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Avoided crossing, the Glossary

In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing, non-crossing or anticrossing) is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable and depending on N continuous real parameters cannot become equal in value ("cross") except on a manifold of N-3 dimensions.^[1]

37 relations: Adiabatic process, Adiabatic theorem, Benzene, Bond hardening, Bond length, Bond softening, Born–Oppenheimer approximation, Bra–ket notation, Christopher Longuet-Higgins, Conical intersection, Coordinate system, Degenerate energy levels, Diagonalizable matrix, Diatomic molecule, Eigenvalues and eigenvectors, Energy minimization, Geometric phase, Hamiltonian (quantum mechanics), Hermitian matrix, Landau–Zener formula, Level repulsion, Linear subspace, Locus (mathematics), Manifold, Molecular Hamiltonian, Observable, Parametrization (geometry), Perturbation theory (quantum mechanics), Physikalische Zeitschrift, Potential energy surface, Quantum chemistry, Quantum mechanics, Resonance (chemistry), Surface (mathematics), Triatomic molecule, Two-state quantum system, Vibronic coupling.

Adiabatic process

An adiabatic process (adiabatic) is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment.

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Adiabatic theorem

The adiabatic theorem is a concept in quantum mechanics.

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Benzene

Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar hexagonal ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, benzene is classed as a hydrocarbon. Benzene is a natural constituent of petroleum and is one of the elementary petrochemicals.

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Bond hardening

Bond hardening is a process of creating a new chemical bond by strong laser fields—an effect opposite to bond softening. Avoided crossing and bond hardening are quantum chemistry.

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Bond length

In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule.

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Bond softening

Bond softening is an effect of reducing the strength of a chemical bond by strong laser fields. Avoided crossing and bond softening are quantum chemistry.

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Born–Oppenheimer approximation

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Avoided crossing and Born–Oppenheimer approximation are quantum chemistry.

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Bra–ket notation

Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case.

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Christopher Longuet-Higgins

Hugh Christopher Longuet-Higgins (11 April 1923 – 27 March 2004) was a British scholar and teacher.

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Conical intersection

In quantum chemistry, a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate (intersect) and the non-adiabatic couplings between these states are non-vanishing. Avoided crossing and conical intersection are quantum chemistry.

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Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Degenerate energy levels

In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Avoided crossing and degenerate energy levels are quantum mechanics.

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Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix.

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Diatomic molecule

Diatomic molecules are molecules composed of only two atoms, of the same or different chemical elements.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation.

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Energy minimization

In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a collection of atoms where, according to some computational model of chemical bonding, the net inter-atomic force on each atom is acceptably close to zero and the position on the potential energy surface (PES) is a stationary point (described later).

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Geometric phase

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian.

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Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Avoided crossing and Hamiltonian (quantum mechanics) are quantum chemistry and quantum mechanics.

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Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: A \text \quad \iff \quad a_.

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Landau–Zener formula

The Landau–Zener formula is an analytic solution to the equations of motion governing the transition dynamics of a two-state quantum system, with a time-dependent Hamiltonian varying such that the energy separation of the two states is a linear function of time. Avoided crossing and Landau–Zener formula are quantum mechanics.

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Level repulsion

Level repulsion is the quantum mechanical equivalent to a repulsion effect in oscillators. Avoided crossing and Level repulsion are quantum mechanics.

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Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspaceThe term linear subspace is sometimes used for referring to flats and affine subspaces.

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Locus (mathematics)

In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Molecular Hamiltonian

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. Avoided crossing and molecular Hamiltonian are quantum chemistry.

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Observable

In physics, an observable is a physical property or physical quantity that can be measured. Avoided crossing and observable are quantum mechanics.

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Parametrization (geometry)

In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

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Perturbation theory (quantum mechanics)

In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one.

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Physikalische Zeitschrift

Physikalische Zeitschrift (English: Physical Journal) was a German scientific journal of physics published from 1899 to 1945 by S. Hirzel Verlag.

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Potential energy surface

A potential energy surface (PES) or energy landscape describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. Avoided crossing and potential energy surface are quantum chemistry and quantum mechanics.

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Quantum chemistry

Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules, materials, and solutions at the atomic level. Avoided crossing and quantum chemistry are quantum mechanics.

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Quantum mechanics

Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms.

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Resonance (chemistry)

In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or polyatomic ions by the combination of several contributing structures (or forms, also variously known as resonance structures or canonical structures) into a resonance hybrid (or hybrid structure) in valence bond theory.

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References

[1] https://en.wikipedia.org/wiki/Avoided_crossing

Also known as Avoided level crossing.

Avoided crossing, the Glossary

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References