⚛Molecular Physics Unit 3 – Atomic Orbitals and Electronic Structure

Fundamental Concepts of Atomic Structure

• Atoms consist of a dense, positively charged nucleus surrounded by negatively charged electrons in motion
• The nucleus contains protons (positively charged) and neutrons (electrically neutral), while electrons orbit the nucleus
• Atomic number represents the number of protons in an atom and determines its chemical identity (e.g., carbon has an atomic number of 6)
• Mass number is the sum of the number of protons and neutrons in an atom
• Isotopes are atoms of the same element with different numbers of neutrons (e.g., carbon-12 and carbon-14)
• Electrons occupy discrete energy levels or shells (designated as n=1, 2, 3, etc.) around the nucleus
• Each energy level can hold a maximum number of electrons: 2n^2, where n is the principal quantum number
  • For example, the first energy level (n=1) can hold up to 2 electrons, while the second level (n=2) can hold up to 8 electrons

Quantum Mechanics and Wave Functions

• Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic levels
• In quantum mechanics, particles exhibit wave-particle duality, meaning they can behave like waves or particles depending on the experiment
• The Heisenberg Uncertainty Principle states that the more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa
• Wave functions, denoted by Ψ(x, y, z), are mathematical descriptions of the quantum state of a particle
• The Schrödinger equation is used to determine the wave function of a system and describes how it changes over time: $-\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi = i\hbar\frac{\partial}{\partial t}\Psi$
  • $\hbar$ is the reduced Planck's constant, m is the mass of the particle, ∇^2 is the Laplacian operator, and V is the potential energy of the system
• Born interpretation of the wave function: The probability of finding a particle at a specific location is proportional to the square of the absolute value of the wave function at that location, $|\Psi(x, y, z)|^2$
• Expectation values of observable quantities (such as position, momentum, and energy) can be calculated using the wave function and the corresponding quantum mechanical operator

Electron Configuration and Energy Levels

• Electron configuration describes the distribution of electrons in an atom's atomic orbitals
• The arrangement of electrons follows the Aufbau principle, which states that electrons fill orbitals in order of increasing energy (1s, 2s, 2p, 3s, 3p, 4s, 3d, etc.)
• Hund's rule: Electrons occupy degenerate orbitals singly before pairing up, and all unpaired electrons have the same spin
• The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms)
• The energy of an electron depends on its principal quantum number (n) and its angular momentum quantum number (l)
• Electrons in higher energy levels are farther from the nucleus and have higher potential energy
• Valence electrons, which are the electrons in the outermost energy level, participate in chemical bonding and determine an element's chemical properties
• Electron configurations can be represented using the noble gas notation, which abbreviates the inner electron shells using the symbol of the preceding noble gas (e.g., [Ne] 3s^2 3p^3 for chlorine)

Shapes and Types of Atomic Orbitals

• Atomic orbitals are regions in space where electrons are most likely to be found
• The shape of an orbital is determined by the angular momentum quantum number (l) and the magnetic quantum number (ml)
• There are four types of orbitals: s, p, d, and f, corresponding to l values of 0, 1, 2, and 3, respectively
• s orbitals are spherical and have no angular dependence (l=0, ml=0)
• p orbitals have dumbbell shapes and are oriented along the x, y, or z axes (l=1, ml=-1, 0, +1)
• d orbitals have more complex shapes, such as cloverleaf or double dumbbell (l=2, ml=-2, -1, 0, +1, +2)
  • The five d orbitals are dxy, dxz, dyz, dx^2-y^2, and dz^2
• f orbitals have even more intricate shapes and are important in the chemistry of lanthanides and actinides (l=3, ml=-3, -2, -1, 0, +1, +2, +3)

Electron Spin and the Pauli Exclusion Principle

• Electron spin is an intrinsic angular momentum of an electron, with two possible values: +1/2 (spin up) and -1/2 (spin down)
• The spin quantum number (ms) represents the electron's spin state
• The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms)
  • This means that each orbital can hold a maximum of two electrons with opposite spins
• Electrons in an atom tend to occupy orbitals in a way that minimizes the total energy of the system
• Hund's rule states that electrons occupy degenerate orbitals singly before pairing up, and all unpaired electrons have the same spin
• The exchange interaction between electrons with parallel spins leads to a lower energy state compared to electrons with opposite spins
• Electron spin plays a crucial role in the formation of chemical bonds and the magnetic properties of materials

Multi-Electron Atoms and Periodic Trends

• In multi-electron atoms, electrons interact with each other and with the nucleus, leading to more complex behavior than in hydrogen-like atoms
• The effective nuclear charge (Zeff) experienced by an electron is lower than the actual nuclear charge due to shielding by inner electrons
• Shielding effect: Inner electrons partially screen the outer electrons from the full attractive force of the nucleus
• Periodic trends in atomic properties arise from the interplay between the effective nuclear charge and the electron configuration
  • Atomic radius generally decreases from left to right across a period and increases down a group
  • Ionization energy increases from left to right across a period and decreases down a group
  • Electron affinity generally increases from left to right across a period and decreases down a group
• The aufbau principle, Hund's rule, and the Pauli exclusion principle govern the electron configuration of multi-electron atoms
• Exceptions to the aufbau principle occur when half-filled or completely filled subshells result in greater stability (e.g., chromium and copper)

Molecular Orbital Theory Basics

• Molecular orbital theory describes the behavior of electrons in molecules using molecular orbitals (MOs)
• MOs are formed by the linear combination of atomic orbitals (LCAO) from the constituent atoms
• Bonding MOs have lower energy than the original atomic orbitals and contribute to the stability of the molecule
  • Electrons in bonding MOs are shared between the nuclei, leading to attractive interactions
• Antibonding MOs have higher energy than the original atomic orbitals and destabilize the molecule
  • Electrons in antibonding MOs are concentrated in regions away from the internuclear axis, leading to repulsive interactions
• The shape and symmetry of MOs depend on the type and orientation of the atomic orbitals involved
• Sigma (σ) MOs are formed by the head-on overlap of atomic orbitals along the internuclear axis (e.g., s+s, s+pz, pz+pz)
• Pi (π) MOs are formed by the sideways overlap of atomic orbitals perpendicular to the internuclear axis (e.g., px+px, py+py)
• The number of MOs formed is equal to the number of atomic orbitals combined

Applications in Spectroscopy and Chemical Bonding

• Understanding atomic orbitals and electronic structure is crucial for interpreting spectroscopic data and predicting chemical bonding
• Absorption and emission spectra arise from transitions between different energy levels or orbitals
  • The frequency of the absorbed or emitted photon corresponds to the energy difference between the levels: $\Delta E = h\nu$
• Selection rules based on quantum numbers govern the allowed transitions between orbitals
• Photoelectron spectroscopy (PES) probes the energies of occupied molecular orbitals by measuring the kinetic energy of ejected electrons
• X-ray absorption spectroscopy (XAS) provides information about the unoccupied molecular orbitals and the local electronic structure around specific atoms
• Chemical bonding involves the sharing or transfer of electrons between atoms to form molecules or ions
• Covalent bonds are formed by the sharing of electrons between atoms, typically resulting in the formation of bonding molecular orbitals
• Ionic bonds involve the transfer of electrons from one atom to another, creating positively and negatively charged ions that attract each other
• The strength and properties of chemical bonds depend on the overlap and energy of the atomic orbitals involved