Module 1: Energy levels of atoms and molecules

Module 1: Energy levels of atoms and molecules#

This module introduces some fundamental concepts in quantum mechanics focusing on the different quantized energy levels in atoms and molecules.

Rather than a systematic development of Quantum theory it covers the analytical simple systems that can later on explain translational, electronic, vibrational, and rotational levels that are needed for statistical thermodynamics in the next module.

A note for those who learned quantum mechanics elsewhere: we will focus on the results of quantum theory applied to atomic and molecular systems rather than the mathematical development. For example, there is no mention of the Hilbert space, the orthogonalization of a base, commuting operators, wavefunction conjugates…etc

The sections in this chapter are:

  1. Bohr’s hydrogen atom

    1. Experimental series

    2. Bohr’s hydrogen model

    3. Calculating wavelengths of the Lymann series

  2. Introduction to Quantum Mechanics. Particle in a box.

    1. The Schrodinger equation for a particle in a box

    2. The quantized energy of the particle

    3. The effect of mass and size of the box. Zero point energy

    4. The wavefunctions of the particle in a box and their meaning

    5. Particle in a box in 2D and 3D. Degeneracy. Plotting 2D wavefunctions

  3. Schrodinger’s hydrogen atom

    1. The Hamiltonian for the hydrogen atom

    2. Solutions to the Schrodinger equation for hydrogen: the orbitals

    3. Energy levels of hydrogen

    4. The radial solution: the size of the atom

    5. Plotting the entire wavefunction: probability and cutoff

  4. Polyelectronic atoms. Electronic correlation and self-consistent field methods

    1. The Hamiltonian for polyelectronic atoms

    2. The Self-Consistent Field solution (SCF) or Hartree-Fock

    3. Quantum chemistry packages: pyscf

    4. Single point calculations. Reading the output

    5. Testing basis sets

    6. Different levels of theory

    7. Ionization energy of atoms

  5. Electronic energy of molecules

    1. The Born-Oppenheimer approximation and potential energy surface

    2. Single point energy of molecules. Assessing the energy of different spin states.

    3. Population analysis

    4. Scanning a distance to find a bond distance

    5. Homolytic and heterolytic breaking of a bond

    6. Structure optimization: optimizing the geometry of water

  6. Vibrational levels of diatomic molecules

    1. The Schrodinger Equation of Nuclear Motion

    2. The harmonic approximation to nuclear motion: The harmonic oscillator

    3. Using the classical harmonic oscillator to assess the “quantumness” of the system

    4. Infrared spectroscopy

    5. Anharmonic corrections

  7. Vibrational levels of polyatomic molecules

    1. Normal modes of vibration

  8. Rotational levels of molecules

    1. The Schrodinger equation for nuclear coordinates: the polar coordinates of the nuclear motion show rotation

    2. The rigid rotor approximation: energy rotational levels and energy spacing: Effect of mass and bond distance

    3. Degeneracy of the rigid rotor

    4. Microwave spectroscopy