Skip to content

The CIS Hamiltonian

In the previous chapter, we expressed the excited-state wavefunction as a linear combination of singly excited Slater determinants,

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

The remaining question is:

How are the coefficients \(c_i^a\) determined?

To answer this, we use the electronic Hamiltonian operator, which contains all the information about the kinetic energy of the electrons, the attraction between electrons and nuclei, and the repulsion between electrons.

The Hamiltonian determines how different excited configurations interact with one another.

The Electronic Hamiltonian

Within the Born–Oppenheimer approximation, the electronic Hamiltonian is

\[ \boxed{ \hat H = \sum_{i=1}^{N} \left( -\frac12\nabla_i^2 - \sum_A \frac{Z_A}{r_{iA}} \right) + \sum_{i<j} \frac1{r_{ij}} } \]

where

  • the first term represents the kinetic energy of the electrons,
  • the second term represents electron–nucleus attraction,
  • the third term represents electron–electron repulsion.

This is the same Hamiltonian used in Hartree–Fock theory.

The difference lies in the basis used to solve the Schrödinger equation.

A Different Basis

In Hartree–Fock theory, the wavefunction is represented by a single Slater determinant.

In CIS, the basis consists of all singly excited determinants,

\[ \Phi_1^6,\; \Phi_2^6,\; \Phi_3^6,\; \Phi_1^7,\; \Phi_2^7,\; ... \]

Instead of optimizing molecular orbitals, CIS calculates how these excited configurations interact.

Matrix Elements of the Hamiltonian

The interaction between two excited determinants is described by the matrix element

\[ \boxed{ H_{ia,jb} = \langle \Phi_i^a | \hat H | \Phi_j^b \rangle } \]

Here,

  • \(\Phi_i^a\) represents one singly excited determinant,
  • \(\Phi_j^b\) represents another singly excited determinant.

Each matrix element measures the coupling between two electronic configurations.

Building the CIS Matrix

Suppose four singly excited determinants are generated,

\[ \Phi_1,\; \Phi_2,\; \Phi_3,\; \Phi_4. \]

The CIS Hamiltonian becomes

\[ \mathbf H= \begin{pmatrix} H_{11} & H_{12} & H_{13} & H_{14}\\ H_{21} & H_{22} & H_{23} & H_{24}\\ H_{31} & H_{32} & H_{33} & H_{34}\\ H_{41} & H_{42} & H_{43} & H_{44} \end{pmatrix} \]

Each row and column corresponds to one singly excited determinant.

The size of this matrix therefore equals the number of possible single excitations.

Physical Meaning of the Matrix Elements

The diagonal elements

\[ H_{ii} \]

represent the energy of an individual excited configuration.

The off-diagonal elements

\[ H_{ij}, \qquad i\neq j, \]

describe the interaction (or coupling) between two different excited configurations.

If two configurations interact strongly, they mix significantly in the final excited-state wavefunction.

If the interaction is small, the configurations remain largely independent.

Why Do Configurations Mix?

Consider two possible excitations,

5 → 6

4 → 6

If these two configurations have similar energies, they can interact through the electronic Hamiltonian.

Instead of producing two completely independent excited states, the Hamiltonian mixes them together.

The resulting excited state becomes

\[ \Psi = c_1\Phi_5^6 + c_2\Phi_4^6. \]

The stronger the coupling between the determinants, the larger the mixing.

Matrix Size

If

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

are available, then the number of singly excited determinants is

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

The CIS Hamiltonian therefore has dimensions

\[ \boxed{ N\times N. } \]

For small molecules, this matrix is relatively small.

For larger molecules, the number of possible excitations increases rapidly, making the Hamiltonian much larger and more computationally demanding.

The Goal of the Hamiltonian

Constructing the Hamiltonian matrix is not the final objective.

The purpose of the matrix is to determine

  • excitation energies,
  • excited-state wavefunctions,
  • and the CI coefficients.

These quantities are obtained by diagonalizing the Hamiltonian matrix.

This process is discussed in the next chapter.

Key Takeaways

  • CIS uses the same electronic Hamiltonian as Hartree–Fock.
  • The basis functions are singly excited Slater determinants.
  • Each matrix element describes the interaction between two excited configurations.
  • Diagonal elements correspond to the energies of individual configurations.
  • Off-diagonal elements describe configuration mixing.
  • The Hamiltonian matrix is diagonalized to obtain the excited-state energies and wavefunctions.

Looking Ahead

The Hamiltonian matrix contains all the information about the interactions between the singly excited configurations.

The next chapter derives the CIS secular equation, showing how diagonalizing this matrix produces the excitation energies and the CI coefficients printed by GAMESS.