In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.
An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.
In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessary and sufficient condition for the manifold to be conformally flat. By contrast, in dimensions n ≥ 4, the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system for three-dimensional Euclidean space, . Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on , or less often on for some other .
In magnetism, the Curie–Weiss law describes the magnetic susceptibility χ of a ferromagnet in the paramagnetic region above the Curie temperature:
In solid-state physics, the tight-binding model is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations.
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine. It provides the theoretical basis for Hückel's rule that cyclic, planar molecules or ions with π-electrons are aromatic. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms). A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method (SHM).
Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital twice, one multiplied by the α spin function and the other multiplied by the β spin function in the Slater determinant, unrestricted Hartree–Fock theory uses different molecular orbitals for the α and β electrons. This has been called a different orbitals for different spins (DODS) method. The result is a pair of coupled Roothaan equations, known as the Pople–Nesbet–Berthier equations.
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.
Bond order potential is a class of empirical (analytical) interatomic potentials which is used in molecular dynamics and molecular statics simulations. Examples include the Tersoff potential, the EDIP potential, the Brenner potential, the Finnis–Sinclair potentials, ReaxFF, and the second-moment tight-binding potentials. They have the advantage over conventional molecular mechanics force fields in that they can, with the same parameters, describe several different bonding states of an atom, and thus to some extent may be able to describe chemical reactions correctly. The potentials were developed partly independently of each other, but share the common idea that the strength of a chemical bond depends on the bonding environment, including the number of bonds and possibly also angles and bond lengths. It is based on the Linus Pauling bond order concept and can be written in the form
In solid state physics, a surface phonon is the quantum of a lattice vibration mode associated with a solid surface. Similar to the ordinary lattice vibrations in a bulk solid, the nature of surface vibrations depends on details of periodicity and symmetry of a crystal structure. Surface vibrations are however distinct from the bulk vibrations, as they arise from the abrupt termination of a crystal structure at the surface of a solid. Knowledge of surface phonon dispersion gives important information related to the amount of surface relaxation, the existence and distance between an adsorbate and the surface, and information regarding presence, quantity, and type of defects existing on the surface.
In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the self-consistent field method.
CONQUEST is a linear scaling, or O(N), density functional theory (DFT) electronic structure open-source code. The code is designed to perform DFT calculations on very large systems containing many thousands of atoms. It can be run at different levels of precision ranging from ab initio tight binding up to full DFT with plane wave accuracy. It has been applied to the study of three-dimensional reconstructions formed by Ge on Si(001), containing over 20,000 atoms. Tests on the UK's national supercomputer HECToR in 2009 demonstrated the capability of the code to perform ground-state calculations on systems of over 1,000,000 atoms.
In theoretical chemistry, the Buckingham potential is a formula proposed by Richard Buckingham which describes the Pauli exclusion principle and van der Waals energy for the interaction of two atoms that are not directly bonded as a function of the interatomic distance . It is a variety of interatomic potentials.
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.
Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.
A phonovoltaic (pV) cell converts vibrational (phonons) energy into a direct current much like the photovoltaic effect in a photovoltaic (PV) cell converts light (photon) into power. That is, it uses a p-n junction to separate the electrons and holes generated as valence electrons absorb optical phonons more energetic than the band gap, and then collects them in the metallic contacts for use in a circuit. The pV cell is an application of heat transfer physics and competes with other thermal energy harvesting devices like the thermoelectric generator.
Multiple scattering theory (MST) is the mathematical formalism that is used to describe the propagation of a wave through a collection of scatterers. Examples are acoustical waves traveling through porous media, light scattering from water droplets in a cloud, or x-rays scattering from a crystal. A more recent application is to the propagation of quantum matter waves like electrons or neutrons through a solid.
