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

**s**of magnitude**which is constant of motion.***√s(s+1).h/2π*The quantum number

**s**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*

**L**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

**L**and the resultant spin angular momentum vector**S**are coupled each other to form total angular momentum vector**J**=>

**J**=**L**+**S**where,

**J**is quantised having value

*√ j (j+1).h/2π*

*where,**j*is inner quantum number.