12.3 Vibrational and rotational states of molecules

Molecules dance to the quantum tune of vibrations and rotations. These motions shape their energy levels and spectra, revealing secrets about molecular structure. Understanding these states is key to unlocking the quantum behavior of molecules.

The Born-Oppenheimer approximation lets us separate electronic and nuclear motions. This simplifies the study of molecular vibrations and rotations, connecting them to the broader picture of molecular quantum mechanics and orbitals.

Vibrational and Rotational Motion of Molecules

Vibrational Modes

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• Diatomic molecules have one vibrational mode along the bond axis
  • The vibration involves the periodic stretching and compression of the bond
• Polyatomic molecules have 3N-6 vibrational modes (3N-5 for linear molecules), where N is the number of atoms
  • Each vibrational mode corresponds to a specific pattern of atomic displacements
  • Examples of vibrational modes in polyatomic molecules include symmetric stretching, asymmetric stretching, and bending modes (H2O, CO2)

Rotational Motion

• Rotational motion in diatomic molecules occurs about an axis perpendicular to the bond axis
  • The molecule rotates as a rigid rotor, with the atoms maintaining a fixed distance from the axis of rotation
• Polyatomic molecules can rotate about three principal axes of rotation
  • The principal axes are determined by the molecule's geometry and correspond to the axes with the minimum, intermediate, and maximum moments of inertia
• The moment of inertia determines the rotational energy levels and depends on the mass and geometry of the molecule
  • Larger moments of inertia result in more closely spaced rotational energy levels (HCl vs. H2)
• Centrifugal distortion arises from the coupling between vibrational and rotational motions
  • As the molecule rotates faster, the bond length increases slightly due to centrifugal forces
  • This leads to a slight increase in the bond length at higher rotational levels and a deviation from the rigid rotor approximation

Quantum Harmonic Oscillator Model for Vibrational Energy

Energy Levels and Wavefunctions

• The quantum harmonic oscillator model approximates the vibrational motion of a diatomic molecule as a simple harmonic oscillator
  • The potential energy is assumed to be a quadratic function of the displacement from equilibrium
• The vibrational energy levels are quantized and given by $E_v = (v + 1/2)hν$, where $v$ is the vibrational quantum number, $h$ is Planck's constant, and $ν$ is the fundamental vibrational frequency
  • The lowest vibrational energy level (v=0) has a non-zero energy of $1/2hν$, known as the zero-point energy
• The vibrational wavefunctions are solutions to the Schrödinger equation for the quantum harmonic oscillator and are expressed in terms of Hermite polynomials
  • The wavefunctions describe the probability distribution of the molecule's displacement from equilibrium for each vibrational level

Anharmonicity

• The anharmonicity of real molecules leads to deviations from the harmonic oscillator model
  • The potential energy curve is not perfectly quadratic but has higher-order terms
• Anharmonicity can be accounted for by introducing anharmonicity constants (e.g., $χ_e$, $γ_e$) in the energy level expression
  • The anharmonicity constants modify the energy level spacing and cause the spacing to decrease with increasing vibrational quantum number
• Anharmonicity also allows for transitions between non-adjacent vibrational levels (overtones) and combination bands in the vibrational spectrum

Selection Rules for Vibrational and Rotational Transitions

Vibrational Transitions

• Vibrational transitions are allowed when the electric dipole moment of the molecule changes during the vibration
  • Mathematically, this is expressed as $(∂μ/∂Q)_0 ≠ 0$, where $μ$ is the dipole moment and $Q$ is the normal coordinate
• The selection rule for vibrational transitions is $Δv = ±1$, known as the fundamental transition
  • This means that transitions occur between adjacent vibrational levels
• Overtone transitions ($Δv = ±2, ±3, ...$) are also possible but have lower intensity
  • Overtones arise due to anharmonicity and involve transitions between non-adjacent vibrational levels

Rotational Transitions

• Rotational transitions are allowed when the molecule has a permanent electric dipole moment
  • Molecules with no permanent dipole moment (e.g., homonuclear diatomics like H2, N2) do not exhibit pure rotational spectra
• The selection rule for rotational transitions is $ΔJ = ±1$, where $J$ is the rotational quantum number
  • Transitions occur between adjacent rotational levels
• In the rigid rotor approximation, the rotational energy levels are given by $E_J = BJ(J+1)$, where $B$ is the rotational constant
  • The rotational constant depends on the moment of inertia and the reduced mass of the molecule

Vibrational-Rotational Transitions

• Vibrational-rotational transitions combine the selection rules for both vibrational and rotational transitions
  • The overall selection rules are $Δv = ±1$ and $ΔJ = 0, ±1$
• The appearance of P, Q, and R branches in the vibrational-rotational spectrum depends on the change in the dipole moment during the transition
  • P branch: $ΔJ = -1$, transitions to lower rotational levels
  • Q branch: $ΔJ = 0$, transitions between the same rotational levels
  • R branch: $ΔJ = +1$, transitions to higher rotational levels

Interpreting Vibrational and Rotational Spectra

Vibrational Spectra

• Vibrational spectra consist of a series of bands corresponding to transitions between different vibrational levels
  • The fundamental band (v=0 → v=1) is the most intense, followed by overtone and combination bands
• The spacing between vibrational bands decreases with increasing vibrational quantum number due to anharmonicity
  • This allows for the determination of anharmonicity constants from the observed band positions
• Rotational fine structure appears within each vibrational band, consisting of closely spaced lines corresponding to transitions between different rotational levels
  • The rotational structure provides information about the molecule's geometry and moment of inertia

Rotational Spectra

• The spacing between rotational lines is determined by the rotational constant $B$, which can be extracted from the spectrum
  • The rotational constant is inversely proportional to the moment of inertia and can be used to calculate bond lengths and angles
• The intensity distribution of rotational lines depends on the population of the rotational levels, which is governed by the Boltzmann distribution
  • The intensity of rotational lines decreases with increasing rotational quantum number

Molecular Constants and Structure Determination

• Molecular constants, such as the equilibrium bond length, vibrational frequency, and anharmonicity constants, can be determined by fitting the observed spectral data to theoretical expressions
  • These constants provide valuable information about the molecule's structure and potential energy surface
• Isotopic substitution leads to changes in the reduced mass and, consequently, the vibrational and rotational constants
  • Comparing the spectra of different isotopomers allows for the identification of isotope effects and the determination of molecular structure
  • Examples include the Birge-Sponer extrapolation and the Dunham expansion for diatomic molecules