Theoretical Chemistry

Theoretical Chemistry is a branch of chemistry that utilizes mathematical models and computational simulations, grounded in the principles of physics, to explain and predict chemical phenomena. By applying quantum mechanics to describe the electronic structure of atoms and molecules, and statistical mechanics to understand the properties of bulk matter, this field seeks to provide a fundamental framework for chemical bonding, reactivity, spectroscopy, and thermodynamics. Rather than conducting physical experiments, theoretical chemists develop and apply theories and computational methods to rationalize experimental observations and to guide the design of new molecules and materials.

1. Foundations of Theoretical Chemistry
   1. Mathematical Preliminaries
      1. Linear Algebra
         1. Vector Spaces and Operations
            1. Vector Definition and Properties
            2. Vector Addition and Scalar Multiplication
            3. Linear Independence and Dependence
            4. Basis Vectors and Spanning Sets
            5. Vector Space Dimension
            6. Subspaces
         2. Inner Product Spaces
            1. Inner Product Definition
            2. Orthogonality and Orthonormality
            3. Gram-Schmidt Orthogonalization
            4. Cauchy-Schwarz Inequality
         3. Matrix Theory
            1. Matrix Operations and Properties
            2. Matrix Addition and Multiplication
            3. Transpose and Conjugate Transpose
            4. Matrix Inverse and Pseudoinverse
            5. Matrix Rank and Nullity
         4. Determinants
            1. Determinant Calculation Methods
            2. Properties of Determinants
            3. Cramer's Rule
         5. Eigenvalue Problems
            1. Eigenvalues and Eigenvectors
            2. Characteristic Polynomial
            3. Diagonalization Process
            4. Hermitian Matrices
            5. Unitary Matrices
            6. Spectral Theorem
      2. Differential Equations
         1. Ordinary Differential Equations
            1. First-Order Linear Equations
            2. Separable Equations
            3. Exact Equations
            4. Second-Order Linear Equations
            5. Homogeneous Solutions
            6. Particular Solutions
            7. Boundary Value Problems
            8. Initial Value Problems
         2. Partial Differential Equations
            1. Classification of PDEs
            2. Separation of Variables Method
            3. Fourier Series Solutions
            4. Schrödinger Equation
            5. Diffusion Equation
            6. Wave Equation
         3. Special Functions
            1. Legendre Polynomials
            2. Associated Legendre Functions
            3. Spherical Harmonics
            4. Hermite Polynomials
            5. Laguerre Polynomials
      3. Calculus of Variations
         1. Functional Derivatives
         2. Euler-Lagrange Equation
         3. Constraints and Lagrange Multipliers
         4. Applications to Physical Systems
      4. Complex Analysis
         1. Complex Numbers and Operations
         2. Analytic Functions
         3. Contour Integration
         4. Residue Theorem
      5. Fourier Analysis
         1. Fourier Series
         2. Fourier Transforms
         3. Discrete Fourier Transform
         4. Applications in Spectroscopy
      6. Group Theory
         1. Symmetry Elements
            1. Identity Element
            2. Rotation Axes
            3. Reflection Planes
            4. Inversion Centers
            5. Improper Rotation Axes
         2. Symmetry Operations
            1. Operation Multiplication
            2. Group Axioms
            3. Order of Groups
         3. Point Groups
            1. Molecular Point Group Determination
            2. Common Point Groups
            3. Character Tables
            4. Mulliken Symbols
         4. Representation Theory
            1. Matrix Representations
            2. Reducible Representations
            3. Irreducible Representations
            4. Great Orthogonality Theorem
            5. Projection Operators
         5. Applications to Molecular Systems
            1. Molecular Orbital Symmetry
            2. Selection Rules
            3. Vibrational Modes
   2. Classical Physics Foundations
      1. Classical Mechanics
         1. Newtonian Mechanics
            1. Newton's Laws of Motion
            2. Force and Acceleration
            3. Conservation of Energy
            4. Conservation of Momentum
            5. Conservation of Angular Momentum
         2. Lagrangian Mechanics
            1. Generalized Coordinates
            2. Lagrangian Function
            3. Principle of Least Action
            4. Euler-Lagrange Equations
            5. Constraints and Degrees of Freedom
         3. Hamiltonian Mechanics
            1. Hamiltonian Function
            2. Hamilton's Equations of Motion
            3. Phase Space Representation
            4. Canonical Transformations
            5. Poisson Brackets
      2. Electromagnetism
         1. Electrostatics
            1. Coulomb's Law
            2. Electric Field and Potential
            3. Gauss's Law
            4. Boundary Conditions
            5. Multipole Expansions
         2. Magnetostatics
            1. Magnetic Fields and Forces
            2. Biot-Savart Law
            3. Ampère's Law
         3. Maxwell's Equations
            1. Gauss's Law for Electricity
            2. Gauss's Law for Magnetism
            3. Faraday's Law of Induction
            4. Ampère-Maxwell Law
         4. Electromagnetic Waves
            1. Wave Equation Derivation
            2. Plane Wave Solutions
            3. Polarization States
            4. Energy and Momentum of EM Waves
         5. Interaction with Matter
            1. Dielectric Materials
            2. Magnetic Materials
            3. Dispersion Relations
      3. Thermodynamics
         1. Laws of Thermodynamics
            1. Zeroth Law and Temperature
            2. First Law and Internal Energy
            3. Second Law and Entropy
            4. Third Law and Absolute Zero
         2. Thermodynamic Potentials
            1. Internal Energy
            2. Enthalpy
            3. Helmholtz Free Energy
            4. Gibbs Free Energy
         3. Phase Transitions
            1. Phase Diagrams
            2. Clausius-Clapeyron Equation
            3. Critical Phenomena