Hund's Rule and Pauli Principle: Exchange Energy and Spin Statistics

Hello! Welcome back to our course on the foundations of organic chemistry.

In our last lesson, we demystified the shapes of atomic orbitals, seeing them as direct mathematical solutions to the Schrödinger equation. We now have a clear picture of the "houses" where electrons reside. Today, we will establish the fundamental rules that govern how multiple electrons occupy these houses. This will lead us to a deep, quantum-mechanical understanding of two principles you are likely familiar with but perhaps haven't explored from first principles: the Pauli exclusion principle and Hund's rule.

By the end of this lesson, you will be able to explain the origin of both of these rules not as mere dictums, but as necessary consequences of electron spin and the symmetry properties of multi-electron wavefunctions.

1. A New Property: Electron Spin

Our description of the electron from the Schrödinger equation gave us three quantum numbers: $n$ , $l$ , and $m_{l}$ . However, experiments in the 1920s (like the Stern-Gerlach experiment) revealed that electrons possess an intrinsic form of angular momentum, as if they were spinning. This property is called spin.

It is quantized and described by a fourth quantum number, the spin magnetic quantum number, $m_{s}$ , which can take one of two values: $+ \frac{1}{2}$ (spin-up, denoted $α$ ) or $- \frac{1}{2}$ (spin-down, denoted $β$ ).

Spin is a purely relativistic quantum phenomenon with no true classical analogue. Its existence is crucial, as it leads to one of the most profound principles in all of science.

2. The Antisymmetry Principle: The Ultimate Rule

In quantum mechanics, all electrons are fundamentally indistinguishable. You cannot label electron A and electron B and track them separately. This indistinguishability has a major consequence for the mathematical form of the multi-electron wavefunction, $Ψ$ .

This is formalized by the Spin-Statistics Theorem:

Particles with integer spin (0, 1, 2, ...), called bosons, must have a total wavefunction that is symmetric upon the exchange of any two particles.
Particles with half-integer spin ( $\frac{1}{2}, \frac{3}{2}, \dots$ ), called fermions, must have a total wavefunction that is antisymmetric upon the exchange of any two particles.

Electrons have spin- $\frac{1}{2}$ and are therefore fermions. This means that for a two-electron system, if we swap the coordinates (both spatial and spin) of electron 1 and electron 2, the wavefunction must flip its sign:

$Ψ (1, 2) = - Ψ (2, 1)$

This is the Antisymmetry Principle. It is the most fundamental rule governing electrons, from which both the Pauli principle and Hund's rule emerge.

Identical Particles

To formalize this concept, please read the first page of the following lecture notes. It concisely introduces the concepts of identical particles, the Spin-Statistics Theorem, and the resulting symmetry requirements for fermions like electrons.

Read the sections 'Introduction' and 'The Spin-Statistics Theorem' (pages 140-141). Focus on understanding why identical particles in quantum mechanics are truly indistinguishable and the core statement of the Spin-Statistics Theorem for fermions.

3. The Pauli Exclusion Principle as a Consequence of Antisymmetry

How does the Antisymmetry Principle lead to the familiar statement that "no two electrons can have the same four quantum numbers"? Let's build a wavefunction and see what happens.

A total wavefunction, $Ψ$ , is a product of a spatial part, $ψ (r_{1}, r_{2})$ , and a spin part, $χ (s_{1}, s_{2})$ . To construct a valid wavefunction, we must combine these parts in a way that makes the total function antisymmetric.

Let's try to violate the Pauli principle by putting two electrons into the exact same state—that is, the same spatial orbital (e.g., 1s) and the same spin state (e.g., spin-up, $α$ ). We denote this spin-orbital as $1 s α$ .

The following video provides a brilliant and clear demonstration of why this is impossible.

Exchange Antisymmetry

The video 'Exchange Antisymmetry' from the 'Physical Chemistry' channel walks through the process of constructing a valid, antisymmetric wavefunction. It directly shows why a state with two electrons in the same spin-orbital cannot exist.

Watch from 0:40 to the end (around 14:57). Pay special attention to: The explanation of symmetric vs. antisymmetric functions (0:40 - 5:14). How an antisymmetric combination is constructed (8:03 - 12:08). The crucial final segment (12:50 - 14:57), where he shows that trying to build an antisymmetric wavefunction for two electrons in the same spin-orbital (e.g., both 1s\alpha) results in the wavefunction being zero.

As the video showed, if you try to construct an antisymmetric wavefunction for two electrons in the same spin-orbital, $ϕ$ , you get:
$Ψ (1, 2) = \frac{1}{2} [ϕ (1) ϕ (2) - ϕ (2) ϕ (1)] = 0$
A wavefunction that is zero everywhere means the probability of finding the electrons is zero—the state is physically impossible. This is the deep origin of the Pauli Exclusion Principle.

A More General Tool: The Slater Determinant

For systems with more than two electrons, this process of manual antisymmetrization becomes cumbersome. A more powerful mathematical tool is the Slater determinant. For a two-electron system in spin-orbitals $χ_{a}$ and $χ_{b}$ , the wavefunction is written as:

\Psi(1,2) = \frac{1}{\sqrt{2!}} \begin{vmatrix} \chi_a(1) & \chi_b(1) \\ \chi_a(2) & \chi_b(2) \endvmatrix} = \frac{1}{\sqrt{2}}[\chi_a(1)\chi_b(2) - \chi_b(1)\chi_a(2)]

Notice two key properties of determinants:

Swapping two rows (exchanging the electrons) multiplies the determinant by -1, automatically satisfying the antisymmetry principle.
If two columns are identical ( $χ_{a} = χ_{b}$ ), the determinant is zero. This provides an elegant, generalized proof of the Pauli Exclusion Principle.

Antisymmetric Wavefunctions: SLATER DETERMINANTS

This document, 'Antisymmetric Wavefunctions: SLATER DETERMINANTS', formalizes these ideas. It's a dense, high-level resource, but it contains the rigorous mathematical framework you're looking for.

Please read the first two pages. Focus on the introduction to antisymmetry and how the Slater determinant is constructed. Pay particular attention to the statement: 'if any two rows (or columns) are identical, the determinant is zero. This guarantees that we can’t violate the Pauli principle by using the same spinorbital twice.'

Test your understanding!

Using the Slater determinant formalism, write down the wavefunction for the ground state of a Helium atom (1s² configuration). What are the spin-orbitals $χ_{a}$ and $χ_{b}$ ?

Show answer

For the He ground state, the two electrons are in the 1s spatial orbital, but by the Pauli principle, they must have opposite spins.
So, the two spin-orbitals are:

$χ_{a} = 1 s α$ (electron in 1s orbital with spin-up)
$χ_{b} = 1 s β$ (electron in 1s orbital with spin-down)

The Slater determinant is:

Ψ (1, 2) = \frac{1}{2} 1 s (1) α (1) 1 s (2) α (2) 1 s (1) β (1) 1 s (2) β (2) = \frac{1}{2} [1 s (1) α (1) \cdot 1 s (2) β (2) - 1 s (1) β (1) \cdot 1 s (2) α (2)]

We can factor out the common spatial part:

Ψ (1, 2) = [1 s (1) 1 s (2)] \cdot \frac{1}{2} [α (1) β (2) - β (1) α (2)]

This shows the ground state has a symmetric spatial part and an antisymmetric spin part.

4. Hund's Rule and the Exchange Interaction

Now we can address Hund's rule: why do electrons occupy degenerate orbitals singly with parallel spins before pairing up? Consider two electrons and two degenerate orbitals, $ϕ_{a}$ and $ϕ_{b}$ (e.g., 2pₓ and 2pᵧ).

From the antisymmetry principle, we have two main possibilities for the total wavefunction $Ψ = ψ_{s p a t ia l} \times χ_{s p in}$ :

Antiparallel Spins (Singlet State): The spin function is antisymmetric. To make $Ψ$ antisymmetric overall, the spatial function $ψ_{s p a t ia l}$ must be symmetric:
$ψ_{sy mm} = \frac{1}{2} [ϕ_{a} (1) ϕ_{b} (2) + ϕ_{b} (1) ϕ_{a} (2)]$
Parallel Spins (Triplet State): The spin function is symmetric. To make $Ψ$ antisymmetric overall, the spatial function $ψ_{s p a t ia l}$ must be antisymmetric:
$ψ_{an t i sy mm} = \frac{1}{2} [ϕ_{a} (1) ϕ_{b} (2) - ϕ_{b} (1) ϕ_{a} (2)]$

The energy difference between these two states comes from the electron-electron Coulomb repulsion term in the Hamiltonian, $H_{12} = \frac{e ^{2}}{4 π ϵ _{0} r _{12}}$ . A detailed derivation shows that the energy splits.

The following video masterfully derives this energy splitting and explains the origin of the exchange energy.

The Exchange Interaction

Professor Nicola Spaldin's lecture on 'The Exchange Interaction' is a rigorous and clear derivation of the energy difference between parallel and antiparallel spin states. It uses the excited state of Helium as an example system.

This video is the core of understanding Hund's rule. Please watch it carefully (up to 28:40, the summary at the end is also helpful). Focus on: The setup of the Hamiltonian for a two-electron system (3:16). The construction of symmetric and antisymmetric spatial wavefunctions based on spin (7:36). The breakdown of the final energy into one-electron energy, a Coulomb integral (which she calls K), and an exchange integral (which she calls J) (15:55). The final result: the state with parallel spins (antisymmetric spatial part) has its energy lowered by the exchange integral J (22:48). Note on notation: The letters used for Coulomb and Exchange integrals can vary. Prof. Spaldin uses K for Coulomb and J for Exchange. Other texts (like resource LINK) use J for Coulomb and K for Exchange. The physics is the same: the exchange term stabilizes the parallel-spin state.

The Physical Meaning of Exchange Energy

As the video derivation shows, the energy for the two states is:

$E_{s in g l e t} = E_{1 e} + K + J$ (Symmetric spatial part)
$E_{t r i pl e t} = E_{1 e} + K - J$ (Antisymmetric spatial part)

Since the exchange integral $J$ is a positive quantity, the triplet state (parallel spins) is lower in energy by an amount $2 J$ . This energy stabilization is the exchange energy.

Why does this happen? Look at the antisymmetric spatial wavefunction:
$ψ_{an t i sy mm} = \frac{1}{2} [ϕ_{a} (1) ϕ_{b} (2) - ϕ_{b} (1) ϕ_{a} (2)]$
If the two electrons approach the same point in space ( $r_{1} \to r_{2}$ ), then $ϕ_{a} (1) \to ϕ_{a} (2)$ and $ϕ_{b} (1) \to ϕ_{b} (2)$ , causing the wavefunction to approach zero. This means that electrons with parallel spins have a very low probability of being found close to each other. By staying farther apart, their average Coulombic repulsion is reduced compared to electrons in the symmetric spatial state. This reduction in repulsion energy is the exchange energy. It is a purely quantum mechanical effect with no classical parallel.

This explains Hund's rule: the configuration with the maximum number of parallel spins is the most stable because it maximizes this stabilizing exchange energy.

Hund's Rule and Exchange Energy — This diagram summarizes the energy contributions. The total pairing energy for electrons in degenerate orbitals includes the repulsive Coulombic energy ($\Pi_c$) and the stabilizing exchange energy ($\Pi_e$). Because exchange energy is negative (stabilizing) and applies only to electrons with parallel spins, the state with maximized parallel spins (unpaired) is lower in energy.

Conclusion

Today we have seen that two of the most important rules for determining electron configurations are not arbitrary. They are direct, unavoidable consequences of the fundamental nature of electrons as identical fermions.

Key Takeaways:

Antisymmetry Principle: The total wavefunction of a system of electrons must change sign upon the exchange of any two electrons. This is the master rule.
Pauli Exclusion Principle: This principle arises because attempting to place two electrons in the same spin-orbital leads to a wavefunction that is identically zero, which is a physical impossibility. The Slater determinant provides an elegant mathematical proof of this.
Hund's Rule: Placing electrons in degenerate orbitals with parallel spins (a triplet state) forces the spatial part of the wavefunction to be antisymmetric. This unique spatial arrangement keeps the electrons further apart on average, reducing their mutual Coulomb repulsion. This energy reduction is known as the exchange energy, which stabilizes the high-spin state.

Preview of the Next Lesson:

Having established the quantum mechanical rules for describing single atoms, we are now ready to build molecules. In the next lesson, we will begin our journey into chemical bonding by applying these principles to construct bonding and antibonding molecular orbitals for simple diatomic molecules using Molecular Orbital (MO) Theory.

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