Hartree Product¶

The electronic Hamiltonian introduced in the previous chapter contains the electron–electron repulsion term, making the many-electron Schrödinger equation impossible to solve exactly for molecules containing more than one electron.

To make progress, early quantum chemists sought approximate descriptions of the many-electron wavefunction. The first successful approximation was proposed by Douglas Hartree in 1928 and is known as the Hartree Product.

The Hartree Product assumes that each electron moves independently in an average electrostatic field generated by all the other electrons. Although this approximation neglects important quantum mechanical effects, it provides the conceptual foundation for Hartree–Fock theory.

The Many-Electron Wavefunction¶

For a molecule containing \(N\) electrons, the exact electronic wavefunction depends simultaneously on the coordinates of every electron,

\[ \Psi(\mathbf r_1,\mathbf r_2,\mathbf r_3,\ldots,\mathbf r_N). \]

This wavefunction describes the probability of finding all electrons simultaneously at specific positions.

Because the motion of every electron is coupled through electron–electron repulsion, the wavefunction cannot be separated into independent one-electron functions.

This coupling is the primary difficulty in solving the electronic Schrödinger equation.

Hartree's Approximation¶

Hartree proposed a simple approximation:

Instead of treating all electrons simultaneously, assume that each electron occupies its own independent orbital.

The total electronic wavefunction is then written as the product of one-electron wavefunctions,

\[ \boxed{ \Psi_H = \phi_1(\mathbf r_1) \phi_2(\mathbf r_2) \cdots \phi_N(\mathbf r_N) } \]

This expression is called the Hartree Product.

Each orbital describes only one electron.

Interpretation¶

The Hartree Product assumes

each electron moves independently,
each electron occupies a molecular orbital,
electrons interact only through an average electrostatic potential.

Graphically,

Electron 1  ───►  Orbital φ₁

Electron 2  ───►  Orbital φ₂

Electron 3  ───►  Orbital φ₃

        ⋮

Electron N  ───►  Orbital φₙ

The total wavefunction is simply the product of these individual orbitals.

Why Is This Useful?¶

The Hartree Product transforms the complicated many-electron problem into a collection of simpler one-electron problems.

Instead of solving

One enormous many-electron equation

we solve

Many independent one-electron equations

This dramatically reduces the computational complexity.

The Independent Electron Approximation¶

Within the Hartree approximation,

each electron experiences an average field generated by all the other electrons.

Instead of considering the instantaneous positions of every electron,

Electron A
        │
        ▼
Average Electrostatic Field
        ▲
Electron B

the interaction is replaced by an average potential.

This approximation removes the explicit coupling between electrons.

Energy of the Hartree Product¶

If the Hartree Product is substituted into the electronic Schrödinger equation,

the total energy becomes approximately

\[ E = \sum_i \langle \phi_i | \hat h | \phi_i \rangle + \text{Average Electron Repulsion}. \]

The total energy therefore consists of

one-electron contributions,
plus an average treatment of electron–electron interactions.

Although approximate, this expression is much easier to evaluate than the exact many-electron energy.

The Major Problem¶

Despite its simplicity, the Hartree Product contains a serious flaw.

Consider two electrons.

The Hartree Product is

\[ \Psi_H = \phi_1(1)\phi_2(2). \]

If we exchange the two electrons,

\[ \Psi_H' = \phi_1(2)\phi_2(1), \]

which is generally not equal to

\[ -\Psi_H. \]

However, quantum mechanics requires that the wavefunction of identical electrons satisfy

\[ \boxed{ \Psi(1,2) = - \Psi(2,1). } \]

The Hartree Product therefore violates one of the most fundamental principles of quantum mechanics.

Violation of the Pauli Principle¶

Electrons are fermions.

According to the Pauli Exclusion Principle,

the electronic wavefunction must be antisymmetric with respect to the exchange of any two electrons.

The Hartree Product does not satisfy this requirement.

As a result,

exchange effects are completely absent,
the Pauli principle is not enforced,
and the resulting wavefunction is physically incorrect.

Why Do We Still Study the Hartree Product?¶

Although it is not used directly in modern quantum chemistry software,

the Hartree Product introduces two important ideas:

molecular orbitals,
independent-electron approximation.

These ideas remain central to Hartree–Fock theory and Density Functional Theory.

Modern methods retain the concept of molecular orbitals while replacing the Hartree Product with a mathematically correct wavefunction.

From Hartree Product to Slater Determinant¶

The deficiency of the Hartree Product can be corrected by constructing a wavefunction that automatically changes sign whenever two electrons are exchanged.

This leads to the Slater Determinant, which satisfies

antisymmetry,
indistinguishability of electrons,
and the Pauli Exclusion Principle.

The Slater Determinant forms the foundation of Hartree–Fock theory and nearly all modern wavefunction-based electronic structure methods.

Summary¶

The Hartree Product is the first approximation to the many-electron wavefunction. It assumes that each electron occupies an independent molecular orbital and that the total wavefunction is simply the product of these orbitals. While this greatly simplifies the many-electron Schrödinger equation, it fails to satisfy the antisymmetry requirement for identical electrons and therefore violates the Pauli Exclusion Principle. These shortcomings motivate the introduction of the Slater Determinant, which provides a mathematically correct description of many-electron wavefunctions.

Next Section¶

The next chapter introduces the Slater Determinant, a wavefunction constructed specifically to satisfy the antisymmetry requirement of identical electrons and to enforce the Pauli Exclusion Principle.