Question

Determine whether the following transitions are allowed or forbidden: a. \(\sqrt[3]{\Pi}_{u} \rightarrow^{3} \Sigma_{g}^{-}\) \(\mathbf{b} .^{1} \Sigma_{g}^{+} \rightarrow^{1} \Pi_{g}\) \(\mathbf{c} .^{3} \Sigma_{g}^{-} \rightarrow^{3} \Pi_{g}\) d. \(^{1} \Pi_{g} \rightarrow^{1} \Delta_{u}\)

Short answer

Transitions a and d are allowed (as all selection rules are satisfied), while transitions b and c are forbidden (as the parity selection rule is violated).

Step-by-step solution

Transition a

For the transition \(\sqrt[3]{\Pi}_{u} \rightarrow^{3} \Sigma_{g}^{-}\), we need to check the selection rules: 1. Change in electronic spin: Initial state: S=1, Final state: S=1 ∆S = 1 - 1 = 0, which satisfies the selection rule. 2. Change in total orbital angular momentum: Initial state: Λ=1 (π), Final state: Λ=0 (Σ) ∆Λ = 1 - 0 = 1, which satisfies the selection rule. 3. Parity: Initial state: u (ungerade), Final state: g (gerade) The parity has changed, which satisfies the selection rule. Since all the selection rules are satisfied, the transition a is allowed.

Transition b

For the transition \(^{1}\Sigma_{g}^{+} \rightarrow^{1}\Pi_{g}\), we need to check the selection rules: 1. Change in electronic spin: Initial state: S=0, Final state: S=0 ∆S = 0 - 0 = 0, which satisfies the selection rule. 2. Change in total orbital angular momentum: Initial state: Λ=0 (Σ), Final state: Λ=1 (π) ∆Λ = 1 - 0 = 1, which satisfies the selection rule. 3. Parity: Initial state: g (gerade), Final state: g (gerade) The parity has not changed, which violates the selection rule. Since the parity selection rule is violated, the transition b is forbidden.

Transition c

For the transition \(^{3}\Sigma_{g}^{-} \rightarrow^{3}\Pi_{g}\), we need to check the selection rules: 1. Change in electronic spin: Initial state: S=1, Final state: S=1 ∆S = 1 - 1 = 0, which satisfies the selection rule. 2. Change in total orbital angular momentum: Initial state: Λ=0 (Σ), Final state: Λ=1 (π) ∆Λ = 1 - 0 = 1, which satisfies the selection rule. 3. Parity: Initial state: g (gerade), Final state: g (gerade) The parity has not changed, which violates the selection rule. Since the parity selection rule is violated, the transition c is forbidden.

Transition d

For the transition \(^{1}\Pi_{g} \rightarrow^{1}\Delta_{u}\), we need to check the selection rules: 1. Change in electronic spin: Initial state: S=0, Final state: S=0 ∆S = 0 - 0 = 0, which satisfies the selection rule. 2. Change in total orbital angular momentum: Initial state: Λ=1 (π), Final state: Λ=2 (Δ) ∆Λ = 2 - 1 = 1, which satisfies the selection rule. 3. Parity: Initial state: g (gerade), Final state: u (ungerade) The parity has changed, which satisfies the selection rule. Since all the selection rules are satisfied, the transition d is allowed. In conclusion, transitions a and d are allowed, while transitions b and c are forbidden.

Key concepts

Selection Rules

In the world of molecular spectroscopy, selection rules serve as vital guidelines to determine whether a particular electronic transition is allowed or forbidden. Fundamentally, these rules are based on the symmetries and properties of an electron's quantum states and the interaction with electromagnetic radiation.

For instance, when examining transitions like \(\sqrt[3]{\Pi}_{u} \rightarrow^{3} \Sigma_{g}^{-}\), we apply selection rules concerning electronic spin, orbital angular momentum, and parity. If all the rules are satisfied, the transition is deemed 'allowed'; otherwise, it's 'forbidden'. This concept is instrumental in predicting the likelihood of a transition occurring, which in turn affects the absorption or emission spectrum of a molecule.

Parity Selection Rule

The parity selection rule is crucial in molecular electronic transitions. It dictates that for an electronic transition to be allowed, the parity must change from even (gerade, 'g') to odd (ungerade, 'u') or vice versa. This is rooted in the fact that electromagnetic radiation (photons) have odd parity.

When we look at a transition like \(^{1}\Sigma_{g}^{+} \rightarrow^{1}\Pi_{g}\), we can quickly determine it violates the parity selection rule as the transition is from a gerade to another gerade state. Hence, conservation of parity in this context results in a forbidden transition. Understanding this rule helps predict the observed spectra and the exact conditions under which an electronic transition may occur.

Orbital Angular Momentum

Another critical concept in molecular transitions is orbital angular momentum, which is determined by the orbital angular momentum quantum number, represented by \(\Lambda\). In electronic transitions, the selection rule states that \(\Delta\Lambda = \pm 1\) is allowed, meaning the quantum number can change by one unit.

For instance, when considering a transition from \(\Pi\) to \(\Sigma\), where \(\Pi\) has \(\Lambda=1\) and \(\Sigma\) has \(\Lambda=0\), such a change satisfies the orbital angular momentum selection rule, signaling an allowed transition. A grasp of this concept is essential for discerning the types of possible transitions within a molecule's electron configuration.

Electronic Spin

Finally, electronic spin plays a determining role in the selection rules for molecular transitions. An electron has a spin quantum number (S), which can impact transitions due to the 'spin selection rule'. This rule specifies that for an electronic transition to be allowed, there should be no change in the multiplicity of the electronic state, which implies \(\Delta S = 0\).

Transitions such as \(\sqrt[3]{\Pi}_{u} \rightarrow^{3} \Sigma_{g}^{-}\) conform to this rule since the initial and final states both have a spin multiplicity of three (\(S=1\)), resulting in \(\Delta S = 0\). Understanding electronic spin and its selection rules is indispensable for predicting the possible electronic transitions and interpreting spectroscopic patterns.