How RHF Works¶

The Restricted Hartree–Fock (RHF) method determines the electronic structure of a closed-shell molecule through an iterative process known as the Self-Consistent Field (SCF) procedure. Instead of solving the many-electron Schrödinger equation exactly, RHF approximates the motion of each electron by allowing it to move in the average electrostatic field created by all the other electrons.

The calculation begins with an approximate description of the molecule and repeatedly improves the molecular orbitals until a self-consistent solution is obtained.

RHF Workflow¶

Input Molecular Geometry
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Choose Basis Set
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Generate Initial Guess Orbitals
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Construct the Fock Matrix
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Solve for Molecular Orbitals
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Update Electron Density
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Converged?
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      │             │
     No            Yes
      │             │
      └────► Repeat ◄────┘
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      Final Energy and Molecular Orbitals

Step 1: Define the Molecular System¶

The calculation begins by reading the molecular geometry provided in the input file along with the chosen basis set. From this information, GAMESS determines the number of atoms, electrons, molecular charge, and spin multiplicity. These define the molecular system that will be studied.

Step 2: Generate an Initial Guess¶

Since the exact molecular orbitals are initially unknown, GAMESS generates an approximate set of orbitals using methods such as the Hückel or Core Hamiltonian approach. These orbitals serve as the starting point for the SCF procedure.

Step 3: Construct the Fock Matrix¶

Using the current electron density, GAMESS constructs the Fock matrix, which represents the effective Hamiltonian experienced by each electron. This matrix incorporates the average interaction between electrons and forms the basis for calculating improved molecular orbitals.

Step 4: Calculate New Molecular Orbitals¶

The Fock matrix is solved to obtain a new set of molecular orbitals and their corresponding orbital energies. These orbitals provide a better approximation to the electronic structure than those from the previous iteration.

Step 5: Update the Electron Density¶

The newly calculated occupied molecular orbitals are used to generate an updated electron density. Since the Fock matrix depends on this density, it must be reconstructed using the new information.

Step 6: Check for Convergence¶

The process of constructing the Fock matrix, solving for molecular orbitals, and updating the electron density is repeated until the changes in the total energy and electron density become smaller than a predefined convergence threshold.

Once convergence is achieved, the calculation produces the final RHF wavefunction, molecular orbital energies, electron density, and total electronic energy.

Why is it Called "Self-Consistent Field"?¶

The term Self-Consistent Field (SCF) refers to the fact that the electron density and the effective field experienced by the electrons depend on each other.

The electron density is used to construct the Fock matrix.
The Fock matrix determines the molecular orbitals.
The molecular orbitals generate a new electron density.

This cycle continues until the input and output electron densities become identical within the chosen convergence criterion. At this point, the electronic structure is said to be self-consistent.

Restricted Hartree–Fock

In RHF, each occupied molecular orbital contains two electrons with opposite spins (α and β) occupying the same spatial orbital. This restriction makes RHF suitable only for closed-shell molecules, where all electrons are paired.

Summary¶

The RHF method is an iterative optimization procedure that starts with an approximate electronic structure and systematically improves it until a self-consistent solution is reached. The converged molecular orbitals obtained from an RHF calculation serve as the foundation for many advanced quantum chemistry methods, including MP2, Configuration Interaction (CI), Coupled Cluster (CC), and CASSCF.