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Inatomic physics, aterm symbol is an abbreviated description of the total spin and orbitalangular momentum quantum numbers of the electrons in a multi-electronatom. So while the wordsymbol suggests otherwise, it represents an actualvalue of aphysical quantity.

For a givenelectron 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 assumeLS 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 aregood quantum numbers.

In the terminology ofatomic spectroscopy,L andS together specify aterm;L,S, andJ specify alevel; andL,S,J and the magnetic quantum numberM_J specify astate. The conventional term symbol has the form^2S+1L_J, whereJ is written optionally in order to specify a level.L is written usingspectroscopic notation: for example, it is written "S", "P", "D", or "F" to representL = 0, 1, 2, or 3 respectively. For coupling schemes other that LS coupling, such as thejj 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 givensubshell or set of subshells, for example to describe eachopen subshell in an atom having more than one. Theground state term symbol for neutral atoms is described, in most cases, byHund's rules. Neutral atoms of the chemical elements have the same term symbolfor each column in thes-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 givenbelow.

Term symbols are also used to describe angular momentum quantum numbers foratomic nuclei and for molecules. Formolecular term symbols, Greek letters are used to designate the component of orbital angular momenta along the molecular axis.

The use of the wordterm for an atom's electronic state is based on theRydberg–Ritz combination principle, an empirical observation that the wavenumbers of spectral lines can be expressed as the difference of twoterms. This was later summarized by theBohr 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 theNational Institute of Standards and Technology (NIST).^[1]

The usual atomic term symbols assumeLS coupling (also known as Russell–Saunders coupling), in which the atom's total spin quantum numberS and the total orbital angular momentum quantum numberL are "good quantum numbers". (Russell–Saunders coupling is named afterHenry Norris Russell andFrederick Albert Saunders, who described it in 1925^[2]). Thespin-orbit interaction then couples the total spin and orbital moments to give the total electronic angular momentum quantum numberJ. Atomic states are then well described by term symbols of the form:

where

• S is the totalspin quantum number for the atom's electrons. The value 2S + 1 written in the term symbol is thespin multiplicity, which is the number of possible values of the spin magnetic quantum numberM_S for a given spinS.
• J is thetotal angular momentum quantum number for the atom's electrons.J has a value in the range from |L −S| toL +S.
• L is the totalorbital quantum number inspectroscopic notation, in which the symbols forL are:

  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, andfundamental; 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 theelectron configuration. For example, 1s²2s²2p^2 3P₀ represents the ground state of a neutralcarbon atom. The superscript 3 indicates that the spin multiplicity 2S + 1 is 3 (it is atriplet state), soS = 1; the letter "P" is spectroscopic notation forL = 1; and the subscript 0 is the value ofJ (in this caseJ =L −S).^[1]

Small letters refer to individual orbitals or one-electron quantum numbers, whereas capital letters refer to many-electron states or their quantum numbers.

For a given electron configuration,

• The combination of an${\displaystyle S}$ value and an${\displaystyle L}$ value is called aterm, and has a statistical weight (i.e., number of possible states) equal to${\displaystyle (2S+1)(2L+1)}$;
• A combination of${\displaystyle S}$,${\displaystyle L}$ and${\displaystyle J}$ is called alevel. A given level has a statistical weight of${\displaystyle 2J+1}$, which is the number of possible states associated with this level in the corresponding term;
• A combination of${\displaystyle S}$,${\displaystyle L}$,${\displaystyle J}$ and${\displaystyle M_J}$ determines a singlestate.

The product⁠${\displaystyle (2S+1)(2L+1)}$⁠ as a number of possible states${\displaystyle |S,M_S,L,M_L⟩}$ with givenS andL is also a number of basis states in the uncoupled representation, where${\displaystyle S}$,${\displaystyle M_S}$,${\displaystyle L}$,${\displaystyle M_L}$ (${\displaystyle M_S}$ and${\displaystyle M_L}$ are z-axis components of total spin and total orbital angular momentum respectively) are good quantum numbers whose corresponding operators mutually commute. With given${\displaystyle S}$ and${\displaystyle L}$, the eigenstates${\displaystyle |S,M_S,L,M_L⟩}$ in this representation span function space of dimension⁠${\displaystyle (2S+1)(2L+1)}$⁠, as${\displaystyle M_S=S,S-1,\dots ,-S+1,-S}$ and${\displaystyle M_L=L,L-1,...,-L+1,-L}$. In the coupled representation where total angular momentum (spin + orbital) is treated, the associated states (oreigenstates) are${\displaystyle |J,M_J,S,L⟩}$ and these states span the function space with dimension of

${\displaystyle \sum _{J=J_{min}=|L-S|}^{J_{max}=L+S}(2J+1)}$

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

As an example, for${\displaystyle S=1,L=2}$, there are(2×1+1)(2×2+1) = 15 different states (= eigenstates in the uncoupled representation) corresponding to the³Dterm, of which(2×3+1) = 7 belong to the³D₃ (J = 3) level. The sum of⁠${\displaystyle (2J+1)}$⁠ 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

${\displaystyle P=(-1{)}^{\sum _i{\ell }_i},}$

where${\displaystyle {\ell }_i}$ is the orbital quantum number for each electron.${\displaystyle P=1}$ means even parity while${\displaystyle P=-1}$ is for odd parity. In fact, only electrons in odd orbitals (with${\displaystyle \ell }$ odd) contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd${\displaystyle \ell }$ 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${\displaystyle 2(2\ell +1)}$ which is even, so the summation of${\displaystyle {\ell }_i}$ in closed subshells is always an even number. The summation of quantum numbers$\sum _i{\ell }_i$ over open (unfilled) subshells of odd orbitals (${\displaystyle \ell }$ odd) determines the parity of the term symbol. If the number of electrons in thisreduced 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 forgerade (German for "even") orungerade ("odd"):

It is relatively easy to predict the term symbol for the ground state of an atom usingHund's rules. It corresponds to a state with maximumS andL.

1. Start with the most stableelectron configuration. Full shells and subshells do not contribute to the overallangular momentum, so they are discarded.
   • If all shells and subshells are full then the term symbol is¹S₀.
2. Distribute the electrons in the availableorbitals, following thePauli exclusion principle.
   • Conventionally, put 1 electron into orbital with highestm_ℓ and then continue filling other orbitals in descendingm_ℓ 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 +1⁄2 of quantum magnetic spin numberm_s.
   • If there are remaining electrons, put them in orbitals in the same order as before, but now assigningm_s = −1⁄2 to them.
3. The overallS is calculated by adding them_s values for each electron. The overallS is then1⁄2 times the number ofunpaired electrons.
4. The overallL is calculated by adding the${\displaystyle m_{\ell }}$ values for each electron (so if there are two electrons in the same orbital, add twice that orbital's${\displaystyle m_{\ell }}$).
5. CalculateJ as
   • if less than half of the subshell is occupied, take the minimum valueJ = |L −S|;
   • if more than half-filled, take the maximum valueJ =L +S;
   • if the subshell is half-filled, thenL will be 0, soJ =S.

As an example, in the case offluorine, 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 (${\displaystyle \ell =1}$).
2. There are three orbitals (${\displaystyle m_{\ell }=1,0,-1}$) that can hold up to${\displaystyle 2(2\ell +1)=6}$ electrons. The first three electrons can takem_s =1⁄2 (↑) but the Pauli exclusion principle forces the next two to havem_s = −1⁄2 (↓) because they go to already occupied orbitals.

   	${\displaystyle m_{\ell }}$
   	+1	0	−1
   ${\displaystyle m_s}$	↑↓	↑↓	↑

3. S =1⁄2 +1⁄2 +1⁄2 −1⁄2 −1⁄2 =1⁄2;
4. L = 1 + 0 − 1 + 1 + 0 = 1, which is "P" in spectroscopic notation.
5. As fluorine 2p subshell is more than half filled,J =L +S =3⁄2. Its ground state term symbol is then^2S+1L_J =²P_3⁄2.

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 (seeblock (periodic table)), all elements may share the same ground state term symbol for the column. Thus, hydrogen and thealkali metals are all²S_1⁄2, thealkaline earth metals are¹S₀, theboron group elements are²P_1⁄2, thecarbon group elements are³P₀, thepnictogens are⁴S_3⁄2, thechalcogens are³P₂, thehalogens are²P_3⁄2, and theinert gases are¹S₀, 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 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. The⁶D_1⁄2 ground state of Nb corresponds to an excited state of V 2112 cm⁻¹ above the⁴F_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 2S1/2																		He 1S0
2	Li 2S1/2	Be 1S0												B 2P1/2	C 3P0	N 4S3/2	O 3P2	F 2P3/2	Ne 1S0
3	Na 2S1/2	Mg 1S0												Al 2P1/2	Si 3P0	P 4S3/2	S 3P2	Cl 2P3/2	Ar 1S0
4	K 2S1/2	Ca 1S0		Sc 2D3/2	Ti 3F2	V 4F3/2	Cr 7S3	Mn 6S5/2	Fe 5D4	Co 4F9/2	Ni 3F4	Cu 2S1/2	Zn 1S0	Ga 2P1/2	Ge 3P0	As 4S3/2	Se 3P2	Br 2P3/2	Kr 1S0
5	Rb 2S1/2	Sr 1S0		Y 2D3/2	Zr 3F2	Nb 6D1/2	Mo 7S3	Tc 6S5/2	Ru 5F5	Rh 4F9/2	Pd 1S0	Ag 2S1/2	Cd 1S0	In 2P1/2	Sn 3P0	Sb 4S3/2	Te 3P2	I 2P3/2	Xe 1S0
6	Cs 2S1/2	Ba 1S0		Lu 2D3/2	Hf 3F2	Ta 4F3/2	W 5D0	Re 6S5/2	Os 5D4	Ir 4F9/2	Pt 3D3	Au 2S1/2	Hg 1S0	Tl 2P1/2	Pb 3P0	Bi 4S3/2	Po 3P2	At 2P3/2	Rn 1S0
7	Fr 2S1/2	Ra 1S0		Lr 2P1/2?	Rf 3F2	Db 4F3/2?	Sg 5D0?	Bh 6S5/2?	Hs 5D4?	Mt 4F9/2?	Ds 3F4?	Rg 2D5/2?	Cn 1S0?	Nh 2P1/2?	Fl 3P0?	Mc 4S3/2?	Lv 3P2?	Ts 2P3/2?	Og 1S0?
				La 2D3/2	Ce 1G4	Pr 4I9/2	Nd 5I4	Pm 6H5/2	Sm 7F0	Eu 8S7/2	Gd 9D2	Tb 6H15/2	Dy 5I8	Ho 4I15/2	Er 3H6	Tm 2F7/2	Yb 1S0
				Ac 2D3/2	Th 3F2	Pa 4K11/2	U 5L6	Np 6L11/2	Pu 7F0	Am 8S7/2	Cm 9D2	Bk 6H15/2	Cf 5I8	Es 4I15/2	Fm 3H6	Md 2F7/2	No 1S0
s-block f-block d-block p-block

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

• First, the total number of possible statesN 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${\displaystyle \ell }$,t is the maximum allowed number of electrons,${\displaystyle t=2(2\ell +1)}$. If there aree electrons in a given subshell, the number of possible states is
  ${\displaystyle N=(\genfrac{}{}{0ex}{}{t}{e})=\frac{t!}{e!(t-e)!}.}$

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

  ${\displaystyle N=\frac{6!}{2!4!}=15}$

  different states.

• Second, all possible states are drawn.M_L andM_S for each state are calculated, with${\displaystyle M=\sum _{i=1}^e{m}_i}$ wherem_i is either${\displaystyle m_{\ell }}$ or${\displaystyle m_s}$ for thei-th electron, andM represents the resultingM_L orM_S respectively:

  	${\displaystyle m_{\ell }}$
  	+1	0	−1	ML	MS
  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

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

  		MS
  		+1	0	−1
  ML	+2		1
  	+1	1	2	1
  	0	1	3	1
  	−1	1	2	1
  	−2		1

• Fourth, smaller tables can be extracted representing each possible term. Each table will have the size (2L+1) by (2S+1), and will contain only "1"s as entries. The first table extracted corresponds toM_L ranging from −2 to +2 (soL = 2), with a single value forM_S (implyingS = 0). This corresponds to a¹D 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 forM_L andM_S both ranging from −1 to +1 (and soS =L = 1, a³P term). The remaining table is a 1×1 table, withL =S = 0, i.e., a¹S term.

  S = 0,L = 2,J = 2 1D2   MS   0 ML +2 1 +1 1 0 1 −1 1 −2 1

  S=1,L=1,J=2,1,0 3P2,3P1,3P0   MS   +1 0 −1 ML +1 1 1 1 0 1 1 1 −1 1 1 1

  S=0,L=0,J=0 1S0   MS   0 ML 0 1

• Fifth, applyingHund'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 atHund's rules § Excited states.)
• 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.

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 fromgroup theory. The configuration 2p² has the symmetry of the following direct product in the full rotation group:

which, using the familiar labelsΓ⁽⁰⁾ = S,Γ⁽¹⁾ = P andΓ⁽²⁾ = D, 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 as¹S,¹D, &¹G), while the anti-symmetric square gives rise to triplets (such as³P &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.^[6]

Basic concepts for all coupling schemes:

• ${\displaystyle \mathit{\ell }}$: individual orbital angular momentum vector for an electron,${\displaystyle \mathbf{s}}$: individual spin vector for an electron,${\displaystyle \mathbf{j}}$: individual total angular momentum vector for an electron,${\displaystyle \mathbf{j}=\mathit{\ell }+\mathbf{s}}$.
• ${\displaystyle \mathbf{L}}$: Total orbital angular momentum vector for all electrons in an atom (${\displaystyle \mathbf{L}=\sum _i{\mathit{\ell }}_i}$).
• ${\displaystyle \mathbf{S}}$: total spin vector for all electrons (${\displaystyle \mathbf{S}=\sum _i{\mathbf{s}}_i}$).
• ${\displaystyle \mathbf{J}}$: total angular momentum vector for all electrons. The way the angular momenta are combined to form${\displaystyle \mathbf{J}}$ depends on the coupling scheme:${\displaystyle \mathbf{J}=\mathbf{L}+\mathbf{S}}$ forLS coupling,${\displaystyle \mathbf{J}=\sum _i{\mathbf{j}}_i}$ forjj coupling, etc.
• A quantum number corresponding to the magnitude of a vector is a letter without an arrow, or without boldface (example:ℓ is the orbital angular momentum quantum number for${\displaystyle \mathit{\ell }}$ and${\displaystyle {\hat{\ell }}^2|\ell ,m_{\ell },\dots ⟩={\hslash }^2\ell \left(\ell +1\right)|\ell ,m_{\ell },\dots ⟩}$)
• The parameter calledmultiplicity represents the number of possible values of the total angular momentum quantum numberJ for certain conditions.
• For a single electron, the term symbol is not written asS is always 1/2, andL is obvious from the orbital type.
• For two electron groupsA andB with their own terms, each term may representS,L andJ which are quantum numbers corresponding to the${\displaystyle \mathbf{S}}$,${\displaystyle \mathbf{L}}$ and${\displaystyle \mathbf{J}}$ vectors for each group. "Coupling" of termsA andB to form a new termC means finding quantum numbers for new vectors${\displaystyle \mathbf{S}={\mathbf{S}}_A+{\mathbf{S}}_B}$,${\displaystyle \mathbf{L}={\mathbf{L}}_A+{\mathbf{L}}_B}$ and${\displaystyle \mathbf{J}=\mathbf{L}+\mathbf{S}}$. This example is forLS 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${\displaystyle X=X_A+X_B,X_A+X_B-1,\dots ,|X_A-X_B|}$ whereX can bes,ℓ,j,S,L,J or any other angular momentum-magnitude-related quantum number.

• Coupling scheme:${\displaystyle \mathbf{L}}$ and${\displaystyle \mathbf{S}}$ are calculated first then${\displaystyle \mathbf{J}=\mathbf{L}+\mathbf{S}}$ is obtained. From a practical point of view, it meansL,S andJ are obtained by using an addition rule of the angular momenta of given electron groups that are to be coupled.
• Electronic configuration + Term symbol:${\displaystyle n{\ell }^N{(}^{(2S+1)}{L}_J)}$.${\displaystyle {(}^{(2S+1)}{L}_J)}$ is a term which is from coupling of electrons in${\displaystyle n{\ell }^N}$group.${\displaystyle n,\ell }$ are principle quantum number, orbital quantum number and${\displaystyle n{\ell }^N}$means there areN (equivalent) electrons in${\displaystyle n\ell }$ subshell. For${\displaystyle L>S}$,⁠${\displaystyle (2S+1)}$⁠ is equal to multiplicity, a number of possible values inJ (final total angular momentum quantum number) from givenS andL. For${\displaystyle S>L}$, multiplicity is⁠${\displaystyle (2L+1)}$⁠ but⁠${\displaystyle (2S+1)}$⁠ is still written in the term symbol. Strictly speaking,${\displaystyle {(}^{(2S+1)}{L}_J)}$ is calledlevel and${\displaystyle {}^{\left(2S+1\right)}L}$ is calledterm. Sometimes right superscripto is attached to the term symbol, meaning the parity${\displaystyle P={\left(-1\right)}^{\sum _i{\ell }_i}}$ of the group is odd (${\displaystyle P=-1}$).
• Example:
  1. 3d^7 4F_7/2:⁴F_7/2 is level of 3d⁷ group in which are equivalent 7 electrons are in 3d subshell.
  2. 3d⁷(⁴F)4s4p(³P⁰) ⁶F⁰
     _9/2:^[7] Terms are assigned for each group (with different principal quantum numbern) and rightmost level⁶F^o
     _9/2 is from coupling of terms of these groups so⁶F^o
     _9/2 represents final total spin quantum numberS, total orbital angular momentum quantum numberL and total angular momentum quantum numberJ in this atomic energy level. The symbols⁴F and³P^o refer to seven and two electrons respectively so capital letters are used.
  3. 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 level⁸F^o
     _13/2 is from coupling of (⁷D^o) and 6p.
  4. 4f(²F⁰) 5d²(¹G) 6s(²G) ¹P⁰
     ₁: There is only one term²F^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 term¹P^o
     ₁.

• Coupling scheme:${\displaystyle \mathbf{J}=\sum _i{\mathbf{j}}_i}$.
• Electronic configuration + Term symbol:${\displaystyle {\left(n_1{{\ell }_1}_{j_1}^{N_1}{n}_2{{\ell }_2}_{j_2}^{N_2}\dots \right)}_J}$
• Example:
  1. ${\displaystyle {{\left({\text{6p}}_{\frac{1}{2}}^2{\text{6p}}_{\frac{3}{2}}^{}\right)}^o}_{3/2}}$: There are two groups. One is${\displaystyle {\text{6p}}_{1/2}^2}$ and the other is${\displaystyle {\text{6p}}_{\frac{3}{2}}^{}}$. In${\displaystyle {\text{6p}}_{1/2}^2}$, there are 2 electrons having${\displaystyle j=1/2}$ in 6p subshell while there is an electron having${\displaystyle j=3/2}$ in the same subshell in${\displaystyle {\text{6p}}_{\frac{3}{2}}^{}}$. Coupling of these two groups results in⁠${\displaystyle 1}$⁠ (coupling ofj of three electrons).
  2. ${\displaystyle {\text{4d}}_{5/2}^3{\text{4d}}_{3/2}^2{\left(\frac{9}{2},2\right)}_{11/2}}$:⁠${\displaystyle 9/2}$⁠ in () is${\displaystyle J_1}$ for 1st group${\displaystyle {\text{4d}}_{5/2}^3}$ and2 in () isJ₂ for 2nd group${\displaystyle {\text{4d}}_{3/2}^2}$. Subscript 11/2 of term symbol is finalJ of${\displaystyle \mathbf{J}={\mathbf{J}}_1+{\mathbf{J}}_2}$.

• Coupling scheme:${\displaystyle \mathbf{K}={\mathbf{J}}_1+{\mathbf{L}}_2}$ and${\displaystyle \mathbf{J}=\mathbf{K}+{\mathbf{S}}_2}$.
• Electronic configuration + Term symbol:${\displaystyle n_1{{\ell }_1}^{N_1}\left({\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}_1\right)n_2{{\ell }_2}^{N_2}\left({\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}_2\right){}^{\left(2S_2+1\right)}{\left[K\right]}_J}$. For⁠${\displaystyle K>S_2,(2S_2+1)}$⁠ is equal to multiplicity, a number of possible values inJ (final total angular momentum quantum number) from givenS₂ andK. For⁠${\displaystyle S_2>K}$⁠, multiplicity is⁠${\displaystyle (2K+1)}$⁠ but⁠${\displaystyle (2S_2+1)}$⁠ is still written in the term symbol.
• Example:
  1. 3p⁵(²P^o
     _1/2)5g ²[9/2]^o
     ₅:${\displaystyle J_1=\frac{1}{2},l_2=4,s_2=1/2}$.⁠${\displaystyle 9/2}$⁠ isK, which comes from coupling ofJ₁ andℓ₂. Subscript 5 in term symbol isJ which is from coupling ofK ands₂.
  2. 4f¹³(²F^o
     _7/2)5d²(¹D) [7/2]^o
     _7/2:${\displaystyle J_1=\frac{7}{2},L_2=2,S_2=0}$.⁠${\displaystyle 7/2}$⁠ isK, which comes from coupling ofJ₁ andL₂. Subscript⁠${\displaystyle 7/2}$⁠ in the term symbol isJ which is from coupling ofK andS₂.

• Coupling scheme:${\displaystyle \mathbf{K}\ell =\mathbf{L}+{\mathbf{S}}_{\mathbf{1}}}$,${\displaystyle \mathbf{J}=\mathbf{K}+{\mathbf{S}}_{\mathbf{2}}}$.
• Electronic configuration + Term symbol:${\displaystyle n_1{{\ell }_1}^{N_1}\left({\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}_1\right)n_2{{\ell }_2}^{N_2}\left({\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}_2\right)L{}^{\left(2S_2+1\right)}{\left[K\right]}_J}$. For⁠${\displaystyle K>S_2,(2S_2+1)}$⁠ is equal to multiplicity, a number of possible values inJ (final total angular momentum quantum number) from givenS₂ andK. For⁠${\displaystyle S_2>K}$⁠, multiplicity is⁠${\displaystyle (2K+1)}$⁠ but⁠${\displaystyle (2S_2+1)}$⁠ is still written in the term symbol.
• Example:
  1. 3d⁷(⁴P)4s4p(³P^o) D^o 3[5/2]^o
     _7/2:${\displaystyle L_1=1,L_2=1,S_1=\frac{3}{2},S_2=1}$.${\displaystyle L=2,K=5/2,J=7/2}$.

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 on[1].

These are notations for describing states of singly excited atoms, especiallynoble gas atoms. Racah notation is basically a combination ofLS or Russell–Saunders coupling andJ₁L₂ coupling.LS coupling is for a parent ion andJ₁L₂ 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.^[8]

In Racah notation, states of excited atoms are denoted as${\displaystyle \left({}^{\left(2S_1+1\right)}{L_1}_{J_1}\right)n\ell {\left[K\right]}_J^o}$. Quantities with a subscript 1 are for the parent ion,n andℓ are principal and orbital quantum numbers for the excited electron,K andJ are quantum numbers for${\displaystyle \mathbf{K}={\mathbf{J}}_1+\mathit{\ell }}$ and${\displaystyle \mathbf{J}=\mathbf{K}+\mathbf{s}}$ where${\displaystyle \mathit{\ell }}$ and${\displaystyle \mathbf{s}}$ 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 areNp⁵nℓ whereN = 2, 3, 4, 5, 6 for Ne, Ar, Kr, Xe, Rn, respectively in order. Since the parent ion can only be²P_1/2 or²P_3/2, the notation can be shortened to${\displaystyle n\ell {\left[K\right]}_J^o}$ or${\displaystyle n{\ell }^{\prime }{\left[K\right]}_J^o}$, wherenℓ means the parent ion is in²P_3/2 whilenℓ′ is for the parent ion in²P_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 asn′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 writingn′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 ofℓ =n′ − 1,n′ − 2, ... , 0 (like a relation betweenn andℓ) 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 givenn′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 givenn′ℓ. An example of Paschen notation is below.

• Quantum number
  • Principal quantum number
  • Azimuthal quantum number
  • Spin quantum number
  • Magnetic quantum number
• Angular quantum numbers
• Angular momentum coupling
• Molecular term symbol

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