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Term symbol (Read 14948 times)
Gerrit-Jan Linker
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Term symbol
02.04.09 at 09:04:07
 
Term symbol
 
The term symbol of an atom is an abbreviated description of the angular momentum quantum numbers in a multi-electron atom.  
 
It is related with the energy level of a given electron configuration.  
 
LS coupling is assumed.  
 
The ground state term symbol for atoms is predicted by Hund's rules.
See: http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1238654610/0#0
Term symbols to molecules are constructued in a similar way but we must pay attention to the orbital angular momentum about the internuclear axis.
 
The term symbol has the form 2S+1LJ
where:

  • S is the total spin
  • 2S+1 is the spin multiplicity, the maximum number of J for a given (L,S) combination.  
    Multiplicity=1 is called a singlet. Multiplicity=2 is called a doublet. Multiplicity=3 is called a triplet.
  • L is the total orbital quantum number in spectroscopic notation. L = 0,1,2,... are S,P,D,...
    For a molecule L is written as a capital Greek letter: 0=∑, 1=∏, ...
  • J is the total angular momentum.

 
For a given electronic configuration there is the following related terminology:
  • a term: the combination of S and L
  • a level: the combination of S, L and J
  • a state: the combination of S, L, J and mj

 
In systems that are non-spherical, states are labeled using symmetry species of the correct point group. When the electronic configuration is e.g. t1g1, the term symbol is 2T1g. For non-degenerate states we use the labels A and B, for doubly degenerate states E, triply degenerate states T, etc.
 
Sources:
http://en.wikipedia.org/wiki/Term_symbol
http://classes.uleth.ca/200403/chem3820a/3820%20lecture%20chapter_6_part2_2004.p df
 
See also:
Hund's rules
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1238654610/0#0
Spin-orbit coupling  
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1204190386
Symmetry properties of many electron state functions  
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1297103707/0#0
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« Last Edit: 12.11.13 at 10:47:38 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Derivation of the term symbols
Reply #1 - 02.04.09 at 09:20:00
 
Derivation of the term symbols in atoms
 
Examples of the derivation of the term symbols in atoms.
 
Helium He: 1s2

  • The 1s subshell is full.
  • S=0 (one up plus one down)
  • Multiplicity, 2S+1 = 1, a singlet
  • L=0 (ml=0 for a s subshell). L=0 is S in spectroscopic notation.
  • J=0
  • Term symbol: 1S0

 
Fluorine F: 1s22s22p5

  • The full sub-shell's are ignored (1s22s2)
  • Consider only 2p5: five electrons in a p sub shell (l=1)
  • There are 3 p orbitals: mj=1,0,-1
    In total they can contain 2(2l+1)= 6 electrons.
  • Place the electrons in the orbitals (for the ground state according to Hund's rules).
    mj= 1: ↑↓
    mj= 0: ↑↓
    mj=-1: ↑
  • S, the total spin quantum number, the sum of all mj values of the electrons (↑=1/2 and ↓=-1/2):
    S= 3x↑ + 2x↓ = 3/2 - 2/2 = 1/2
  • The multiplicity, 2S+1 = 2, a doublet
  • L, the total orbital quantum number, the sum of all ml values of the electrons:
    L= 2x1 + 2x0 + 1x(-1) = 1, 1=P in spectroscopic notation
  • Since the 2p subshell in fluorine is more than half filled J=L+S=1+(1/2)=3/2
  • The term symbol is therefore: 2P3/2
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« Last Edit: 02.04.11 at 21:18:33 by Gerrit-Jan Linker »  

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Derivation of the term symbols in molecules
Reply #2 - 02.04.11 at 21:19:19
 
Derivation of the term symbols in molecules
 
Examples of the derivation of the term symbols in molecules.
 
H2+: 1sσ1

  • A single electron in a σ orbital has zero orbital angular momentum: the orbital is cylindrically symmetrical and has no angular nodes.
  • Since the orbital angular momentum = 0 the term symbol is Σ
  • The multiplicity is 2S+1 = 2(1/2)+1 = 2. A doublet. The term symbol becomes 2Σ
  • The parity of the occupied σ is gerade (g) as it is invariant under inversion symmetry. The term symbol becomes 2Σg.

 
O2: (1sσ)2(1sσ*)2(2sσ)2(2sσ*)2(2pπ)4(2pσ)0(2pπ*)0

  • Valence electrons: 2s22px12py1  
  • A π electron has one unit of orbital angular momentum about the intermolecular axis.
  • If the two π electrons orbit in the same direction the term symbol becomes ∆ and ir they orbit in opposite directions then the term symbol becomes Σ. The Σ state of O2 is lower in energy.
  • There are two unpaired electrons in the 2pxπ*2 and 2pyπ*2 MO's. O2 is a triplet. The term symbol becomes. 3Σ
  • The overall parity is (closed shell) x g x g = g. The term symbol becomes 3Σg
  • A final subscript is added to reflect the behaviour of the molecular wavefunction under reflection in a plane containing the nuclei.
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« Last Edit: 22.07.11 at 21:34:56 by Gerrit-Jan Linker »  

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