The Configuration Interaction Wavefunction

In the previous chapter, we generated a set of singly excited Slater determinants by promoting one electron from an occupied molecular orbital to a virtual molecular orbital.

Each determinant represents one possible electronic configuration of the excited molecule.

However, a real excited state is rarely described by a single electronic configuration.

Instead, the true excited-state wavefunction is constructed as a linear combination of many singly excited determinants.

This idea is the foundation of Configuration Interaction (CI).

---

Why Combine Many Determinants?

Suppose a molecule has several possible single excitations,

Occupied → Virtual

5 → 6
5 → 7
4 → 6
4 → 7
3 → 6
...

Each excitation corresponds to a different Slater determinant,

\[ \Phi_5^6,\; \Phi_5^7,\; \Phi_4^6,\; \Phi_4^7,\; ... \]

If we chose only one of these determinants, we would ignore all the others.

In reality, the excited electronic state is influenced by several electronic configurations simultaneously.

Therefore, we combine all important configurations into a single wavefunction.

---

Linear Combination of Configurations

The CIS wavefunction is written as

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

where

• \(\Phi_i^a\) is a singly excited Slater determinant,
• \(c_i^a\) is the corresponding CI coefficient.

The coefficients determine how strongly each configuration contributes to the excited state.

---

Meaning of the CI Coefficients

The coefficients

\[ c_i^a \]

are obtained by solving the Configuration Interaction equations.

Each coefficient represents the importance of one electronic excitation.

For example,

\[ \Psi = 0.92\Phi_5^6 + 0.24\Phi_4^6 + 0.08\Phi_3^7 \]

indicates that

• the 5 → 6 excitation dominates the excited state,
• the 4 → 6 excitation contributes moderately,
• the 3 → 7 excitation contributes only weakly.

Thus, the magnitude of each coefficient provides direct insight into the electronic character of the excited state.

---

Normalization of the Wavefunction

Like every quantum mechanical wavefunction,

the CIS wavefunction must satisfy the normalization condition,

\[ \boxed{ \langle\Psi_{\mathrm{CIS}} | \Psi_{\mathrm{CIS}} \rangle = 1 } \]

Consequently,

the CI coefficients satisfy

\[ \boxed{ \sum_{i,a} |c_i^a|^2 = 1 } \]

The squared coefficient,

\[ |c_i^a|^2, \]

can be interpreted as the fractional contribution of that excited configuration to the overall wavefunction.

---

Interpreting the Coefficients

Consider the following example.

Excitation	Coefficient
5 → 6	0.91
4 → 6	0.27
5 → 7	0.09

The corresponding excited-state wavefunction is

\[ \Psi = 0.91\Phi_5^6 + 0.27\Phi_4^6 + 0.09\Phi_5^7 \]

The first configuration contributes most strongly to the excited state.

This is why quantum chemistry programs often list only the largest CI coefficients in the output.

Small coefficients usually have only a minor influence on the character of the excited state.

---

Why Only Single Excitations?

The complete Configuration Interaction wavefunction can be written as

\[ \Psi = c_0\Phi_0 + \sum_i c_i\Phi_i + \sum_{ij}c_{ij}\Phi_{ij} + \cdots \]

where

• \(\Phi_0\) is the Hartree–Fock reference determinant,
• \(\Phi_i\) represents singly excited determinants,
• \(\Phi_{ij}\) represents doubly excited determinants,
• higher terms correspond to triple and quadruple excitations.

In Full Configuration Interaction (FCI), all of these terms are included.

However, the computational cost increases rapidly with system size.

Configuration Interaction Singles simplifies the expansion by retaining only the singly excited determinants,

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

This approximation makes the method computationally efficient while still providing a useful description of many excited states.

---

Why Is the Hartree–Fock Determinant Not Included?

For pure excited-state CIS calculations, the excited-state wavefunction is constructed entirely from singly excited determinants.

The Hartree–Fock reference determinant,

\[ \Phi_0, \]

represents the ground state and is therefore excluded from the excited-state expansion.

As a result,

\[ c_0 = 0. \]

This means that the excited-state wavefunction is composed solely of excited electronic configurations.

Note

Some Configuration Interaction methods include the reference determinant in the wavefunction expansion. However, standard CIS for excited-state calculations uses only singly excited determinants.

---

Connection with GAMESS Output

During a CIS calculation, GAMESS reports the dominant electronic transitions together with their CI coefficients.

For example,

5 -> 6    0.87
4 -> 6    0.23
5 -> 7    0.08

This output corresponds directly to the CIS wavefunction,

\[ \Psi = 0.87\Phi_5^6 + 0.23\Phi_4^6 + 0.08\Phi_5^7. \]

Thus, the printed coefficients are the numerical values of the expansion coefficients obtained after solving the Configuration Interaction equations.

---

Key Takeaways

• A real excited state is generally a mixture of several electronic configurations.
• The CIS wavefunction is a linear combination of singly excited Slater determinants.
• The CI coefficients determine the contribution of each excitation.
• Large coefficients correspond to dominant electronic transitions.
• The coefficients reported in GAMESS are the expansion coefficients of the CIS wavefunction.

---

Looking Ahead

Now that we have established the mathematical form of the CIS wavefunction, the next step is to determine how the coefficients are calculated.

This requires constructing the Configuration Interaction Hamiltonian matrix, whose elements describe the interaction between different singly excited determinants.

Diagonalizing this matrix yields the excitation energies and the CI coefficients discussed in this chapter.

---