Roothaan–Hall Equations¶

In the previous chapter, the molecular orbitals were expressed as linear combinations of atomic basis functions using the Linear Combination of Atomic Orbitals (LCAO) approximation,

\[ \psi_i=\sum_{\mu}C_{\mu i}\chi_\mu. \]

This transformation converts the Hartree–Fock differential equations into a matrix problem that can be solved numerically. The resulting equations are known as the Roothaan–Hall Equations, developed independently by Clemens Roothaan and George Hall in the early 1950s.

These equations form the computational foundation of almost every modern electronic structure method, including Hartree–Fock, Density Functional Theory (DFT), MP2, Coupled Cluster, and many multireference methods.

From Orbitals to Matrices¶

Recall the Hartree–Fock equation,

\[ \hat F\psi_i=\varepsilon_i\psi_i. \]

The unknown quantity is the molecular orbital \(\psi_i\).

Using the LCAO approximation,

\[ \psi_i=\sum_{\mu}C_{\mu i}\chi_\mu, \]

each molecular orbital is written as a linear combination of basis functions.

Substituting this expression into the Hartree–Fock equation gives

\[ \hat F \left( \sum_\mu C_{\mu i}\chi_\mu \right) = \varepsilon_i \left( \sum_\mu C_{\mu i}\chi_\mu \right). \]

The problem is now expressed entirely in terms of basis functions and unknown expansion coefficients.

Projection onto the Basis¶

To obtain algebraic equations, each side is multiplied by another basis function and integrated over all space,

\[ \int \chi_\nu^* \hat F \psi_i \,d\tau = \varepsilon_i \int \chi_\nu^* \psi_i \,d\tau. \]

This operation projects the differential equation onto the finite basis set.

After substituting the LCAO expansion, the result becomes a matrix equation.

Matrix Form¶

The Hartree–Fock equations become

\[ \boxed{ \mathbf F\mathbf C = \mathbf S\mathbf C\mathbf\varepsilon } \]

This is known as the Roothaan–Hall Equation.

Instead of solving differential equations,

Gaussian solves this matrix eigenvalue problem.

Meaning of Each Matrix¶

The Roothaan–Hall equation contains four matrices.

Fock Matrix¶

\[ \mathbf F \]

The Fock matrix contains all information about the electronic Hamiltonian.

Its elements are

\[ F_{\mu\nu} = \langle \chi_\mu | \hat F | \chi_\nu \rangle. \]

It includes

kinetic energy,
electron–nucleus attraction,
Coulomb interactions,
exchange interactions.

Coefficient Matrix¶

\[ \mathbf C \]

Each column of the coefficient matrix contains the expansion coefficients for one molecular orbital.

For example,

          Molecular Orbitals

         ψ₁   ψ₂   ψ₃

χ₁      0.71 0.12 ...

χ₂      0.42 0.53 ...

χ₃     -0.28 0.64 ...

Every molecular orbital is therefore represented by one column of the matrix.

Overlap Matrix¶

Unlike ordinary vectors,

basis functions are not orthogonal.

Therefore,

\[ \boxed{ S_{\mu\nu} = \langle \chi_\mu | \chi_\nu \rangle } \]

The overlap matrix measures how much two basis functions overlap in space.

If two basis functions are identical,

\[ S=1. \]

If they are completely orthogonal,

\[ S=0. \]

The overlap matrix distinguishes quantum chemistry from ordinary eigenvalue problems.

Orbital Energy Matrix¶

\[ \mathbf\varepsilon \]

The diagonal matrix

\[ \mathbf\varepsilon = \begin{pmatrix} \varepsilon_1&0&0\\ 0&\varepsilon_2&0\\ 0&0&\ddots \end{pmatrix} \]

contains the molecular orbital energies.

These values later appear as

occupied orbital energies,
virtual orbital energies,
HOMO,
LUMO,

in the Gaussian output file.

Generalized Eigenvalue Problem¶

The Roothaan–Hall equation

\[ \mathbf F\mathbf C = \mathbf S\mathbf C\mathbf\varepsilon \]

is called a generalized eigenvalue problem because the overlap matrix appears on the right-hand side.

If the basis functions were perfectly orthogonal,

\[ \mathbf S=\mathbf I, \]

the equation would reduce to the familiar form

\[ \mathbf F\mathbf C = \mathbf C\mathbf\varepsilon. \]

Because Gaussian basis functions overlap,

the overlap matrix must always be included.

Why Isn't the Problem Solved Immediately?¶

Although the Roothaan–Hall equation looks like a standard matrix equation,

the Fock matrix is not known beforehand.

This is because

Coefficient Matrix
        │
        ▼
Electron Density
        │
        ▼
Fock Matrix
        │
        ▼
New Coefficient Matrix

The Fock matrix depends on the electron density,

while the electron density depends on the molecular orbitals,

which themselves are obtained by solving the Roothaan–Hall equation.

This circular dependence prevents a direct solution.

The Need for Iteration¶

Since the Fock matrix depends on the unknown molecular orbitals,

Gaussian proceeds iteratively.

The program

Makes an initial guess for the molecular orbitals.
Constructs the Fock matrix.
Solves the Roothaan–Hall equation.
Obtains new molecular orbitals.
Builds a new electron density.
Reconstructs the Fock matrix.
Repeats the cycle until convergence.

This iterative process is called the Self-Consistent Field (SCF) procedure.

Connection to Density Functional Theory¶

Although derived from Hartree–Fock theory,

the same mathematical framework is used in Kohn–Sham Density Functional Theory.

The only major difference is that

the Hartree–Fock Fock matrix is replaced by the Kohn–Sham matrix,
while the matrix equation itself remains almost identical.

Consequently,

the same numerical algorithms are employed for both HF and DFT calculations in Gaussian.

Computational Workflow¶

The complete computational procedure is now

Molecular Geometry
        │
        ▼
Choose Basis Set
        │
        ▼
Construct Basis Functions
        │
        ▼
Build Fock Matrix
        │
        ▼
Solve Roothaan–Hall Equation
        │
        ▼
Obtain Molecular Orbitals
        │
        ▼
Build Electron Density
        │
        ▼
Repeat Until Converged

This is essentially what Gaussian performs during every SCF calculation.

Summary¶

The Roothaan–Hall equations convert the Hartree–Fock differential equations into a matrix eigenvalue problem that can be solved numerically. The molecular orbitals are represented as linear combinations of basis functions, and the unknown quantities become the expansion coefficients rather than continuous functions. Because the Fock matrix depends on the electron density, the equations must be solved iteratively using the Self-Consistent Field procedure. This matrix formulation forms the computational core of both Hartree–Fock and Kohn–Sham Density Functional Theory.

Next Section¶

The next chapter introduces the Self-Consistent Field (SCF) Algorithm, which describes the iterative procedure Gaussian uses to repeatedly construct the Fock matrix, solve the Roothaan–Hall equations, update the electron density, and achieve convergence.