Molecular Physics, Chemical Bonding & Molecular Spectroscopy

1. Classification of Molecular Chemical Bonds

Molecules are stable aggregates of two or more atoms bound together by electromagnetic interactions. Chemical bonds are categorized by their physical bonding mechanisms:

• Ionic Bonds: Formed by complete electrostatic electron transfer between atoms of widely differing electronegativities (e.g., $\text{NaCl}, \text{KBr}$). The cohesive energy is governed by Coulomb attraction balanced by Pauli short-range electron core repulsion:
  $$U(R) = - \frac{e^2}{4\pi\varepsilon_0 R} + \frac{A}{R^n} \quad (n \approx 8\text{ to }10)$$
• Covalent Bonds: Formed between atoms of similar electronegativities by the quantum mechanical sharing of valence electrons (e.g., $\text{H}_2, \text{O}_2, \text{CH}_4$). The shared electron density accumulates in the internuclear region, electrostatically screening the positive nuclei from mutual repulsion and lowering total quantum energy.
• Van der Waals Bonds: Weak intermolecular bonds ($0.01\text{ to }0.1\,\text{eV}$) arising from fluctuating dipole-induced dipole electrostatic attractions ($U(R) \propto -1/R^6$).
• Hydrogen Bonds: Intermediate dipole-dipole attractions ($0.1\text{ to }0.5\,\text{eV}$) formed between an electropositive hydrogen atom covalently bound to an electronegative atom (N, O, F) and an adjacent lone pair.

2. The Molecular Orbital Concept & LCAO Approximation

In molecular orbital theory, electrons are not confined to individual atoms, but occupy delocalized Molecular Orbitals (MOs) extending across the entire molecule.

The simplest mathematical approximation is the Linear Combination of Atomic Orbitals (LCAO). For a diatomic molecule with nuclei $A$ and $B$, a molecular orbital $\psi_{MO}$ is constructed from atomic orbitals $\phi_A$ and $\phi_B$:

For homonuclear diatomics ($c_A = \pm c_B$), this produces two distinct spatial distributions:

1. Bonding Molecular Orbital ($\sigma_g$): Symmetric linear combination $\psi_+ = N_+ (\phi_A + \phi_B)$. Electron density builds up constructively between the nuclei, lowering the electrostatic potential energy and forming a stable chemical bond.
2. Antibonding Molecular Orbital ($\sigma_u^*$): Antisymmetric linear combination $\psi_- = N_- (\phi_A - \phi_B)$. A nodal plane where $\psi = 0$ exists midway between the nuclei. Electron density is pushed away from the internuclear region, producing net nuclear repulsion.

1. The Hydrogen Molecular Ion ($H_2^+$)

The simplest molecular system in nature is the hydrogen molecular ion $H_2^+$, consisting of two positive protons ($A$ and $B$) separated by internuclear distance $R$, and a single electron.

The electronic Hamiltonian in atomic units ($\hbar = m_e = e = 1$) is:

Using the LCAO trial functions formed from hydrogen $1s$ orbitals $\phi_A$ and $\phi_B$:

where the Overlap Integral $S$ is:

The electronic expectation energy $E_{\pm}(R) = \langle \psi_{\pm} | \hat{H} | \psi_{\pm} \rangle$ evaluates to:

where $J$ is the Coulomb Integral (classical electrostatic interaction between the charge cloud around nucleus $A$ and nucleus $B$) and $K$ is the Exchange (Resonance) Integral (representing quantum electron tunneling between the two protons):

Because $K$ is negative, the symmetric state $E_+(R)$ develops a pronounced potential energy minimum at equilibrium bond length $R_e = 1.06\,\text{\AA}$ with a binding dissociation energy $D_e = 2.79\,\text{eV}$.

2. The Neutral Hydrogen Molecule ($H_2$) & Heitler-London Theory

In the neutral hydrogen molecule ($H_2$), there are two electrons. By the Pauli exclusion principle, the total two-electron wavefunction must be antisymmetric under exchange of electrons $1$ and $2$:

• Singlet Ground State ($S = 0$, Antiparallel Spins): The spin state is antisymmetric $\chi_{singlet} = \frac{1}{\sqrt{2}}(\alpha_1 \beta_2 - \beta_1 \alpha_2)$. Consequently, the spatial wavefunction must be symmetric:
  $$\psi_S(\vec{r}_1, \vec{r}_2) = \frac{1}{\sqrt{2(1 + S^2)}} [ \phi_A(1) \phi_B(2) + \phi_B(1) \phi_A(2) ]$$
  This generates a deep potential well with equilibrium bond length $R_e = 0.74\,\text{\AA}$ and strong covalent dissociation energy $D_e = 4.75\,\text{eV}$.
• Triplet Excited State ($S = 1$, Parallel Spins): The spin state is symmetric, requiring an antisymmetric spatial function $\psi_T \propto [\phi_A(1)\phi_B(2) - \phi_B(1)\phi_A(2)]$. Spatial electron density vanishes between the nuclei, producing purely repulsive forces for all $R$.

1. The Born-Oppenheimer Approximation (1927)

Max Born and J. Robert Oppenheimer recognized that atomic nuclei are vastly more massive than electrons ($M_{nucleus} / m_e \sim 1836\text{ to }10^5$). Consequently, electrons move at velocities hundreds of times faster than nuclei.

On the timescale of electronic orbital motion, the heavy nuclei can be treated as essentially stationary fixed points in space. The full molecular Hamiltonian separates into:

1. An Electronic Schrödinger Equation solved at fixed nuclear configurations $R$, generating potential energy curves $V_{el}(R)$.
2. A Nuclear Schrödinger Equation governing the vibrational and rotational motions of the nuclei moving on the effective potential energy surface $V_{el}(R)$.

2. Molecular Energy Hierarchy

To excellent approximation, the total internal energy of a molecule is the sum of three independent contributions:

Because $\Delta E_{rot} \ll \Delta E_{vib} \ll \Delta E_{el}$, molecular spectra exhibit fine structure: electronic bands contain closely spaced vibrational progressions, which in turn contain ultra-dense rotational lines.

1. The Rigid Rotor Model of a Diatomic Molecule

Consider a diatomic molecule consisting of two masses $m_1$ and $m_2$ separated by a fixed equilibrium bond length $r_0$. The classical moment of inertia about the center of mass axis is:

The classical kinetic energy of rotation is $E = \frac{L^2}{2 I}$. In quantum mechanics, orbital angular momentum is quantized: $L^2 = J(J+1) \hbar^2$, where $J = 0, 1, 2, 3, \dots$ is the Rotational Quantum Number.

The quantized rotational energy levels are:

where the Rotational Constant $B$ (expressed in wavenumber units $\text{cm}^{-1}$) is:

2. Selection Rules & Rotational Spectra

For an electric dipole transition to occur in pure rotational spectroscopy:

1. Gross Selection Rule: The molecule must possess a permanent electric dipole moment ($\mu_{el} \neq 0$). Homonuclear molecules ($\text{H}_2, \text{N}_2, \text{O}_2$) have zero dipole moment and are completely microwave inactive. Heteronuclear molecules ($\text{CO}, \text{HCl}, \text{NO}$) have permanent dipoles and exhibit intense rotational absorption.
2. Specific Selection Rule:
   $$\Delta J = \pm 1 \quad (+1 \text{ for absorption, } -1 \text{ for emission})$$

The wavenumber of the transition from level $J$ to $J + 1$ is:

Evaluating for consecutive transitions:

• $J = 0 \to 1$: $\bar{\nu} = 2B$
• $J = 1 \to 2$: $\bar{\nu} = 4B$
• $J = 2 \to 3$: $\bar{\nu} = 6B$
• $J = 3 \to 4$: $\bar{\nu} = 8B$

Fundamental Experimental Signature: The pure rotational absorption spectrum consists of a series of equidistant spectral lines separated by constant spacing $2B$:

By measuring this line separation $\Delta \bar{ u} = 2B$ in the microwave laboratory, one calculates the moment of inertia $I = \frac{h}{8\pi^2 c B}$ and determines the internuclear bond distance $r_0 = \sqrt{I / \mu}$ to four decimal places of precision!

1. Harmonic vs. Anharmonic Morse Potential

Near equilibrium separation $r_0$, the molecular potential energy can be approximated as a simple harmonic oscillator with bond force constant $k$:

The quantized vibrational energy levels are:

where $\bar{\nu}_0 = \frac{1}{2\pi c} \sqrt{\frac{k}{\mu}}$. Even in the ground state ($v = 0$), the molecule possesses irreducible zero-point vibrational energy $E_0 = \frac{1}{2} \hbar \omega_0$.

Real chemical bonds dissociate at large separations. A far more realistic model is the Morse Potential:

where $D_e$ is the depth of the potential well. The energy levels of an anharmonic Morse oscillator are:

where $x_e$ is the anharmonicity constant. Anharmonicity relaxes the strict harmonic selection rule $\Delta v = \pm 1$, permitting weaker overtone transitions ($\Delta v = \pm 2, \pm 3$).

2. Vibration-Rotation Spectra & P/R Branch Architecture

Because rotational energy levels are densely packed within each vibrational state, a vibrational transition ($v = 0 \to 1$) is always accompanied by simultaneous rotational transitions ($J \to J'$). The combined energy of a vibration-rotation state is:

For heteronuclear diatomic molecules with zero electronic angular momentum ($\Sigma$ states), the selection rules are:

The resulting spectrum splits into two distinct symmetric branches flanking the missing fundamental vibrational frequency $\bar{ u}_0$:

1. The $R$-Branch ($\Delta J = +1$, $J' = J + 1$): Rotational energy increases during vibrational absorption:
   $$\bar{\nu}_R(J) = \bar{\nu}_0 + B(J+1)(J+2) - B J(J+1) = \bar{\nu}_0 + 2 B (J + 1) \quad (J = 0, 1, 2, \dots)$$
   Lines appear at higher frequencies: $\bar{\nu}_0 + 2B, \bar{\nu}_0 + 4B, \bar{\nu}_0 + 6B, \dots$
2. The $P$-Branch ($\Delta J = -1$, $J' = J - 1$): Rotational energy decreases:
   $$\bar{\nu}_P(J) = \bar{\nu}_0 + B(J-1)J - B J(J+1) = \bar{\nu}_0 - 2 B J \quad (J = 1, 2, 3, \dots)$$
   Lines appear at lower frequencies: $\bar{\nu}_0 - 2B, \bar{\nu}_0 - 4B, \bar{\nu}_0 - 6B, \dots$
3. The Missing $Q$-Branch ($\Delta J = 0$): A line at $\bar{ u} = \bar{ u}_0$ would correspond to $\Delta J = 0$. Because $\Delta J = 0$ is forbidden, there is a distinct central gap of width $4B$ at the fundamental origin!

1. Electronic Transitions & The Franck-Condon Principle

Electronic transitions involve major rearrangements of electron clouds, shifting the equilibrium internuclear distance from $r_0$ to $r_0'$.

The Franck-Condon Principle states: Because atomic nuclei are vastly heavier than electrons, an electronic transition occurs so rapidly ($\sim 10^{-15}\text{ s}$) that the nuclei do not have time to change their positions or momenta during the transition.

On a potential energy diagram, electronic transitions are represented as strictly vertical lines. The transition probability (intensity) between vibrational state $v$ in the ground electronic state and $v'$ in the excited state is proportional to the Franck-Condon Factor—the square of the vibrational overlap integral:

2. The Raman Effect: Classical Polarizability Model (1928)

Sir C.V. Raman discovered that when a transparent substance is irradiated with intense monochromatic light of frequency $ u_0$, a small fraction ($\sim 10^{-6}$) of the scattered radiation emerges with altered frequencies ($ u_0 \pm u_v$).

Classically, the incident electric field $\vec{E}(t) = \vec{E}_0 \cos(2\pi u_0 t)$ induces an electric dipole moment in the molecule:

where $\alpha$ is the molecular polarizability. As the molecule vibrates with natural frequency $ u_v$, its polarizability fluctuates periodically:

Substituting $\alpha(t)$ into the induced dipole equation:

The oscillating dipole radiates electromagnetic waves at three distinct frequencies:

1. Rayleigh Scattering ($\nu_0$): Elastic scattering at the unshifted incident frequency.
2. Stokes Lines ($\nu_0 - \nu_v$): Inelastic scattering red-shifted to lower frequency. The incident photon transfers energy to excite a molecular vibration.
3. Anti-Stokes Lines ($\nu_0 + \nu_v$): Inelastic scattering blue-shifted to higher frequency. The incident photon absorbs energy from an already-vibrating molecule.

3. Quantum Interpretation & The Rule of Mutual Exclusion

In quantum theory, an incident photon of energy $h u_0$ promotes the molecule to a transient virtual state:

• If it de-excites to a higher vibrational level ($v = 0 \to 1$), the scattered photon carries reduced energy $h(\nu_0 - \nu_v)$ (Stokes).
• If it de-excites from an initial excited level to the ground level ($v = 1 \to 0$), the scattered photon carries increased energy $h(\nu_0 + \nu_v)$ (Anti-Stokes).

Because the thermal population of the excited state $v = 1$ is governed by the Boltzmann distribution $N_1 / N_0 = e^{-h u_v / k_B T} \ll 1$, Stokes lines are always vastly more intense than Anti-Stokes lines:

The Rule of Mutual Exclusion: For molecules with a center of inversion symmetry (such as $\text{CO}_2, \text{C}_2\text{H}_4, \text{N}_2$), vibrations that are Infrared active (change in dipole moment, $\partial\mu/\partial q \neq 0$) are Raman inactive, and vibrations that are Raman active (change in polarizability, $\partial\alpha/\partial q \neq 0$) are Infrared inactive.