Fundamentals of Quantum Chemistry

Working note — Part of my personal research notebook. Entries may be incomplete and are revised over time.

• Physical system: consists of atoms and molecules, which are composed of nuclei (protons and neutrons) and electrons. The system is characterized by the interactions between these particles, primarily governed by electromagnetic forces.
  • physical systems in Quantum Chemistry:
    • a single atom (e.g., a hydrogen atom or a carbon atom).
    • a molecule (e.g., water $H_2O$ or methane $CH_4CH_4$).
    • a chemical reaction system (e.g., the reaction between hydrogen and oxygen to form water).
  • key features of a physical system are the positions and charges of the nuclei and the distribution of electrons around them.
• State of the physical system: state of a physical system in quantum chemistry is described by the wave function of the system, which encodes all the information about the positions and momenta of the electrons and nuclei.
  • the wavefunction $\psi(r_1, r_2, …, r_N, t)$ depends the coordinates of all electrons $(r_1, r_2, …, r_N,)$ and time ($t$), and it ivolves according to the Schrödinger equation.
  • states in Quantum Chemistry:
    • ground state, which is the lowest energy state of the system
    • excited state, which is a higher energy state of the system
    • molecular orbital, which describes the distribution of an electron in a molecule.
    • transition state, which represents the highest energy state along a reaction pathway.
  • state of the system can also be described using density functional theory (DFT), where the electron density $\rho(r)$ is the fundamental quantity.
  • state of the system determines its chemical properties, reactivity, and spectroscopic behavior.
  • Changes in the state (e.g., electronic transitions) are responsible for chemical reactions and light absorption/emission.
• Physical law: the fundamental principles and equations that govern the behavior of electrons and nuclei in atoms and molecules. These laws are derived from quantum mechanics and electrodynamics.
  • physical laws in Quantum Chemistry include:
    • Schrodinger equation: \(i \hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N, t\right)=\hat{H} \Psi\left(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N, t\right),\) where $\hat{H}$ is the Hamiltonian operator for the system. - when solved, it gives you the wave function or electron density, which defines the state of the system
    • Born-Oppenheimer approximation: \(\hat{H}=\hat{T}_e+\hat{V}_{e e}+\hat{V}_{e n}+\hat{V}_{n n}\), where \(\hat{T}_e\) is the KE of electron, \(\hat${V}_{ee}\) is the electron-electron interactions, \(\hat{V}_{e n}\) is the electron-nucleus interaction, and \(\hat{V}_{n n}\) is the nucleaus-nucleus interaction. - simplified version of the Schrodinger equation, making it feasible (computationally) to calculate the electronic structure of molecules
    • other approximation methods for solving the Schrodinger equation, providing practical tools for quantum chemical calculations
      • Hartree-Fock equation: \(\hat{F} \psi_i(\mathbf{r})=\epsilon_i \psi_i(\mathbf{r})\) where $\hat{F} $ is the Fock operator, $\psi_i(\mathbf{r})$ are the molecular orbitals, and $\epsilon_i $ are the orbital energies.
      • Kohn-Sham equations: \(\left[-\frac{\hbar^2}{2 m} \nabla^2+V_{\mathrm{eff}}(\mathbf{r})\right] \phi_i(\mathbf{r})=\epsilon_i \phi_i(\mathbf{r})\)
      • Configuration interactions
      • Couple-Pair theories
      • Many-body perturbation theory
    • Pauli exclusion principle
  • Physical laws in quantum chemistry are subject to conditions of applicability. For example:
    • Born-Oppenheimer approximation is valid when the nuclear motion is much slower than the electronic motion.
    • Hartree-Fock method provides accurate results for systems with weak electron correlation but fails for strongly correlated systems.

1. Szabo, A., & Ostlund, N. S. (1996). Modern Quantum Chemistry. Courier.