Lecture Notes: Approximations in Molecular Hamiltonian and Quantum Chemistry

The Challenge of Solving the Schrödinger Equation

• Problem Statement: The Schrödinger equation for an arbitrary molecule cannot be solved exactly with finite computing resources in a finite time.
• Solvability: Only solvable exactly for hydrogen atom (1 electron, 1 nucleus).
• Reason: Due to the many-body problem with n charged particles.
  • Typical molecular systems have multiple nuclei and electrons.
  • Leads to a quadratic number of interactions: ( \frac{n^2}{2} - \frac{n}{2} ).
• Complexity: Classified as an NP-complete problem, exponentially more difficult as the number of particles increases.

Introducing Approximations

• Need: Approximations are necessary to handle molecular Hamiltonians and find approximate solutions.
• Hartree-Fock Theory: An eventual goal of these approximations.
• Key Observation: The mass of nuclei is much greater than that of electrons.
  • E.g., Hydrogen nucleus is ~2000 times the mass of an electron.
  • Nuclei move slower relative to electrons.

The Born-Oppenheimer Approximation

• Concept: Treat nuclei positions as fixed relative to electrons.
  • Kinetic energy of nuclei approximated as zero.
  • Nuclei treated as classical point particles.
• Implications:
  • Nuclear Kinetic Energy: Goes to zero, as nuclei are fixed and not moving.
  • Nuclear-Nuclear Repulsion: Becomes constant, as nuclei have specific locations.
  • Electron Calculations: Focus shifts to electronic Hamiltonian and wave functions.

Electronic Hamiltonian & Wave Functions

• Separation of Calculations:
  • Focus on electronic Hamiltonian and electronic wave function for electron interactions.
  • Electronic Hamiltonian Equation: ( \hat{H}{\text{electron}} \psi{\text{electron}} = E_{\text{electron}} \psi_{\text{electron}} ).
  • Components:
    • Electron kinetic energy.
    • Electron-nuclear attraction.
    • Electron-electron repulsion.

Constant Nuclear-Nuclear Repulsion Term

• Characteristics: Constant term due to fixed positions.
• Role: Can be added to electronic energy; it affects total energy but not wave functions.
  • Energy Shift: Always shifts energy upward but does not alter orbital shapes or wave functions.

Conclusion

• The lecture establishes the need for approximations to study complex molecular systems.
• Highlights the critical role of the Born-Oppenheimer approximation in simplifying calculations by fixing nuclear positions.
• Sets a foundation for focusing on electronic interactions and energies in quantum chemistry.