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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 explained by the Bohr quantum theory, 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.[1] 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.

Contents

LS coupling and symbolEdit

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 or spin–orbit coupling. Atomic states are then well described by term symbols of the form

 

where

S is the total spin quantum number. 2S + 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 2L + 1).[2] This is easily proven by using Jmax = L + S and Jmin = |L - S|, so that the number of possible J with given L and S is simply Jmax - Jmin + 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:
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 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 1s22s22p2 3P2. The superscript 3 indicates that the spin state is a triplet, and therefore S = 1 (2S + 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 1s22s22p2 3P0.[1]

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 statesEdit

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 (2S+1)(2L+1);
  • A combination of S, L and J is called a level. A given level has a statistical weight of (2J+1), which is the number of possible microstates associated with this level in the corresponding term;
  • A combination of S, L, J and MJ determines a single state.

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 (mS and mL 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   in this representation span function space of dimension  , as   and  . In the coupled representation where total angular momentum (spin + orbital) is treated, the associated microstates (or eigenstates) 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 (2×1+1)(2×2+1) = 15 different microstates (= eigenstates in the uncoupled representation) corresponding to the 3D term, of which (2×3+1) = 7 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 parityEdit

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:

2Po
½
has odd parity, but 3P0 has even parity.

Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for gerade (German for "even") or ungerade ("odd"):

2P½,u for odd parity, and 3P0,g for even.

Ground state term symbolEdit

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.
    • If all shells and subshells are full then the term symbol is 1S0.
  2. 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 ms to them (i.e. +½). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning ms = −½ to them.
  3. 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 (so if there are two electrons in the same orbital, add twice that orbital's  ).
  4. Calculate J as
    • if less than half of the subshell is occupied, take the minimum value J = |LS|;
    • if more than half-filled, take the maximum value J = L + S;
    • if the subshell is half-filled, then L will be 0, so J = S.

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   electrons. The first three electrons can take ms = ½ (↑) but the Pauli exclusion principle forces the next two to have ms = −½ (↓) because they go to already occupied orbitals.
     
    +1 0 −1
      ↑↓ ↑↓
  3. S = ½ + ½ + ½ − ½ − ½ = ½; and L = 1 + 0 − 1 + 1 + 0 = 1, which is "P" in spectroscopic notation.
  4. As fluorine 2p subshell is more than half filled, J = L + S = 32. Its ground state term symbol is then 2S+1LJ = 2P32.

Atomic term symbols of the chemical elementsEdit

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 2S12, the alkali earth metals are 1S0, the boron column elements are 2P12, the carbon column elements are 3P0, the pnictogens are 4S32, the chalcogens are 3P2, the halogens are 2P32, 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[3] are given in the collapsed table below (with citations for the heaviest elements here). 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.[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
2S12

He
1S0
2 Li
2S12
Be
1S0

B
2P12
C
3P0
N
4S32
O
3P2
F
2P32
Ne
1S0
3 Na
2S12
Mg
1S0

Al
2P12
Si
3P0
P
4S32
S
3P2
Cl
2P32
Ar
1S0
4 K
2S12
Ca
1S0
Sc
2D3/2
Ti
3F2
V
4F3/2
Cr
7S3
Mn
6S5/2
Fe
5D4
Co
4F9/2
Ni
3F4
Cu
2S12
Zn
1S0
Ga
2P12
Ge
3P0
As
4S32
Se
3P2
Br
2P32
Kr
1S0
5 Rb
2S12
Sr
1S0
Y
2D3/2
Zr
3F2
Nb
6D1/2
Mo
7S3
Tc
6S5/2
Ru
5F5
Rh
4F9/2
Pd
1S0
Ag
2S12
Cd
1S0
In
2P12
Sn
3P0
Sb
4S32
Te
3P2
I
2P32
Xe
1S0
6 Cs
2S12
Ba
1S0
La
2D3/2
  Hf
3F2
Ta
4F3/2
W
5D0
Re
6S5/2
Os
5D4
Ir
4F9/2
Pt
3D3
Au
2S12
Hg
1S0
Tl
2P12
Pb
3P0
Bi
4S32
Po
3P2
At
2P32
Rn
1S0
7 Fr
2S12
Ra
1S0
Ac
2D3/2
  Rf
3F2
Db
4F3/2?
Sg
5D0?
Bh
6S5/2?
Hs
 
Mt
 
Ds
 
Rg
 
Cn
 
Nh
 
Fl
 
Mc
 
Lv
 
Ts
 
Og
 

  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
Lu
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
Lr
2P1/2?

Term symbols for an electron configurationEdit

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

  • First, the total number of possible microstates N 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  , t is the maximum allowed number of electrons,  . If there are e 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 ( ), so there are
     
    different microstates.
  • Second, all possible microstates are drawn. ML and MS for each microstate are calculated, with   where mi is either   or   for the i-th electron, and M represents the resulting ML or MS respectively:
         
      +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 microstates for each MLMS 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 to ML ranging from −2 to +2 (so L = 2), with a single value for MS (implying S = 0). 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 ML and MS both ranging from −1 to +1 (and so S = L = 1, a 3P term). The remaining table is a 1×1 table, with L = S = 0, i.e., a 1S term.
    S = 0, L = 2, J = 2
    1D2
      Ms
      0
      +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
      +1 1 1 1
    0 1 1 1
    −1 1 1 1
    S=0, L=0, J=0
    1S0
      Ms
      0
      0 1
  • 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 Hund'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.

Case of three equivalent electronsEdit

  • For three equivalent electrons (with the same orbital quantum number  ), there is also a general formula (denoted by   below) to count the number of any allowed terms with total orbital quantum number "L" and total spin quantum number "S".
 
where the floor function   denotes the greatest integer not exceeding x.
The detailed proof can be found in Renjun Xu's original paper.[4]
  • For a general electronic configuration of  , namely k equivalent electrons occupying one subshell, the general treatment and computer code can also be found in this paper.[4]

Alternative method using group theoryEdit

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:

Γ(1) × Γ(1) = Γ(0) + [Γ(1)] + Γ(2),

which, using the familiar labels Γ(0) = S, Γ(1) = P and Γ(2) = D, can be written as

P × P = S + [P] + D.

The square brackets enclose the anti-symmetric square. Hence the 2p2 configuration has components with the following symmetries:

S + D (from the symmetric square and hence having symmetric spatial wavefunctions);
P (from the anti-symmetric square and hence having an anti-symmetric spatial wavefunction).

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:

1S + 1D (spatially symmetric, spin anti-symmetric)
3P (spatially anti-symmetric, spin symmetric).

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

Γ(j) × Γ(j) = Γ(2j) + Γ(2j-2) + ... + Γ(0) + [Γ(2j-1) + ... + Γ(1)].

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

Γ(j) × Γ(k) = Γ(j+k) + Γ(j+k−1) + ... + Γ(|jk|)

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.[5]

Summary of various coupling schemes and corresponding term symbolsEdit

Basic concepts for all coupling schemes:

  •  : individual orbital angular momentum vector for an electron,  : individual spin vector for an electron,  : individual total angular momentum vector for an electron,  .
  •  : Total orbital angular momentum vector for all electrons in an atom ( ).
  •  : total spin vector for all electrons ( ).
  •  : total angular momentum vector for all electrons. The way the angular momenta are combined to form   depends on the coupling scheme:   for LS coupling,   for jj coupling, etc.
  • A quantum number corresponding to the magnitude of a vector is a letter without an arrow (ex: l is the orbital angular momentum quantum number for   and  )
  • 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  ,  and   vectors for each group. "Coupling" of terms A and B to form a new term C means finding quantum numbers for new vectors  ,   and  . 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  where X can be s, l, j, S, L, J or any other angular momentum-magnitude-related quantum number.

LS coupling (Russell-Saunders coupling)Edit

  • Coupling scheme:   and   are calculated first then   is obtained. In practical point of view, it means L, S and J are obtained by using addition rule of angular momentums with given electronics groups that are to be coupled.
  • Electronic configuration + Term symbol:  .   is a Term which is from coupling of electrons in  group.   are principle quantum number, orbital quantum number and  means there are N (equivalent) electrons in   subshell. For  ,   is equal to multiplicity, a number of possible values in J (final total angular momentum quantum number) from given S and L. For  , 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  of group is odd (P = -1).
  • Example:
    1.  :   is Level of 3d7 group in which are equivalent 7 electrons are in 3d subshell.
    2.  :[6] Terms are assigned for each group (with different principal quantum number n) and rightmost Level   is from coupling of Terms of these groups so   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 3Po refer to seven and two electrons respectively so capital letters are used.
    3.  : There is a space between 5d and  . It means   and   are coupled to get  . Final level   is from coupling of   and  .
    4.  : There is only one Term   which is isolated in the left of the leftmost space. It means   is coupled lastly;   and   are coupled to get   then   and   are coupled to get final Term  .

jj CouplingEdit

  • Coupling scheme:  .
  • Electronic configuration + Term symbol:  
  • Example:
    1.  : There are two groups. One is   and the other is  . In  , there are 2 electrons having   in   subshell while there is an electron having   in the same subshell in  .Coupling of these two groups results in   (coupling of j of three electrons).
    2.  :   in () is   for 1st group   and 2 in () is J2 for 2nd group  . Subscript 11/2 of Term symbol is final J of  .

J1L2 couplingEdit

  • Coupling scheme:   and  .
  • Electronic configuration + Term symbol:  . For   is equal to multiplicity, a number of possible values in J (final total angular momentum quantum number) from given S2 and K. For  , multiplicity is   but   is still written in the Term symbol.
  • Example:
    1.  :  .   is K, which comes from coupling of J1 and l2. Subscript 5 in Term symbol is J which is from coupling of K and s2.
    2.  :  .   is K, which comes from coupling of J1 and L2. Subscript   in Term symbol is J which is from coupling of K and S2.

LS1 couplingEdit

  • Coupling scheme: ,  .
  • Electronic configuration + Term symbol:  . For   is equal to multiplicity, a number of possible values in J (final total angular momentum quantum number) from given S2 and K. For  , multiplicity is   but   is still written in the Term symbol.
  • Example:
    1.  :  .  .

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 [1].

Racah notation and Paschen notationEdit

These are notations for describing states of singly excited atoms, especially noble gas atoms. Racah notation is basically a combination of LS or Russell-Saunder 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.[7]

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 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 (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
1s22s22p6 Ground state [Ne]3s23p6 Ground state
1s22s22p53s1 1s [Ne]3s23p54s1 1s
1s22s22p53p1 2p [Ne]3s23p54p1 2p
1s22s22p54s1 2s [Ne]3s23p55s1 2s
1s22s22p54p1 3p [Ne]3s23p55p1 3p
1s22s22p55s1 3s [Ne]3s23p56s1 3s

See alsoEdit

NotesEdit

  1. ^ There is no official convention for naming angular momentum values greater than 20 (symbol Z). Many authors begin using Greek letters at this point (  ...). The occasions for which such notation is necessary are few and far between, however.

ReferencesEdit

  1. ^ a b c d NIST Atomic Spectrum Database To read neutral carbon atom levels for example, type "C I" in the Spectrum box and click on Retrieve data.
  2. ^ Levine, Ira N., Quantum Chemistry (4th ed., Prentice-Hall 1991), ISBN 0-205-12770-3
  3. ^ "NIST Atomic Spectra Database Ionization Energies Form". NIST Physical Measurement Laboratory. National Institute of Standards and Technology (NIST). October 2018. Retrieved 28 January 2019. This form provides access to NIST critically evaluated data on ground states and ionization energies of atoms and atomic ions.
  4. ^ a b Xu, Renjun; Zhenwen, Dai (2006). "Alternative mathematical technique to determine LS spectral terms". Journal of Physics B: Atomic, Molecular and Optical Physics. 39 (16): 3221–3239. arXiv:physics/0510267. Bibcode:2006JPhB...39.3221X. doi:10.1088/0953-4075/39/16/007.
  5. ^ McDaniel, Darl H. (1977). "Spin factoring as an aid in the determination of spectroscopic terms". Journal of Chemical Education. 54 (3): 147. Bibcode:1977JChEd..54..147M. doi:10.1021/ed054p147.
  6. ^ "Atomic Spectroscopy - Different Coupling Scheme 9. Notations for Different Coupling Schemes". NIST Physical Measurement Laboratory. National Institute of Standards and Technology (NIST). 1 November 2017. Retrieved 31 January 2019.
  7. ^ "APPENDIX 1 - Coupling Schemes and Notation" (PDF). University of Toronto: Advanced Physics Laboratory - Course Homepage. Retrieved 5 Nov 2017.