Term symbol


In quantum mechanics, the term symbol is an abbreviated description of the angular momentum quantum numbers in a multi-electron atom. Each energy level of an atom with a given electron configuration is described by not only the electron configuration but also its own term symbol, as the energy level also depends on the total angular momentum including spin. The usual atomic term symbols assume LS coupling. The ground state term symbol is predicted by Hund's rules.
The use of the word term for an energy level 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 with photon energies.
Tables of atomic energy levels identified by their term symbols have been compiled by the National Institute of Standards and Technology. In this database, neutral atoms are identified as I, singly ionized atoms as II, etc. Neutral atoms of the chemical elements have the same term symbol for each column in the s-block and p-block elements, but may differ in d-block and f-block elements, if the ground state electron configuration changes within a column. Ground state term symbols for chemical elements are given below.

LS coupling and symbol

For light atoms, the spin–orbit interaction is small so that the total orbital angular momentum L and total spin S are good quantum numbers. The interaction between L and S is known as LS coupling, Russell–Saunders coupling or spin–orbit coupling. Atomic states are then well described by term symbols of the form
where
The nomenclature is derived from the characteristics of the spectroscopic lines corresponding to orbitals: sharp, principal, diffuse, and fundamental; the rest being named in alphabetical order from G onwards, except that J is omitted. When used to describe electron states in an atom, the term symbol usually follows the electron configuration. For example, one low-lying energy level of the carbon atom state is written as 1s22s22p2 3P2. The superscript 3 indicates that the spin state is a triplet, and therefore S = 1, the P is spectroscopic notation for L = 1, and the subscript 2 is the value of J. Using the same notation, the ground state of carbon is 1s22s22p2 3P0.
Small letters refer to individual orbitals or one-electron quantum numbers, whereas capital letters refer to many-electron states or to their quantum numbers.

Terms, levels, and states

The term symbol is also used to describe compound systems such as mesons or atomic nuclei, or molecules. For molecules, Greek letters are used to designate the component of orbital angular momenta along the molecular axis.
For a given electron configuration
The product as a number of possible microstates with given S and L is also a number of basis states in the uncoupled representation, where S, mS, L, mL are good quantum numbers whose corresponding operators mutually commute. With given S and L, the eigenstates in this representation span function space of dimension, as and. In the coupled representation where total angular momentum is treated, the associated microstates are and these states span the function space with dimension of
as. Obviously the dimension of function space in both representation must be the same.
As an example, for, there are different microstates corresponding to the 3D term, of which belong to the 3D3 level. The sum of for all levels in the same term equals 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.

Term symbol parity

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 contribute to the total parity: an odd number of electrons in odd orbitals 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 subshells of odd orbitals determines the parity of the term symbol. If the number of electrons in this reduced summation is odd then the parity is also odd.
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 or ungerade :

Ground state term symbol

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.
  1. Start with the most stable electron configuration. Full shells and subshells do not contribute to the overall angular momentum, so they are discarded.
  2. *If all shells and subshells are full then the term symbol is 1S0.
  3. Distribute the electrons in the available orbitals, following the Pauli exclusion principle. First, fill the orbitals with highest magnetic quantum number| value with one electron each, and assign a maximal ms to them. Once all orbitals in a subshell have one electron, add a second one, assigning to them.
  4. The overall S is calculated by adding the ms values for each electron. According to Hund's first rule, the ground state has all unpaired electron spins parallel with the same value of ms, conventionally chosen as +½. The overall S is then ½ times the number of unpaired electrons. The overall L is calculated by adding the values for each electron.
  5. Calculate J as
  6. *if less than half of the subshell is occupied, take the minimum value ;
  7. *if more than half-filled, take the maximum value ;
  8. *if the subshell is half-filled, then L will be 0, so .
As an example, in the case of fluorine, the electronic configuration is 1s22s22p5.

  1. Discard the full subshells and keep the 2p5 part. So there are five electrons to place in subshell p.
  2. There are three orbitals that can hold up to. The first three electrons can take but the Pauli exclusion principle forces the next two to have because they go to already occupied orbitals.
  3. ; and, which is "P" in spectroscopic notation.
  4. As fluorine 2p subshell is more than half filled,. Its ground state term symbol is then.

Atomic term symbols of the chemical elements

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 "s-block" and "p-block" elements, all elements may share the same ground state term symbol for the column. Thus, hydrogen and the alkali metals are all 2S, the alkali earth metals are 1S0, the boron column elements are 2P, the carbon column elements are 3P0, the pnictogens are 4S, the chalcogens are 3P2, the halogens are 2P, and the inert gases are 1S0, per the rule for full shells and subshells stated above.
Term symbols for the ground states of most chemical elements are given in the collapsed table below. In the d-block and f-block, 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 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 ground-state term symbols are V and Nb. The 6D1/2 ground state of Nb corresponds to an excited state of V 2112 cm−1 above the 4F3/2 ground state of V, which in turn corresponds to an excited state of Nb 1143 cm−1 above the Nb ground state. These energy differences are small compared to the 15158 cm−1 difference between the ground and first excited state of Ca, which is the last element before V with no d electrons.

Term symbols for an electron configuration

The process to calculate all possible term symbols for a given electron configuration is somewhat longer.

Case of three equivalent electrons

For configurations with at most two electrons per subshell, an alternative and much quicker method of arriving at the same result can be obtained from group theory. The configuration 2p2 has the symmetry of the following direct product in the full rotation group:
which, using the familiar labels, and, can be written as
The square brackets enclose the anti-symmetric square. Hence the 2p2 configuration has components with the following symmetries:
The Pauli principle and the requirement for electrons to be described by anti-symmetric 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 3d2, using the general formula
The symmetric square will give rise to singlets, while the anti-symmetric square gives rise to triplets.
More generally, one can use
where, since the product is not a square, it is not split into symmetric and anti-symmetric parts. Where two electrons come from inequivalent orbitals, both a singlet and a triplet are allowed in each case.

Summary of various coupling schemes and corresponding term symbols

Basic concepts for all coupling schemes:
Most famous coupling schemes are introduced here but these schemes can be mixed together to express energy state of atom. This summary is based on .

Racah notation and Paschen notation

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 J1L2 coupling. LS coupling is for a parent ion and J1L2 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...3p6 to an excited state...3p54p in electronic configuration, 3p5 is for the parent ion while 4p is for the excited electron.
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 gas atom, usual excited states are Np5nl where N = 2, 3, 4, 5, 6 for Ne, Ar, Kr, Xe, Rn, respectively in order. Since the parent ion can only be 2P1/2 or 2P3/2, the notation can be shortened to or, where nl means the parent ion is in 2P3/2 while nl′ is for the parent ion in 2P1/2 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 hydrogen-like 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, 2p for, 2s for, 3p for, 3s for, etc. Rules of writing n′l from the lowest electronic configuration of the excited electron are: l is written first, n′ is consecutively written from 1 and the relation of l = n′ − 1, n′ − 2,..., 0 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. # 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 Neonn′lElectronic configuration of Argonn′l
1s22s22p6Ground state3s23p6Ground state
1s22s22p53s11s3s23p54s11s
1s22s22p53p12p3s23p54p12p
1s22s22p54s12s3s23p55s12s
1s22s22p54p13p3s23p55p13p
1s22s22p55s13s3s23p56s13s