The term symbol for the ground dinitrogen cation radical (\(N^+_2\)) is

Concept:

• The term symbol describes the (total) angular momentum quantum numbers in a multi-electron atom.
• Each energy level of an atom with a given electron configuration is described by not only the electron configuration but also its own term symbol, as the energy level also depends on the total angular momentum including spin.
• The molecular term symbol is represented by

\(^{2S + 1}\Lambda _{(g/u)}^{( + / - )}\)

where S is the total spin quantum number, and it is given by

\({n \over 2}\), where n is number of unpaired electrons. 

\(\Lambda \) is the projection of the orbital angular momentum along the internuclear axis,

(g/u) represents the symmetry or parity with respect to inversion (i) through a center of symmetry,
(+/-) is the reflection symmetry along an arbitrary plane containing the internuclear axis.

Explanation:

• N2+ contains a total of 13 electrons and the electronic configuration of N₂⁺ is \(1\sigma _g^21{\sigma ^ * }_u^22\sigma _g^22{\sigma ^ * }_u^21\pi _u^43\sigma _g^1\)
• The number of unpaired electron in N2+ is 1, and it is present in a \(\sigma \) orbital.  
• The total spin quantum number (S) for N2+ is \({1 \over 2}\), and the value of 2S+1 is,

\(2 \times {1 \over 2} + 1\)

 \( = 2\)

• The value of \(\Lambda \) for a \(\sigma \) orbital is 0, and it is coded by \({\sum\nolimits }\)
• As the symmetry of the  \(\sigma \) orbital is gerade, it is represented as g.
• The  \(\sigma \) orbital is symmetrical with respect to the reflection symmetry, thus it will be represented by a +ve sign.
• Thus, the molecular term symbol is \(^2\sum _g^ + \)    

Conclusion:

• Hence, the term symbol for the ground dinitrogen cation radical (\(N^+_2\)) is \(^2\sum _g^ + \)