Term Symbol

In quantum mechanics, the term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can be described by a term symbol). 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 (also known as Russell– Saunders coupling or spin-orbit 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 (multiplied by ''hc'', where ''h'' is the Planck constant and ''c'' the speed of light) with quantized energy levels and the spectral wavenumbers (again multiplied by ''hc'') 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.

Term symbols with ''LS'' coupling

For light atoms, the spin–orbit interaction (or coupling) 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'' (named after Henry Norris Russell and Frederick Albert Saunders, who described this in 1925) or ''spin-orbit coupling''. Atomic states are then well described by term symbols of the form

where

• ''S'' is the total spin quantum number. 2''S'' + 1 is the spin multiplicity, which represents the number of possible states of ''J'' for a given ''L'' and ''S'', provided that ''L'' ≥ ''S''. (If ''L'' < ''S'', the maximum number of possible ''J'' is 2''L'' + 1.) This is easily proven by using ''J''_max = ''L'' + ''S'' and ''J''_min = , ''L'' − ''S'', , so that the number of possible ''J'' with given ''L'' and ''S'' is simply ''J''_max − ''J''_min + 1 as ''J'' varies in unit steps.


• ''J'' is the total angular momentum quantum number


• ''L'' is the total orbital quantum number in spectroscopic notation. The first 17 symbols of L are:

The nomenclature (S, P, D, F) is derived from the characteristics of the spectroscopic lines corresponding to (s, p, d, f) 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 1s²2s²2p^2 3P₂. The superscript 3 indicates that the spin state is a triplet, and therefore ''S'' = 1 (2''S'' + 1 = 3), 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 1s²2s²2p^2 3P₀.NIST Atomic Spectrum Database
To read neutral carbon atom levels for example, type "C I" in the Spectrum box and click on 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.

Terms, levels, and states

The term symbol is also used to describe compound systems such as mesons or atomic nuclei, or molecules (see molecular term symbol). For molecules, Greek letters are used to designate the component of orbital angular momenta along the molecular axis.

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) equal to (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 ''S'', ''L'', ''J'' and ''M_J'' determines a single state.

The product as a number of possible microstates $, S,m_,L,m_\rangle$ with given ''S'' and ''L'' is also a number of basis states in the uncoupled representation, where ''S, m_S, L, m_L'' (''m_S'' and ''m_L'' are z-axis components of total spin and total orbital angular momentum respectively) are good quantum numbers whose corresponding operators mutually commute. With given ''S'' and ''L'', the eigenstates $, S,m_,L,m_\rangle$ in this representation span function space of dimension , as $m_=S,S-1,\dots, -S+1, -S$ and $m_=L,L-1,...,-L+1,-L$. In the coupled representation where total angular momentum (spin + orbital) is treated, the associated microstates (or eigenstates) are $, J,M_,S,L\rangle$ and these states span the function space with dimension of

as $m_=J,J-1,\dots,-J+1,-J$. Obviously, the dimension of function space in both representations must be the same.

As an example, for $S = 1, L = 2$, there are different microstates (= 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 (2''S''+1)(2''L''+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 $\ell_i$ is the orbital quantum number for each electron. $P=1$ means even parity while $P=-1$ is for odd parity. In fact, only electrons in odd orbitals (with $\ell$ odd) contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd $\ell$ 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 $2(2\ell+1)$ which is even, so the summation of $\ell_i$ in closed subshells is always an even number. The summation of quantum numbers $\sum_\ell_$ over open (unfilled) subshells of odd orbitals ($\ell$ 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 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''.
#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. First, fill the orbitals with highest value with one electron each, and assign a maximal ''m_s'' to them (i.e. +). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning to them.
#The overall ''S'' is calculated by adding the ''m_s'' values for each electron. According to Hund's first rule, the ground state has all unpaired electron spins parallel with the same value of m_s, conventionally chosen as +. The overall ''S'' is then times the number of unpaired electrons. The overall ''L'' is calculated by adding the $m_\ell$ values for each electron (so if there are two electrons in the same orbital, add twice that orbital's $m_\ell$).
#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.

1. Discard the full subshells and keep the 2p⁵ part. So there are five electrons to place in subshell p ($\ell = 1$).
   


2. There are three orbitals ($m_\ell = 1, 0, -1$) 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 (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 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 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 symbols for an electron configuration

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

• First, the total number of possible microstates is calculated for a given electron configuration. As before, the filled (sub)shells are discarded, and only the partially filled ones are kept. For a given orbital quantum number $\ell$, is the maximum allowed number of electrons, . If there are electrons in a given subshell, the number of possible microstates is

  As an example, consider the carbon electron structure: 1s²2s²2p². After removing full subshells, there are 2 electrons in a p-level ($\ell =1$), so there are

  different microstates.
  



• Second, all possible microstates are drawn. ''M_L'' and ''M_S'' for each microstate are calculated, with $M=\sum_^e m_i$ where ''m_i'' is either $m_\ell$ or $m_s$ for the ''i''-th electron, and ''M'' represents the resulting ''M_L'' or ''M_S'' respectively:



• Third, the number of microstates for each (''M_L'',''M_S'') possible combination is counted:



• Fourth, smaller tables can be extracted representing each possible term. Each table will have the size (2''L''+1) by (2''S''+1), and will contain only "1"s as entries. The first table extracted corresponds to ''M_L'' ranging from −2 to +2 (so ), with a single value for ''M_S'' (implying ). This corresponds to a ¹D term. The remaining terms fit inside the middle 3×3 portion of the table above. Then a second table can be extracted, removing the entries for ''M_L'' and ''M_S'' both ranging from −1 to +1 (and so , a ³P term). The remaining table is a 1×1 table, with , i.e., a ¹S term.



• Fifth, applying Hund's rules, the ground state can be identified (or the lowest state for the configuration of interest). Hund's rules should not be used to predict the order of states other than the lowest for a given configuration. (See examples at .)
  
• If only two equivalent electrons are involved, there is an "Even Rule" which states that, for two equivalent electrons, the only states that are allowed are those for which the sum (L + S) is even.
  

Case of three equivalent electrons

Alternative method using group theory

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 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 (such as ¹S, ¹D, & ¹G), while the anti-symmetric square gives rise to triplets (such as ³P & ³F).

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:
* $\boldsymbol$: individual orbital angular momentum vector for an electron, $\mathbf$: individual spin vector for an electron, $\mathbf$: individual total angular momentum vector for an electron, $\mathbf = \boldsymbol + \mathbf$.
* $\mathbf$: Total orbital angular momentum vector for all electrons in an atom ($\mathbf=\sum_\boldsymbol_$).
* $\mathbf$: total spin vector for all electrons ($\mathbf=\sum_\mathbf_$).
* $\mathbf$: total angular momentum vector for all electrons. The way the angular momenta are combined to form $\mathbf$ depends on the coupling scheme: $\mathbf = \mathbf + \mathbf$ for ''LS'' coupling, $\mathbf = \sum_\mathbf_$ for ''jj'' coupling, etc.
* A quantum number corresponding to the magnitude of a vector is a letter without an arrow, or without boldface (ex: ''ℓ'' is the orbital angular momentum quantum number for $\boldsymbol$ and $\left, \ell,m,\ldots \right\rangle =\ell\left( \ell+1 \right)\left, \ell,m,\ldots \right\rangle$)
* The parameter called multiplicity represents the number of possible values of the total angular momentum quantum number ''J'' for certain conditions.
* For a single electron, the term symbol is not written as ''S'' is always 1/2, and ''L'' is obvious from the orbital type.
* For two electron groups ''A'' and ''B'' with their own terms, each term may represent ''S'', ''L'' and ''J'' which are quantum numbers corresponding to the $\mathbf$, $\mathbf$ and $\mathbf$ vectors for each group. "Coupling" of terms ''A'' and ''B'' to form a new term ''C'' means finding quantum numbers for new vectors $\mathbf= \mathbf_ + \mathbf_$, $\mathbf=\mathbf_+ \mathbf_$ and $\mathbf = \mathbf + \mathbf$. This example is for ''LS'' coupling and which vectors are summed in a coupling is depending on which scheme of coupling is taken. Of course, the angular momentum addition rule is that $X= X_+X_,X_+X_-1, \dots, , X_-X_,$ where ''X'' can be ''s'', ''ℓ'', ''j'', ''S'', ''L'', ''J'' or any other angular momentum-magnitude-related quantum number.

''LS'' coupling (Russell–Saunders coupling)

* Coupling scheme: $\mathbf$ and $\mathbf$ are calculated first then $\mathbf=\mathbf+\mathbf$ is obtained. From a practical point of view, it means ''L'', ''S'' and ''J'' are obtained by using an addition rule of the angular momenta of given electron groups that are to be coupled.
* Electronic configuration + Term symbol: $n)$. $)$ is a Term which is from coupling of electrons in $n$group. $n,\ell$ are principle quantum number, orbital quantum number and $n$means there are ''N'' (equivalent) electrons in $n \ell$ subshell. For $L > S$, is equal to multiplicity, a number of possible values in ''J'' (final total angular momentum quantum number) from given ''S'' and ''L''. For $S>L$, multiplicity is but is still written in the Term symbol. Strictly speaking, $)$ is called ''Level'' and $$ is called ''Term''. Sometimes superscript ''o'' is attached to the Term, means the parity $P=$ of group is odd ($P = -1$).
* Example:
*# 3d^7 4F_7/2: ⁴F_7/2 is Level of 3d⁷ group in which are equivalent 7 electrons are in 3d subshell.
*# 3d⁷(⁴F)4s4p(³P⁰) ⁶F: Terms are assigned for each group (with different principal quantum number ''n'') and rightmost Level  ⁶F is from coupling of Terms of these groups so ⁶F represents final total spin quantum number ''S'', total orbital angular momentum quantum number ''L'' and total angular momentum quantum number ''J'' in this atomic energy level. The symbols ⁴F and ³P^''o'' refer to seven and two electrons respectively so capital letters are used.
*# 4f⁷(⁸S⁰)5d (⁷D^''o'')6p ⁸F_13/2: There is a space between 5d and (⁷D^''o''). It means (⁸S⁰) and 5d are coupled to get (⁷D^''o''). Final level ⁸F is from coupling of (⁷D^''o'') and 6p.
*# 4f(²F⁰) 5d²(¹G) 6s(²G) ¹P: There is only one Term ²F^''o'' which is isolated in the left of the leftmost space. It means (²F^''o'') is coupled lastly; (¹G) and 6s are coupled to get (²G) then (²G) and (²F^''o'') are coupled to get final Term ¹P.

''jj'' Coupling

* Coupling scheme: $\mathbf = \sum_ \mathbf_$.
* Electronic configuration + Term symbol: $$
* Example:
*# $_$: There are two groups. One is $\text^_$ and the other is $\text_^$. In $\text^_$, there are 2 electrons having $j=1/2$ in 6p subshell while there is an electron having $j=3/2$ in the same subshell in $\text_^$. Coupling of these two groups results in (coupling of ''j'' of three electrons).
*# $\text_^\text_^~\$: in () is $J_1$ for 1st group $\text^_$ and in () is ''J''₂ for 2nd group $\text^_$. Subscript 11/2 of Term symbol is final ''J'' of $\mathbf=\mathbf_+\mathbf_$.

''J''₁''L''₂ coupling

* Coupling scheme: $\mathbf=\mathbf_1+\mathbf_2$ and $\mathbf= \mathbf+\mathbf_2$.
* Electronic configuration + Term symbol: $^\left( \mathrm_1 \right)^\left( \mathrm_2 \right)~\$. For is equal to multiplicity, a number of possible values in ''J'' (final total angular momentum quantum number) from given ''S''₂ and ''K''. For , multiplicity is but is still written in the Term symbol.
* Example:
*# 3p⁵(²P)5g ² /2 $= \frac,=4,~=1/2$. is ''K'', which comes from coupling of ''J''₁ and ''ℓ''₂. Subscript 5 in Term symbol is ''J'' which is from coupling of ''K'' and ''s''₂.
*# 4f¹³(²F)5d²(¹D)  /2 $= \frac,=2,~=0$. is ''K'', which comes from coupling of ''J''₁ and ''L''₂. Subscript in Term symbol is ''J'' which is from coupling of ''K'' and ''S''₂.

''LS''₁ coupling

* Coupling scheme:$\mathbf \ell= \mathbf +\mathbf$, $\mathbf = \mathbf+\mathbf$.
* Electronic configuration + Term symbol: $^\left( \mathrm_1\right) ^ \left(\mathrm_2\right)\ ~L~\$. For is equal to multiplicity, a number of possible values in ''J'' (final total angular momentum quantum number) from given ''S''₂ and ''K''. For , multiplicity is but is still written in the Term symbol.
* Example:
*# 3d⁷(⁴P)4s4p(³P^''o'') D^''o'' 3 /2 $=1,~=1,~=\frac, ~=1$. $L=2, K=5/2, J=7/2$.
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 o

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 ''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 ...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 \left( ^_ \right)n\ell\left K \right^. Quantities with a subscript 1 are for the parent ion, and are principal and orbital quantum numbers for the excited electron, ''K'' and ''J'' are quantum numbers for $\mathbf=\mathbf_+\boldsymbol$ and $\mathbf=\mathbf+\mathbf$ where $\boldsymbol$ and $\mathbf$ 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 where ''N'' = 2, 3, 4, 5, 6 for Ne, Ar, Kr, Xe, Rn, respectively in order. Since the parent ion can only be ²P_1/2 or ²P_3/2, the notation can be shortened to n\ell\left K \right^ or n\ell'\left K \right^, where means the parent ion is in ²P_3/2 while is for the parent ion in ²P_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 hydrogen-like theory. It has a rather simple structure to indicate energy levels of an excited atom. The energy levels are denoted as . is just an orbital quantum number of the excited electron. is written in a way that 1s for , 2p for , 2s for , 3p for , 3s for , etc. Rules of writing from the lowest electronic configuration of the excited electron are: (1) is written first, (2) is consecutively written from 1 and the relation of (like a relation between and ) is kept. 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 (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 . An example of Paschen notation is below.

See also

* Quantum number
** Principal quantum number
** Azimuthal quantum number
** Spin quantum number
** Magnetic quantum number
* Angular quantum numbers
* Angular momentum coupling
* Molecular term symbol
* Hole formalism

Notes

References

{{DEFAULTSORT:Term Symbol
Atomic physics
Theoretical chemistry
Quantum chemistry