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.
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)
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).
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.