• Good Books:
  • Chapter from a bioinorganic chemistry book:
    • PDF Link
  • Peter Atkins Physical Chemistry QMC

Basis

• Photons carry information in both Angular momentum by the de Broglie relation: $p=\frac{h}{\lambda }$.
  • A photon has both an electric moment and magnetic moment that rotate about the axis of propagation.
  • When photons interact with a substance, they transfer angular momentum to the nuclei and electrons. Assuming the Born-Oppenheimer approximation holds, electrons are primarily affected by these momentum transfers (the Heller formalism contradicts this).
  • The oscillating motion of electrons results in the generation of electromagnetic radiation at the same frequency used to excite them.
  • In IR Spectroscopy, the energy of a photon is absorbed, exciting the molecule from the lowest vibrational state (${\psi }_i$) to a higher vibrational state (${\psi }_f$). The photon's energy must match the energy difference between these states (resonance condition).
    • A transition requires a change in dipole moment between states, which is the transition dipole moment ($\overrightarrow{\mu }$) given by $\hat{\mu }$.

The Classical Oscillator and Quantization

• A chemical bond can be modeled as two point masses connected by a spring.
• Hooke's law describes this as:$${\nu }_{osc}=\frac{1}{2\pi }\sqrt{\frac{k}{\mu }}\mu =\frac{m_1{m}_2}{m_1+m_2}$$
  • The restoring force: $\kappa =-k\mathrm{\Delta }R$, where $R$ is the bond length.
  • If a boundary is enforced on a molecule (like a bond), it becomes quantized. The more narrow the well of quantization, the more quantized it becomes.
    • Transitions between states can be described as $⟨f|\hat{\mu }|i⟩$, where $\hat{\mu }$ is the dipole moment operator.
  • A quantum mechanical oscillator can only exist in discrete states:
  $$E_{vib}=h{\nu }_{osc}(v+\frac{1}{2})=\frac{h}{2\pi }\sqrt{\frac{k}{m_{eff}}}(v+\frac{1}{2})v=0,1,2,\dots$$
  • Solving the time-independent Schrodinger equation gives the total energy of an oscillator ($E_{vib}$) as a sum of kinetic and potential energy:
  $$\frac{-{\hslash }^2}{2m_{eff}}\frac{d^2\psi }{d{x}^2}+\frac{1}{2}{k}_f{x}^2\psi =E\psi$$
  • The potential can be approximated with a Taylor series:
  $$V_x=V_{(x=0)}+{\left(\frac{dV}{dx}\right)}_{(x=0)}x+\frac{1}{2}{\left(\frac{d^{2}V}{d{x}^2}\right)}_{(x=0)}{x}^2+\dots$$
  • Near the equilibrium bond length, the potential well approximates a parabola:
  $$k={\left(\frac{d^{2}V}{d{x}^2}\right)}_{(x=0)}$$
  • The potential well is anharmonic, but the model of a harmonic oscillator holds well for small vibrations.

Dipole Moments

• Electric dipole moments:

  • A dipole forms when two monopoles of opposite charge are in proximity. The strength of the Dipole moment is given by:
  $$\mu =qR$$
  • The electric field generated by the charges is:
  $$\overrightarrow{E}=\frac{1}{4\pi {ϵ}_0}\frac{\overrightarrow{\mu }}{R^2}$$
  • The dipole moment operator is:
  $$\hat{\mu }={\mu }_{(x=0)}+{\left(\frac{d\mu }{dx}\right)}_{(x=0)}x+\frac{1}{2}{\left(\frac{d^2\mu }{d{x}^2}\right)}_{(x=0)}{x}^2+\dots$$

• IR Spectroscopy:

  • Transition between states is described by the transition dipole moment:
  $${\mu }_{fi}=⟨vf|\hat{\mu }|vi⟩$$
  • Replacing $\hat{\mu }$ with its expansion:
  $${\mu }_{fi}=⟨vf|({\mu }_{(x=0)}+{\left(\frac{d\mu }{dx}\right)}_{(x=0)}x+\frac{1}{2}{\left(\frac{d^2\mu }{d{x}^2}\right)}_{(x=0)}{x}^2)|vi⟩$$
  • The selection rule for IR transitions is $\mathrm{\Delta }v=\pm 1$.

• Raman Spectroscopy:

  • Polarizability ($\alpha$) measures how a molecule responds to an oscillating electric field.
  • The Induced dipole moment operator is:
  $$\widehat{{\mu }_{in}}={\alpha }_x\overrightarrow{E_t}$$
  • The selection rule for Raman scattering is also $\mathrm{\Delta }v=\pm 1$, but unlike IR, it depends on the change in polarizability.