The wavefunction of a many-electron closed-shell atom can expressed as a Slater determinant (Section 10.4b). A useful property of determinants is that interchanging any two rows or columns changes their sign and therefore, if any two rows or columns are identical, then the determinant vanishes.

The Pauli exclusion principle is a fundamental principle in quantum mechanics that has profound implications for the structure of atoms and the behavior of particles. Simply put, it states that no two fermions (particles like electrons that have half-integer spin) can occupy the same quantum state simultaneously. In other words, each electron within an atom must have a unique set of quantum numbers, which include the principal quantum number, orbital angular momentum quantum number, magnetic quantum number, and spin quantum number.

Applying this principle to atomic and molecular systems gives rise to complex electron configurations that determine chemical properties. For instance, it explains the distinctive arrangement of electrons in different atomic orbitals and the limited capacity of each orbital. It is also why atoms have unique electron configurations leading to the periodic table's structure.

When forming a Slater determinant for a many-electron system, the exclusion principle ensures that no two rows (or columns) are identical, because if they were, the determinant would be zero, indicating that the wavefunction does not represent a permissible state for the system.

The wavefunction is a cornerstone concept of quantum mechanics and represents the state of a quantum system. It contains all the information necessary to describe the system's properties and behaviors. Mathematically, wavefunctions are complex functions of the coordinates and spins of all the particles in the system. In many-electron systems, each electron's wavefunction is a product of spatial variables (which describe where the electron is likely to be found) and spin variables (which describe the electron's spin orientation).

Wavefunctions are not directly observable, but their absolute squares give the probability of finding particles in certain states or locations — a core concept known as the probability density function. For example, when you calculate the square of the absolute value of an electron's wavefunction at a particular point in space, you find the probability of detecting the electron at that point.

The shape and amplitude of a wavefunction can tell us about the energy levels, stability, and reactivity of the quantum system. In chemical reactions, wavefunctions help predict how atoms or molecules will interact, bond, and change.

A quantum state is the set of all information that describes a quantum system completely. For an electron in an atom, this includes parameters such as energy, angular momentum, magnetic orientation, and spin. A specific quantum state can be designated by a set of quantum numbers, and according to quantum mechanics, these states are typically quantized, meaning that only certain discrete states are allowed.

In a multi-electron system, each electron must occupy a distinct quantum state due to the Pauli exclusion principle. The overall state of the system is represented by the wavefunction, which is a composite of the quantum states of all the individual electrons. The uniqueness of quantum states in a given system is responsible for the vast diversity of chemical phenomena we observe because it governs how electrons are distributed amongst the atomic and molecular orbitals.

Quantization of Energy

Quantization is a distinctive feature of quantum states, indicating that certain properties, such as energy levels in an atom, can only take on specific values. This is what allows electrons to only exist in certain energy levels around the nucleus of an atom, leading to the well-known structure of the electron shells and subshells that determine an atom's chemical behavior.

Many-electron systems, such as atoms with more than one electron or molecules composed of multiple atoms, present complex interactions that quantum mechanics aims to describe. The behavior and interactions within these systems are governed by the principles of wavefunctions and quantum states, as each electron in the system is in constant interaction with the nucleus and other electrons.

In such systems, not only is the repulsion between electrons significant, but also the quantum nature of these particles becomes particularly important. As aforementioned, the Pauli exclusion principle requires that electrons must occupy unique quantum states, and this has a large impact on how the electrons are arranged and how they bond with each other.

Electron Correlation

One of the key challenges in studying many-electron systems is considering electron correlation — the interaction between electrons as they avoid each other due to their like charges. Accurate representations of these systems, such as those provided by the Slater determinant, must account for this correlation to predict properties like bond strength, ionization energies, and magnetic properties correctly. Understanding the complex wavefunctions of many-electron systems is essential in fields ranging from chemistry to solid-state physics, as they underpin the behavior of all molecular and condensed matter.