Mathematics for Nerds¶
The previous sections explained how to perform a Restricted Hartree–Fock (RHF) calculation in GAMESS. In this section, we shift our focus from using the method to understanding the theory behind it.
Beginning with the electronic Schrödinger equation, we will gradually derive the Hartree–Fock method and show how it ultimately leads to the matrix equations solved by quantum chemistry software.
The chapters are arranged in a logical order, with each building upon concepts introduced in the previous one.
Recommended Background
A basic understanding of undergraduate quantum mechanics, linear algebra, and atomic orbitals is helpful, but each derivation is presented step by step.
Learning Roadmap¶
Electronic Schrödinger Equation
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Born–Oppenheimer Approximation
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Slater Determinants
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Hartree–Fock Approximation
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Fock Operator
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Roothaan–Hall Equations
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Self-Consistent Field Algorithm
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Worked Example: Water Molecule
Chapters¶
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Electronic Schrödinger Equation
Begin with the fundamental equation of quantum mechanics and develop the electronic Hamiltonian used in quantum chemistry.
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Born–Oppenheimer Approximation
Learn why separating nuclear and electronic motion makes molecular quantum mechanics computationally feasible.
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Slater Determinants
Understand why electrons require antisymmetric wavefunctions and how Slater determinants satisfy the Pauli Exclusion Principle.
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Hartree–Fock Approximation
Introduce the mean-field approximation, variational principle, Coulomb interaction, and exchange interaction.
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The Fock Operator
Derive the Fock operator and understand the physical meaning of each term in the Hartree–Fock equations.
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Roothaan–Hall Equations
Transform the Hartree–Fock equations into the matrix form solved by modern quantum chemistry programs.
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SCF Algorithm
Explore how RHF calculations converge through the Self-Consistent Field procedure implemented in GAMESS.
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Worked Example: Water Molecule
Follow an RHF calculation for water from input geometry to the final converged electronic structure.