Quantum Chemistry I

70 lessons, pages 2901–2970.

Why Chemistry Needs Quantum Mechanics — Bonding, spectra and stability beyond classical physics
Failures of Classical Physics — Black-body radiation, the photoelectric effect and line spectra
Wave–Particle Duality and the de Broglie Wavelength — λ = h/p for electrons and molecules
The Wavefunction and Its Meaning — ψ as the complete description of a quantum state
The Born Interpretation and Probability Density — |ψ|² dτ as the probability of finding a particle
Normalisation of Wavefunctions — Choosing the constant so total probability equals one
Acceptable Wavefunctions — Single-valued, continuous, finite and square-integrable
Operators and Observables — Position, momentum and energy operators
Eigenvalue Equations — Eigenfunctions, eigenvalues and measurement outcomes
The Time-Independent Schrödinger Equation — Ĥψ = Eψ and stationary states
The Hamiltonian Operator — Kinetic and potential energy terms
Expectation Values — Average results of repeated measurements
The Uncertainty Principle in Quantum Chemistry — Δx Δp ≥ ħ/2 and non-commuting operators
The Free Particle — Plane waves and a continuous range of energies
Setting Up the Particle in a One-Dimensional Box — Infinite walls, zero potential inside and the Schrödinger equation
Boundary Conditions and Quantisation — Why ψ must vanish at the walls and only certain waves fit
Energy Levels of the Particle in a Box — Eₙ = n²h²/8mL² and level spacing
Normalised Box Wavefunctions — ψₙ = (2/L)^½ sin(nπx/L)
Nodes and the Shapes of Box Wavefunctions — n − 1 interior nodes and increasing curvature
Zero-Point Energy — Why a confined particle can never be at rest
Probability Distributions in the Box — Where the particle is most likely to be found
Orthogonality of Box Wavefunctions — Overlap integrals of different eigenfunctions vanish
Expectation Values for the Particle in a Box — ⟨x⟩, ⟨p⟩ and ⟨p²⟩ calculated
The Correspondence Principle — Recovering classical behaviour at large n and large mass
Particle in a Two-Dimensional Box — Separation of variables and two quantum numbers
Particle in a Three-Dimensional Box — Energy levels in a cuboid and translational states of gases
Degeneracy and Symmetry — Equal energies in square and cubic boxes
Conjugated Polyenes as Boxes: The Free-Electron Model — π electrons delocalised along a carbon chain
Predicting Absorption Wavelengths of Dyes — HOMO–LUMO gaps and colour from box length
Finite Wells and Quantum Tunnelling — Wavefunctions penetrating classically forbidden regions
Quantum Dots and Nanoparticle Colour — Size-dependent band gaps from confinement
Particle in a Box: Problem-Solving Workshop — Energies, wavelengths and probabilities in practice
Particle on a Ring — Cyclic boundary conditions and quantised angular momentum
Angular Momentum in Quantum Mechanics — L² and L_z operators and their eigenvalues
Particle on a Sphere and Spherical Harmonics — Y(l, mₗ) functions and space quantisation
The Hydrogen Atom Hamiltonian — Coulomb potential, reduced mass and kinetic energy
Separation of Variables in Spherical Coordinates — Splitting ψ into radial and angular parts
Energy Levels of the Hydrogen Atom — Eₙ = −R_H/n² and the ionisation energy
The Hydrogen Spectrum from Quantum Theory — Lyman, Balmer and Paschen series derived
Quantum Numbers n, l and mₗ — Allowed values and what each one controls
Radial Wavefunctions — Exponential decay and polynomial factors
Radial Distribution Functions — P(r) = 4πr²ψ² and shell structure
Radial and Angular Nodes — Counting n − l − 1 radial and l angular nodes
Shapes of s, p and d Orbitals — Boundary surfaces and phase
Real and Complex Orbitals — Combining mₗ = ±1 functions into pₓ and p_y
The Most Probable Radius and the Bohr Radius — Maximum of the 1s radial distribution at a₀
Mean Radius and Expectation Values in Hydrogen — ⟨r⟩, ⟨1/r⟩ and average potential energy
Hydrogen-like Ions and Nuclear Charge — He⁺ and Li²⁺ with energies scaling as Z²
Electron Spin and the Spin Quantum Number — mₛ = ±½ and the Stern–Gerlach experiment
Degeneracy in the Hydrogen Atom — n² spatial states per shell and its breaking
Selection Rules for Hydrogen Transitions — Δl = ±1 and transition dipole moments
Atomic Units — Hartree, bohr and simplifying the equations
The Virial Theorem for Hydrogen — ⟨T⟩ = −½⟨V⟩ and energy partitioning
From Hydrogen to Many-Electron Atoms — Orbital approximation, shielding and the Pauli principle
Hydrogen Atom: Problem-Solving Workshop — Energies, nodes and radial probabilities in practice
Why Approximation Methods Are Needed — Electron repulsion makes exact solutions impossible
The Variation Principle — Any trial energy lies at or above the true ground-state energy
Proof of the Variation Theorem — Expanding a trial function in exact eigenfunctions
Trial Wavefunctions and Variational Parameters — Minimising the energy with respect to adjustable constants
Variational Treatment of the Hydrogen Atom — An exponential trial function recovering the exact result
Gaussian Trial Functions for Hydrogen — An approximate energy and why Gaussians are still used
The Helium Atom and Electron Repulsion — The two-electron Hamiltonian and the independent-electron guess
Effective Nuclear Charge from the Variation Method — Optimising Z for helium and the meaning of screening
Linear Variation Functions — Trial functions built from combinations of basis functions
Secular Equations and Secular Determinants — Coulomb, resonance and overlap integrals
The Hydrogen Molecule Ion H₂⁺ — Bonding and antibonding combinations from the variation method
Basis Sets and the Road to Computational Chemistry — From simple trial functions to modern calculations
Perturbation Theory versus the Variation Method — Two complementary approximation strategies
Variation Principle: Problem-Solving Workshop — Trial functions, energy minimisation and secular determinants
Quantum Chemistry I: Unit Review — Particle in a box, the hydrogen atom and the variation principle together