The World of Quantum
Chemistry

What is Quantum Chemistry?

Quantum Chemistry is the application of quantum mechanical principles to chemical systems. It provides the theoretical framework for understanding atomic structure, chemical bonding, molecular properties, and spectroscopy at the subatomic level.

Classical (Newtonian) mechanics fails completely at the atomic scale. Quantum mechanics, born in the early 20th century, arose from experimental anomalies that classical physics could not explain.

Why Classical Mechanics Falls Short

Classical physics predicts that electrons orbiting a nucleus should continuously radiate energy and spiral into the nucleus within ~10⁻¹¹ s — yet atoms are stable. Classical thermodynamics predicts infinite energy for hot bodies — the ultraviolet catastrophe. These failures necessitated an entirely new framework.

The Blackbody Problem

A blackbody is an ideal object that absorbs all incident radiation and re-emits it as thermal radiation. The spectral distribution of emitted radiation depends only on temperature, not on the material's nature.

Experimentally, the emitted spectrum shows a characteristic peak that shifts to shorter wavelengths as temperature increases (Wien's displacement law). Classical physics, using the Rayleigh–Jeans law, predicted intensity ∝ T/λ⁴, which correctly described long wavelengths but diverged to infinity at short wavelengths — the infamous ultraviolet catastrophe.

Planck's Quantum Hypothesis (1900)

Max Planck resolved the catastrophe by postulating that oscillators in the blackbody wall cannot have arbitrary energies — they are quantized in discrete units.

Photoelectric Effect (Einstein, 1905)

When light shines on a metal surface, electrons are ejected only if the light frequency exceeds a threshold frequency ν₀, regardless of intensity. Classical wave theory predicted intensity alone should determine electron emission — a prediction spectacularly falsified.

Einstein explained this by proposing light itself is quantized into photons, each carrying energy hν.

Compton Effect (1923)

X-rays scattered by electrons showed a wavelength shift depending on scattering angle — confirming photons carry momentum like particles.

The Gold Foil Experiment (1909–1911)

Hans Geiger and Ernest Marsden, under Rutherford's direction, fired α-particles (helium nuclei) at a thin gold foil and detected them on a surrounding fluorescent screen.

Key Observations vs. Thomson's "Plum Pudding"

Spectral Series of Hydrogen

When hydrogen gas is excited electrically, it emits light at discrete wavelengths forming distinct series. These lines were unexplained by classical physics.

Bohr's Model of the Hydrogen Atom (1913)

Niels Bohr combined Rutherford's nuclear model with Planck's quantum ideas and postulated that electrons orbit the nucleus only in specific allowed orbits of quantized angular momentum, without radiating energy.

Sommerfeld's Elliptical Orbits

Arnold Sommerfeld extended Bohr's model by allowing elliptical orbits with two quantum numbers: principal quantum number n and azimuthal quantum number k (now ℓ). This explained the fine structure of hydrogen spectral lines using relativistic corrections.

The Principle

Werner Heisenberg (1927) demonstrated a fundamental limit on the precision with which complementary pairs of observables can be simultaneously known. This is not a limitation of instruments — it is intrinsic to quantum nature.

de Broglie Hypothesis (1924)

If light (classically a wave) can behave as a particle (photon), then matter (classically a particle) should exhibit wave properties. Louis de Broglie proposed that any particle of momentum p has an associated wavelength.

Experimental Verification

Davisson and Germer (1927) observed electron diffraction from a nickel crystal, confirming de Broglie's hypothesis. G.P. Thomson independently confirmed electron diffraction through thin foils.

The Wave Equation of Quantum Mechanics

Erwin Schrödinger (1926) formulated a differential equation governing the time evolution of the wave function ψ, which encodes all quantum mechanical information about a system.

Physical Meaning of the Wave Function

The wave function ψ(r) is not directly observable. Its physical significance is given by the Born interpretation: |ψ(r)|² gives the probability density of finding the particle at position r.

The Six Fundamental Postulates

Perturbative Scattering Theory

The Born approximation treats scattering as a first-order perturbation. When a particle scatters from a weak potential V(r), the scattered wave function is approximated using the incident plane wave as the zeroth-order solution.

1D Infinite Square Well

A particle of mass m confined between rigid walls at x=0 and x=L (V=0 inside; V=∞ outside). This simple model demonstrates how confinement alone causes quantization.

3D Cubic Box

Free Rotation in 2D — The Rigid Rotor

A particle of mass m constrained to move on a ring of radius R (V=0 everywhere on the ring). This is the simplest model for rotational quantization and directly applies to electrons in cyclic π systems (Hückel theory).

Solution of the Schrödinger Equation

The hydrogen atom (one proton + one electron) is the only atom with exact analytical solutions. Solving the TISE in spherical polar coordinates separates into radial and angular parts.

Shape, Size, and Orientation

Atomic orbitals are the one-electron wave functions ψₙₗₘ of the hydrogen-like atom. The boundary surface encloses 90–95% of the electron probability density.

Radial Distribution Function

The Multi-Electron Problem

For atoms with two or more electrons, the Schrödinger equation cannot be solved exactly due to electron–electron repulsion. The Hamiltonian becomes:

Pauli Exclusion Principle & Electron Configuration

No two electrons in an atom can have identical quantum numbers (n, ℓ, mₗ, mₛ). This forces electrons into distinct orbitals and governs the periodic table's structure.

Hund's Rules

Aufbau Principle — Orbital Filling Order

Intrinsic Angular Momentum

Spin is an intrinsic quantum mechanical property of particles with no classical analog. Electrons have spin quantum number s = ½. Spin angular momentum does not arise from physical rotation — it is a fundamentally quantum property.

Antisymmetry & the Pauli Principle

The wave function of a many-fermion system must be antisymmetric under exchange of any two electrons — this is the quantum mechanical origin of the Pauli exclusion principle.

The Hartree–Fock Approximation

In the Hartree–Fock (HF) method, each electron is treated as moving in the average field of all other electrons. The exact wavefunction is approximated by a single Slater determinant.

Valence Bond (VB) Theory

Proposed by Heitler and London (1927), VB theory views a bond as the overlap of atomic orbitals from two atoms, with electrons shared (not delocalized). The wave function is written as a product of overlapping atomic orbitals.

Molecular Orbital (MO) Theory

MO theory (Hund, Mulliken, Lennard-Jones) delocalizes electrons over the entire molecule. Molecular orbitals are formed as linear combinations of atomic orbitals (LCAO).

MO Diagram — Homonuclear Diatomics (Li₂ to Ne₂)

Heteronuclear Diatomics (e.g., HF, CO, NO)

When the two atoms differ, AO energies differ. MOs are unequal mixtures of AOs — more weight on the atom whose AO is closest in energy. This leads to polar bonds (unequal electron distribution) and non-zero dipole moments.

Valence Shell Electron Pair Repulsion

VSEPR (Gillespie–Nyholm theory) predicts molecular geometry by minimizing repulsion between electron pairs (bonding pairs + lone pairs) around a central atom.

Symmetry and Group Theory

For polyatomic molecules, group theory is used to classify MOs by symmetry. MOs are formed from Symmetry-Adapted Linear Combinations (SALCs) of atomic orbitals that transform as irreducible representations of the molecular point group.

Water (H₂O) — A Case Study

Water (C₂ᵥ symmetry) has four molecular orbital types: a₁ (totally symmetric), b₁ (perpendicular to molecular plane), b₂ (in plane). The lone pairs occupy a₁ and b₁ MOs, explaining the bent geometry and lone-pair reactivity.

VB Resonance Structures

When a single Lewis structure is inadequate (e.g., benzene, ozone, SO₄²⁻), the true structure is a resonance hybrid of contributing VB structures. The actual bond lengths and energies lie between those of individual contributors.

The Concept of Hybridization

Hybridization is a mathematical mixing of pure atomic orbitals on the same atom to form new hybrid orbitals that have better directionality for bonding. It is a VB concept; MO theory does not require it.

Hückel Approximations for π Systems

Erich Hückel (1930) developed a simplified MO treatment for planar conjugated π systems by making three key approximations: (1) σ and π electrons treated separately; (2) all Coulomb integrals (α) equal; (3) resonance integrals (β) only between adjacent carbons.

Ethylene (2 AOs)

Benzene (6 AOs — Cyclic)

Why Approximations?

The Schrödinger equation is exactly solvable only for hydrogen-like atoms. All other chemically interesting systems require approximation methods. Two pillars: Perturbation Theory and the Variational Principle.

1. Variational Principle

The variational method provides an upper bound to the ground state energy: any trial wave function will give an energy equal to or higher than the true ground state energy.

2. Time-Independent Perturbation Theory

When the Hamiltonian can be written as Ĥ = Ĥ⁰ + λĤ', where Ĥ' is a small perturbation of the exactly solvable Ĥ⁰, energy and wave function are expanded in a power series of λ.

3. WKB (Wentzel–Kramers–Brillouin) Approximation

Applies to slowly varying potentials; semi-classical approximation connects classical and quantum regimes. Widely used in tunneling calculations.