Unit 3: Infrared & Vibrational Spectroscopy of Diatomic & Polyatomic Molecules

For small displacements \(q = r - r_e\) around equilibrium bond length \(r_e\), the molecular potential energy can be expanded in a Taylor series: \(V(q) = V(0) + \left(\frac{dV}{dq}\right)_0 q + \frac{1}{2}\left(\frac{d^2V}{dq^2}\right)_0 q^2 + \dots\). Taking \(V(0) = 0\) and noting that \((dV/dq)_0 = 0\) at equilibrium, the harmonic approximation retains only the quadratic term: \(V(q) = \frac{1}{2} k q^2\), where \(k = \left(\frac{d^2V}{dq^2}\right)_0\) is the bond force constant.

The time-independent Schrödinger equation for the harmonic oscillator is:

\[ -\frac{\hbar^2}{2\mu}\frac{d^2\psi_v}{dq^2} + \frac{1}{2} k q^2 \psi_v = E_v \psi_v \]

The normalized eigenfunctions are expressed in terms of Hermite polynomials \(H_v(\xi)\):

\[ \psi_v(q) = \left(\frac{\alpha}{\pi}\right)^{1/4} \frac{1}{\sqrt{2^v v!}} H_v(\sqrt{\alpha} q) e^{-\alpha q^2 / 2}, \quad \alpha = \frac{\sqrt{\mu k}}{\hbar} = \frac{\mu \omega}{\hbar} \]

The quantized energy eigenvalues are:

\[ E_v = \hbar \omega \left(v + \frac{1}{2}\right) = h c \bar{\omega}_e \left(v + \frac{1}{2}\right), \quad v = 0, 1, 2, \dots \]

where the harmonic vibrational wavenumber is \(\bar{\omega}_e = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}\). Even at absolute zero (\(T = 0\text{ K}\)), the molecule possesses zero-point energy (ZPE): \(E_0 = \frac{1}{2}\hbar\omega = \frac{1}{2}hc\bar{\omega}_e\), a direct consequence of the Heisenberg uncertainty principle.

The transition dipole moment integral between vibrational states \(|v\rangle\) and \(|v'\rangle\) is \(\mu_{v'v} = \langle \psi_{v'} | \hat{\mu}(q) | \psi_v \rangle\). Expanding the molecular dipole moment as a Taylor series about equilibrium:

\[ \hat{\mu}(q) = \mu_0 + \left(\frac{d\mu}{dq}\right)_0 q + \frac{1}{2}\left(\frac{d^2\mu}{dq^2}\right)_0 q^2 + \dots \]

Substituting this into the transition moment:

\[ \mu_{v'v} = \mu_0 \langle \psi_{v'} | \psi_v \rangle + \left(\frac{d\mu}{dq}\right)_0 \langle \psi_{v'} | q | \psi_v \rangle + \dots \]

By orthogonality of Hermite functions, \(\langle \psi_{v'} | \psi_v \rangle = 0\) for \(v' \neq v\). Therefore:

1. Gross Selection Rule: The dipole moment must vary with bond displacement: \(\left(\frac{d\mu}{dq}\right)_0 \neq 0\). Homonuclear diatomics have zero dipole derivative at all distances and are strictly infrared inactive. Heteronuclear diatomics (\(\text{HCl}, \text{CO}, \text{NO}\)) have non-zero dipole derivatives and are infrared active.
2. Specific Selection Rule: Using the Hermite polynomial recurrence relation \(q |v\rangle = \sqrt{\frac{\hbar}{2\mu\omega}} (\sqrt{v+1}|v+1\rangle + \sqrt{v}|v-1\rangle)\), the transition moment is non-zero only when: \[ \Delta v = v' - v = \pm 1 \]

In the harmonic approximation, all transitions occur at the exact same wavenumber \(\tilde{\nu} = \bar{\omega}_e\).

### Advanced Mathematical Formulation: Dunham Expansion and the RKR Potential Inversion Beyond simple second-order Morse potential expansions, high-resolution rovibrational transitions spanning dozens of vibrational levels require Dunham's semiclassical representation of diatomic energy levels. #### 1. The Dunham Series Expansion J. L. Dunham expressed the rovibrational term values \(T(v, J)\) as a double power series in \((v + 1/2)\) and \(J(J+1)\): \[ T(v, J) = \sum_{l=0}^\infty \sum_{j=0}^\infty Y_{lj} \left( v + \frac{1}{2} \right)^l [J(J+1)]^j \] where \(Y_{lj}\) are the Dunham coefficients. The leading Dunham coefficients relate directly to standard spectroscopic constants: \[ Y_{10} \approx \omega_e + \frac{B_e^3}{4\omega_e^2} (a_1^2 \dots) \approx \omega_e \] \[ Y_{20} \approx -\omega_e x_e \] \[ Y_{01} \approx B_e \] \[ Y_{11} \approx -\alpha_e \] \[ Y_{02} \approx -D_e = -\frac{4 B_e^3}{\omega_e^2} \] #### 2. The Dimensionless Born-Oppenheimer Potential The internuclear potential is written in terms of the dimensionless reduced coordinate \(\xi = \frac{R - R_e}{R_e}\): \[ V(\xi) = a_0 \xi^2 \left( 1 + a_1 \xi + a_2 \xi^2 + a_3 \xi^3 + \dots \right) \] where \(a_0 = \frac{\omega_e^2}{4 B_e}\). Dunham derived explicit analytical expressions connecting the potential coefficients \(a_k\) to the spectroscopic constants: \[ a_1 = -1 - \frac{\alpha_e \omega_e}{6 B_e^2} \] \[ a_2 = \frac{5}{4} a_1^2 - \frac{2 \omega_e x_e}{3 B_e} \] #### 3. Semiclassical Rydberg-Klein-Rees (RKR) Inversion The RKR method utilizes the Bohr-Sommerfeld WKB quantization condition to directly determine the classical turning points \(R_{\text{min}}(v)\) and \(R_{\text{max}}(v)\) from experimental \(G(v)\) and \(B_v\) data without assuming an analytical potential shape: \[ R_{\text{max}}(v) - R_{\text{min}}(v) = 2 f(v) = 2 \sqrt{\frac{\hbar}{2\pi \mu c}} \int_{-1/2}^v \frac{dv'}{\sqrt{G(v) - G(v')}} \] \[ \frac{1}{R_{\text{min}}(v)} - \frac{1}{R_{\text{max}}(v)} = 2 g(v) = 2 \sqrt{\frac{2\pi \mu c}{\hbar}} \int_{-1/2}^v \frac{B_{v'}}{\sqrt{G(v) - G(v')}} dv' \] Solving these two simultaneous algebraic relations gives: \[ R_{\text{min}}(v) = \sqrt{f(v)^2 + \frac{f(v)}{g(v)}} - f(v) \] \[ R_{\text{max}}(v) = \sqrt{f(v)^2 + \frac{f(v)}{g(v)}} + f(v) \] This inversion algorithm constructs the true quantum potential energy curve \(V(R)\) directly from high-resolution IR Fourier-transform spectra.

Real chemical bonds deviate significantly from harmonic behavior at larger displacements: the potential energy steepens at short distances due to core electron repulsion and flattens asymptotically to the bond dissociation limit at large internuclear separations. The Morse potential accurately models this physical behavior:

\[ V(r) = D_e \left[ 1 - e^{-a(r - r_e)} \right]^2 \]

where \(D_e\) is the spectroscopic dissociation energy relative to the potential minimum, and \(a = \sqrt{\frac{k_e}{2 D_e}}\) governs the curvature of the well.

Anharmonic Vibrational Term Values

Solving the Schrödinger equation with the Morse potential yields the vibrational term values:

\[ G(v) = \bar{\omega}_e \left(v + \frac{1}{2}\right) - \bar{\omega}_e x_e \left(v + \frac{1}{2}\right)^2 + \bar{\omega}_e y_e \left(v + \frac{1}{2}\right)^3 + \dots \]

where \(x_e = \frac{h c \bar{\omega}_e}{4 D_e}\) is the dimensionless anharmonicity constant. Anharmonicity has profound physical consequences:

• Energy level spacing contracts with increasing \(v\): \(\Delta G_{v+1/2} = G(v+1) - G(v) = \bar{\omega}_e - 2\bar{\omega}_e x_e(v + 1)\).
• The harmonic selection rule \(\Delta v = \pm 1\) breaks down: weak overtone transitions become allowed:
  • Fundamental (\(v = 0 \to 1\)): \(\tilde{\nu}_1 = G(1) - G(0) = \bar{\omega}_e(1 - 2x_e)\)
  • First Overtone (\(v = 0 \to 2\)): \(\tilde{\nu}_2 = G(2) - G(0) = 2\bar{\omega}_e(1 - 3x_e)\)
  • Second Overtone (\(v = 0 \to 3\)): \(\tilde{\nu}_3 = G(3) - G(0) = 3\bar{\omega}_e(1 - 4x_e)\)
• Hot bands (\(v = 1 \to 2\)) appear at elevated temperatures at slightly lower frequencies: \(\tilde{\nu}_{\text{hot}} = \bar{\omega}_e(1 - 4x_e)\).

The Birge-Sponer extrapolation provides an elegant experimental methodology for determining the bond dissociation energy \(D_e\) directly from observed vibrational line spacings.

Linear Birge-Sponer Formulation

The separation between consecutive vibrational levels is:

\[ \Delta G_{v+1/2} = G(v+1) - G(v) = \bar{\omega}_e - 2\bar{\omega}_e x_e (v + 1) \]

Plotting \(\Delta G_{v+1/2}\) linearly against \((v+1)\) produces a straight line with intercept \(\bar{\omega}_e\) and negative slope \(-2\bar{\omega}_e x_e\). The dissociation limit occurs at maximum quantum number \(v_{\max}\) where the energy difference reaches zero: \(\Delta G_{v_{\max}+1/2} = 0\):

\[ v_{\max} + 1 = \frac{\bar{\omega}_e}{2\bar{\omega}_e x_e} = \frac{1}{2x_e} \]

The total dissociation energy \(D_e\) is the area under the Birge-Sponer curve:

\[ D_e = \sum_{v=0}^{v_{\max}} \Delta G_{v+1/2} \approx \int_0^{v_{\max}+1} [\bar{\omega}_e - 2\bar{\omega}_e x_e (v+1)] d(v+1) = \frac{\bar{\omega}_e^2}{4\bar{\omega}_e x_e} \]

The ground state chemical dissociation energy \(D_0\) accounts for zero-point energy:

\[ D_0 = D_e - G(0) = D_e - \left(\frac{1}{2}\bar{\omega}_e - \frac{1}{4}\bar{\omega}_e x_e\right) \]

In the gas phase, molecules undergo simultaneous vibration and rotation. In the Born-Oppenheimer approximation, the total rovibrational term value is the sum of vibrational and rotational energies:

\[ T(v, J) = G(v) + F(J) = \left[\bar{\omega}_e\left(v + \frac{1}{2}\right) - \bar{\omega}_e x_e\left(v + \frac{1}{2}\right)^2\right] + B_v J(J+1) \]

For a heteronuclear diatomic molecule in a \(^1\Sigma\) electronic ground state (zero electronic orbital angular momentum), the selection rules are:

\[ \Delta v = \pm 1, \quad \Delta J = \pm 1 \quad (\Delta J = 0 \text{ is forbidden!}) \]

R-Branch and P-Branch Manifolds

Consider the fundamental absorption transition from \(v = 0\) to \(v = 1\), with band origin \(\tilde{\nu}_0 = G(1) - G(0)\):

1. R-Branch (\(\Delta J = +1\), \(J' = J'' + 1\)): \[ \tilde{\nu}_R(J'') = \tilde{\nu}_0 + F_1(J''+1) - F_0(J'') = \tilde{\nu}_0 + 2B_1 + (3B_1 - B_0)J'' + (B_1 - B_0)J''^2 \] If \(B_1 \approx B_0 = B\), this simplifies to: \(\tilde{\nu}_R(J'') = \tilde{\nu}_0 + 2B(J''+1)\) for \(J'' = 0, 1, 2, \dots\). Lines appear at \(\tilde{\nu}_0 + 2B, \tilde{\nu}_0 + 4B, \tilde{\nu}_0 + 6B, \dots\).
2. P-Branch (\(\Delta J = -1\), \(J' = J'' - 1\)): \[ \tilde{\nu}_P(J'') = \tilde{\nu}_0 + F_1(J''-1) - F_0(J'') = \tilde{\nu}_0 - (B_1 + B_0)J'' + (B_1 - B_0)J''^2 \] If \(B_1 \approx B_0 = B\), this simplifies to: \(\tilde{\nu}_P(J'') = \tilde{\nu}_0 - 2B J''\) for \(J'' = 1, 2, 3, \dots\). Lines appear at \(\tilde{\nu}_0 - 2B, \tilde{\nu}_0 - 4B, \tilde{\nu}_0 - 6B, \dots\).

Because \(\Delta J = 0\) is forbidden, no absorption line appears at the exact band origin \(\tilde{\nu}_0\). The separation between the first R-line (\(R(0) = \tilde{\nu}_0 + 2B\)) and the first P-line (\(P(1) = \tilde{\nu}_0 - 2B\)) is \(4B\), leaving a prominent zero-gap in the center of the band.

In real molecules, the average bond length in the excited vibrational state \(v=1\) is slightly larger than in the ground state \(v=0\) due to anharmonicity of the Morse potential: \(\langle r \rangle_{v=1} > \langle r \rangle_{v=0}\). Consequently, the moment of inertia increases and the rotational constant decreases in the upper vibrational state:

\[ B_v = B_e - \alpha_e \left(v + \frac{1}{2}\right) \]

where \(B_e\) is the equilibrium rotational constant and \(\alpha_e\) is the vibration-rotation coupling constant (\(\alpha_e > 0\)). Therefore, \(B_1 < B_0\), and \(B_1 - B_0 = -\alpha_e\).

Line Spacing Asymmetry & Band Head Formation

Incorporating \(\alpha_e\), the rovibrational transition frequencies become:

\[ \tilde{\nu}_R(J) = \tilde{\nu}_0 + (2B_e - 3\alpha_e) + (2B_e - 4\alpha_e)J - \alpha_e J^2 \] \[ \tilde{\nu}_P(J) = \tilde{\nu}_0 - (2B_e - 2\alpha_e)J - \alpha_e J^2 \]

Notice the quadratic term \(-\alpha_e J^2\):

• In the R-branch, the quadratic term opposes the linear term. As \(J\) increases, the spacing between consecutive R-lines progressively contracts, eventually converging to a reversal point known as a band head where \(\frac{d\tilde{\nu}_R}{dJ} = 0\).
• In the P-branch, the quadratic term adds to the linear term in the negative direction, causing the spacing between consecutive P-lines to steadily widen with increasing \(J\).

The fundamental infrared absorption band of carbon monoxide (\(^{12}\text{C}^{16}\text{O}\)) centered at \(\tilde{\nu}_0 = 2143.27\text{ cm}^{-1}\) serves as the quintessential benchmark for high-resolution rovibrational spectroscopy.

Method of Combination Differences

To extract \(B_0\) and \(B_1\) independently without relying on assumptions about the band origin \(\tilde{\nu}_0\), spectroscopists use combination differences:

1. Common Lower State Combination Differences (\(\Delta_2 F''(J)\)): Consider transitions starting from the same lower state \(J''\) and terminating at \(J'+1\) (via R-branch) and \(J'-1\) (via P-branch): \[ \Delta_2 F'(J) = R(J) - P(J) = F_1(J+1) - F_1(J-1) = 4B_1\left(J + \frac{1}{2}\right) \] Plotting \(R(J) - P(J)\) against \((J + 1/2)\) gives a straight line with slope \(4B_1\).
2. Common Upper State Combination Differences (\(\Delta_2 F''(J)\)): Consider transitions terminating on the same upper state \(J'\) originating from \(J''-1\) and \(J''+1\): \[ \Delta_2 F''(J) = R(J-1) - P(J+1) = F_0(J+1) - F_0(J-1) = 4B_0\left(J + \frac{1}{2}\right) \] Plotting \(R(J-1) - P(J+1)\) against \((J + 1/2)\) gives a straight line with slope \(4B_0\).

For \(^{12}\text{C}^{16}\text{O}\), this analysis yields \(B_0 = 1.9225\text{ cm}^{-1}\), \(B_1 = 1.9050\text{ cm}^{-1}\), \(\alpha_e = 0.0175\text{ cm}^{-1}\), and equilibrium bond length \(r_e = 112.83\text{ pm}\).

### Advanced Research Monograph: Ultrafast Two-Dimensional Infrared (2D-IR) Spectroscopy While linear FTIR measures time-averaged vibrational transitions, ultrafast 2D-IR spectroscopy provides structural correlations and sub-picosecond dynamical timescales analogous to 2D NMR, but with a \(10^9\times\) faster temporal window. #### 1. Three-Pulse Coherent Infrared Photon Echo 2D-IR employs three femtosecond mid-IR laser pulses (\(\sim 100\text{ fs}\) duration) in a non-collinear boxcar geometry: \[ \text{Pulse 1 } (t = 0) \xrightarrow{\tau} \text{Pulse 2 } (t = \tau) \xrightarrow{T_w} \text{Pulse 3 } (t = \tau + T_w) \xrightarrow{t} \text{Echo Detection} \] - **Coherence time (\(\tau\)):** Encodes the initial excitation frequency \(\omega_{\text{pump}}\). - **Waiting time (\(T_w\)):** Population time during which structural evolution, vibrational energy transfer, and chemical exchange take place. - **Detection time (\(t\)):** Radiates the four-wave mixing third-order nonlinear polarization \(\mathbf{P}^{(3)}(t)\), detected via spectral interferometry to yield the probe frequency \(\omega_{\text{probe}}\). #### 2. Spectral Signatures in 2D-IR Contour Plots At each waiting time \(T_w\), a 2D plot of \(\omega_{\text{probe}}\) versus \(\omega_{\text{pump}}\) displays two distinct features for each vibrational mode: 1. **Ground State Bleach / Stimulated Emission (Diagonal Peak, negative sign):** Located at \(\omega_{\text{pump}} = \omega_{01}\) and \(\omega_{\text{probe}} = \omega_{01}\). 2. **Excited-State Absorption (Off-diagonal / Anharmonic Shift, positive sign):** Corresponds to the \(v = 1 \rightarrow 2\) transition, shifted downwards along the probe axis by the vibrational mechanical anharmonicity: \[ \Delta = \omega_{01} - \omega_{12} = 2 \omega_e x_e \] #### 3. Spectral Diffusion and Hydrogen-Bond Dynamics At \(T_w = 0\), solvent fluctuations cause inhomogeneous broadening, creating an elongated elliptical diagonal peak. - As \(T_w\) increases, molecular reorientation and rapid hydrogen-bond breaking/re-forming randomize the local electrostatic field felt by the oscillator. - This **spectral diffusion** causes the 2D contour to round into a symmetric circle. - The decay rate of the peak ellipse eccentricity (Center Line Slope, CLS) directly quantifies the frequency-frequency correlation function (FFCF) \(\langle \delta\omega(t)\delta\omega(0)\rangle\), revealing that water hydrogen-bond networks rearrange on a blistering timescale of \(\sim 1.5\text{ ps}\).

For a non-linear polyatomic molecule with \(N\) atoms, there exist \(3N - 6\) independent vibrational degrees of freedom (\(3N - 5\) for linear molecules). E. Bright Wilson formalized the exact mathematical treatment of polyatomic vibrations using the Wilson FG matrix method.

The Wilson FG Secular Matrix Equation

Internal coordinates \(S\) (bond stretches \(\Delta r\), valence angle bends \(\Delta\theta\), out-of-plane wags \(\Delta\gamma\), and torsions \(\Delta\tau\)) are related to Cartesian displacements through kinetic energy matrix \(\mathbf{G}\) and potential energy matrix \(\mathbf{F}\):

\[ 2 T = \dot{\mathbf{S}}^T \mathbf{G}^{-1} \dot{\mathbf{S}}, \quad 2 V = \mathbf{S}^T \mathbf{F} \mathbf{S} \]

The vibrational frequencies \(\lambda_k = 4\pi^2 c^2 \tilde{\nu}_k^2\) are the eigenvalues of the secular determinant:

\[ |\mathbf{F}\mathbf{G} - \lambda \mathbf{E}| = 0 \]

where \(\mathbf{G}\) elements depend purely on atomic masses and equilibrium molecular geometry, while \(\mathbf{F}\) contains the quadratic harmonic force constants (diagonal stretching force constants \(f_r\), bending force constants \(f_\theta\), and off-diagonal stretch-bend interaction constants).

Diagnostic Characteristic Infrared Group Frequencies

Because many chemical functional groups involve high force constants or light atoms (such as \(\text{-H}\)), their vibrational modes are mechanically decoupled from the rest of the molecular framework, producing highly characteristic absorption bands regardless of the surrounding molecular structure:

The region below \(1500\text{ cm}^{-1}\), designated the fingerprint region, contains complex coupled skeletal vibrations unique to each individual molecule.