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Term symbol
In quantum mechanics, the term symbol 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 is predicted by Hund's rules.
The term symbol has the form
where S is the total spin quantum number. 2 S +1 is the spin multiplicity : the maximum number of different possible states of J for a given ( L , S ) combination. L is the total orbital quantum number in spectroscopic notation. The symbols for L = 0,1,2,3,4,5 are S,P,D,F,G,H respectively. J is the total angular momentum quantum number.
When used to describe electron states in an atom, the term symbol usually follows the electron configuration, e.g., in the case of carbon, the ground state is 1s^2 2s^2 2p^2 3 P 0. The 3 indicates that 2S+1=3 and so S=1, the P is spectroscopic notation for L=1, and 0 is the value of J.
The term symbol is also used to describe compound systems such as mesons or atomic nuclei, or even molecules (see molecular term symbol). In that last case, Greek letters are used to designate the (molecular) orbital angular momenta.
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) of (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 L , S , J and MJ determines a single state.
As an example, for S = 1, L = 2, there are (2ร1+1)(2ร2+1) = 15 different microstates corresponding to the 3 D term, of which (2ร3+1) = 7 belong to the 3 D 3 (J=3) level. The sum of (2 J +1) for all levels in the same term equals (2 S +1)(2 L +1). 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. In fact, only electrons in odd orbitals contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd l such as in p, f,...) will make an odd term symbol, while an even number of electrons in odd orbitals will make an even term symbol, irrespective of the number of electrons in even orbitals.
When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted:
has odd parity, but has even parity.
Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for gerade (German for 'even') or ungerade ('odd'):
for odd parity and for even.
Ground state term symbol
It is relatively easy to calculate the term symbol for the ground state of an atom. It corresponds with a state with maximal 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. o If all shells and subshells are full then the term symbol is.
- Distribute the electrons in the available orbitals, following the Pauli exclusion principle. First, we fill the orbitals with highest ml value with one electron each, and assign a maximal ms to them (i.e. +1/2). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning ms = โ1/2 to them.
- The overall S is calculated by adding the ms values for each electron. That is the same as multiplying ยฝ 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, then we add twice that orbital's ml ).
- Calculate J as: o if less than half of the subshell is occupied, take the minimum value J = | L โ S | ; o if more than half-filled, take the maximum value J = L + S ; o 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: 1s^2 2s^2 2p^5.
- Second, draw all possible microstates. Calculate ML and MS for each microstate, with
where mi is either ml or ms for the i -th electron, and M represents the resulting ML or MS respectively:
ml +1 0 โ 1 ML MS โ โ 1 1 all up โ โ 0 1 โ โ โ 1 1 โ โ 1 โ 1 all down โ โ 0 โ 1 โ โ โ 1 โ 1 โโ 2 0 โ โ 1 0 โ โ 0 0 โ โ 1 0 โโ 0 0 โ โ โ 1 0 โ โ 0 0 โ โ โ 1 0
one up
one down
- Third, count the number of microstates for each ML โ MS possible combination
MS
โ 1^1 2
ML
- Fourth, extract smaller tables representing each possible term. Each table will be (2 L +1)(2 S +1), and will contain "1"s as entries. The first table extracted corresponds to ML ranging from โ2 to +2 (so L = 2), with a single value for MS (implying S = 0). This corresponds to a 1 D term. The remaining table is 3ร3. Then we extract a second table, removing the entries for ML and MS both ranging from โ1 to +1 (and so S = L = 1, a 3 P term). The remaining table is a 1ร1 table, with L = S = 0, i.e., a 1 S term.
S =0, L =2, J =
1 D
2
Ms 0 +2 1 +1 1 0 1 โ 1 1
Ml
โ 2^1
S =1, L =1, J =2,1,
3 P
2 ,^
3 P
1 ,^
3 P
0
Ms +1 0 โ 1 +1 1 1 1 Ml 0 1 1 1 โ 1 1 1 1
S =0, L =0, J =
1 S
0
Ms 0 Ml 0 1
- Fifth, applying Hund's rules, deduce which is the ground state (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.
