The correct match for column A with column B is:

​

​

Correct option is C

​The notation for a full term symbol is:

​​The energy and the orbital angular momentum of a multielectron species are determined by a quantum number, L. Energy states for which L=0, 1, 2, 3, 4... are known as S, P, D, F, G... terms, respectively.
For any system containing more than one electron, the energy of an electron with principal quantum number n depends on the value of l, and this also determines the orbital angular momentum which is given by the equation:$$\text{Orbital angular momentum} = \left( \sqrt{l(l + 1)} \right) \frac{h}{2\pi}$$

​​Planck constant is expressed in SI units, it has the exact value h=6.626×10^-34JHz^-1​

The spin quantum number, s, determines the magnitude of the spin angular momentum of an electron and has a value of 1/2. For a 1-electron species, m_s is the magnetic spin angular momentum and has a value of +1/2 or -1/2. The spin angular momentum for a multielectron species is given by the following equation, where S is the total spin quantum number. $$\text{Spin angular momentum} = \left( \sqrt{S(S+1)} \right) \frac{h}{2\pi}$$ The quantum number M_S is obtained by algebraic summation of the m_s values for individual electrons: $$M_S = \sum m_s$$

​For each value of S, there are (2S+1) values of M_S. 

​The interaction between the total angular orbital momentum, L, and the total spin angular momentum, S is defined by the total angular momentum quantum number, J.
​The following equation gives the relationship for the total angular momentum for a multi-electron species. ​

$$\text{Total angular momentum} = \left( \sqrt{J(J+1)} \right) \frac{h}{2\pi}$$

The quantum number J takes values (L+S), (L+S-1).....L-S, and these values fall into the series 0,1,2....or 1/2,3/2,5/2. The method of obtaining J from L and S is based on LS (or Russell-Saunders) coupling, that is spin-orbit coupling. The value of J for the ground state is given by (L-S) for a sub-shell that is less than half-filled, and by (L+S) for a sub-shell that is more than half-filled.

​Lu³⁺ion ([Xe]4f¹⁴) has completely filled orbitals and the f-f transition is spin forbidden (ΔS is not equal to zero). Hence, there is no transition.

The Eu³⁺ ion has a 4f⁶ electron configuration and S = 3. The term with the lowest energy is ⁷F with L = 3. Strong spin-orbit coupling results in a J = 0 ground state. As the Eu³⁺ ion has a less than half-filled electronic configuration, the value of J will be |3-3| = 0. Thus, an energy state with the term symbol ⁵D₀, is possible for the Eu³⁺ ion.

⁵D₀ → ⁷F_n, emission is observed in the Tb³⁺ ion. The Tb³⁺ ion has a 4f⁸ electronic configuration and S = 3. The term with the lowest energy is ⁷F with L = 3. Strong spin-orbit coupling results in a J = 6 ground state. As the Tb³⁺ ion has a more than half-filled electronic configuration, one possible value of J can be = L+S-2 =3+3-2 = 4. Thus, an energy state with the term symbol ⁵D₄ is possible for the Tb³⁺ ion. 

 Sm³⁺ ion has a 4f⁵ electronic configuration. At room temperature, the first, second and third excited state are found to be populated. Mixing of these states with higher J values causes the larger deviation in observed magnetic moment because of overlapping J levels.