Intro to Quantum Chem

Introduction to Quantum Chemistry

• Prepared by: IC Duay, PhD
• References:
  • McQuarrie, D. A. (2008). Quantum Chemistry (2nd ed.). University Science Books.
  • Levine, I. N. (2014). Quantum Chemistry (7th ed.). Pearson Education, Inc.

Quantum Mechanics

• Mechanics is the study of motion and forces acting on objects.
• Classical Mechanics:
  • Deals with macroscopic objects.
  • Unable to accurately describe behavior at the atomic level.
• Quantum Mechanics:
  • Addresses the behavior of very small particles (e.g., electrons, atoms).
  • Quantum chemistry applies quantum mechanics to chemical problems.

Comparison of Classical and Quantum Mechanics

• Classical Mechanics:
  • Large, heavy, continuous objects.
  • Uses Newton's equations, deterministic trajectories, intuitive.
• Quantum Mechanics:
  • Small, light, discrete/quantized objects.
  • Utilizes Schrödinger’s equations, incorporates wavefunctions, probabilistic, often non-intuitive.

Applications of Quantum Chemistry

• Biochemistry:
  • Quantum calculations enhance prediction of biomolecular conformations.
• Nanomaterials:
  • Properties can be accurately described with quantum mechanics, significantly differ at nanoscale.
  • Example: Quantum dots.
• Analytical Chemistry:
  • Spectral lines in spectrophotometry explained by quantum mechanics as electron-light interactions.

Wave Nature of Light

• Young's interference experiment demonstrates the wave nature of light.
• Points of same phase add constructively.

Electromagnetic Waves

• Maxwell’s equations indicate oscillating electric charges radiate energy as electromagnetic waves.
• Light speed in vacuum: $\lambda \nu = c$
  • Where:
  • $\lambda$ = wavelength
  • $\nu$ = frequency
  • $c$ = speed of light ($3.00 imes 10^8$ m/s).

Blackbody Radiation

• A blackbody absorbs and emits all frequencies.
• Emitted radiation follows Rayleigh-Jeans Law:
  $\rho(v,T) = \frac{8\pi k_B T}{c^3} v^2$
• This leads to the ultraviolet catastrophe.

Planck’s Quantum Hypothesis

• Introduced energy quantization for blackbody radiation: $E = nh\nu, \quad n \in \mathbb{Z}$
  • Where $h$ is Planck's constant $6.626 imes 10^{-34}$ J∙s.
  • Modified Rayleigh-Jeans Law to Planck’s distribution law:
    $\rho(v,T) = \frac{8\pi h}{c^3} \frac{v^3}{e^{\frac{hv}{k_BT}} - 1}$

The Photoelectric Effect

• Photons can eject electrons from metal surfaces:
  $KE = \frac{1}{2} mv^2$
• Experimentally confirmed by Hertz (1886) and explained by Einstein (1905).
• Light behaves as particles (photons) with quantized energies:
  $E = nh<br />\nu$

Classical vs Quantum Theories of the Photoelectric Effect

• Classical Theory: Kinetic energy depends on intensity (amplitude) of radiation.
• Quantum Theory:
  • Kinetic energy of ejected electrons depends on frequency.
  • There exists a threshold frequency $\nu_0$ to eject electrons.
  • Kinetic energy is proportional to frequency above $\nu_0$:
    $KE = h\nu - \phi$
  • $\phi$ is the work function.

Emission Spectrum of Hydrogen

• Lyman Series (UV), Balmer Series (VIS), Paschen Series (IR)
• Balmer formula: \nu = R_H \left(\frac{1}{2^2} - \frac{1}{n^2}\right), \quad n>2
  • Where $R_H = 109677.57 cm^{-1}$.

Bohr’s Model (Hydrogen Atom)

• Electrons orbit a fixed nucleus with specific radii derived from Coulomb's force:
  $\frac{e^2}{4\pi \epsilon<em>0 r^2} = m</em>e v^2 / r$
• Bohr’s quantization condition:
  L = m_e v r = n\h
• Energy levels defined by:
  $E<em>n = -\frac{m</em>e e^4}{8 \epsilon_0^2 h^2 n^2}$

Quantum Harmonic Oscillator

• Describes molecular vibrations.
• Energy levels:
  $E_n = \left(n + \frac{1}{2}\right)h<br />\nu$
• Zero-point energy concept introduced: atoms vibrate even at zero temperature.

Summary on Pertinent Quantum Theory Concepts

• Wave-particle duality: Light and matter exhibit both wave-like and particle-like properties.
• Uncertainty Principle: Position and momentum cannot be simultaneously known precisely.
• Probability densities: Represent likelihood of finding particles in specific locations within a quantum system.