☀️Photochemistry Unit 3 – Electronic Transitions and Selection Rules

Key Concepts

• Electronic transitions involve the promotion of an electron from a lower energy level to a higher energy level
• Absorption of a photon with sufficient energy can cause an electronic transition
• Selection rules govern which electronic transitions are allowed or forbidden based on symmetry and spin considerations
• The Franck-Condon principle states that electronic transitions occur much faster than nuclear motion, resulting in vertical transitions on potential energy surfaces
• Born-Oppenheimer approximation separates electronic and nuclear motion, simplifying the treatment of electronic transitions
• Absorption and emission spectra provide information about the energy levels and electronic structure of molecules
• Molecular orbitals, formed by the combination of atomic orbitals, play a crucial role in determining the nature of electronic transitions

Quantum Mechanical Foundations

• The Schrödinger equation describes the wave function and energy levels of a quantum system
  • The time-independent Schrödinger equation is used to determine the stationary states and energy levels
  • The time-dependent Schrödinger equation describes the evolution of the wave function over time
• The wave function, denoted as $\Psi(x, t)$, contains all the information about a quantum system
  • The probability of finding a particle at a specific location is proportional to the square of the absolute value of the wave function, $|\Psi(x, t)|^2$
• Operators in quantum mechanics correspond to observable quantities (position, momentum, energy)
  • The Hamiltonian operator represents the total energy of the system
• Eigenvalues and eigenfunctions are obtained by solving the Schrödinger equation
  • Eigenvalues correspond to the allowed energy levels of the system
  • Eigenfunctions represent the wave functions associated with each energy level
• The Pauli exclusion principle states that no two electrons in an atom or molecule can have the same set of quantum numbers

Types of Electronic Transitions

• $\sigma \rightarrow \sigma^*$ transitions involve the excitation of an electron from a bonding $\sigma$ orbital to an antibonding $\sigma^*$ orbital
  • These transitions typically require high energy and occur in the far UV region
• $\pi \rightarrow \pi^*$ transitions involve the excitation of an electron from a bonding $\pi$ orbital to an antibonding $\pi^*$ orbital
  • These transitions require lower energy compared to $\sigma \rightarrow \sigma^*$ transitions and occur in the near UV or visible region
  • Molecules with conjugated $\pi$ systems (benzene) exhibit $\pi \rightarrow \pi^*$ transitions
• $n \rightarrow \pi^*$ transitions involve the excitation of a non-bonding electron (lone pair) to an antibonding $\pi^*$ orbital
  • These transitions require relatively low energy and occur in the UV or visible region
  • Molecules with heteroatoms containing lone pairs (carbonyl compounds) exhibit $n \rightarrow \pi^*$ transitions
• Charge-transfer transitions involve the transfer of an electron from one molecular entity (donor) to another (acceptor)
  • These transitions can occur in complexes or between separate molecules
• Metal-to-ligand charge transfer (MLCT) transitions involve the transfer of an electron from a metal orbital to a ligand orbital in coordination compounds

Selection Rules

• Selection rules determine whether an electronic transition is allowed or forbidden based on the symmetry and spin of the initial and final states
• The Laporte selection rule states that transitions between states of the same parity (gerade to gerade or ungerade to ungerade) are forbidden in centrosymmetric molecules
  • Vibronic coupling can relax the Laporte selection rule, making certain forbidden transitions weakly allowed
• The spin selection rule states that transitions between states of different spin multiplicities are forbidden
  • Singlet-to-singlet and triplet-to-triplet transitions are allowed, while singlet-to-triplet and triplet-to-singlet transitions are forbidden
  • Spin-orbit coupling can relax the spin selection rule, making spin-forbidden transitions weakly allowed (phosphorescence)
• The symmetry selection rule states that transitions are allowed if the direct product of the irreducible representations of the initial state, the transition moment operator, and the final state contains the totally symmetric representation
• The overlap integral between the initial and final state wave functions must be non-zero for a transition to be allowed
• Selection rules can be used to predict the intensity and polarization of electronic transitions

Absorption and Emission Spectra

• Absorption spectra show the wavelengths or frequencies of light absorbed by a molecule, corresponding to electronic transitions from the ground state to excited states
  • Absorption bands appear as peaks or broad features in the spectrum
  • The intensity of absorption bands depends on the transition probability and the concentration of the absorbing species
• Emission spectra show the wavelengths or frequencies of light emitted by a molecule, corresponding to electronic transitions from excited states to the ground state
  • Emission bands appear as peaks or broad features in the spectrum
  • The intensity of emission bands depends on the transition probability and the population of the emitting excited state
• The Stokes shift is the difference between the maximum absorption and emission wavelengths
  • It arises from the rapid relaxation of excited states to lower vibrational levels and solvent reorganization
• Kasha's rule states that emission typically occurs from the lowest excited state of a given spin multiplicity, regardless of the initial excited state populated
• Fluorescence is the emission of light from a singlet excited state to the singlet ground state
  • Fluorescence lifetimes are typically on the order of nanoseconds
• Phosphorescence is the emission of light from a triplet excited state to the singlet ground state
  • Phosphorescence lifetimes are typically longer than fluorescence lifetimes (microseconds to seconds) due to the spin-forbidden nature of the transition

Molecular Orbitals and Transitions

• Molecular orbitals are formed by the linear combination of atomic orbitals (LCAO) from the constituent atoms
  • Bonding orbitals have lower energy than the original atomic orbitals and contribute to the stability of the molecule
  • Antibonding orbitals have higher energy than the original atomic orbitals and contribute to the instability of the molecule
• The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) play crucial roles in electronic transitions
  • The HOMO-LUMO gap determines the energy required for the lowest-energy electronic transition
• Molecular orbital diagrams provide a visual representation of the relative energies and electron occupancies of molecular orbitals
  • The ordering of molecular orbitals depends on the symmetry and energy of the constituent atomic orbitals
• Electronic transitions in molecules involve the promotion of an electron from an occupied molecular orbital to an unoccupied molecular orbital
  • The symmetry and overlap of the initial and final molecular orbitals determine the probability and intensity of the transition
• Hückel molecular orbital theory is a simplified method for calculating the energies and coefficients of $\pi$ molecular orbitals in conjugated systems
  • It provides qualitative insights into the electronic structure and spectroscopic properties of aromatic compounds

Applications in Spectroscopy

• UV-Vis spectroscopy measures the absorption of ultraviolet and visible light by molecules
  • It provides information about the electronic transitions and the presence of chromophores (functional groups that absorb light)
  • The Beer-Lambert law relates the absorbance to the concentration and molar absorptivity of the sample
• Fluorescence spectroscopy measures the emission of light from excited states populated by absorption
  • It is highly sensitive and can detect low concentrations of fluorescent molecules
  • Fluorescence quantum yield and lifetime measurements provide information about the efficiency and kinetics of the emission process
• Circular dichroism (CD) spectroscopy measures the differential absorption of left and right circularly polarized light
  • It is sensitive to the chirality and conformation of molecules
  • CD spectra can provide information about the secondary structure of proteins and the stereochemistry of organic compounds
• Transient absorption spectroscopy measures the time-resolved absorption changes following excitation by a pump pulse
  • It can monitor the dynamics of excited states, charge transfer processes, and photochemical reactions on timescales ranging from femtoseconds to microseconds
• Resonance Raman spectroscopy measures the enhancement of Raman scattering when the excitation wavelength coincides with an electronic transition
  • It provides vibrational information about the excited state and the structural changes associated with the electronic transition

Practice Problems and Examples

1. Consider a molecule with a HOMO-LUMO gap of 3.5 eV. What is the wavelength of light required to induce the lowest-energy electronic transition?

   • Using the equation $E = hc/\lambda$, where $h$ is Planck's constant and $c$ is the speed of light, we can calculate the wavelength: $\lambda = hc/E = (6.626 \times 10^{-34} \text{ J s}) \times (2.998 \times 10^8 \text{ m/s}) / (3.5 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV}) = 354 \text{ nm}$
   • The wavelength required for the lowest-energy electronic transition is 354 nm, which falls in the UV region of the electromagnetic spectrum.

2. A molecule has a molar absorptivity of $5000 \text{ M}^{-1}\text{cm}^{-1}$ at 400 nm. If a 1 cm cuvette is filled with a 0.001 M solution of this molecule, what is the absorbance at 400 nm?

   • Using the Beer-Lambert law, $A = \varepsilon bc$, where $A$ is the absorbance, $\varepsilon$ is the molar absorptivity, $b$ is the path length, and $c$ is the concentration: $A = (5000 \text{ M}^{-1}\text{cm}^{-1}) \times (1 \text{ cm}) \times (0.001 \text{ M}) = 5$
   • The absorbance of the solution at 400 nm is 5.

3. A molecule undergoes a $\pi \rightarrow \pi^*$ transition. Which of the following statements is true? a) The transition is spin-forbidden b) The transition is Laporte-forbidden in centrosymmetric molecules c) The transition occurs between a bonding and an antibonding molecular orbital d) The transition requires higher energy than an $n \rightarrow \pi^*$ transition

   • The correct answer is c) The transition occurs between a bonding and an antibonding molecular orbital.
   • $\pi \rightarrow \pi^*$ transitions involve the excitation of an electron from a bonding $\pi$ orbital to an antibonding $\pi^*$ orbital. These transitions are spin-allowed and Laporte-allowed in non-centrosymmetric molecules. They generally require lower energy than $\sigma \rightarrow \sigma^*$ transitions but higher energy than $n \rightarrow \pi^*$ transitions.

4. The fluorescence spectrum of a molecule shows a maximum emission at 450 nm, while its absorption spectrum shows a maximum absorption at 400 nm. What is the Stokes shift for this molecule?

   • The Stokes shift is the difference between the maximum absorption and emission wavelengths: Stokes shift = Emission maximum - Absorption maximum = 450 nm - 400 nm = 50 nm
   • The Stokes shift for this molecule is 50 nm.

5. A molecule has a singlet ground state (S0) and a singlet excited state (S1). Which of the following transitions is spin-allowed? a) S0 → S1 b) S0 → T1 (triplet excited state) c) T1 → S0 d) T1 → T2 (higher triplet excited state)

   • The correct answer is a) S0 → S1.
   • According to the spin selection rule, transitions between states of the same spin multiplicity are allowed. Singlet-to-singlet transitions (S0 → S1) and triplet-to-triplet transitions (T1 → T2) are spin-allowed, while singlet-to-triplet (S0 → T1) and triplet-to-singlet (T1 → S0) transitions are spin-forbidden.