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Lecture Content Overview

Lecture 4: Harmonic Oscillation

• Harmonic Oscillation is fundamental in discussing molecular dynamics.

• The focus is on electromagnetic (EM) waves and spectroscopy related to oscillation states in molecules.

  • Key Equation for Harmonic Oscillation:

    • Displacement equation: ( L = kx )

    • This illustrates how an object, when displaced, aims to return to its equilibrium.

• The energy states in oscillation are constant and quantized, leading to standing wave solutions.

  • Wave Function Solutions:

    • ( Y = \frac{B}{T} H_n(q) e^{(-\frac{q^2}{2})} )

      • Where ( H_n ) are Hermite polynomials, representing the energy states.

• Quantized Energy Levels:

  • The lowest energy state is above zero, termed Zero Point Energy (ZPE).

  • Particle tunneling potential is discussed as particles move beyond classical limits at high energy states.

• Equal Energy States demonstrate symmetry and quantized harmonic motion.

Lecture 5: Hydrogen Atom

• The model of the hydrogen atom explores the quantum behavior of particles confined in smaller dimensions.

  • Quantum Mechanical Properties:

    • Wave equation: ( H\Psi = E\Psi )

• Quantum Numbers and Orbitals:

  • Quantum numbers define the state of electrons in an atom:

    • Principal Quantum Number (n): Determines energy levels.

    • Angular Momentum Quantum Number (l): Defines shape of orbital

    • Magnetic Quantum Number (m): Orientation of the orbital in space.

• For each principal quantum number (n=1,2,3,...), a series of orbitals (s, p, d) exist:

  • Examples of quantum states:

    • n=2, l=1, m=-1,0,1 (2p orbital)

    • n=3, l=2, m=-2,-1,0,1,2 (3d orbital)

Radial Probability Density Function

• The radial probability density function for electrons in orbitals is given by:

  • ( 4\pi r^2 [R(r)]^2 )

• This allows calculation of the likelihood of finding an electron in a spherical volume.

• Shapes of Orbitals:

  • Spherical harmonics describe spatial arrangements and distributions.

  • Each orbital has distinctive shapes based on quantized angular momentum (l).

Banding & Molecular Geometry

• Molecular geometry is analyzed through symmetry and group theory concepts.

  • Molecular Geometry Examples:

    • Trigonal Planar (e.g., BCl3) can be represented with coordinate transformations.

• Discussion includes the implications on electronic configurations and shapes of molecules in relation to bonding.