Why Do Two Electrons in the Same Orbital Have Antiparallel Spins?

Understanding the behavior of electrons in atomic orbitals is crucial to the field of chemistry and quantum mechanics. The requirement for two electrons in the same orbital to have antiparallel spins is a fundamental principle in physics, often stemming from the Pauli Exclusion Principle. This article explores the underlying reasons for this phenomenon, its implications, and the mathematical foundations behind it.

The Role of the Pauli Exclusion Principle

The Pauli Exclusion Principle is a cornerstone of quantum mechanics, stating that no two fermions, such as electrons, can occupy the same quantum state simultaneously. Since electrons are fermions, they must differ in at least one of their four quantum numbers—in this case, the spin quantum number—to coexist in the same orbital.

The Quantum Numbers

Each electron in an atom is described by a set of four quantum numbers:

When two electrons occupy the same orbital, they share the same values of n, l, and ml; hence, their spins must differ. The spin quantum number (ms) can take values of ±1/2.

The Implications

The Stability and Chemical Properties of Atoms

The arrangement of electrons in orbitals and their spins significantly contributes to the stability and chemical properties of atoms. Antiparallel spins help minimize electron-electron repulsion within an atom. This is essential for the stability of atoms, as repulsion between electrons in the same orbital can disrupt the atom's structure.

Magnetic Properties

The spins of electrons also play a crucial role in the magnetic properties of materials. Electrons with parallel spins can lead to paramagnetism, while those with antiparallel spins result in diamagnetism. Understanding these properties is essential for the development of new materials and technologies.

Mathematical Foundations and Examples

Consider the case of helium, a neutral atom with two electrons. Each electron can be placed in a hydrogenic orbital, such as the 1s orbital. However, the overall wavefunction must be antisymmetric to reflect the Pauli Exclusion Principle.

Single-Particle States and Symmetry

Let's take electron 1 in the 1s orbital and electron 2 in the 2s orbital. The overall wavefunction must be:

ψ(1s, 2s) 1s1(2s2) - 1s2(2s1)

Ignoring spin, the spatial parts of the states are already antisymmetric, so the spin part gets included. For instance:

1s1 2s2 X1Y2 - X2Y1

Here, X and Y represent the spin states of the electrons. The product of spinors must be antisymmetric, meaning that the wavefunction changes sign when the indices are interchanged. This antisymmetry ensures that the total wavefunction remains valid under the Pauli Exclusion Principle.

Four-Fermi States

The wavefunction can be expanded using four-fermi states. For helium with two electrons, we can put both electrons in the 1s orbital provided we antisymmetrize the product of spin states:

1s1(1s2) X1Y2 - X2Y1

This state is called a "singlet," meaning the electrons have opposite spins in an antisymmetric manner. The triplet state, on the other hand, has symmetric products of spinors, such as:

1s1(1s2) X1X2 X2X1

The singlet and triplet states are eigenstates of the Total Spin operator and represent the irreducible group of the direct-product of the single spin operators.

In conclusion, the requirement for two electrons in the same orbital to have antiparallel spins is a fundamental aspect of quantum mechanics, reflecting the Pauli Exclusion Principle and ensuring the stability and unique properties of atomic matter.