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Worked Example – RHF Calculation of Water

Throughout the previous chapters, we developed the mathematical framework underlying the Restricted Hartree–Fock (RHF) method. We began with the electronic Schrödinger equation, introduced the Born–Oppenheimer approximation, constructed antisymmetric wavefunctions using Slater determinants, derived the Hartree–Fock equations, transformed them into the Roothaan–Hall matrix equation, and finally examined the Self-Consistent Field (SCF) algorithm.

In this chapter, we connect these theoretical concepts to a real RHF calculation performed using GAMESS.

The objective is to identify where each mathematical concept appears during an actual calculation.

The Problem

Consider the water molecule

\[ \mathrm{H_2O} \]

Our goal is to determine

  • the electronic energy,
  • the molecular orbitals,
  • the electron density,

using the Restricted Hartree–Fock method.

Step 1 — Molecular Geometry

The first requirement is the molecular geometry.

The Cartesian coordinates define the positions of the nuclei.

Once the geometry is supplied,

the nuclei are assumed to remain fixed during the electronic calculation according to the Born–Oppenheimer approximation.

At this point,

the electronic Hamiltonian is completely determined.

Step 2 — Choosing a Basis Set

The molecular orbitals are unknown.

Instead of solving for them directly,

they are expanded as

\[ \phi_i = \sum_\mu C_{\mu i}\chi_\mu \]

where

  • \(\chi_\mu\) are basis functions,
  • \(C_{\mu i}\) are unknown coefficients.

In this example,

the STO-3G basis set is used.

The basis functions define the size of the matrices used throughout the Hartree–Fock calculation.

Step 3 — Initial Guess

The SCF procedure begins with an initial estimate of the molecular orbitals.

In the example calculation,

the Hückel guess is employed.

Although this initial guess is approximate,

it provides the starting point for constructing the first density matrix.

Step 4 — Density Matrix

Using the occupied molecular orbitals,

the density matrix is calculated as

\[ P_{\mu\nu} = 2 \sum_i C_{\mu i} C_{\nu i} \]

The density matrix describes how the electrons are distributed throughout the molecule.

Every subsequent step of the RHF procedure depends on this matrix.

Step 5 — Constructing the Fock Matrix

Using the density matrix,

GAMESS constructs the Fock matrix

\[ \mathbf F = \mathbf H + \mathbf G(\mathbf P) \]

The Fock matrix contains

  • kinetic energy,
  • electron–nucleus attraction,
  • Coulomb interaction,
  • exchange interaction.

It represents the effective one-electron Hamiltonian used during each SCF iteration.

Step 6 — Solving the Roothaan–Hall Equations

The next step is solving

\[ \mathbf{FC} = \mathbf{SC} \boldsymbol{\varepsilon} \]

Diagonalization produces

  • updated molecular orbital coefficients,
  • orbital energies.

These orbitals are then used to generate a new density matrix.

Step 7 — The SCF Cycle

The complete computational cycle is

Guess Orbitals
      │
      ▼
Density Matrix
      │
      ▼
Fock Matrix
      │
      ▼
Roothaan–Hall Equation
      │
      ▼
New Orbitals
      │
      ▼
New Density Matrix
      │
      ▼
Convergence Check

This cycle repeats until the electron density no longer changes between successive iterations.

At convergence,

the molecular orbitals are said to be self-consistent.

Step 8 — Final Electronic Energy

Once convergence is achieved,

the total Hartree–Fock energy is computed.

The final energy consists of

  • one-electron contributions,
  • Coulomb interactions,
  • exchange interactions.

This energy corresponds to the minimum energy obtainable using a single Slater determinant.

Because electron correlation is neglected,

the RHF energy remains higher than the exact non-relativistic energy.

Step 9 — Molecular Orbitals

Diagonalization of the Fock matrix produces a complete set of molecular orbitals.

These orbitals are classified as

  • occupied orbitals,
  • virtual orbitals.

Among them,

two orbitals are particularly important.

HOMO

The

Highest Occupied Molecular Orbital

contains the highest-energy electrons present in the ground state.

LUMO

The

Lowest Unoccupied Molecular Orbital

is the first available orbital that can accept electrons during an excitation.

The HOMO–LUMO energy difference provides useful qualitative information regarding

  • molecular stability,
  • chemical reactivity,
  • electronic excitations.

Connecting Theory with GAMESS Output

Throughout an RHF calculation, each section of the GAMESS output corresponds to a concept introduced in the previous chapters.

GAMESS Output Mathematical Concept
Molecular Geometry Born–Oppenheimer Approximation
Basis Set LCAO Expansion
Initial Guess Starting Molecular Orbitals
Density Matrix Electron Density
Fock Matrix Effective Hamiltonian
SCF Iterations Self-Consistent Field Algorithm
Orbital Energies Eigenvalues of the Fock Operator
Molecular Orbitals Eigenvectors of the Roothaan–Hall Equation
Total Energy Hartree–Fock Energy

This illustrates how the theoretical framework developed throughout this tutorial is reflected in an actual electronic structure calculation.

Summary

The Restricted Hartree–Fock method follows a logical sequence:

  1. Define the molecular geometry.
  2. Apply the Born–Oppenheimer approximation.
  3. Expand molecular orbitals in a basis set.
  4. Construct the density matrix.
  5. Build the Fock matrix.
  6. Solve the Roothaan–Hall equations.
  7. Repeat until self-consistency is achieved.
  8. Compute the final electronic energy and molecular orbitals.

Although RHF neglects electron correlation, it provides a robust and computationally efficient description of molecular electronic structure. It also serves as the starting point for many advanced electronic structure methods, including MP2, Configuration Interaction (CI), Coupled Cluster (CC), CASSCF, and Density Functional Theory (DFT).

Conclusion

You have now completed the mathematical journey behind the Restricted Hartree–Fock method.

Starting from the fundamental principles of quantum mechanics, we developed the approximations and equations that ultimately lead to the SCF procedure implemented in modern quantum chemistry software.

Understanding these concepts not only helps interpret GAMESS output but also provides the foundation required for studying more advanced electronic structure methods.