Quantum Chemistry: Hartree-Fock Method

Hartree-Fock Theory

• Hartree-Fock energy is a method to optimize the wavefunction using molecular orbital coefficients as variational parameters.
• Molecular orbitals are expressed as linear combinations of basis functions.
• Basis functions can be atomic orbitals or any other suitable mathematical function.

Linear Combination of Basis Functions

• Hartree-Fock molecular orbitals are expressed as linear combinations of basis functions.
• K molecular orbitals are generated from a set of K atomic orbitals.
• The linear combination of atomic orbitals is used to create molecular orbitals.

Hartree-Fock Energy

• The energy equation is: E = ∑ haa + ∑∑ (Jab - Kab) / 2
• haa is the one-electron energy.
• Jab is the Coulomb integral.
• Kab is the exchange integral.

One-Electron Energy

• haa = ∫ dx1 χa(x1) [(-∑(1/2)∇12 - ∑(Zi/rIi)] χa(x1)
• It represents the kinetic energy of an electron in molecular orbital a.
• It also represents the Coulombic attraction between the nuclei and the electron in molecular orbital a.

Coulomb Integral

• Jab = ∫∫ dx1dx2 χa*(x1) χa(x1) χb*(x2) χb(x2) / r12
• It represents the Coulomb interaction between two charge clouds of electrons in molecular orbitals a and b.
• It is a two-electron integral.

Exchange Integral

• Kab = ∫∫ dx1dx2 χa*(x1) χb(x1) χb*(x2) χa(x2) / r12
• It represents the exchange energy resulting from the Pauli repulsion between electrons in molecular orbitals a and b.
• It is a two-electron integral.

Hartree-Fock Calculations

• The basic idea is to solve the Schrödinger equation for a system of M nuclei and N electrons.
• The full Hamiltonian within the Born-Oppenheimer approximation is used.
• A trial wavefunction consisting of one Slater determinant is used.
• Molecular orbitals are expressed as linear combinations of basis functions.

Variational Minimization of the Hartree-Fock Energy

• The energy equation is minimized by altering the molecular orbital coefficients.
• The constraint is that the molecular orbitals are orthogonal.
• The 'best' set of molecular orbitals is obtained by solving the eigenvalue problem.

Fock Operator

• The Fock operator is: fˆa(x1) = hˆa(x1) + ∑ Jˆb(x1) - ∑ Kˆb(x1)
• It represents the effective Hamiltonian for the electron in molecular orbital a.
• The Fock operator acts on the molecular orbital a.
• The eigenfunctions of the Fock operator are the Hartree-Fock molecular orbitals.