric and antisymmetric parts. Where two electrons come from inequivalent orbitals, both a singlet and a triplet are allowed in each case.^{[6]}
Basic concepts for all coupling schemes:
Most famous coupling schemes are introduced here but these schemes can be mixed to express the energy state of an atom. This summary is based on [1].
These are notations for describing states of singly excited atoms, especially noble gas atoms. Racah notation is basically a combination of LS or Russell–Saunders coupling and J_{1}L_{2} coupling. LS coupling is for a parent ion and J_{1}L_{2} coupling is for a coupling of the parent ion and the excited electron. The parent ion is an unexcited part of the atom. For example, in Ar atom excited from a ground state ...3p^{6} to an excited state ...3p^{5}4p in electronic configuration, 3p^{5} is for the parent ion while 4p is for the excited electron.^{[8]}
In Racah notation, states of excited atoms are denoted as . Quantities with a subscript 1 are for the parent ion, n and ℓ are principal and orbital quantum numbers for the excited electron, K and J are quantum numbers for and where and are orbital angular momentum and spin for the excited electron respectively. “o” represents a parity of excited atom. For an inert (noble) gas atom, usual excited states are Np^{5}nℓ where N = 2, 3, 4, 5, 6 for Ne, Ar, Kr, Xe, Rn, respectively in order. Since the parent ion can only be ^{2}P_{1/2} or ^{2}P_{3/2}, the notation can be shortened to or , where nℓ means the parent ion is in ^{2}P_{3/2} while nℓ′ is for the parent ion in ^{2}P_{1/2} state.
Paschen notation is a somewhat odd notation; it is an old notation made to attempt to fit an emission spectrum of neon to a hydrogenlike theory. It has a rather simple structure to indicate energy levels of an excited atom. The energy levels are denoted as n′ℓ#. ℓ is just an orbital quantum number of the excited electron. n′ℓ is written in a way that 1s for (n = N + 1, ℓ = 0), 2p for (n = N + 1, ℓ = 1), 2s for (n = N + 2, ℓ = 0), 3p for (n = N + 2, ℓ = 1), 3s for (n = N + 3, ℓ = 0), etc. Rules of writing n′ℓ from the lowest electronic configuration of the excited electron are: (1) ℓ is written first, (2) n′ is consecutively written from 1 and the relation of ℓ = n′ − 1, n′ − 2, ... , 0 (like a relation between n and ℓ) is kept. n′ℓ is an attempt to describe electronic configuration of the excited electron in a way of describing electronic configuration of hydrogen atom. # is an additional number denoted to each energy level of given n′ℓ (there can be multiple energy levels of given electronic configuration, denoted by the term symbol). # denotes each level in order, for example, # = 10 is for a lower energy level than # = 9 level and # = 1 is for the highest level in a given n′ℓ. An example of Paschen notation is below.
Electronic configuration of Neon  n′ℓ  Electronic configuration of Argon  n′ℓ 

1s^{2}2s^{2}2p^{6}  Ground state  [Ne]3s^{2}3p^{6}  Ground state 
1s^{2}2s^{2}2p^{5}3s^{1}  1s  [Ne]3s^{2}3p^{5}4s^{1}  1s 
1s^{2}2s^{2}2p^{5}3p^{1}  2p  [Ne]3s^{2}3p^{5}4p^{1}  2p 
1s^{2}2s^{2}2p^{5}4s^{1}  2s  [Ne]3s^{2}3p^{5}5s^{1}  2s 
1s^{2}2s^{2}2p^{5}4p^{1}  3p  [Ne]3s^{2}3p^{5}5p^{1}  3p 
1s^{2}2s^{2}2p^{5}5s^{1}  3s  [Ne]3s^{2}3p^{5}6s^{1}  3s 