The **key difference between Hartree and Hartree-Fock method** is that Hartree method uses a bosonic wave function, whereas Hartree-Fock method uses a fermionic wave function.

Both methods Hartree and Hartree-Fock are self-consistent field methods similar to the density functional theory. However, these two methods focus on two different wave function types.

### CONTENTS

1. Overview and Key Difference

2. What is Hartree Method

3. What is Hartree-Fock Method

4. Hartree vs Hartree-Fock Method in Tabular Form

5. Summary – Hartree vs Hartree-Fock Method

## What is Hartree Method?

Hartree method is a method used in approximating the values of energy and the wavefunction of quantum multi-electron systems that are in a stationary state. It assumes that the exact N-body wavefunction of the system is able to be approximated by a product of the single-electron wavefunctions. Using this technique, we can use N spin orbitals to derive different N-coupled equations for them. Further, a solution to these equations can yield the Hartree wavefunction and the energy of the system.

According to the previous quantum mechanical systems, we can predict the exact solutions of multi-electrons. Schrodinger equation contains a group of multi-electron wavefunctions. Each of these wavefunctions has an associated energy eigenvalue. Moreover, these wavefunctions and energy values tend to describe the ground state and excited state of the multi-electron atom because the hydrogen wavefunctions and the associated energies are used to explain the ground state and excited state of an atom of hydrogen. Therefore, we can predict that the quantum numbers are involved as well.

The Hartree method was introduced by Douglas Hartree in 1948 as the best method to find the best possible one-electron wavefunctions. Two years later, Vladimir Fock also experimented with this method, which led to the discovery of the Hartree-Fock method.

## What is Hartree-Fock Method?

Hartree-Fock method is a method that approximates the ability to determine the energy and the wave function of a quantum many-body system. This is determined concerning a stationary state. This term is mainly used in computational physics and chemistry. Moreover, this method often tends to assume that the exact N-body wave function of a particular system is able to be approximated using a determinant of a single slater of N-spin orbitals. Further, the solution of these equations can yield the Hartree-Fock wave function and the energy of the system.

The typical applications of the Hartree-Fock method include deriving the solution of the Schrodinger equation for atoms, molecules, nanostructures, and solids. However, it also has widespread uses in nuclear physics. Further, in atomic structure theory, calculations involving a spectrum with many excited energy levels also have applications of the Hartree-Fock method. Furthermore, it is the central starting point for many methods that describe the many-electron system far more accurately.

## What is the Difference Between Hartree and Hartree-Fock Method?

Hartree method and Hartree-Fock method are important concepts that are related to each other. The key difference between Hartree and Hartree-Fock method is that Hartree method uses a bosonic wave function, whereas Hartree-Fock method uses a fermionic wave function.

The below infographic presents the differences between Hartree and Hartree-Fock method in tabular form for side by side comparison.

## Summary – Hartree vs Hartree-Fock Method

Hartree method is a method used in approximating the values of energy and the wavefunction of quantum multi-electron systems which are in a stationary state. Hartree-Fock method is a method that approximates the ability to determine the energy and the wave function of a quantum many-body system. The key difference between Hartree and Hartree-Fock method is that Hartree method uses a bosonic wave function, whereas Hartree-Fock method uses a fermionic wave function.

##### Reference:

1. “8.3: Hartree-Fock Equations Are Solved by the Self-Consistent Field Method.” *Chemistry LibreTexts*, Libretexts, 11 Aug. 2020.

##### Image Courtesy:

1. “Hartree-Fock” By Edsanville at English Wikipedia – Own work by the original uploader (CC BY-SA 3.0) via Commons Wikimedia

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