Singly Excited Slater Determinants

In the previous chapter, we learned that the Hartree–Fock wavefunction consists of a single Slater determinant representing the electronic ground state. We also saw that this determinant cannot describe electronically excited states because the electronic configuration changes when an electron is promoted to a higher-energy orbital.

To describe these excited configurations, Configuration Interaction Singles (CIS) constructs new Slater determinants by exciting one electron from an occupied molecular orbital to a virtual molecular orbital.

These new determinants form the basis of the CIS wavefunction.

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The Hartree–Fock Reference Determinant

For a closed-shell molecule, the Hartree–Fock ground-state wavefunction is

\[ \Psi_0 = \Phi_0 \]

where

\[ \Phi_0 = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(1) & \chi_2(1) & \cdots & \chi_N(1) \\ \chi_1(2) & \chi_2(2) & \cdots & \chi_N(2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(N) & \chi_2(N) & \cdots & \chi_N(N) \end{vmatrix} \]

This determinant contains only the occupied molecular orbitals.

Every electron occupies one of these orbitals, producing the lowest-energy electronic configuration.

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Occupied and Virtual Orbitals

After an RHF calculation, the molecular orbitals can be divided into two groups.

Higher Energy

Virtual Orbitals

φa
φa+1
φa+2
------------------------
LUMO

------------------------
HOMO

φi
φi−1
φi−2

Occupied Orbitals

Lower Energy

The occupied orbitals contain electrons, whereas the virtual orbitals are initially empty.

Electronic excitation consists of transferring an electron from an occupied orbital into one of the virtual orbitals.

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Constructing a Single Excitation

Suppose an electron occupying orbital

\[ \phi_i \]

is promoted into a virtual orbital

\[ \phi_a. \]

The original Hartree–Fock determinant

\[ \Phi_0 \]

is transformed into a new determinant,

\[ \boxed{ \Phi_i^a } \]

where

• \(i\) denotes the occupied orbital from which the electron is removed,
• \(a\) denotes the virtual orbital into which the electron is placed.

The notation

\[ \Phi_i^a \]

is read as

"the determinant obtained by exciting one electron from orbital \(i\) to orbital \(a\)."

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Visual Representation

Consider four occupied orbitals and three virtual orbitals.

Ground state

Virtual

7
6
5

--------------
4  ↑↓
3  ↑↓
2  ↑↓
1  ↑↓

Occupied

Now excite one electron

Virtual

7
6
5  ↑

--------------
4  ↑
3  ↑↓
2  ↑↓
1  ↑↓

Occupied

This represents

\[ \Phi_4^5 \]

because one electron has been promoted from orbital 4 to orbital 5.

The Hartree–Fock determinant has therefore changed into a singly excited determinant.

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Many Possible Single Excitations

If a molecule has

• \(n_{\text{occ}}\) occupied orbitals
• \(n_{\text{virt}}\) virtual orbitals

then every occupied orbital can be excited into every virtual orbital.

The total number of singly excited determinants is therefore

\[ \boxed{ N_{\text{single}} = n_{\text{occ}} \times n_{\text{virt}} } \]

For example,

if a molecule has

• 5 occupied orbitals
• 10 virtual orbitals

then

\[ N_{\text{single}} = 5 \times 10 = 50 \]

possible singly excited determinants can be generated.

Each of these determinants becomes a basis function in the CIS calculation.

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Why Are Slater Determinants Used?

Each excited configuration must still satisfy the fundamental properties of the electronic wavefunction.

In particular,

• electrons are indistinguishable,
• the wavefunction must remain antisymmetric,
• exchanging two electrons must change the sign of the wavefunction.

Because a Slater determinant automatically satisfies these requirements, every excited configuration generated in CIS is also represented by a Slater determinant.

Thus,

both the Hartree–Fock reference state and all singly excited configurations obey the Pauli Exclusion Principle.

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The CIS Basis

The Hartree–Fock determinant serves as the reference configuration.

From this reference,

all singly excited determinants are generated.

The CIS basis therefore consists of

Reference Determinant

Φ₀

↓

Single Excitations

Φ₁⁶
Φ₂⁶
Φ₃⁶
Φ₄⁶
...

Φ₁⁷
Φ₂⁷
Φ₃⁷
...

Each determinant represents one possible electronic excitation.

The next step is to combine these determinants to construct the excited-state wavefunction.

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Key Takeaways

• The Hartree–Fock wavefunction is a single Slater determinant.
• A singly excited determinant is obtained by promoting one electron from an occupied orbital to a virtual orbital.
• The notation

\[ \Phi_i^a \]

represents an excitation from occupied orbital \(i\) to virtual orbital \(a\). - The number of possible single excitations is

\[ n_{\text{occ}} \times n_{\text{virt}}. \]

• These singly excited determinants form the basis of the CIS method.

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Looking Ahead

We have now constructed the basis functions used in Configuration Interaction Singles.

The next chapter combines these determinants into a single mathematical expression known as the Configuration Interaction wavefunction, where each determinant is assigned a coefficient describing its contribution to the excited state.

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