FTIR Spectroscopy & Diatomic Force Constants theory

For a vibrating and rotating heteronuclear diatomic molecule modeled as an anharmonic oscillator and non-rigid rotor, the electric dipole moment must change during vibrational displacement (dμ/dr ≠ 0) for infrared activity. In the electric dipole approximation, the vibrational selection rule for the harmonic oscillator is Δv = ±1 (the fundamental band). For real anharmonic molecules, weak overtones (Δv = ±2, ±3) also appear. Concurrently, conservation of angular momentum imposes the rotational selection rule ΔJ = ±1 for diatomic molecules with zero electronic orbital angular momentum in the electronic ground state (¹Σ⁺, such as HCl). The transition with ΔJ = J' − J'' = +1 (where the molecule gains one quantum of rotational angular momentum) constitutes the R-branch (high-frequency side), while ΔJ = −1 (where the molecule loses rotational angular momentum) forms the P-branch (low-frequency side). A Q-branch (ΔJ = 0) is strictly forbidden for ¹Σ diatomic molecules because photon emission/absorption involves intrinsic spin-1 angular momentum that must be compensated by a change in molecular rotational state.

The central gap at 2885.90 cm⁻¹ corresponds to the hypothetical pure vibrational transition (v = 0, J = 0 ⟶ v = 1, J = 0), which would have ΔJ = 0 (the Q-branch origin). Because HCl in its ground electronic state (¹Σ⁺) possesses no electronic orbital angular momentum about the internuclear axis, a pure vibrational transition with zero change in rotational state (ΔJ = 0) cannot satisfy conservation of total angular momentum when absorbing an electric dipole photon (spin quantum number S = 1). Consequently, the transition probability is identically zero, leaving an empty frequency gap of width Δν̃ ≈ 4B₀ ≈ 41.8 cm⁻¹ between the first R-branch line R(0) at 2906.17 cm⁻¹ and the first P-branch line P(1) at 2865.02 cm⁻¹.

Natural chlorine consists of two stable isotopes: ³⁵Cl (75.76% natural abundance) and ³⁷Cl (24.24% natural abundance), yielding an isotopic ratio of approximately 3.12:1. Because isotopic substitution changes the nuclear mass without altering the electronic potential energy surface (Born-Oppenheimer approximation), the reduced mass of H³⁷Cl (μ = 1.63155 × 10⁻²⁷ kg) is slightly greater than that of H³⁵Cl (μ = 1.62665 × 10⁻²⁷ kg). Because the vibrational frequency scales inversely with the square root of reduced mass (ν̃₀ ∝ μ⁻¹/²), the band origin of H³⁷Cl is shifted toward lower wavenumbers by Δν̃₀ ≈ 2.1 cm⁻¹ (ν̃₀ = 2883.8 cm⁻¹ for H³⁷Cl vs 2885.9 cm⁻¹ for H³⁵Cl). Similarly, the rotational constant B = h / (8π²cμr²) is slightly smaller for H³⁷Cl. Under high-resolution FTIR spectroscopy (≤ 0.5 cm⁻¹), every single R-branch and P-branch line is cleanly resolved into a doublet: a taller peak for H³⁵Cl and a closely adjacent shorter peak for H³⁷Cl whose intensity ratio mirrors the 3:1 natural isotopic abundance.

In a real diatomic molecule, the internuclear bond is not a rigid rod but an anharmonic oscillator. Because the potential energy curve is asymmetric (Morse potential), as the vibrational quantum number increases from v = 0 to v = 1, the average internuclear bond distance increases (r₁ > r₀) due to anharmonic repulsion at short distances. Because the rotational constant is inversely proportional to the moment of inertia (B_v = h / (8π²cμ⟨r²⟩_v)), the effective rotational constant in the excited vibrational state is smaller than in the ground state: B₁ < B₀. The difference is parameterized by the vibration-rotation coupling constant: B_v = B_e − α_e (v + 1/2), where α_e = B₀ − B₁ > 0. For transition lines parameterized by running index m: ν̃(m) = ν̃₀ + (B₁ + B₀)m + (B₁ − B₀)m² = ν̃₀ + (2B_e − 2α_e)m − α_e m². In the R-branch (m > 0), the negative quadratic term −α_e m² opposes m, causing successive line spacings Δν̃ = 2B₀ − 2α_e(m + 1) to contract progressively toward higher frequencies. In the P-branch (m < 0), the quadratic term reinforces the spacing, causing successive lines to spread progressively further apart.

The rotational lines are indexed by integer m: for the R-branch, transitions originate at J'' = m − 1 and terminate at J' = m, so ν̃_R(m−1) = ν̃₀ + (B₁ + B₀)m + (B₁ − B₀)m². For the P-branch, transitions originate at J'' = m and terminate at J' = m − 1, corresponding to replacing m with −m: ν̃_P(m) = ν̃₀ − (B₁ + B₀)m + (B₁ − B₀)m². Adding the two corresponding transition wavenumbers: ν̃_R(m−1) + ν̃_P(m) = [ν̃₀ + (B₁ + B₀)m + (B₁ − B₀)m²] + [ν̃₀ − (B₁ + B₀)m + (B₁ − B₀)m²] = 2ν̃₀ + 2(B₁ − B₀)m². Dividing by 2 yields the mean wavenumber: ν̃_mean(m) = [ν̃_R(m−1) + ν̃_P(m)] / 2 = ν̃₀ + (B₁ − B₀)m². Because B₁ − B₀ = −α_e, this simplifies identically to: ν̃_mean(m) = ν̃₀ − α_e · m². Thus, plotting ν̃_mean(m) versus m² yields an exact straight line with slope equal to −α_e and Y-intercept equal to the fundamental vibrational band origin ν̃₀.

In classical and quantum mechanics, the fundamental vibrational frequency of a harmonic oscillator is given by ω₀ = √(k/μ) in radians per second, or ν₀ = (1 / 2π) √(k/μ) in Hertz. Converting frequency to spectroscopic wavenumber in units of cm⁻¹ via ν̃₀ = ν₀ / c (where c is the speed of light in cm/s): ν̃₀ = (1 / 2πc) √(k/μ). Squaring both sides: ν̃₀² = k / (4π²c²μ). Rearranging explicitly for the force constant: k = 4π²c² ν̃₀² μ. For H³⁵Cl, using ν̃₀ = 2885.90 cm⁻¹, c = 2.99792458 × 10¹⁰ cm/s, and μ = 1.626652 × 10⁻²⁷ kg: k = 4π² × (2.997925 × 10¹⁰)² × (2885.90)² × (1.626652 × 10⁻²⁷) = 480.68 N/m ≈ 481.0 N/m. This force constant measures the curvature (second derivative) of the electronic potential energy surface at the equilibrium internuclear distance: k = (d²V/dr²)_{r=r_e}.

The rotational constant B_e represents the rigid-rotor rotational energy parameter: B_e = h / (8π² c I_e), where I_e = μ r_e² is the equilibrium moment of inertia about the center of mass. Solving for the equilibrium internuclear bond distance r_e: r_e = √[ h / (8π² c μ B_e) ]. In terms of the experimental constants B₀ and B₁ obtained from combination differences or line positions: B_e = B₀ + 0.5 α_e = (3 B₀ − B₁) / 2. For H³⁵Cl, B_e = 10.5934 cm⁻¹ and μ = 1.62665 × 10⁻²⁷ kg yield: r_e = √[ 6.62607 × 10⁻³⁴ / (8π² × 2.997925 × 10¹⁰ × 1.62665 × 10⁻²⁷ × 10.5934) ] = 1.2746 × 10⁻¹⁰ m = 1.275 Å. This establishes the equilibrium separation between the hydrogen nucleus and chlorine nucleus in the electronic ground state with sub-picometer precision.