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. 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.
Contents
- LS coupling and symbol
- Terms levels and states
- Term symbol parity
- Ground state term symbol
- Term symbols for an electron configuration
- Summary of various coupling schemes and corresponding term symbols
- LS coupling Russell Saunders coupling
- jj Coupling
- J1L2 coupling
- LS1 coupling
- Racah notation and Paschen notation
- References
LS coupling and symbol
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). This is easily proved 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: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, 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.
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 product (2S+1)(2L+1) as a number of possible microstates
As an example, for S = 1, L = 2, 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 (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.
Term symbol parity
The parity of a term symbol is calculated as
where li is the orbital quantum number for each electron.
When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted:
2Po½ 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 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 ml 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.
- The overall S is calculated by adding the ms values for each electron. That is the same as multiplying ½ times the number of unpaired electrons. The overall L is calculated by adding the ml values for each electron (so if there are two electrons in the same orbital, add twice that orbital's ml).
- Calculate J as
- if less than half of the subshell is occupied, take the minimum value J = |L − S|;
- 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 (l = 1).
2. There are three orbitals (ml = 1, 0, −1) that can hold up to 2(2l + 1) = 6 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.
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 = 3⁄2. Its ground state term symbol is then 2S+1LJ = 2P 3⁄2.
Term symbols for an electron configuration
The process to calculate all possible term symbols for a given electron configuration is a bit longer.
Case of three equivalent electrons
where the floor function
The detailed proof could be found in Renjun Xu's original paper.
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:
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(|j−k|)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:
LS coupling (Russell-Saunders coupling)
-
3 d 7 4 F 7 / 2 4 F 7 / 2 -
3 d 7 ( 4 F ) 4 s 4 p ( 3 p 0 ) 6 F 9 / 2 0 6 F 9 / 2 o 6 F 9 / 2 o -
4 f 7 ( 8 S 0 ) 5 d ( 7 D o ) 6 p 8 F 13 / 2 0 ( 7 D o ) . In means( 8 S 0 ) and 5d are coupled to get( 7 D o ) . Final level8 F 13 / 2 o ( 7 D o ) and 6p. -
4 f ( 2 F 0 ) 5 d 2 ( 1 G ) 6 s ( 2 G ) 1 P 1 0 ( 2 F o ) which is isolated in the left of the leftmost space. It means( 2 F o ) is coupled lastly;( 1 G ) and 6s are coupled to get( 2 G ) then( 2 G ) and( 2 F o ) are coupled to get final Term1 P 1 o
jj Coupling
-
( 6 p 1 2 2 6 p 3 2 ) o 3 / 2 6 p 1 / 2 2 6 p 3 2 6 p 1 / 2 2 j = 1 / 2 in 6p subshell while there is an electron havingj = 3 / 2 in the same subshell in6 p 3 2 -
4 d 5 / 2 3 4 d 3 / 2 2 ( 9 2 , 2 ) 11 / 2 J 1 4 d 5 / 2 3 4 d 3 / 2 2 J → = J 1 → + J 2 →
J1L2 coupling
-
3 p 5 ( 2 P 1 2 o ) 5 g 2 [ 9 / 2 ] 5 o J 1 = 1 2 , l 2 = 4 , s 2 = 1 / 2 . 9/2 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. -
4 f 13 ( 2 F 7 2 o ) 5 d 2 ( 1 D ) 1 [ 7 / 2 ] 7 / 2 o J 1 = 7 2 , L 2 = 2 , S 2 = 0 . 7/2 is K, which comes from coupling of J1 and L2. Subscript 7/2 in Term symbol is J which is from coupling of K and S2.
LS1 coupling
-
3 d 7 ( 4 P ) 4 s 4 p ( 3 P o ) D o 3 [ 5 / 2 ] 7 / 2 o L 1 = 1 , L 2 = 1 , S 1 = 3 2 , S 2 = 1 .L = 2 , K = 5 / 2 , J = 7 / 2 .
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 notation
They’re notations for describing states of singly excited atoms, especially rare 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 an 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
Paschen notation is somewhat weird notation; it is an old notation made to attempt to fit an emission spectrum of Ne to a hydrogen-like theory. It has rather a simple structure to indicate energy levels of 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), and 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 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. Example of Paschen notation is below.
See also
Notes
References
Summary of various coupling schemes and corresponding term symbols
Basic concepts for all coupling schemes:
LS coupling (Russell-Saunders coupling)
-
3 d 7 4 F 7 / 2 4 F 7 / 2 -
3 d 7 ( 4 F ) 4 s 4 p ( 3 p 0 ) 6 F 9 / 2 0 6 F 9 / 2 o 6 F 9 / 2 o -
4 f 7 ( 8 S 0 ) 5 d ( 7 D o ) 6 p 8 F 13 / 2 0 ( 7 D o ) . In means( 8 S 0 ) and 5d are coupled to get( 7 D o ) . Final level8 F 13 / 2 o ( 7 D o ) and 6p. -
4 f ( 2 F 0 ) 5 d 2 ( 1 G ) 6 s ( 2 G ) 1 P 1 0 ( 2 F o ) which is isolated in the left of the leftmost space. It means( 2 F o ) is coupled lastly;( 1 G ) and 6s are coupled to get( 2 G ) then( 2 G ) and( 2 F o ) are coupled to get final Term1 P 1 o
jj Coupling
-
( 6 p 1 2 2 6 p 3 2 ) o 3 / 2 6 p 1 / 2 2 6 p 3 2 6 p 1 / 2 2 j = 1 / 2 in 6p subshell while there is an electron havingj = 3 / 2 in the same subshell in6 p 3 2 -
4 d 5 / 2 3 4 d 3 / 2 2 ( 9 2 , 2 ) 11 / 2 J 1 4 d 5 / 2 3 4 d 3 / 2 2 J → = J 1 → + J 2 →
J1L2 coupling
-
3 p 5 ( 2 P 1 2 o ) 5 g 2 [ 9 / 2 ] 5 o J 1 = 1 2 , l 2 = 4 , s 2 = 1 / 2 . 9/2 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. -
4 f 13 ( 2 F 7 2 o ) 5 d 2 ( 1 D ) 1 [ 7 / 2 ] 7 / 2 o J 1 = 7 2 , L 2 = 2 , S 2 = 0 . 7/2 is K, which comes from coupling of J1 and L2. Subscript 7/2 in Term symbol is J which is from coupling of K and S2.
LS1 coupling
-
3 d 7 ( 4 P ) 4 s 4 p ( 3 P o ) D o 3 [ 5 / 2 ] 7 / 2 o L 1 = 1 , L 2 = 1 , S 1 = 3 2 , S 2 = 1 .L = 2 , K = 5 / 2 , J = 7 / 2 .
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 notation
They’re notations for describing states of singly excited atoms, especially rare 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 an 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
Paschen notation is somewhat weird notation; it is an old notation made to attempt to fit an emission spectrum of Ne to a hydrogen-like theory. It has rather a simple structure to indicate energy levels of 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), and 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 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. Example of Paschen notation is below.