Why Hartree–Fock Fails¶

Hartree–Fock (RHF) is the foundation of many electronic structure methods and often provides an excellent description of molecules near their equilibrium geometry. It assumes that all electrons occupy a single electronic configuration, represented by a single Slater determinant.

For many systems, this approximation works remarkably well. However, there are important situations where the electronic structure cannot be described by a single configuration alone. In these cases, RHF becomes inadequate, and more advanced multiconfigurational methods such as CASSCF are required.

This chapter explains the limitations of RHF and motivates the need for the CASSCF method.

The Single-Configuration Approximation¶

In RHF, every electron occupies one molecular orbital, and the complete electronic wavefunction is constructed from a single arrangement of these occupied orbitals.

Conceptually, the wavefunction looks like

Ground-State Configuration

Orbital 1   ↑↓

Orbital 2   ↑↓

Orbital 3   ↑↓

Orbital 4   __

Orbital 5   __

Only this single arrangement contributes to the wavefunction.

This approximation assumes that electrons remain in these orbitals throughout the calculation.

When Does RHF Work Well?¶

The RHF approximation performs well when one electronic configuration dominates the molecular wavefunction.

Examples include

stable closed-shell molecules,
molecules near their equilibrium geometry,
systems with large energy gaps between occupied and virtual orbitals,
ground-state energy calculations where electron correlation is relatively small.

Typical examples are

methane (CH₄),
ammonia (NH₃),
water (H₂O),
carbon dioxide (CO₂).

For these molecules, the electronic structure is largely described by a single configuration.

Situations Where RHF Begins to Fail¶

Many chemically important systems cannot be described by a single electronic configuration.

These include

excited electronic states,
bond dissociation,
transition states,
diradicals,
conical intersections,
transition-metal complexes,
photochromic molecules,
molecules with nearly degenerate orbitals.

In these situations, several electronic configurations contribute almost equally to the wavefunction.

Example: Bond Breaking¶

Consider stretching a simple chemical bond.

Near equilibrium,

Configuration A

≈ 100%

Configuration B

≈ 0%

A single electronic configuration dominates.

As the bond is stretched,

Configuration A

≈ 60%

Configuration B

≈ 40%

Now both configurations become important.

A single-determinant description is no longer sufficient.

RHF cannot describe this situation accurately because it always assumes that Configuration A contributes 100%.

Example: Excited States¶

Excited electronic states involve promoting electrons from occupied molecular orbitals to virtual molecular orbitals.

Ground State

Occupied → Occupied

↓

Excitation

↓

Electron promoted

↓

Excited Configuration

The excited-state wavefunction is therefore naturally composed of multiple electronic configurations.

Although CIS introduces excited configurations, it does not re-optimize the molecular orbitals. Consequently, its accuracy is limited, particularly for systems with strong electron correlation.

Near-Degenerate Orbitals¶

Sometimes two molecular orbitals have very similar energies.

Orbital A

────────────

Orbital B

────────────

Because the energy difference is small, electrons can occupy either orbital with nearly equal probability.

The true wavefunction becomes a mixture of multiple electronic configurations.

RHF forces the electrons into only one of these possibilities, introducing significant errors.

Static Electron Correlation¶

The primary limitation of RHF is its inability to describe static electron correlation.

Static correlation arises when several electronic configurations contribute significantly to the exact wavefunction.

Instead of

Wavefunction

↓

Configuration A

the true wavefunction resembles

Configuration A

+

Configuration B

+

Configuration C

+

...

Each configuration contributes to the overall electronic structure.

RHF ignores all but one of these configurations.

Why CIS Is Not Enough¶

Configuration Interaction Singles (CIS) improves upon RHF by introducing singly excited configurations.

Ground Configuration

+

Single Excitations

However,

the molecular orbitals remain fixed,
only single excitations are included,
higher excitations are neglected,
static correlation is still poorly described.

For many photochemical systems, this approximation is insufficient.

What Is Needed?¶

To accurately describe strongly correlated systems, we need a method that

includes multiple electronic configurations,
allows electrons to redistribute among important orbitals,
optimizes the molecular orbitals during the calculation,
provides a balanced description of ground and excited states.

This is precisely the goal of CASSCF.

Instead of assuming

One Configuration

CASSCF considers

Configuration 1

+

Configuration 2

+

Configuration 3

+

...

↓

Final Wavefunction

The contribution of each configuration is determined automatically during the calculation.

From RHF to CASSCF¶

The progression of electronic structure methods can be summarized as

RHF

↓

Single Configuration

↓

CIS

↓

Single Excitations

↓

CASSCF

↓

All Important Configurations

Each method builds upon the previous one while addressing its limitations.

Key Takeaways¶

RHF assumes that one electronic configuration dominates the wavefunction.
This approximation is excellent for many closed-shell ground-state molecules.
RHF fails when multiple configurations contribute significantly.
Bond breaking, excited states, diradicals, and photochemical systems require a multiconfigurational description.
CIS introduces excited configurations but does not optimize the orbitals.
CASSCF overcomes these limitations by optimizing both the molecular orbitals and the multiconfigurational wavefunction.