Detail study of Hartree Fock and its assumptions

 

Hartree–Fock (HF) Method

Scientists asked:

“How can we simplify the many-electron problem?”

Replace complex interactions with an average field. This is the Hartree–Fock idea.

Main Assumption

Each electron moves in the average field of all other electrons instead of tracking exact electron–electron motion.

The Hartree–Fock method is the foundation of modern quantum chemistry. It is an approximate ab initio method used to determine the electronic structure of atoms and molecules by solving the Schrödinger equation for many-electron systems.

The method was independently developed by Douglas Hartree and Vladimir Fock.

Daily life Example

Track every student talking to every other student → impossible

Hartree–Fock approach:

Assume each student feels an average noise level. Much easier to handle.

Why Do We Need Hartree–Fock?

For a molecule with many electrons, the exact electronic Schrödinger equation is:

 

The Hamiltonian contains electron–electron repulsion terms:

 

This makes the equation impossible to solve exactly for systems with more than one electron.

Hartree–Fock simplifies the problem by:

• Approximating the many-electron wavefunction
• Treating electron–electron interaction in an average way
• Using the variational principle

Basic Assumptions of Hartree–Fock

Independent Particle Approximation

Each electron moves in the average field of all other electrons.

Instead of solving for all electrons simultaneously, HF solves one-electron equations.

 Slater Determinant Wavefunction

To satisfy the Pauli exclusion principle and anti-symmetry requirement, HF uses a Slater determinant.

For N electrons:

Where:

• x = space + spin coordinates
• ψ_i​ = spin orbitals

 This automatically ensures:

• antisymmetry
• proper spin behavior
• Pauli principle

Variational Principle in Hartree–Fock

HF uses the variational method:

 

We minimize the energy with respect to orbitals under orthonormality constraints. This leads to the Hartree–Fock equations.

Hartree–Fock Equations

The central HF equation is:

Where

Structure of the Fock Operator

      contains

• kinetic energy
• nuclear attraction

(b) Coulomb Operator $J$

Represents classical electron–electron repulsion.

Electron cloud repels another electron.

(c) Exchange Operator $K$

Purely quantum mechanical effect arising from antisymmetry.

Important properties:

• no classical analogue
• depends on spin
• leads to exchange stabilization

Self-Consistent Field (SCF) Procedure

Hartree–Fock equations are solved iteratively.

SCF Steps

Step 1: Guess orbitals

Start with initial molecular orbitals.

Step 2: Build Fock matrix

Using current orbitals.

Step 3: Solve Roothaan equations 

FC = SC_ε

Get new orbitals.

Step 4: Check convergence

If energy change is small → stop
Otherwise → repeat

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Because the field depends on orbitals and orbitals depend on field → self-consistent.

Types of Hartree–Fock

(i) Restricted Hartree–Fock (RHF)

• Used for closed-shell systems
• α and β electrons share same spatial orbital

Example: H₂, H₂O

Most common for organic molecules.

 (ii) Unrestricted Hartree–Fock (UHF)

• Used for open-shell systems
• α and β orbitals are different

Example: radicals

Problem is spin contamination

 (iii) Restricted Open-Shell HF (ROHF)

• For open-shell but controlled spin

Independent Electrons

Hartree–Fock assumes

1.      Electrons move independently

2.      But in an average potential

3.      Not completely free

4.      Not fully interacting

This is called mean field approximation

Molecular Orbitals Concept

HF describes electrons using molecular orbitals (MOs).

Each electron occupies one orbital.

Properties:

✅ Orbitals are orthogonal
✅ Each holds max 2 electrons
✅ Follow Pauli principle

Electron cloud is spread in molecular orbitals around nuclei.

Why Is Iteration Needed? (SCF Intuition)

Important conceptual step.

The circular problem:

• Orbitals depend on electron field
• Electron field depends on orbitals

Circular dependency!

Self-Consistent Field (SCF)

Procedure:

1. Guess orbitals
2. Calculate average field
3. Solve for new orbitals
4. Repeat until stable

Simple Analogy

Like adjusting mirror:

You adjust → check → adjust → check → until perfect.

What Hartree–Fock Does Well

✅ Good molecular geometries
✅ Reasonable orbital picture
✅ Foundation of quantum chemistry
✅ Starting point for advanced methods

What Hartree–Fock lacks

HF is not perfect because:

• It misses electron correlation
• Energies are slightly high
• Some weak interactions poorly described

Later DFT improves this.

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