Restricted Hartree–Fock (RHF)¶

The Restricted Hartree–Fock (RHF) method is the simplest self-consistent field (SCF) approach for solving the electronic Schrödinger equation. In RHF, electrons are described by a single Slater determinant, where each occupied spatial molecular orbital contains two electrons with opposite spins. The molecular orbitals are optimized iteratively until the total electronic energy becomes self-consistent.

RHF serves as the foundation for many electronic structure methods because it provides the reference wavefunction and molecular orbitals required by correlated methods such as MP2, CCSD, CASSCF, CASPT2, and NEVPT2. Although RHF neglects electron correlation beyond the mean-field approximation, it remains an essential first step in many quantum chemistry calculations.

In ORCA, RHF calculations are straightforward to perform and provide molecular orbital energies, electron density, total electronic energy, and other quantities that form the basis for more sophisticated wavefunction methods.

Learning Objectives¶

After completing this tutorial you will be able to:

Understand the principles of the Restricted Hartree–Fock method.
Prepare an RHF input file in ORCA.
Execute an RHF calculation from the command line.
Interpret the important sections of the ORCA output.
Understand the SCF convergence process.
Visualize molecular orbitals generated by an RHF calculation.
Use the RHF wavefunction as the starting point for multireference calculations.

Calculation Workflow¶

The RHF calculation follows the workflow illustrated below.

┌──────────────────────┐
│ Molecular Geometry   │
└──────────┬───────────┘
           ▼
┌──────────────────────┐
│ Prepare RHF Input    │
└──────────┬───────────┘
           ▼
┌──────────────────────┐
│ Run ORCA Calculation │
└──────────┬───────────┘
           ▼
┌──────────────────────┐
│ SCF Iterations       │
└──────────┬───────────┘
           ▼
┌──────────────────────┐
│ Converged RHF        │
│ Wavefunction         │
└──────────┬───────────┘
           ▼
┌──────────────────────┐
│ Analyze Results      │
└──────────┬───────────┘
           ▼
┌──────────────────────┐
│ Reference for        │
│ CASSCF / CASPT2 /    │
│ NEVPT2               │
└──────────────────────┘

Tutorial Contents¶

This tutorial is organized into the following sections:

Introduction – Overview of the Restricted Hartree–Fock method.
Input File – Construction of an RHF input file in ORCA.
Running the Calculation – Executing the calculation from the terminal.
Output Analysis – Understanding the important sections of the output file.
Mathematical Background – Hartree–Fock equations and the self-consistent field procedure.
Complete RHF Algorithm – Step-by-step overview of the RHF computational workflow.

Example System¶

Throughout this tutorial we use the water molecule (H₂O) as a simple closed-shell example. The calculation employs the def2-SVP basis set together with the Restricted Hartree–Fock method.

The example demonstrates the complete RHF workflow, from preparing the input file to analyzing the converged wavefunction, and serves as the starting point for the multireference methods discussed in later chapters.

Prerequisites¶

Before beginning this tutorial, you should be familiar with:

Basic molecular quantum mechanics.
Atomic and molecular orbitals.
Electronic configurations of closed-shell molecules.
Running ORCA from the command line.

Knowledge of electron correlation methods is not required, since these topics are introduced in the subsequent CASSCF, CASPT2, and NEVPT2 tutorials.