The possible terms arising from a p¹d¹ configuration are

Correct option is D

​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 equation 

$$\text{Orbital angular momentum} = \sqrt{l(l+1)} \frac{h}{2\pi}$$

​The energy and the orbital angular momentum of a multielectron species are determined by a new quantum number, L, which is related to the values of l for the individual electrons. Since the orbital angular momentum has magnitude and (2l+1) spatial orientations with respect to the z axis (i.e. the number of values of m_l), vectorial summation of individual l values is necessary. The value of m_l for any electron denotes the component of its orbital angular momentum, m_l(h/2π), along the z axis. Summation of m_l​ values for individual electrons in a multi-electron system therefore gives the resultant orbital magnetic quantum number M_L:

$$M_L = \sum m_l$$

​Just as m_l may have the (2l+1) values l, (l-1) ... 0 ...-(l-1), -l, so M_L can have (2L+1) values L, (L-1) ... 0 ...-(L-1), -L. The allowed values of L can be determined from l for the individual electrons in the multi-electron system. For two electrons with values of l₁ and l₂:

$$L = (l_1 + l_2), (l_1 + l_2 - 1), \dots, |l_1 - l_2|$$

​The modulus sign around the last term indicates that 
allowed values of L are 4, 3, 2, 1 or 0. For systems with three or more electrons, the electron–electron coupling must be considered in sequential steps: couple l₁ and l₂ as above to give a resultant L, and then couple L with l₃, and so on. Energy states for which L=0, 1, 2, 3, 4... are known as S,P, D, F, G... terms, respectively. These are analogous to the s, p, d, f, g... labels used to denote atomic orbitals with l=0, 1, 2, 3, 4... in the 1-electron case. The resultant orbital angular momentum for a multi-electron system is given as$$|l_1 - l_2|$$ may only be zero or a positive value. As an example, consider a p² configuration. Each electron has l=1, and so the allowed values of L are 2, 1 or 0. Similarly, for a d² configuration, each electron has l=2, and so the

$$\text{Orbital angular momentum} = \sqrt{L(L+1)} \frac{h}{2\pi}$$

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. We now need to define the quantum numbers S and M_S for multielectron species. The spin angular momentum for a multielectron species is given by equation, where S is the total spin quantum number.

$$\text{Spin angular momentum} = \sqrt{S(S+1)} \frac{h}{2\pi}$$

​The quantum number M_S is obtained by algebraic summation of the m_s values for individual electrons: $$M_S = \sum m_s$$

​For a system with n electrons, each having s=1/2, possible values of S fall into two series depending on the total number of electrons: S=1/2,3/2,5/2....(for an odd number of electrons) and S=0,1,2...(for an even number of electrons)

S cannot take negative values. The case of S=1/2 clearly corresponds to a 1-electron system, for which values of m_s are +1/2 or -1/2, and values of M_S are also +1/2 or -1/2. For each value of S, there are (2S+1) values of M_S : Allowed values of M_S : S, (S-1), ... -(S-1), -S
Thus, for S=0,M_S=0, for S=1,M_S=1, 0 or -1, and for S=3/2, M_S=3/2, 1/2, -1/2 or -3/2.

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