Quantum chemistry

Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry that applies the principles of quantum mechanics to chemical systems. Its central aim is to describe and predict the electronic structure of atoms, molecules and condensed phases, and to relate this electronic structure to observable physical and chemical properties such as molecular geometry, spectra, thermodynamic quantities and reaction rates. Because exact quantum-mechanical treatments are only possible for very simple systems, quantum chemistry relies on systematically controlled approximations and computational methods that make the solution of the Schrödinger equation tractable while preserving essential physical features.

Definition and Scope

Quantum chemistry focuses primarily on the electronic contributions to molecular properties. At the atomic level, it seeks approximate solutions to the time-independent Schrödinger equation (or, in relativistic cases, the Dirac equation) for systems containing multiple nuclei and electrons. These solutions yield an electronic wave function, from which quantities such as energy, charge distribution, dipole moments, polarisabilities and spectroscopic transition probabilities can be derived.
Most applications assume the Born–Oppenheimer approximation, in which the much heavier nuclei are treated as fixed while solving for the electrons. The electronic wave function is then considered a function of nuclear coordinates, and its changes as the nuclei move define a potential energy surface on which nuclear motion occurs.
Quantum chemistry also extends to:

• the prediction and interpretation of vibrational, rotational and electronic spectra
• the study of reaction pathways, transition states and activation energies
• the description of quantum effects in molecular dynamics and chemical kinetics
• the modelling of solutions, solids and materials using electronic-structure methods adapted to extended systems.

Because the computational cost of accurate methods rises steeply with system size (often scaling as a high power of the number of basis functions or electrons), an important practical objective is to balance accuracy against feasibility.

Historical Development

The conceptual foundations of quantum chemistry were laid soon after the birth of quantum mechanics. The Schrödinger equation provided a rigorous framework for atomic and molecular structure, and its early application to the hydrogen atom established the correspondence between quantum numbers and electronic shells.
A crucial conceptual step predating formal quantum mechanics was Gilbert N. Lewis’s 1916 work The Atom and the Molecule, which introduced the idea of electron pairs and valence electrons. This provided a qualitative picture of chemical bonding that later gained a quantitative basis.
In 1927 Walter Heitler and Fritz London produced what is often regarded as the first truly quantum-chemical paper by applying quantum mechanics to the hydrogen molecule. Their treatment demonstrated how a covalent bond could arise from the quantum-mechanical superposition of electron-exchange configurations, inaugurating what became valence bond theory.
During the 1930s Linus Pauling, together with other contributors such as Sugiura, Wang and Slater, developed the valence-bond approach into a coherent theoretical framework. Pauling’s 1939 book The Nature of the Chemical Bond became a standard reference, introducing generations of chemists to quantum theory of bonding.
In 1937 Hans Hellmann published one of the first textbooks explicitly devoted to quantum chemistry, and in the following decades major contributions came from scientists including Langmuir, Mulliken, Born, Oppenheimer, Hückel, Hartree, Lennard-Jones and Fock. Their work laid the foundations of molecular orbital theory, approximate solution techniques and the formalism of electronic structure.

Electronic Structure and the Schrödinger Equation

The electronic structure of a system is defined by the quantum state of its electrons, subject to the electrostatic potential created by the nuclei. In quantum chemistry, this structure is obtained by solving
H^elΨ=EΨ\hat{H}_{\text{el}} \Psi = E \PsiH^el​Ψ=EΨ
where H^el\hat{H}_{\text{el}}H^el​ is the electronic Hamiltonian and Ψ\PsiΨ the electronic wave function. Under the Born–Oppenheimer approximation, nuclear coordinates enter as parameters rather than dynamical variables.
An exact analytical solution of the non-relativistic Schrödinger equation is only possible for one-electron systems such as the hydrogen atom. Even for the simplest multi-electron entities, such as helium or H₂, approximations or numerical methods are required. In the case of the dihydrogen cation, exact expressions for the bound-state energies within the Born–Oppenheimer framework can be written in terms of special functions, but this is an exception rather than the rule.
The impossibility of closed-form solutions for general molecules defines a central task of quantum chemistry: devising approximate yet systematically improvable computational methods that approach the true solution in a controlled manner. This activity forms a large part of computational chemistry.

Valence Bond Theory

The valence bond (VB) method emerged directly from the Heitler–London picture. In VB theory, emphasis is placed on pairwise interactions between atoms and on the formation of localised electron-pair bonds, often mirroring classical Lewis structures.
Key ideas in VB theory include:

• orbital hybridisation – the mixing of atomic s, p (and sometimes d) orbitals to form directed hybrid orbitals (e.g. sp³, sp², sp) aligned with bond directions
• resonance – the description of a molecule as a superposition of several contributing structures that differ in electron distribution but share the same nuclear framework.

VB theory connects naturally with traditional chemical intuition, especially in describing local bonding patterns, electron pairing and bond energies. However, its heavily localised perspective can make it less convenient for describing delocalised π-systems and extended conjugation, and its quantitative implementation can be computationally demanding for large systems.

Molecular Orbital Theory

An alternative viewpoint is provided by molecular orbital (MO) theory, developed mainly by Friedrich Hund and Robert S. Mulliken. Here, electrons are not associated with specific bonds but occupy orbitals delocalised over the entire molecule. Molecular orbitals are constructed as linear combinations of atomic orbitals (LCAO), and electrons fill these orbitals according to the Pauli principle and Hund’s rules.
MO theory forms the backbone of the Hartree–Fock (HF) approximation, in which each electron moves in an average field generated by all other electrons. HF introduces the concept of self-consistent field (SCF) iterations: starting from an initial guess for the orbitals, the equations are solved repeatedly until convergence is reached.
Post-Hartree–Fock methods such as configuration interaction, Møller–Plesset perturbation theory and coupled cluster theory systematically improve on HF by including electron correlation effects beyond the mean-field approximation. MO-based approaches have proved particularly successful in predicting spectroscopic properties, excitation energies and potential energy surfaces.

Density Functional Theory

Density functional theory (DFT) provides a different route to the many-electron problem by taking the electron density rather than the many-electron wave function as the primary variable. Early attempts by Thomas and Fermi modelled electrons as a uniform gas and were too crude for molecular applications, but they established the idea that all ground-state properties are functionals of the density.
Modern DFT rests on the Hohenberg–Kohn theorems and the Kohn–Sham formulation, in which an interacting system is mapped to a fictitious non-interacting one with the same density. The total energy functional is decomposed into the Kohn–Sham kinetic energy, the classical Coulomb term, an external potential term, and an exchange–correlation functional that embodies the remaining quantum-mechanical effects.
The practical performance of DFT depends critically on the choice of this exchange–correlation functional, and a large part of contemporary research focuses on its improvement. DFT methods often scale roughly as n3n^3n3 with respect to the number of basis functions, making them considerably more efficient than many post-Hartree–Fock methods while offering comparable accuracy for many ground-state properties. This favourable balance has made DFT one of the most widely used tools in modern quantum and computational chemistry, especially for large molecules and materials.

Quantum Chemistry and Spectroscopy

Quantum chemistry is closely tied to spectroscopy, which probes the quantisation of energy levels in atoms and molecules. Common techniques include:

• infrared (IR) spectroscopy for vibrational transitions
• nuclear magnetic resonance (NMR) spectroscopy for nuclear spin environments and molecular structure
• scanning probe microscopy, which can be interpreted with quantum-chemical models of surface electronic states.

Theoretical calculations are used to predict vibrational frequencies, NMR chemical shifts, electronic absorption spectra and other observables, allowing direct comparison with experiment. In many cases, agreement between calculated and measured spectra provides stringent validation of the underlying electronic-structure method.

Chemical Dynamics and Potential Energy Surfaces

Beyond static properties, quantum chemistry contributes to the understanding of chemical dynamics, the time-dependent motion of nuclei and electrons during reactions.
Within the adiabatic Born–Oppenheimer framework, interatomic forces are described by potential energy surfaces (PES), which relate molecular energy to nuclear configuration. Reaction paths, transition states and intermediate complexes appear as features on these surfaces. Early work by Rice, Ramsperger and Kassel, later refined into RRKM theory and related to transition state theory, made it possible to estimate unimolecular reaction rates from PES characteristics.
In adiabatic dynamics, nuclear motion is confined to a single electronic surface. Nonadiabatic dynamics introduces couplings between multiple surfaces belonging to different electronic states; these couplings, termed vibronic couplings, allow transitions between states. The Landau–Zener model, developed by Stueckelberg, Landau and Zener, provides a simple expression for the probability of such transitions near avoided crossings. Spin-forbidden processes are an important class of nonadiabatic reactions where electronic spin changes during the transition.
Practical simulations encompass:

• fully quantum quantum dynamics
• semiclassical dynamics, where nuclear motion is treated classically with quantum corrections
• purely classical molecular dynamics (MD) on quantum-derived PES
• mixed quantum–classical dynamics and path-integral molecular dynamics for including nuclear quantum effects
• statistical descriptions using Monte Carlo methods for equilibrium and thermodynamic properties.

Significance and Current Challenges

Quantum chemistry provides the microscopic foundation for much of modern theoretical and computational chemistry. It links fundamental quantum-mechanical laws to experimentally measurable quantities such as molecular structures, reaction barriers, rate constants and spectra. Its methods are indispensable in areas as diverse as drug design, catalysis, materials science, surface chemistry and atmospheric chemistry.