Quantum Chemistry

The equation that cannot be solved, and the approximations chemistry is built on

About this course

Quantum chemistry from the variational principle and antisymmetry through Born-Oppenheimer and the potential energy surface, Hartree-Fock as a mean field, the basis-set expansion that turns calculus into linear algebra, and what the mean field gets wrong — correlation, configuration interaction and size consistency.

What you will be able to do

  • Use the variational principle as an error bar that points one way
  • Explain why antisymmetry, not repulsion alone, gives matter its structure
  • Separate nuclear and electronic motion, and read a potential energy surface
  • Set up Hartree-Fock as a mean field and expand it in a finite basis
  • Quantify what the mean field misses, and the size-consistency trap in configuration interaction

What this course covers

  • The Born-Oppenheimer Approximation
  • The Potential Energy Surface
  • Isotope Test of the Shared Surface
  • Zero-Point Energy and the Kinetic Isotope Effect
  • H₂⁺ as the First Computed Chemical Bond
  • Conical Intersections and Surface Breakdown
  • Electron density vs. wave function as a description
  • Hohenberg-Kohn theorems
  • Kohn-Sham construction and the exchange-correlation functional
  • Approximations to the XC functional: LDA, GGAs, and hybrids
  • Self-interaction error and delocalization error
  • DFT's failures, strengths, and corrected misconception
  • Correlation energy: definition and atomic sizes
  • Why correlation controls chemistry despite being ~1% of total energy
  • The correlation hole vs. the Fermi hole
  • Mean-field failure for bond breaking: stretched H₂
  • Dynamic vs. static correlation and the module-three roadmap
  • The dimensional impossibility of solving the Schrödinger equation
  • The Hartree energy unit and the energy scale of chemistry
  • Hydrogen as a metrological control: reduced-mass correction
  • Helium ionization energy, mass polarization, and the full error budget
  • The ladder of approximations for helium's ground-state energy
  • Correlation energy and why 1.45% error is chemically fatal
  • The variational principle as the course's central theorem
  • Basis Set Expansion and the Roothaan–Hall Matrix Equation
  • Basis Set as a Variational Bound and Named Surrender
  • SCF Loop in Matrix Form: Hydrogen and Helium Hydride
  • Partial Charges as Scheme-Dependent Conventions
  • Cost Scaling and Near-Linear Dependence of the Basis
  • Electron repulsion and the mean-field approximation
  • Self-consistent field and the Hartree-Fock method
  • Effective nuclear charge and orbital screening
  • SCF convergence, variational equivalence, and correlation energy
  • Koopmans' theorem and its cancellation of errors
  • Double-counting, correlation, and limits of Hartree-Fock
  • The Nuclear Cusp and Gaussian Shape Error
  • Convergence of Gaussian Expansions Toward Exact Energy
  • Basis Set Construction: Contraction, Split Valence, Polarization, Diffuse
  • Variational Principle Limits: Bigger Basis Is Not Uniformly Better
  • Basis Set Superposition Error and the Counterpoise Correction
  • Dispersion as Physics No Basis Can Supply at Hartree–Fock Level
  • Configuration Interaction: Core Idea and Vocabulary
  • Factorial Scaling: The Determinant Count Wall
  • Full CI Within a Basis: Exactness and Its Limits
  • Size Consistency Failure of Truncated CI (CISD)
  • Variational Bound vs. Size Consistency as Competing Criteria
  • Davidson Correction and the Path to Coupled Cluster
  • Orbital energy gap picture and its failure
  • Hole attraction and the Rydberg series
  • CIS excitation energies benchmarked on helium
  • Singlet-triplet splitting from exchange
  • Hilleras-Undheim-McDonald variational bound for CIS
  • TD-DFT: linear response formulation and failures
  • Oscillator strengths, spin selection, and double excitations
  • Rayleigh's Variational Principle for Pitch
  • Proof of the Variational Principle
  • Variational Method Applied to Hydrogen
  • Limits of the Variational Bound: Properties and Differences
  • Split-Shell Trial Function and Radial Correlation in Helium
  • Using the Bound to Detect Errors in Published Results
  • Antisymmetry postulate for identical particles
  • Exchange integral, singlet–triplet splitting, and Hund's rule
  • Slater determinant and the Pauli exclusion principle
  • Fermi hole and the stability of matter
  • Bosonic lithium thought experiment
  • The non-crossing rule and seam dimensionality
  • Conical intersection geometry
  • Failure of the Born–Oppenheimer approximation at a cone
  • Landau–Zener formula and funnel width
  • Vision and retinal isomerization through a conical intersection
  • Geometric phase sign change as a cone signature
  • Møller–Plesset perturbation theory setup
  • The MP2 formula and its practical advantages
  • Radius of convergence and complex singularities
  • Static correlation, intruder states, and when to trust MP
  • The exponential ansatz and size consistency
  • Disconnected vs. connected excitations
  • Coupled cluster doubles for minimal-basis H₂
  • CCSD(T): the gold-standard method and its cost
  • Chemical accuracy: target, protocol, and basis requirements
  • Failures of CCSD(T) and the coupled-cluster ladder
  • Relativistic electron speeds in heavy atoms
  • Dirac equation vs. textbook mass heuristic for 1S contraction
  • Relativistic contraction reaching the valence shell
  • Gold's colour, ionisation energy, and electron affinity
  • Mercury's anomalously low melting point
  • Relativistic origin of the lead-acid battery voltage
  • Practical methods for including relativity in calculations
  • Observables versus conventions in quantum chemistry
  • One wave function, three atomic charges
  • Historical landscape of charge-partitioning schemes
  • Variational stationarity: why energy is robust and charge is not
  • Three labels every computed number must carry
  • The corrected sentence about atomic charges
  • Capstone synthesis: the course as one argument

Syllabus

  1. The Problem and the Principle

    1. L1 · The Equation That Cannot Be Solved
    2. L2 · The Variational Principle: an Error Bar That Points One Way
    3. L3 · Antisymmetry, and Why Matter Has Structure

  2. The Surface and the Mean Field

    1. L4 · Born-Oppenheimer, and the Surface Chemistry Thinks With
    2. L5 · Hartree-Fock: Each Electron in the Average of the Others

  3. Making It Computable

    1. L6 · Turning Calculus into Linear Algebra
    2. L7 · Basis Sets, and an Attraction That Is Not Physics

  4. Beyond the Mean Field

    1. L8 · What the Mean Field Gets Wrong, Exactly
    2. L9 · Configuration Interaction, and the Size-Consistency Trap
    3. L10 · Perturbation Theory, and a Series That Need Not Converge
    4. L11 · Coupled Cluster: the Exponential Ansatz

  5. Density Functionals, Excited States and Relativity

    1. L12 · Density Functional Theory: Exact, with an Unknown Ingredient
    2. L13 · Excited States, and Why They Are Harder Than They Look
    3. L14 · Conical Intersections: Where the Surface Stops Existing
    4. L15 · Relativity in a Chemistry Course: Gold, Mercury and the Battery
    5. L16 · What a Wavefunction Is For, and the Course as One Argument