From RHF to CIS¶

In the previous section on Hartree–Fock (RHF) theory, we learned that the electronic wavefunction is approximated by a single Slater determinant constructed from the occupied molecular orbitals.

This approximation provides an excellent description of the ground electronic state for many molecules. The molecular orbitals are optimized through the Self-Consistent Field (SCF) procedure until the total electronic energy is minimized.

Mathematically, the Hartree–Fock wavefunction is written as

\[ \Psi_{\mathrm{HF}} = \Phi_0 \]

where

\(\Phi_0\) is the ground-state Slater determinant.

Although this wavefunction describes the ground state reasonably well, it cannot adequately represent electronically excited states.

Why Hartree–Fock Cannot Describe Excited States¶

Consider a closed-shell molecule in its ground state.

The molecular orbitals are filled according to the Aufbau principle.

Higher Energy

Virtual Orbitals
------------------
      LUMO
------------------
      LUMO+1
------------------

Occupied Orbitals
------------------
      HOMO
------------------
      HOMO−1
------------------

Lower Energy

The Hartree–Fock wavefunction assumes that all electrons remain in this lowest-energy arrangement.

However, when a molecule absorbs light, one or more electrons may be promoted to higher-energy orbitals.

For example,

Ground State

HOMO      ↑↓

LUMO

becomes

Excited State

HOMO      ↑

LUMO       ↓

The electronic configuration has changed.

Because the Hartree–Fock wavefunction contains only the ground-state determinant, it cannot represent this excited configuration.

A New Electronic Configuration¶

Suppose an electron is promoted from an occupied orbital

\[ \phi_i \]

to a virtual orbital

\[ \phi_a \]

The resulting electronic configuration is no longer the Hartree–Fock determinant.

Instead, it becomes a new Slater determinant,

\[ \Phi_i^a \]

where

electron removed from occupied orbital \(i\),
electron placed into virtual orbital \(a\).

This determinant represents one possible excited electronic configuration.

Many Possible Excitations¶

A molecule usually contains several occupied and virtual orbitals.

Consequently,

many different single excitations are possible.

For example,

Occupied      Virtual

1  →  6
2  →  6
3  →  6
4  →  6
5  →  6

1  →  7
2  →  7
3  →  7
...

Each excitation corresponds to a different excited Slater determinant.

Some of these excitations contribute strongly to an excited state, while others contribute only weakly.

Beyond a Single Determinant¶

Instead of describing the excited state using just one determinant,

Configuration Interaction combines many excited determinants into a single wavefunction.

The excited-state wavefunction is therefore expressed as

\[ \Psi = c_1\Phi_1 + c_2\Phi_2 + c_3\Phi_3 +\cdots \]

where

each \(\Phi_i\) represents a different excited determinant,
each coefficient \(c_i\) determines the contribution of that configuration.

The coefficients are determined by solving the Configuration Interaction equations.

Why Only Single Excitations?¶

In principle, electrons may undergo

one-electron excitations,
two-electron excitations,
three-electron excitations,
and higher-order excitations.

Configuration Interaction can therefore be expanded as

Ground-State Determinant
        │
        ├── Single Excitations
        ├── Double Excitations
        ├── Triple Excitations
        └── Higher Excitations

Including all possible excitations leads to the Full Configuration Interaction (FCI) method.

Although FCI provides the exact solution within a given basis set, its computational cost increases exponentially with system size and quickly becomes impractical.

To reduce the computational cost, the CIS method includes only single excitations.

The CIS Approximation¶

Configuration Interaction Singles assumes that the excited-state wavefunction can be constructed entirely from singly excited determinants.

Mathematically,

\[ \boxed{ \Psi_{\mathrm{CIS}} = \sum_{i}^{occ} \sum_{a}^{virt} c_i^a \Phi_i^a } \]

where

\(i\) represents occupied orbitals,
\(a\) represents virtual orbitals,
\(\Phi_i^a\) is a singly excited Slater determinant,
\(c_i^a\) is its CI coefficient.

Unlike Hartree–Fock, which optimizes molecular orbitals, CIS keeps the orbitals fixed and determines the best combination of excited configurations.

Comparing RHF and CIS¶

Hartree–Fock (RHF)	Configuration Interaction Singles (CIS)
Describes the ground state	Describes excited states
Single Slater determinant	Linear combination of singly excited determinants
Optimizes molecular orbitals	Uses fixed RHF orbitals
Solves the SCF equations	Solves the CI eigenvalue problem

The RHF wavefunction serves as the reference for the CIS calculation, making CIS a natural extension of the Hartree–Fock method.

Key Takeaways¶

Hartree–Fock accurately describes the electronic ground state but cannot represent excited states.
Excited states arise when electrons are promoted from occupied to virtual molecular orbitals.
Each promotion generates a new Slater determinant.
CIS constructs the excited-state wavefunction as a linear combination of singly excited determinants.
The Hartree–Fock molecular orbitals remain fixed throughout the CIS calculation.

Looking Ahead¶

In this chapter, we introduced the idea of generating excited electronic configurations from the Hartree–Fock reference wavefunction.

The next chapter explains how these singly excited Slater determinants are constructed and why they form the basis of the Configuration Interaction Singles method.