In atomic physics, a term symbol is an abbreviated description of the spin, orbital and total angular momentum quantum numbers of the electrons in a multi-electron 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 all-electron 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 MJ specify a state. The conventional term symbol has the form 2S+1LJ, 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 s-block and p-block elements, but differ in d-block and f-block elements where the ground-state 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).
Term symbols with LS coupling 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). The spin-orbit 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 S is the total spin quantum number for the atom's electrons. The value 2S + 1 written in the term symbol is the spin multiplicity, which is the number of possible values of the spin magnetic quantum number MS for a given spin S. J is the total angular momentum quantum number for the atom's electrons. J has a value in the range from |L − S| to L + S. L is the total orbital quantum number in spectroscopic notation, in which the symbols for L are: L = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... (continued alphabetically)There is no official convention for naming orbital angular momentum values greater than 20 (symbol ) but they are rarely needed. Some authors use Greek letters (α, β, γ, ...) after .
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, 1s22s22p2 3P0 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).NIST Atomic Spectrum Database For example, to display the levels for a neutral carbon atom, enter "C I" or "C 0" in the "Spectrum" box and click "Retrieve data".
Small letters refer to individual orbitals or one-electron quantum numbers, whereas capital letters refer to many-electron states or their quantum numbers.
Terminology: terms, levels, and states For a given electron configuration, The combination of an value and an value is called a term, and has a statistical weight (i.e., number of possible states) equal to ; A combination of , and is called a level. A given level has a statistical weight of , which is the number of possible states associated with this level in the corresponding term; A combination of , , and determines a single state.
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 z-axis 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 different states (= eigenstates in the uncoupled representation) corresponding to the 3D term, of which belong to the 3D3 (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.
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 (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"):
Ground state term symbol It is relatively easy to predict the term symbol for the ground state of an atom using Hund's rules. It corresponds to a state with maximum 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 1S0. Distribute the electrons in the available orbitals, following the Pauli exclusion principle. Conventionally, put 1 electron into orbital with highest and then continue filling other orbitals in descending order with one electron each, until you are out of electrons, or all orbitals in the subshell have one electron. Assign, again conventionally, all these electrons a value + of quantum magnetic spin number . If there are remaining electrons, put them in orbitals in the same order as before, but now assigning to them. The overall S is calculated by adding the ms values for each electron. The overall S is then times the number of unpaired electrons. The overall L is calculated by adding the values for each electron (so if there are two electrons in the same orbital, add twice that orbital's ). Calculate J as if less than half of the subshell is occupied, take the minimum value ; if more than half-filled, take the maximum value ; 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. Discard the full subshells and keep the 2p5 part. So there are five electrons to place in subshell p (). 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. +10−1↑↓↑↓↑ <li> ; , which is "P" in spectroscopic notation. 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 (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 2S, the alkaline earth metals are 1S0, the boron group elements are 2P, the carbon group 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.For the sources for these term symbols in the case of the heaviest elements, see Template:Infobox element/symbol-to-electron-configuration/term-symbol. 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 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 ground-state term symbols are V and Nb. The 6D ground state of Nb corresponds to an excited state of V 2112 cm−1 above the 4F 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 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 2 3 4 5 6 7 colspan="4" colspan="4"
Term symbols for an electron configuration The process to calculate all possible term symbols for a given electron configuration is somewhat longer.