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IR/UV-Vis

Field: Spectroscopy

Infrared and Ultraviolet-Visible spectroscopy.

Sequence of Expressions

Definition

Stokes Shift

Define νabs\nu_{abs} as the frequency of maximum absorption and νem\nu_{em} as the frequency of maximum emission for a given transition. The Stokes Shift (ΔνS\Delta \nu_S) is defined by the difference in these frequencies:\nΔνS=νem−νabs\Delta \nu_S = \nu_{em} - \nu_{abs}
Consider a molecular vibration described by a normal coordinate QQ. A vibrational mode is IR active if the derivative of the molecular dipole moment μ\boldsymbol{\mu} with respect to QQ is non-zero:\n∂μ∂Q∣Q=0≠0\frac{\partial \boldsymbol{\mu}}{\partial Q} \bigg|_{Q=0} \neq \mathbf{0}
Let A(ν)\text{A}(\nu) be the absorbance spectrum of a substance. The UV-Vis Absorption Maxima (λmax\lambda_{max} or νmax\nu_{max}) are the frequencies νmax\nu_{max} that maximize the molar absorptivity Eˉ(ν)\bar{\text{E}}(\nu): \nνmax=arg⁡max⁡ν(Eˉ(ν))\nu_{max} = \arg \max_{\nu} \left( \bar{\text{E}}(\nu) \right) \nEquivalently, λmax=c/νmax\lambda_{max} = c / \nu_{max}, where cc is the speed of light.
The population ratio Ni/NjN_i / N_j of two energy states EiE_i and EjE_j at thermal equilibrium temperature TT is determined by the Boltzmann factor: NiNj=e−(Ei−Ej)kBT\frac{N_i}{N_j} = e^{-\frac{(E_i - E_j)}{k_B T}}. For a general energy level EiE_i, the population NiN_i is given by the Boltzmann distribution: Ni=Ntotale−Ei/kBTN_i = N_{total} e^{-E_i / k_B T}, where kBk_B is the Boltzmann constant and NtotalN_{total} is the total number of particles.
For a rigid rotor molecule, the rotational energy levels E(J)E(J) are quantized by the rotational quantum number JJ. The energy levels are given by:\nE(J)=BJ(J+1)E(J) = B J(J+1) \nwhere BB is the rotational constant (in units of energy) and J∈{0,1,2,… }J \in \{0, 1, 2, \dots\}. The dispersion relation describes the spacing between these levels.
Define the wavenumber kk (in cm−1\text{cm}^{-1}) as the reciprocal of the wavelength λ\lambda (in cm\text{cm}): k=1λk = \frac{1}{\lambda}. The relationship between wavenumber kk, energy EE, and Planck's constant hh and the speed of light cc is given by: E=hckE = h c k. This proportionality establishes that kk is directly proportional to the energy of the absorbed photon.
The Beer-Lambert Law relates the measured absorbance AA to the concentration cc of the absorbing species, the path length ll of the cuvette, and the molar absorptivity ϵ\epsilon at a specific wavelength λ\lambda: A=ϵclA = \epsilon c l. Here, AA is unitless, cc has units of mol⋅L−1\text{mol}\cdot\text{L}^{-1}, ll has units of cm\text{cm}, and ϵ\epsilon has units of L⋅mol−1⋅cm−1\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}.
Let AA be the absorbance, cc be the concentration of the absorbing species, ll be the path length, and Eˉ(ν)\bar{\text{E}}(\nu) be the molar absorptivity coefficient at frequency ν\nu. The Beer-Lambert Law is formally stated as:\nA(ν)=Eˉ(ν)⋅c⋅l\text{A}(\nu) = \bar{\text{E}}(\nu) \cdot c \cdot l
Let EvE_v be the quantized vibrational energy levels of a molecule, where v∈Z≥0v \in \mathbb{Z}_{\ge 0}. The energy levels are approximated by the harmonic oscillator model: Ev=hcνˉ(v+1/2)E_v = h c \bar{\nu} (v + 1/2). The transition probability between levels v′′v'' and v′v' is governed by the transition dipole moment ⟨v′∣μ∣v′′⟩\langle v' | \boldsymbol{\mu} | v'' \rangle, where μ\boldsymbol{\mu} is the molecular dipole moment operator. For fundamental transitions, the selection rule dictates Δv=v′−v′′=±1\Delta v = v' - v'' = \pm 1.
For an infrared transition between vibrational states v′′→v′v'' \to v', the transition is allowed if and only if the transition dipole moment integral ⟨v′∣μ∣v′′⟩\langle v' | \boldsymbol{\mu} | v'' \rangle is non-zero. Mathematically, this requires the derivative of the dipole moment with respect to the normal coordinate QQ: ∂μ∂Q∣Q=0≠0\left. \frac{\partial \boldsymbol{\mu}}{\partial Q} \right|_{Q=0} \neq \mathbf{0}. This condition ensures that the change in dipole moment during the vibration is non-zero.