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Roothaan–Hall Equations

In the previous chapter, we derived the Hartree–Fock equation

\[ \hat F\phi_i=\varepsilon_i\phi_i \]

Although this equation resembles the Schrödinger equation, it is still an integro-differential equation and cannot be solved analytically for most molecules.

The key idea behind the Roothaan–Hall method is to transform this continuous equation into a matrix eigenvalue problem, which can be solved efficiently using numerical linear algebra.

This transformation forms the computational foundation of nearly every modern Hartree–Fock program, including GAMESS, Gaussian, and ORCA.

Why Do We Need Basis Functions?

A molecular orbital is an unknown function.

Instead of solving directly for this function, we approximate it as a linear combination of known functions called basis functions.

Mathematically,

\[ \boxed{ \phi_i = \sum_{\mu=1}^{K} C_{\mu i}\chi_\mu } \]

where

Symbol Meaning
\(\phi_i\) Molecular orbital
\(\chi_\mu\) Basis function
\(C_{\mu i}\) Expansion coefficient
\(K\) Number of basis functions

The basis functions are fixed, while the coefficients are unknown.

The goal of the Hartree–Fock calculation is therefore to determine the optimal values of the coefficients.

Linear Combination of Atomic Orbitals

Most quantum chemistry programs use atomic orbitals as basis functions.

This approximation is known as the Linear Combination of Atomic Orbitals (LCAO).

For example,

\[ \phi_i = C_{1i}\chi_1 + C_{2i}\chi_2 + \cdots + C_{Ki}\chi_K \]

Each molecular orbital is therefore constructed by combining many atomic basis functions.

Note

In practical calculations, the basis functions are usually Gaussian-type orbitals (GTOs), such as STO-3G, 6-31G, or cc-pVDZ.

Substituting into the Hartree–Fock Equation

We substitute

\[ \phi_i=\sum_\mu C_{\mu i}\chi_\mu \]

into

\[ \hat F\phi_i=\varepsilon_i\phi_i \]

to obtain

\[ \hat F \sum_\mu C_{\mu i}\chi_\mu = \varepsilon_i \sum_\mu C_{\mu i}\chi_\mu \]

This equation is still continuous.

To convert it into matrix form, we multiply by another basis function and integrate over all space.

Matrix Elements

The Fock matrix is defined as

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

Each element describes the interaction between two basis functions through the Fock operator.

The Overlap Matrix

Unlike ordinary vectors,

basis functions are generally not orthogonal.

Therefore,

we introduce the overlap matrix

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

Properties of the overlap matrix:

  • symmetric
  • positive definite
  • diagonal elements are equal to one for normalized basis functions

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

The Coefficient Matrix

The unknown expansion coefficients are collected into a matrix,

\[ \mathbf C = \begin{bmatrix} C_{11}&C_{12}&\cdots\\ C_{21}&C_{22}&\cdots\\ \vdots&\vdots&\ddots \end{bmatrix} \]

Each column corresponds to one molecular orbital.

The Orbital Energy Matrix

The orbital energies are placed into a diagonal matrix,

\[ \boldsymbol{\varepsilon} = \begin{bmatrix} \varepsilon_1&0&0\\ 0&\varepsilon_2&0\\ 0&0&\ddots \end{bmatrix} \]

Each diagonal element represents the energy of one molecular orbital.

The Roothaan–Hall Equation

Combining all these matrices yields

\[ \boxed{ \mathbf{FC} = \mathbf{SC} \boldsymbol{\varepsilon} } \]

This equation is known as the Roothaan–Hall equation.

It is the matrix form of the Hartree–Fock equations.

This is the equation solved by RHF programs.

Meaning of Each Matrix

Matrix Description
\(\mathbf F\) Fock matrix
\(\mathbf S\) Overlap matrix
\(\mathbf C\) Molecular orbital coefficients
\(\boldsymbol{\varepsilon}\) Orbital energies

The calculation proceeds by determining the coefficient matrix that satisfies this equation.

A Generalized Eigenvalue Problem

The Roothaan–Hall equation differs from the ordinary eigenvalue equation

\[ \mathbf{Ax}=\lambda\mathbf{x} \]

because of the overlap matrix.

Instead,

we solve

\[ \mathbf{FC} = \mathbf{SC} \boldsymbol{\varepsilon} \]

This is called a generalized eigenvalue problem.

Since the overlap matrix is known,

it can be transformed into an ordinary eigenvalue problem using matrix orthogonalization techniques.

Building the Fock Matrix

An important point is that the Fock matrix is not fixed.

It depends on the electron density,

which depends on the occupied molecular orbitals.

Therefore,

the computational procedure becomes

  1. Guess molecular orbitals.
  2. Build the density matrix.
  3. Construct the Fock matrix.
  4. Solve the Roothaan–Hall equation.
  5. Obtain new molecular orbitals.
  6. Repeat until convergence.

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

Why the Roothaan–Hall Equation is Important

Without the Roothaan–Hall formulation,

the Hartree–Fock equations would remain continuous integro-differential equations.

The matrix formulation enables

  • efficient numerical algorithms,
  • large molecular calculations,
  • implementation on modern computers,
  • use of optimized linear algebra libraries.

Virtually every Hartree–Fock implementation uses this approach.

Key Takeaways

  • Molecular orbitals are expanded as linear combinations of basis functions.
  • The unknown quantities are the molecular orbital coefficients.
  • The Hartree–Fock equations become a matrix equation.
  • The overlap matrix appears because basis functions are generally non-orthogonal.
  • Solving the Roothaan–Hall equation yields the molecular orbitals and their orbital energies.

Looking Ahead

Although we now have a matrix equation,

the Fock matrix still depends on the electron density,

which itself depends on the molecular orbitals.

Consequently,

the Roothaan–Hall equations must be solved repeatedly until the input and output electron densities become identical.

This iterative procedure is known as the Self-Consistent Field (SCF) algorithm, which is the topic of the next chapter.