Term Symbol for N₂⁺

To determine the term symbol for the ground state dinitrogen cation radical (\(N^+_2\)), we first need to find its electronic configuration. Dinitrogen (\(N_2\)) has 14 electrons (7 from each nitrogen atom). Its molecular orbital configuration is:

\((\sigma_{1s})^2 (\sigma^*_{1s})^2 (\sigma_{2s})^2 (\sigma^*_{2s})^2 (\pi_{2p})^4 (\sigma_{2p})^2\)

This configuration accounts for 14 electrons. The highest occupied molecular orbital (HOMO) is \(\sigma_{2p}\).

For the dinitrogen cation radical (\(N^+_2\)), we remove one electron from the HOMO of \(N_2\). So, \(N^+_2\) has 13 electrons and its configuration is:

\((\sigma_{1s})^2 (\sigma^*_{1s})^2 (\sigma_{2s})^2 (\sigma^*_{2s})^2 (\pi_{2p})^4 (\sigma_{2p})^1\)

Now, let's derive the term symbol \(^{2S+1}\Lambda_{\Omega}\) for the ground state of \(N^+_2\) based on this configuration.

Spin Multiplicity (2S+1)

The spin multiplicity is determined by the total spin angular momentum (S). The configuration \((\sigma_{2p})^1\) has one unpaired electron in the \(\sigma_{2p}\) orbital. The spin of this electron is s = 1/2. Since this is the only unpaired electron, the total spin S = 1/2.

Spin Multiplicity = \(2S + 1 = 2(1/2) + 1 = 1 + 1 = 2\)

The term symbol will have a superscript of 2, indicating a doublet state.

Orbital Angular Momentum (\(\Lambda\))

The projection of the total orbital angular momentum along the internuclear axis is denoted by \(\Lambda\). For diatomic molecules, \(\Lambda\) is the absolute value of the sum of the individual electron's \(\lambda\) values.

• For \(\sigma\) orbitals, \(\lambda = 0\).
• For \(\pi\) orbitals, \(\lambda = \pm 1\).
• For \(\delta\) orbitals, \(\lambda = \pm 2\).

Filled shells or subshells (\((\sigma)^2\), \((\pi)^4\)) have a total \(\Lambda = 0\). We only need to consider the unpaired electron in the \(\sigma_{2p}\) orbital.

For the \(\sigma_{2p}^1\) configuration, the unpaired electron is in a \(\sigma\) orbital, so its \(\lambda = 0\). The total \(\Lambda\) for the molecule is the sum of \(\lambda\) values, but since only one electron in an incomplete shell contributes, \(\Lambda = |\lambda| = |0| = 0\).

The value of \(\Lambda\) corresponds to letters:

• \(\Lambda = 0 \implies \Sigma\) state
• \(\Lambda = 1 \implies \Pi\) state
• \(\Lambda = 2 \implies \Delta\) state
• etc.

Since \(\Lambda = 0\), the state is a \(\Sigma\) state.

Parity (g/u)

The parity (gerade (g) or ungerade (u)) indicates the symmetry of the molecular orbital with respect to inversion through the center of symmetry. Gerade (g) means the wavefunction is symmetric upon inversion (\(f(-x, -y, -z) = f(x, y, z)\)), while ungerade (u) means it is antisymmetric (\(f(-x, -y, -z) = -f(x, y, z)\)). For molecular states derived from configurations, the overall parity is determined by the product of the parities of the individual electrons in the incomplete shells. The parity of an orbital is g if the atomic orbitals forming it have the same parity and combine in-phase (bonding sigma from pz, antibonding pi from p), or u if they have different parity or combine out-of-phase (antibonding sigma from pz, bonding pi from p).

The \(\sigma_{2p}\) bonding molecular orbital is formed primarily from the overlap of \(2p_z\) atomic orbitals (which are ungerade) along the internuclear axis. The bonding \(\sigma_{2p}\) orbital is gerade (g). The unpaired electron is in the \(\sigma_{2p}\) orbital, which is g.

Therefore, the overall parity of the state is g.

Symmetry (\(+/-)\) for \(\Sigma\) States

For \(\Sigma\) states (\(\Lambda = 0\)), we need to determine the symmetry with respect to reflection in a plane containing the internuclear axis. A \(\Sigma^+\) state is symmetric upon reflection, while a \(\Sigma^-\) state is antisymmetric. The \(\sigma\) molecular orbital is symmetric with respect to any plane containing the internuclear axis.

The \(\sigma_{2p}\) orbital is symmetric upon reflection, so the state is \(^+\).

Putting it Together

Combining the components:

• Spin Multiplicity: 2
• \(\Lambda\) state: \(\Sigma\) (\(\Lambda = 0\))
• Parity: g
• Symmetry for \(\Sigma\): \(+\)

The term symbol for the ground state \(N^+_2\) radical is \(^2\Sigma_g^+\).

Comparing this with the given options, the correct term symbol is ² \({\sum\nolimits_g^ + {} }\).