Basis Set Expansion (LCAO Approximation)¶

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

\[ \hat F \chi_i = \varepsilon_i \chi_i, \]

where the unknown quantities are the molecular orbitals \(\chi_i\).

Although these equations are mathematically elegant, they cannot be solved directly for real molecules because the molecular orbitals are unknown continuous functions.

To make the problem computationally tractable, quantum chemistry replaces the unknown molecular orbitals with a finite expansion of known mathematical functions called basis functions.

This approximation is known as the Linear Combination of Atomic Orbitals (LCAO) and forms the foundation of nearly all modern electronic structure calculations.

Why Do We Need Basis Functions?¶

Suppose we wish to determine an unknown function.

Instead of solving for the function directly, we approximate it as a weighted sum of known functions.

For example,

\[ f(x) = c_1 g_1(x) + c_2 g_2(x) + c_3 g_3(x) +\cdots \]

where

\(g_i(x)\) are known functions,
\(c_i\) are unknown coefficients.

Instead of solving for an entire function, we now only need to determine a finite set of coefficients.

Quantum chemistry applies exactly the same idea to molecular orbitals.

Linear Combination of Atomic Orbitals¶

Each molecular orbital is written as

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

where

\(\psi_i\) is a molecular orbital,
\(\chi_\mu\) is an atomic basis function,
\(C_{\mu i}\) is the expansion coefficient,
\(K\) is the total number of basis functions.

Rather than solving directly for the orbital,

Gaussian solves for the coefficients

\[ C_{\mu i}. \]

Physical Interpretation¶

The LCAO approximation states that

A molecular orbital is constructed by combining many atomic orbitals.

Graphically,

Atomic Orbitals

χ₁
χ₂
χ₃
χ₄
...
χₖ

        │
        │  Linear Combination
        ▼

ψ₁  Molecular Orbital

ψ₂  Molecular Orbital

ψ₃  Molecular Orbital

Each molecular orbital is therefore a mixture of several atomic basis functions.

What Are Basis Functions?¶

A basis function is a mathematical function centered on an atom that is used to describe the distribution of electrons.

Modern quantum chemistry generally employs Gaussian-type orbitals (GTOs) because they allow molecular integrals to be evaluated efficiently.

Typical basis functions resemble

s orbitals
p orbitals
d orbitals
f orbitals

but they are mathematical approximations rather than exact atomic orbitals.

Gaussian-Type Orbitals¶

A Gaussian basis function has the form

\[ \boxed{ \chi(r) = N e^{-\alpha r^2} } \]

where

\(N\) is the normalization constant,
\(\alpha\) controls the width of the orbital,
\(r\) is the distance from the nucleus.

Gaussian uses these functions because products of Gaussian functions can be evaluated analytically, making calculations much faster.

Why Not Use Exact Atomic Orbitals?¶

The exact hydrogenic orbitals have the form

\[ e^{-r}, \]

which correctly reproduce the physical shape of atomic orbitals.

However,

integrals involving exponential functions are computationally very expensive.

Gaussian functions,

\[ e^{-\alpha r^2}, \]

allow many molecular integrals to be computed analytically.

Although each Gaussian function is less accurate individually, combinations of many Gaussian functions provide excellent approximations while remaining computationally efficient.

Contracted Basis Functions¶

A single Gaussian function is usually not flexible enough to describe an atomic orbital accurately.

Instead,

several primitive Gaussian functions are combined,

\[ \boxed{ \chi = \sum_i d_i g_i } \]

where

\(g_i\) are primitive Gaussian functions,
\(d_i\) are contraction coefficients.

The resulting function is called a contracted Gaussian function.

Most basis sets supplied with Gaussian consist of contracted Gaussian functions.

Example: The 6-31G Basis Set¶

In this documentation, we use the 6-31G basis set.

Its name describes how contracted Gaussian functions are constructed.

6: Six primitive Gaussian functions are combined to describe the core orbitals.
3: Three primitive Gaussian functions form the first part of the valence orbital.
1: One primitive Gaussian function forms the second part of the valence orbital.

This is known as a split-valence basis set because the valence orbitals are represented by two functions of different flexibility.

Compared with minimal basis sets, 6-31G provides a more accurate description of chemical bonding while remaining computationally efficient.

Expansion Coefficients¶

The unknown quantities in the LCAO expansion are the coefficients

\[ C_{\mu i}. \]

Each coefficient indicates how much a particular basis function contributes to a molecular orbital.

For example,

\[ \psi_1 = 0.72\chi_1 + 0.48\chi_2 - 0.32\chi_3. \]

Large coefficients indicate strong contributions, while small coefficients indicate only minor contributions.

Determining these coefficients is the primary objective of the Hartree–Fock and Kohn–Sham procedures.

From Differential Equations to Matrix Equations¶

Substituting the LCAO expansion into the Hartree–Fock equations transforms

\[ \hat F\psi = \varepsilon\psi \]

into a matrix eigenvalue problem.

Instead of solving differential equations,

Gaussian now solves for the coefficient matrix

\[ \mathbf C. \]

This transformation allows the electronic structure problem to be solved using standard numerical linear algebra techniques.

Why Is the Choice of Basis Set Important?¶

The basis set controls the balance between computational cost and accuracy.

A small basis set

requires less memory,
converges quickly,
but provides lower accuracy.

A larger basis set

improves the description of electron density,
produces more reliable energies,
but increases computational cost.

Choosing an appropriate basis set is therefore one of the most important decisions in any quantum chemical calculation.

Summary¶

The Linear Combination of Atomic Orbitals (LCAO) approximation replaces the unknown molecular orbitals with finite expansions of known basis functions. By expressing each molecular orbital as a weighted sum of Gaussian basis functions, the Hartree–Fock equations are transformed from differential equations into algebraic matrix equations. The unknown quantities become the expansion coefficients rather than the orbitals themselves, making electronic structure calculations practical on modern computers.

Next Section¶

Substituting the LCAO expansion into the Hartree–Fock equations leads directly to the Roothaan–Hall Equations, where the electronic structure problem is expressed as a matrix eigenvalue equation that can be solved numerically.