Electron Correlation & Configuration Interaction (CI)

 

Electron Correlation & Configuration Interaction (CI)

Do electrons in a molecule really move independently?

Answer: No
They repel each other and try to avoid each other all the time.

What Hartree–Fock (HF) Assumes

In Hartree–Fock (SCF):

• Each electron moves in an average field of others
• Instant repulsion ignored
• Only one Slater determinant is used

So, HF is like “Students in a class are treated as if they don’t interact with each other in real-time.

Definition:

Electron correlation = real interaction between electrons

Mathematical Idea

Difference between exact energy and HF energy

Since HF ignores correlation

• E_HF​ is too high
• True energy is lower

Example

Imagine:

• Two people walking in a narrow hallway
• They adjust their positions to avoid collision

That adjustment = electron correlation

HF assumption:

“They walk randomly without caring about each other” 

Reality:

“They coordinate movement to avoid collision” 

Types:

Dynamic Correlation

Due to electron motion

Important in all systems

Example:

• Electrons in a stable molecule (like H₂O)
• They continuously avoid each other

Like:

People slightly shifting while walking

Static (Non-dynamic) Correlation

·         Important when multiple configurations contribute

·         Example: bond breaking

✔ Happens when one structure is not enough

Example: Bond Breaking (H₂ molecule)

At equilibrium:

• One configuration works fine ✔

When bond stretches:

• Electron can be on either atom
• Need multiple configurations

Why HF Fails?

Example: H₂ Dissociation

At large distance:

• Electron 1 → Atom A

Electron 2 → Atom B HF says:

Both electrons are shared equally

Reality:

Electrons localize separately

So HF gives wrong energy

Configuration Interaction (CI)

Basic Idea:

Wavefunction expressed as a linear combination of determinants:

${\mathrm{<o:p></o:p>}}_{}$

${\mathrm{<o:p></o:p>}}_{}$

•  Φ_o​: ground state (HF)
•  Φ₁​: Excited state (one electron moved)
•  Φ₂​: Two electrons excited

${\mathrm{}}_{}$

CI = mixing multiple electron arrangements

Imagine 2 electrons in 2 orbitals:

HF Picture:

• Both electrons stay in lowest orbital

CI Picture:

• Electron can:
• Stay in ground state
• Jump to higher orbital

${\mathrm{}}_{}$

Final wavefunction = mixture of possibilities

Types of CI

Type	Meaning	Example
CIS	1 electron excited	HOMO → LUMO
CID	2 electrons excited	Both move
CISD	Singles + Doubles	Most common
Full CI	All possibilities	Exact but expensive

Think of CI like:

A student solving a problem using multiple methods together

HF:

Only one method

CI:

Combines many approaches

Advantages:

Because it:
✔ Includes electron correlation
✔ Improves energy
✔ Gives better accuracy

Limitations:

• Very expensive 💻
• Slow for large molecules

Full CI is only possible for small systems