Spectroscopy Types

Every photon carries energy E = hν = hc/λ, where h is Planck's constant (6.626 × 10⁻³⁴ J·s), ν is frequency (Hz), c is the speed of light, and λ is wavelength. Absorption occurs only when this photon energy exactly equals the gap ΔE between two quantized molecular energy levels. This is the fundamental resonance condition of spectroscopy.

Molecules rotate about their centre of mass. Quantum mechanics restricts rotational energy to discrete levels: E_J = hcB·J(J+1), where B is the rotational constant (B = h/8π²Ic, I = moment of inertia) and J = 0, 1, 2, … is the rotational quantum number. Transitions between adjacent levels (selection rule: ΔJ = ±1) require microwave photons. For a diatomic molecule like HCl, absorptions appear as a series of equally spaced lines at frequencies ν = 2Bc(J+1). Microwave spectroscopy measures B with extraordinary precision, giving bond lengths to five significant figures. A key requirement: the molecule must have a permanent electric dipole moment — so homonuclear diatomics (N₂, O₂) are microwave-inactive.

Chemical bonds vibrate like quantum harmonic oscillators. The vibrational energy levels are E_v = hν_vib(v + ½), where v = 0, 1, 2, … is the vibrational quantum number and ν_vib = (1/2π)√(k/μ) is the fundamental vibrational frequency (k = force constant, μ = reduced mass). The ground state (v = 0) has zero-point energy ½hν_vib — bonds never truly stop vibrating. The selection rule ΔV = ±1 means that IR absorption produces the fundamental band at wavenumber ν̃ = (1/2πc)√(k/μ). Heavier atoms and weaker bonds vibrate at lower wavenumbers (lower energy). IR absorption also requires a change in dipole moment during the vibration: CO₂ (O=C=O) has an IR-active asymmetric stretch at ~2349 cm⁻¹ but an IR-inactive symmetric stretch. Functional groups have characteristic IR absorptions: O–H stretch ~3200–3600 cm⁻¹, C=O stretch ~1700–1750 cm⁻¹, C–H stretch ~2850–3000 cm⁻¹. Chemists use IR spectra as molecular fingerprints to identify functional groups.

UV-Vis photons (energies ~1.8–6.2 eV) promote electrons from occupied molecular orbitals to higher-energy vacant orbitals. The most common transitions in organic molecules are π → π (conjugated systems, ~200–400 nm) and n → π (lone pair to antibonding π orbital, ~270–330 nm). The Beer–Lambert Law, A = εcl, links the measured absorbance A to the molar absorptivity ε (L mol⁻¹ cm⁻¹), concentration c (mol L⁻¹), and path length l (cm). The wavelength of maximum absorption (λ_max) shifts as conjugation increases — each additional C=C unit in a polyene red-shifts λ_max by ~30 nm (the basis of colour in carotenoids). Transition metal complexes show d–d transitions and charge-transfer bands; [Cu(H₂O)₆]²⁺ absorbs red and orange light (~800 nm), appearing blue. UV-Vis is widely used in quantitative analysis and for studying reaction kinetics.

Not every energy-level transition is allowed. Selection rules arise from the mathematics of quantum mechanical transition dipole moments. For rotational spectroscopy: ΔJ = ±1 and the molecule needs a permanent dipole. For vibrational spectroscopy: ΔV = ±1 and the dipole moment must change. For electronic spectroscopy: spin must be conserved (ΔS = 0), so singlet → triplet transitions (intersystem crossing) are formally forbidden but can occur weakly via spin-orbit coupling. Raman spectroscopy (not covered in detail here) is complementary to IR: vibrations that are IR-inactive may be Raman-active if there is a change in polarisability, making the two techniques together a complete picture of molecular vibrations.

• E — Photon energy (Joule (J)): The energy carried by one photon of electromagnetic radiation.
• h — Planck's constant (J·s): Universal constant equal to 6.626 × 10⁻³⁴ J·s; relates energy to frequency.
• \nu — Frequency (Hertz (Hz = s⁻¹)): Number of wave cycles per second; higher frequency means higher energy.
• c — Speed of light (m s⁻¹): Approximately 3.00 × 10⁸ m s⁻¹ in a vacuum.
• \lambda — Wavelength (Metre (m) or nm, µm, cm): Distance between successive wave crests; inversely proportional to energy.

Problem. An IR absorption band for the C=O stretch of ethanal (acetaldehyde, CH₃CHO) appears at a wavenumber of 1730 cm⁻¹. (a) Calculate the frequency (in Hz) and the energy (in J) of a photon at this wavenumber. (b) If the reduced mass of the C=O oscillator is 1.14 × 10⁻²⁶ kg, estimate the force constant k of the C=O bond.

Answer. Frequency = 5.19 × 10¹³ Hz; Photon energy = 3.44 × 10⁻²⁰ J; Force constant k ≈ 1210 N m⁻¹.

• Molecular Orbital Theory — UV-Vis spectroscopy directly probes transitions between molecular orbitals (e.g., HOMO → LUMO); understanding MO theory explains why different molecules absorb at different wavelengths.
• Polarity and Dipole Moments — IR and microwave selection rules require a change in dipole moment; only polar bonds or molecules with permanent dipoles are IR/microwave active.
• Molecular Vibrations and Bond Energies — The force constant extracted from IR frequencies is directly related to bond strength and order, connecting spectroscopy to the concept of bond energy.
• Quantization of Energy — The entire premise of spectroscopy — that only specific photon energies are absorbed — rests on the quantization of molecular energy levels derived from quantum mechanics.
• Beer–Lambert Law — The Beer–Lambert Law (A = εcl) is the mathematical foundation of quantitative UV-Vis spectroscopy, used in analytical chemistry to determine concentrations.