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The L-S coupling is also known as ‘Russel-Saunders’ coupling after the two astronomers who first used it in studying atomic spectra emitted by stars. In atoms which obey this coupling we introduce the various perturbations in the order :

(a) spin-spin correlation
(b) orbital-orbital interaction
(c)spin-orbit interaction
a) Spin-Spin Correlation :
 
As a result of spin spin corelation effect, the indivdual spin angular momentum vector of the optical electrons are strongly coupled with one another to form a resultant spin angular momentum vector of magnitude √s(s+1).h/2π which is constant of motion.
The quantum number takes values from  s = (s1- s2) , (s1 -s2)+1 ,………
 
 
Let us first combine s1 and s2 as s’
=>              s’ = ( 0, 1 )
Now, let us combine these values with s3:
If we couple s3 = 1/2 with s’ = 0 ,we get 
=>          s’ = 0 ,s3 = 1/2
i.e. it behave like a single electron system.
Therefore,  S = s3 = 1/2
If we couple  s’ = 1 and s3 = 1/2 
Hence,  S = 1/2 ,1/2 , 3/2
 
Hence, For a three electron system there are two doublets. 
Multiplicity for ( s = 1/2 ) = 2s + 1
                                            = 2(1/2) + 1
                                            = 2 (Doublets) 
For s = 3/2 
         Multiplicity = 2s + 1
                             = 2(3/2) + 1 
                             = 4
Hence, we get a quadrat.
b) Orbital-Orbital Interaction :-
 
As a result of electrostatic interaction, the individual orbital angular momentum vectors of optical electrons are strongly coupled with one another to form a resultant orbital angular momentum vector of magnitude √l(l+1).h/2π
 
has following values :
For 2p, 3p, 4d electrons. Let us first consider two p-electrons ( 2p, 3p)
This is the condition for orbital-orbital interaction.
c) Spin-Orbital Interaction :-
 
As a result of Spin-orbital interaction,  the resultant orbital angular momentum vector and the resultant spin angular momentum vector S  are coupled each other to form total angular momentum vector J
=>          J = L + S
where,  is quantised having value
   √ j (j+1).h/2π
where, is inner quantum number.