Skip to content

The CASSCF Hamiltonian

In the previous chapter, we constructed the CASSCF wavefunction as a linear combination of Configuration State Functions (CSFs),

\[ \boxed{ \Psi=\sum_I c_I\Phi_I } \]

The next question is:

How do we calculate the energy of this wavefunction?

The answer lies in the electronic Hamiltonian operator, which contains all of the physical interactions between the electrons and nuclei in the molecule.

In this chapter, we derive the CASSCF Hamiltonian and show how it is transformed into a matrix that can be solved numerically.

The Time-Independent Schrödinger Equation

Every electronic structure method begins with the time-independent Schrödinger equation,

\[ \boxed{ \hat H \Psi = E\Psi } \]

where

  • \(\hat H\) is the Hamiltonian operator,
  • \(\Psi\) is the electronic wavefunction,
  • \(E\) is the electronic energy.

Our goal is therefore simple:

Find the wavefunction that satisfies this equation while producing the lowest possible energy.

The Electronic Hamiltonian

Within the Born–Oppenheimer approximation, the nuclei are treated as fixed, and only the electrons are considered explicitly.

The electronic Hamiltonian becomes

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

This equation contains every interaction experienced by the electrons.

Understanding Each Term

The Hamiltonian consists of three contributions.

1. Electron Kinetic Energy

\[ -\frac12\nabla_i^2 \]

This operator represents the kinetic energy of electron \(i\).

The Laplacian

\[ \nabla^2 \]

describes the curvature of the electronic wavefunction.

Greater curvature corresponds to larger kinetic energy.

2. Electron–Nucleus Attraction

\[ - \sum_A \frac{Z_A}{r_{iA}} \]

This term represents the electrostatic attraction between electron \(i\) and nucleus \(A\).

where

  • \(Z_A\) is the nuclear charge,
  • \(r_{iA}\) is the distance between electron and nucleus.

This interaction lowers the total energy because opposite charges attract.

3. Electron–Electron Repulsion

\[ \sum_{i<j} \frac1{r_{ij}} \]

Every electron repels every other electron.

This term makes the Schrödinger equation difficult to solve because the motion of every electron depends on every other electron.

Almost every approximation in quantum chemistry exists because of this term.

Why Is the Schrödinger Equation Difficult?

Suppose there is only one electron.

Electron

↓

Nucleus

The Schrödinger equation can be solved exactly.

Now consider ten electrons.

Electron 1

↔

Electron 2

↔

Electron 3

↔

...

↔

Electron 10

Now every electron interacts with every other electron simultaneously.

There is no exact analytical solution.

This is why approximate electronic structure methods such as RHF and CASSCF are required.

Applying the Hamiltonian to the CASSCF Wavefunction

The CASSCF wavefunction is

\[ \Psi=\sum_I c_I\Phi_I \]

Substituting this into the Schrödinger equation gives

\[ \boxed{ \hat H \sum_I c_I\Phi_I = E \sum_I c_I\Phi_I } \]

Since the Hamiltonian is a linear operator,

it acts independently on every Configuration State Function.

Projecting onto the CSF Basis

To determine the unknown coefficients \(c_I\),

we multiply the equation by another Configuration State Function,

\[ \Phi_J \]

giving

\[ \boxed{ \left< \Phi_J \right| \hat H \left| \Psi \right> = E \left< \Phi_J | \Psi \right> } \]

Substituting the CASSCF wavefunction,

\[ \sum_I c_I \left< \Phi_J | \hat H | \Phi_I \right> = E \sum_I c_I \left< \Phi_J | \Phi_I \right> \]

Orthonormality of the CSFs

Configuration State Functions are orthonormal,

meaning

\[ \boxed{ \langle \Phi_J | \Phi_I \rangle = \delta_{IJ} } \]

where

\[ \delta_{IJ} = \begin{cases} 1,& I=J\\ 0,& I\neq J \end{cases} \]

This greatly simplifies the equations.

The Hamiltonian Matrix

We now define the Hamiltonian matrix elements as

\[ \boxed{ H_{IJ} = \langle \Phi_I | \hat H | \Phi_J \rangle } \]

Each matrix element represents the interaction between two Configuration State Functions.

For example,

CSF 1

↓

CSF 2

↓

Interaction Energy

If two configurations interact strongly,

the corresponding Hamiltonian element becomes large.

Constructing the Hamiltonian Matrix

For three CSFs,

the Hamiltonian matrix becomes

\[ \mathbf H = \begin{pmatrix} H_{11} & H_{12} & H_{13}\\ H_{21} & H_{22} & H_{23}\\ H_{31} & H_{32} & H_{33} \end{pmatrix} \]

where

  • diagonal elements describe the energy of each CSF,
  • off-diagonal elements describe the coupling between different configurations.

Physical Interpretation

The Hamiltonian matrix tells us

  • how much energy each configuration possesses,
  • how strongly different configurations interact,
  • how electrons move between configurations.

This interaction is what produces the multiconfigurational wavefunction.

Why Is the Matrix Important?

Once the Hamiltonian matrix has been constructed,

the entire CASSCF problem reduces to solving

\[ \boxed{ \mathbf H \mathbf c = E \mathbf c } \]

This is simply an eigenvalue problem.

The eigenvalues become the electronic energies,

while the eigenvectors become the CI coefficients.

Everything now depends on solving this matrix equation efficiently.

This is the Configuration Interaction (CI) problem, discussed in the next chapter.

Key Takeaways

  • Every electronic structure method begins with the Schrödinger equation.
  • The electronic Hamiltonian contains the kinetic energy, electron–nucleus attraction, and electron–electron repulsion.
  • Applying the Hamiltonian to the CASSCF wavefunction generates a matrix representation.
  • Each matrix element describes the interaction between two Configuration State Functions.
  • Solving the Hamiltonian matrix yields the energies and CI coefficients of the electronic states.