6.3 Energies and selection rules for vibrational and rotational transitions

Energies and selection rules for vibrational and rotational transitions are key to understanding molecular motion. These rules determine which energy jumps are allowed, shaping the unique spectral fingerprints of different molecules.

By studying these transitions, we can unlock secrets about molecular structure and behavior. This knowledge helps us interpret spectra, identify compounds, and predict how molecules will interact in various situations.

Selection Rules for Vibrational and Rotational Transitions

Vibrational Selection Rule

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• The selection rule for vibrational transitions is Δv = ±1, where v is the vibrational quantum number
• Transitions can only occur between adjacent vibrational energy levels (v = 0 → v = 1 or v = 2 → v = 1)
• This selection rule is based on the conservation of energy and the properties of the vibrational wavefunctions
• Example: In a diatomic molecule like CO, the allowed vibrational transitions are from v = 0 to v = 1, v = 1 to v = 2, and so on

Rotational Selection Rules

• The selection rule for rotational transitions depends on the type of molecule and the nature of the transition
• For electric dipole transitions in heteronuclear diatomic molecules (CO, HCl), the selection rule is ΔJ = ±1, where J is the rotational quantum number
• For homonuclear diatomic molecules (N2, O2), electric dipole transitions are forbidden, and the selection rule for quadrupole transitions is ΔJ = ±2
• The specific rotational selection rule is determined by the symmetry of the molecule and the presence of a permanent electric dipole moment
• The parity of the rotational wavefunction, related to the symmetry of the molecule, also influences the allowed rotational transitions

Allowed Transitions in Vibrational-Rotational Spectra

Vibrational Transitions

• Only transitions between adjacent vibrational energy levels (Δv = ±1) are allowed due to the vibrational selection rule
• Examples of allowed vibrational transitions:
  • v = 0 → v = 1 (fundamental transition)
  • v = 1 → v = 2 (first overtone)
  • v = 2 → v = 1 (hot band transition)
• Vibrational transitions with Δv > ±1 are forbidden but can sometimes be observed with lower intensity due to anharmonicity or other mechanisms

Rotational Transitions

• In heteronuclear diatomic molecules, rotational transitions with ΔJ = ±1 are allowed (P-branch: ΔJ = -1, R-branch: ΔJ = +1)
• In homonuclear diatomic molecules, only quadrupole transitions with ΔJ = ±2 are allowed (O-branch: ΔJ = -2, S-branch: ΔJ = +2)
• Examples of allowed rotational transitions in heteronuclear molecules:
  • J = 0 → J = 1 (R-branch)
  • J = 2 → J = 1 (P-branch)
• The allowed transitions for more complex molecules depend on their symmetry and the presence of a permanent electric dipole moment
• Forbidden transitions that violate the selection rules can sometimes be observed with lower intensity due to mechanisms like vibronic coupling or magnetic dipole transitions

Energies of Vibrational and Rotational Transitions

Vibrational Transition Energies

• The energy of a vibrational transition is given by ΔE_vib = hν(v + 1) - hν(v) = hν, where h is Planck's constant and ν is the vibrational frequency
• The vibrational frequency (ν) depends on the reduced mass of the molecule and the force constant of the bond
• The energy spacing between vibrational levels is constant in the harmonic oscillator approximation but decreases slightly with increasing v due to anharmonicity
• Example: In a diatomic molecule like HCl, the vibrational frequency is determined by the reduced mass of H and Cl and the strength of the H-Cl bond

Rotational Transition Energies

• The energy of a rotational transition is given by ΔE_rot = hcB[J(J + 1) - J'(J' + 1)], where h is Planck's constant, c is the speed of light, B is the rotational constant, and J and J' are the initial and final rotational quantum numbers
• The rotational constant (B) depends on the moment of inertia of the molecule and can be calculated using the reduced mass and the bond length
• The energy spacing between rotational levels increases with increasing J, resulting in a series of equally spaced lines in the rotational spectrum
• Example: In a diatomic molecule like CO, the rotational constant is determined by the reduced mass of C and O and the C-O bond length

Vibrational-Rotational Transition Energies

• The energy of a vibrational-rotational transition is the sum of the vibrational and rotational transition energies: ΔE_total = ΔE_vib + ΔE_rot
• The vibrational and rotational transitions occur simultaneously, leading to a series of bands in the vibrational-rotational spectrum, each corresponding to a specific vibrational transition and containing a series of rotational lines
• The energy spacing between the vibrational bands is determined by the vibrational frequency, while the spacing between the rotational lines within a band is determined by the rotational constant

Interpreting Vibrational-Rotational Spectra

Vibrational-Rotational Bands

• Vibrational-rotational spectra consist of a series of bands, each corresponding to a specific vibrational transition (Δv) and containing a series of rotational lines
• The spacing between the vibrational bands is determined by the vibrational frequency (ν) of the molecule
• The intensity of the bands depends on the population of the initial vibrational state (governed by the Boltzmann distribution) and the transition dipole moment
• Example: In the vibrational-rotational spectrum of HCl, the fundamental band (v = 0 → v = 1) is the most intense, followed by the first overtone (v = 0 → v = 2) and the hot band (v = 1 → v = 2)

Rotational Structure within Vibrational Bands

• Each vibrational band contains a series of rotational lines, corresponding to transitions between different rotational levels (ΔJ)
• The spacing between the rotational lines within a band is determined by the rotational constant (B) of the molecule
• The intensity of the rotational lines follows a Boltzmann distribution, with the most intense lines corresponding to the most populated rotational states
• The presence of a Q-branch (ΔJ = 0) indicates a molecule with a non-zero angular momentum in the excited vibrational state (linear molecules), while the absence of a Q-branch and the presence of only P-branch (ΔJ = -1) and R-branch (ΔJ = +1) transitions indicate a molecule with zero angular momentum in the excited vibrational state (diatomic molecules in a Σ electronic state)

Extracting Molecular Information from Spectra

• The analysis of vibrational-rotational spectra provides valuable information about the molecular structure, bond lengths, force constants, and vibrational frequencies
• The vibrational frequency can be determined from the spacing between the vibrational bands, while the rotational constant can be obtained from the spacing between the rotational lines within a band
• The intensity distribution of the rotational lines can be used to determine the temperature of the sample and the population of different rotational states
• Example: The vibrational-rotational spectrum of CO can be used to determine the C-O bond length, the force constant of the bond, and the vibrational frequency of the molecule