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109-399: The Jahn–Teller effect , sometimes also referred to as Jahn–Teller distortion , describes the geometrical distortion of molecules and ions that results from certain electron configurations. The Jahn–Teller theorem essentially states that any non-linear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy, because

218-572: A Euclidean space ) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O( d ). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules . Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx . Each element of

327-622: A chemical compound composed of more than one element, e.g. water (two hydrogen atoms and one oxygen atom; H 2 O). In the kinetic theory of gases , the term molecule is often used for any gaseous particle regardless of its composition. This relaxes the requirement that a molecule contains two or more atoms, since the noble gases are individual atoms. Atoms and complexes connected by non-covalent interactions , such as hydrogen bonds or ionic bonds , are typically not considered single molecules. Concepts similar to molecules have been discussed since ancient times, but modern investigation into

436-440: A plane , e.g. graphene ; or three-dimensionally e.g. diamond , quartz , sodium chloride . The theme of repeated unit-cellular-structure also holds for most metals which are condensed phases with metallic bonding . Thus solid metals are not made of molecules. In glasses , which are solids that exist in a vitreous disordered state, the atoms are held together by chemical bonds with no presence of any definable molecule, nor any of

545-532: A '+' exponent, for example [3,3,3] has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example [[3,3,3]] with its order doubled to 240. The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups . Related chiral groups exist for each with half

654-609: A JT distortion has been documented in the literature for ground or excited electronic states. A somewhat special role is played by tetrahedral systems like CH 4 and P 4 . Here threefold degenerate electronic states and vibrational modes come into play. Nevertheless, also twofold degeneracies continue to be important. The dynamics of Jahn-Teller distortion in CH 4 has been characterized by transient X-ray absorption spectroscopy, revealing that symmetry breaking occurs within ten femtoseconds in this prototypical system. Among larger systems,

763-569: A certain number of these atoms united by attraction to form a single molecule . In coordination with these concepts, in 1833 the French chemist Marc Antoine Auguste Gaudin presented a clear account of Avogadro's hypothesis, regarding atomic weights, by making use of "volume diagrams", which clearly show both semi-correct molecular geometries, such as a linear water molecule, and correct molecular formulas, such as H 2 O: In 1917, an unknown American undergraduate chemical engineer named Linus Pauling

872-399: A dimension of a few angstroms (Å) to several dozen Å, or around one billionth of a meter. Single molecules cannot usually be observed by light (as noted above), but small molecules and even the outlines of individual atoms may be traced in some circumstances by use of an atomic force microscope . Some of the largest molecules are macromolecules or supermolecules . The smallest molecule

981-522: A finite number of point groups that are symmetric over some lattice or grid with that number of dimensions. These are the crystallographic point groups . Point groups can be classified into chiral (or purely rotational) groups and achiral groups. The chiral groups are subgroups of the special orthogonal group SO( d ): they contain only orientation-preserving orthogonal transformations, i.e., those of determinant +1. The achiral groups contain also transformations of determinant −1. In an achiral group,

1090-608: A focus in the literature has been on benzene and its radical cation, as well as on their halo (especially fluoro) derivatives. Already in the early 1980s, a wealth of information emerged from the detailed analysis of experimental emission spectra of 1,3,5- trifluoro- and hexafluoro (and chloro) benzene radical cations. For the parent benzene cation one has to rely on photoelectron spectra with comparatively lower resolution because this species does not fluoresce (see also section § Spectroscopy and reactivity ). Rather detailed ab initio calculations have been carried out which document

1199-435: A graphical type of formula called a structural formula may be needed. Structural formulas may in turn be represented with a one-dimensional chemical name, but such chemical nomenclature requires many words and terms which are not part of chemical formulas. Molecules have fixed equilibrium geometries—bond lengths and angles— about which they continuously oscillate through vibrational and rotational motions. A pure substance

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1308-462: A molecule is inherently an operational definition. Philosophically, therefore, a molecule is not a fundamental entity (in contrast, for instance, to an elementary particle ); rather, the concept of a molecule is the chemist's way of making a useful statement about the strengths of atomic-scale interactions in the world that we observe. Point groups In geometry , a point group is a mathematical group of symmetry operations ( isometries in

1417-487: A point group is either a rotation ( determinant of M = 1 ), or it is a reflection or improper rotation (determinant of M = −1 ). The geometric symmetries of crystals are described by space groups , which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems , each number of dimensions has only

1526-401: A regular arrangement of molecules (as in a crystal). Microwave spectroscopy commonly measures changes in the rotation of molecules, and can be used to identify molecules in outer space. Infrared spectroscopy measures the vibration of molecules, including stretching, bending or twisting motions. It is commonly used to identify the kinds of bonds or functional groups in molecules. Changes in

1635-511: A shortening of these bonds instead (the Jahn–Teller theorem does not predict the direction of the distortion, only the presence of an unstable geometry). When such an elongation occurs, the effect is to lower the electrostatic repulsion between the electron-pair on the Lewis basic ligand and any electrons in orbitals with a z component, thus lowering the energy of the complex. The inversion centre

1744-408: A triplet T 1 electronic state, the spin–orbit coupling operator λ L . S {\displaystyle \lambda \mathbf {L} .\mathbf {S} } can be replaced by γ λ L . S {\displaystyle \gamma \lambda \mathbf {L} .\mathbf {S} } , where γ {\displaystyle \gamma } is a function of

1853-421: Is a double group representation when the antisymmetric part {Γ ⊗ Γ} A is considered instead. Modes which do couple are said to be JT-active. As an example, consider a doublet electronic state E in cubic symmetry. The symmetric part of E ⊗ E is A 1 + E. Therefore, the state E will couple to vibrational modes Q i {\displaystyle Q_{i}} transforming as a 1 and e. However,

1962-402: Is a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are termed shared pairs or bonding pairs , and the stable balance of attractive and repulsive forces between atoms, when they share electrons, is termed covalent bonding . Ionic bonding is a type of chemical bond that involves the electrostatic attraction between oppositely charged ions, and

2071-455: Is a very simple type of chemical formula. It is the simplest integer ratio of the chemical elements that constitute it. For example, water is always composed of a 2:1 ratio of hydrogen to oxygen atoms, and ethanol (ethyl alcohol) is always composed of carbon, hydrogen, and oxygen in a 2:6:1 ratio. However, this does not determine the kind of molecule uniquely – dimethyl ether has the same ratios as ethanol, for instance. Molecules with

2180-435: Is composed of molecules with the same average geometrical structure. The chemical formula and the structure of a molecule are the two important factors that determine its properties, particularly its reactivity . Isomers share a chemical formula but normally have very different properties because of their different structures. Stereoisomers , a particular type of isomer, may have very similar physico-chemical properties and at

2289-483: Is derived from French molécule (1678), from Neo-Latin molecula , diminutive of Latin moles "mass, barrier". The word, which until the late 18th century was used only in Latin form, became popular after being used in works of philosophy by Descartes . The definition of the molecule has evolved as knowledge of the structure of molecules has increased. Earlier definitions were less precise, defining molecules as

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2398-486: Is essential for the understanding of the chemical bond. The simplest of molecules is the hydrogen molecule-ion , H 2 , and the simplest of all the chemical bonds is the one-electron bond . H 2 is composed of two positively charged protons and one negatively charged electron , which means that the Schrödinger equation for the system can be solved more easily due to the lack of electron–electron repulsion. With

2507-454: Is highly "fluxional") which can be attributed to the fact that the 2nd-order coupling terms vanish by symmetry and the leading higher-order terms are of 4th order. The JTE is usually stronger where the electron density associated with the degenerate orbitals is more concentrated. This effect therefore plays a large role in determining the structure of transition metal complexes with active internal 3d orbitals. The most iconic and prominent of

2616-525: Is immeasurable. Indeed, for electrons in non-bonding or weakly bonding molecular orbitals , the effect is expected to be weak. However, in many situations the JT effect is important. Interest in the JTE increased after its first experimental verification. Various model systems were developed probing the degree of degeneracy and the type of symmetry. These were solved partly analytically and partly numerically to obtain

2725-444: Is known as a Mexican Hat potential. Here, ω {\displaystyle \omega } is the frequency of the vibrational e mode, μ {\displaystyle \mu } is its mass and k {\displaystyle k} is a measure of the strength of the JT coupling. The conical shape near the degeneracy at the origin makes it immediately clear that this point cannot be stationary , that is,

2834-473: Is manifested in the UV-VIS absorbance spectra of some compounds, where it often causes splitting of bands. It is readily apparent in the structures of many copper(II) complexes. Additional, detailed information about the anisotropy of such complexes and the nature of the ligand binding can be however obtained from the fine structure of the low-temperature electron spin resonance spectra. The underlying cause of

2943-480: Is not just restricted to Cu(II) octahedral complexes. There are many other configurations, involving changes both in the initial structure and electronic configuration of the metal that yield degenerate states and, thus, JTE. However, the amount of distortion and stabilisation energy of the effect is strongly dependent on the particular case. In octahedral Cu(II), the JTE is particularly strong because In other configurations involving π or δ bonding, like for example when

3052-464: Is preserved after the distortion. In octahedral complexes, the Jahn–Teller effect is most pronounced when an odd number of electrons occupy the e g orbitals. This situation arises in complexes with the configurations d , low-spin d or high-spin d complexes, all of which have doubly degenerate ground states. In such compounds the e g orbitals involved in the degeneracy point directly at

3161-556: Is the diatomic hydrogen (H 2 ), with a bond length of 0.74 Å. Effective molecular radius is the size a molecule displays in solution. The table of permselectivity for different substances contains examples. The chemical formula for a molecule uses one line of chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, and plus (+) and minus (−) signs. These are limited to one typographic line of symbols, which may include subscripts and superscripts. A compound's empirical formula

3270-617: Is the primary interaction occurring in ionic compounds . The ions are atoms that have lost one or more electrons (termed cations ) and atoms that have gained one or more electrons (termed anions ). This transfer of electrons is termed electrovalence in contrast to covalence . In the simplest case, the cation is a metal atom and the anion is a nonmetal atom, but these ions can be of a more complicated nature, e.g. molecular ions like NH 4 or SO 4 . At normal temperatures and pressures, ionic bonding mostly creates solids (or occasionally liquids) without separate identifiable molecules, but

3379-437: Is valid only for sufficiently strong JT couplings, that is, when several or many vibrational energy quanta fit into the energy window between the conical intersection and the minimum of the lower JT-split APES. For the many cases of small to intermediate JT couplings this energy window and the corresponding adiabatic low-energy regime does not exist. Rather, the levels on both JT-split APES are intricately mixed for all energies and

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3488-460: The Q i {\displaystyle Q_{i}} . The adiabatic potential energy surfaces (APES) are then obtained as the eigenvalues of this matrix. In the original paper it is proven that there are always linear terms in the expansion. It follows that the degeneracy of the wave function cannot correspond to a stable structure. In mathematical terms, the APESs characterising

3597-594: The 3 z 2 − r 2 {\displaystyle 3z^{2}-r^{2}} and x 2 − y 2 {\displaystyle x^{2}-y^{2}} orbitals, that are degenerate and free to hybridise in the octahedral geometry, will mix to produce appropriate equivalent orbitals in each direction like 3 x 2 − r 2 {\displaystyle 3x^{2}-r^{2}} or 3 y 2 − r 2 {\displaystyle 3y^{2}-r^{2}} . The JTE

3706-534: The Avogadro constant using three different methods, all involving liquid phase systems. First, he used a gamboge soap-like emulsion, second by doing experimental work on Brownian motion , and third by confirming Einstein's theory of particle rotation in the liquid phase. In 1927, the physicists Fritz London and Walter Heitler applied the new quantum mechanics to the deal with the saturable, nondynamic forces of attraction and repulsion, i.e., exchange forces, of

3815-587: The Hamiltonian of a polyatomic system define PESs as functions of normal modes Q i {\displaystyle Q_{i}} of the system (i.e. linear combinations of the nuclear displacements with specific symmetry properties). At the reference point of high symmetry, where the symmetry-induced degeneracy occurs, several of the eigenvalues coincide. By a detailed and laborious analysis, Jahn and Teller showed that – excepting linear molecules – there are always first-order terms in an expansion of

3924-607: The crystallographic restriction theorem to these groups yields the 32 crystallographic point groups . The reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The [3,3] group can be doubled, written as [[3,3]], mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group. The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith, Section 4, Tables 4.1–4.3. The following list gives

4033-492: The irreducible representations (irreps) that apply to the symmetry of the electronic and vibrational states. For example, E ⊗ e would refer to an electronic doublet state transforming as E coupled to a vibrational doublet state transforming as e. In general, a vibrational mode transforming as Λ will couple to an electronic state transforming as Γ if the symmetric part of the Kronecker product [Γ ⊗ Γ] S contains Λ, unless Γ

4142-590: The t 2g orbitals, which do not point directly at the ligands (see the table below). The same is true in tetrahedral complexes (e.g. manganate : distortion is very subtle because there is less stabilisation to be gained because the ligands are not pointing directly at the orbitals. The expected effects for octahedral coordination are given in the following table: w: weak Jahn–Teller effect ( t 2g orbitals unevenly occupied) s: strong Jahn–Teller effect expected ( e g orbitals unevenly occupied) blank: no Jahn–Teller effect expected. The Jahn–Teller effect

4251-476: The Chemical Bond" in which he used quantum mechanics to calculate properties and structures of molecules, such as angles between bonds and rotation about bonds. On these concepts, Pauling developed hybridization theory to account for bonds in molecules such as CH 4 , in which four sp³ hybridised orbitals are overlapped by hydrogen 's 1s orbital, yielding four sigma (σ) bonds . The four bonds are of

4360-506: The JT distortion arise as the eigenvalues of the potential energy matrix. Generally, the APESs take the characteristic appearance of a double cone, circular or elliptic, where the point of contact, i.e. degeneracy, denotes the high-symmetry configuration for which the JT theorem applies. For the above case of the linear E ⊗ e JT effect the situation is illustrated by the APES displayed in the figure, with part cut away to reveal its shape, which

4469-435: The JT stabilization energies for the various (four) JT active modes and also quantify the moderate barriers for the JT pseudorotation. Finally, a somewhat special role is played by systems with a fivefold symmetry axis like the cyclopentadienyl radical. Careful laser spectroscopic investigations have shed useful light on the JT interactions. In particular they reveal that the barrier to pseudorotation almost vanishes (the system

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4578-407: The JT systems in coordination chemistry is probably the case of Cu(II) octahedral complexes. While in perfectly equivalent coordination, like a CuF 6 complex associated to a Cu(II) impurity in a cubic crystal like KMgF 3 , perfect octahedral (O h ) symmetry is expected. In fact a lower tetragonal symmetry is usually found experimentally. The origin of this JTE distortion it revealed by examining

4687-509: The JT theorem follows from the theory of molecular symmetry ( point group theory). A less rigorous but more intuitive explanation is given in section § Coordination chemistry . To arrive at a quantitative description of the JT effect, the forces appearing between the component wave functions are described by expanding the Hamiltonian in a power series in the Q i {\displaystyle Q_{i}} . Owing to

4796-489: The JT-split APES has specific consequences for the nuclear dynamics, here considered in the fully quantum sense. For sufficiently strong JT coupling, the minimum points are sufficiently far (at least by a few vibrational energy quanta) below the JT intersection. Two different energy regimes are then to be distinguished, those of low and high energy. As already stated above, the distinction of low and high energy regimes

4905-451: The JTE in E′ and E″ states, also the pseudo JTE between an E state and a nearby A state may play a role. The JT distortion reduces the symmetry from D 3h to C 2v (see figure), and it depends on the details of the interactions whether the isosceles triangle has an acute or an obtuse-angled (such as Na 3 ) minimum energy structure. Natural extensions are systems like NO 3 and NH 3 where

5014-568: The Jahn–Teller effect is the presence of molecular orbitals that are both degenerate and open shell (i.e., incompletely occupied). This situation is not unique to coordination complexes and can be encountered in other areas of chemistry. In organic chemistry the phenomenon of antiaromaticity has the same cause and also often sees molecules distorting; as in the case of cyclobutadiene and cyclooctatetraene (COT). The JT theorem can be stated in different forms, two of which are given here: Alternatively and considerably shorter: Spin-degeneracy

5123-475: The a 1 modes will result in the same energy shift to all states and therefore do not contribute to any JT splitting. They can therefore be neglected. The result is an E ⊗ e JT effect. This JT effect is experienced by triangular molecules X 3 , tetrahedral molecules ML 4 , and octahedral molecules ML 6 when their electronic state has E symmetry. Components of a given vibrational mode are also labelled according to their transformation properties. For example,

5232-472: The advent of modern (" ab initio ") electronic structure calculations whereby the relevant parameters characterising JT systems can be reliably determined from first principles. Thus one could go beyond studies of model systems that explore the effect of parameter variations on the PES and vibronic energy levels; one could also go on beyond fitting these parameters to experimental data without clear knowledge about

5341-441: The arrangements of electrons yield absorption or emission lines in ultraviolet, visible or near infrared light, and result in colour. Nuclear resonance spectroscopy measures the environment of particular nuclei in the molecule, and can be used to characterise the numbers of atoms in different positions in a molecule. The study of molecules by molecular physics and theoretical chemistry is largely based on quantum mechanics and

5450-432: The bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3] has six 3-fold gyration points and symmetry order 20160. The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups . Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by

5559-430: The bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3] has five 3-fold gyration points and symmetry order 2520. The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups . Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by

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5668-418: The complex would distort into a compressed geometry. Experimentally elongated geometries are overwhelmingly observed and this fact has been attributed both to metal-ligand anharmonic interactions and 3d-4s hybridisations. Given that all the directions containing a fourfold axis are equivalent the distortion is equally likely to happen in any of these orientations. From the electronic point of view this means that

5777-433: The configuration space: its inclusion extends the applicability of JT-related models to symmetry breaking in a far broader range of molecular and solid-state systems. Chronology: A given JT problem will have a particular point group symmetry , such as T d symmetry for magnetic impurity ions in semiconductors or I h symmetry for the fullerene C 60 . JT problems are conventionally classified using labels for

5886-436: The degenerate state is associated to the t 2g orbitals of an octahedral configuration, the distortion and stabilisation energies are usually much smaller and the possibility of not observing the distortion due to dynamic JT effects is much higher. Similarly for rare-earth ions where covalency is very small, the distortions associated to the JTE are usually very weak. Importantly, the JTE is associated with strict degeneracy in

5995-484: The development of fast digital computers, approximate solutions for more complicated molecules became possible and are one of the main aspects of computational chemistry . When trying to define rigorously whether an arrangement of atoms is sufficiently stable to be considered a molecule, IUPAC suggests that it "must correspond to a depression on the potential energy surface that is deep enough to confine at least one vibrational state". This definition does not depend on

6104-601: The different electronic interactions with axial X ligands and equatorial Y ligands . In this and other similar cases some remaining vibronic effects related to the JTE are still present but are quenched with respect to the case with degeneracy due to the splitting of the orbitals. From spectra with rotational resolution, moments of inertia and hence bond lengths and angles can be determined "directly" (at least in principle). From less well-resolved spectra one can still determine important quantities like JT stabilization energies and energy barriers (e.g. to pseudorotation). However, in

6213-450: The distortion lowers the overall energy of the species. For a description of another type of geometrical distortion that occurs in crystals with substitutional impurities see article off-center ions . The Jahn–Teller effect is most often encountered in octahedral complexes of the transition metals. The phenomenon is very common in six-coordinate copper (II) complexes. The d electronic configuration of this ion gives three electrons in

6322-620: The double-cone topology of the linear E ⊗ e JT system directly reflects the high underlying symmetry. It is one of the earliest (if not the earliest) examples in the literature of a conical intersection of potential energy surfaces. Conical intersections have received wide attention in the literature starting in the 1990s and are now considered paradigms of nonadiabatic excited-state dynamics, with far-reaching consequences in molecular spectroscopy, photochemistry and photophysics. Some of these will be commented upon further below. In general, conical intersections are far less symmetric than depicted in

6431-484: The electron occupies the mainly 3 z 2 − r 2 {\displaystyle 3z^{2}-r^{2}} level, which antibonding orbital the final geometry of the complex would be elongated as the axial ligands will be pushed away to reduce the global energy of the system. On the other hand, if the electron went into the mainly x 2 − y 2 {\displaystyle x^{2}-y^{2}} antibonding orbital

6540-416: The electronic configuration of the undistorted complex. For an octahedral geometry, the five 3d orbitals partition into t 2g and e g orbitals (see diagram). These orbitals are occupied by nine electrons corresponding to the d 9 {\displaystyle d^{9}} electronic configuration of Cu(II). Thus, the t 2g shell is filled, and the e g shell contains 3 electrons. Overall

6649-635: The electronic subsystem and so it cannot appear in systems without this property. For example, the JTE is often associated to cases like quasi-octahedral CuX 2 Y 4 complexes where the distances to X and Y ligands are clearly different. However, the intrinsic symmetry of these complexes is already tetragonal and no degenerate e g orbital exists, having split into a 1g (mainly 3 z 2 − r 2 {\displaystyle 3z^{2}-r^{2}} ) and b 1g (mainly x 2 − y 2 {\displaystyle x^{2}-y^{2}} ) orbitals due to

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6758-445: The figure. They can be tilted and elliptical in shape etc., and also peaked and sloped intersections have been distinguished in the literature. Furthermore, for more than two degrees of freedom, they are not point-like structures but instead they are seams and complicated, curved hypersurfaces, also known as intersection space. The coordinate sub-space displayed in the figure is also known as a branching plane. The characteristic shape of

6867-469: The first-order effects have been significantly reduced. Reduction factors are particularly useful for describing experimental results, such as EPR and optical spectra, of paramagnetic impurities in semiconducting , dielectric , diamagnetic and ferrimagnetic hosts. For a long time, applications of JT theory consisted mainly in parameter studies (model studies) where the APES and dynamical properties of JT systems have been investigated as functions on

6976-419: The four-dimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lower-dimensional reflection groups). Each group is specified as a Coxeter group , and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope . Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with

7085-464: The heavenly bodies. The viewpoint of Leucippus and Empedocles, along with the aether, was accepted by Aristotle and passed to medieval and renaissance Europe. In a more concrete manner, however, the concept of aggregates or units of bonded atoms, i.e. "molecules", traces its origins to Robert Boyle 's 1661 hypothesis, in his famous treatise The Sceptical Chymist , that matter is composed of clusters of particles and that chemical change results from

7194-589: The hydrogen molecule. Their valence bond treatment of this problem, in their joint paper, was a landmark in that it brought chemistry under quantum mechanics. Their work was an influence on Pauling, who had just received his doctorate and visited Heitler and London in Zürich on a Guggenheim Fellowship . Subsequently, in 1931, building on the work of Heitler and London and on theories found in Lewis' famous article, Pauling published his ground-breaking article "The Nature of

7303-735: The laws governing their structure and properties. In practice, however, this distinction is vague. In molecular sciences, a molecule consists of a stable system ( bound state ) composed of two or more atoms. Polyatomic ions may sometimes be usefully thought of as electrically charged molecules. The term unstable molecule is used for very reactive species, i.e., short-lived assemblies ( resonances ) of electrons and nuclei , such as radicals , molecular ions , Rydberg molecules , transition states , van der Waals complexes , or systems of colliding atoms as in Bose–Einstein condensate . Molecules as components of matter are common. They also make up most of

7412-420: The ligands, so distortion can result in a large energetic stabilisation. Strictly speaking, the effect also occurs when there is a degeneracy due to the electrons in the t 2g orbitals ( i.e. configurations such as d or d , both of which are triply degenerate). In such cases, however, the effect is much less noticeable, because there is a much smaller lowering of repulsion on taking ligands further away from

7521-430: The literature) the advent of density functional theory (DFT) opened up new avenues to treat larger systems including solids. This allowed details of JT systems to be characterised and experimental findings to be reliably interpreted. It lies at the heart of most developments addressed in section § Applications . Two different strategies are conceivable and have been used in the literature. One can Naturally,

7630-404: The mass of a neutral carbon-12 ( C isotope ) atom. For network solids , the term formula unit is used in stoichiometric calculations. For molecules with a complicated 3-dimensional structure, especially involving atoms bonded to four different substituents, a simple molecular formula or even semi-structural chemical formula may not be enough to completely specify the molecule. In this case,

7739-414: The matrix elements of the Hamiltonian in terms of symmetry-lowering (in the language of group theory : non-totally symmetric) normal modes. These linear terms represent forces that distort the system along these coordinates and lift the degeneracy. The point of degeneracy can thus not be stationary, and the system distorts toward a stationary point of lower symmetry where stability can be attained. Proof of

7848-452: The minerals that make up the substance of the Earth, sand, clay, pebbles, rocks, boulders, bedrock , the molten interior , and the core of the Earth . All of these contain many chemical bonds, but are not made of identifiable molecules. No typical molecule can be defined for salts nor for covalent crystals , although these are often composed of repeating unit cells that extend either in

7957-510: The more accurate approach (2) may be limited to smaller systems, while the simpler approach (1) lends itself to studies of larger systems. The JT distortion of small molecules (or molecular ions) is directly deduced from electronic structure calculations of their APES (through DFT and/or ab initio computations). These molecules / ions are often radicals, such as trimers of alkali atoms (Li 3 and Na 3 ), that have unpaired spins and in particular in (but not restricted to) doublet states. Besides

8066-552: The nature of molecules and their bonds began in the 17th century. Refined over time by scientists such as Robert Boyle , Amedeo Avogadro , Jean Perrin , and Linus Pauling , the study of molecules is today known as molecular physics or molecular chemistry. According to Merriam-Webster and the Online Etymology Dictionary , the word "molecule" derives from the Latin " moles " or small unit of mass. The word

8175-440: The nature of the interaction between the atoms, but only on the strength of the interaction. In fact, it includes weakly bound species that would not traditionally be considered molecules, such as the helium dimer , He 2 , which has one vibrational bound state and is so loosely bound that it is only likely to be observed at very low temperatures. Whether or not an arrangement of atoms is sufficiently stable to be considered

8284-403: The nuclear motion always proceeds on both JT split APES simultaneously. In 1965, Frank Ham proposed that the dynamic JTE could reduce the expected values of observables associated with the orbital wavefunctions due to the superposition of several electronic states in the total vibronic wavefunction. This effect leads, for example, to a partial quenching of the spin–orbit interaction and allowed

8393-517: The oceans and atmosphere. Most organic substances are molecules. The substances of life are molecules, e.g. proteins, the amino acids of which they are composed, the nucleic acids (DNA and RNA), sugars, carbohydrates, fats, and vitamins. The nutrient minerals are generally ionic compounds, thus they are not molecules, e.g. iron sulfate. However, the majority of familiar solid substances on Earth are made partly or completely of crystals or ionic compounds, which are not made of molecules. These include all of

8502-427: The order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3] has four 3-fold gyration points and symmetry order 360. The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups . Related pure rotational groups exist for each with half the order, and can be represented by

8611-540: The orientation-preserving transformations form a (chiral) subgroup of index 2. Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram . Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral (except for

8720-411: The rearrangement of the clusters. Boyle argued that matter's basic elements consisted of various sorts and sizes of particles, called "corpuscles", which were capable of arranging themselves into groups. In 1789, William Higgins published views on what he called combinations of "ultimate" particles, which foreshadowed the concept of valency bonds . If, for example, according to Higgins, the force between

8829-460: The reflectional groups can be doubled by an isomorphism , mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order. Point groups in three dimensions , sometimes called molecular point groups after their wide use in studying symmetries of molecules . They come in 7 infinite families of axial groups (also called prismatic), and 7 additional polyhedral groups (also called Platonic). In Schönflies notation , Applying

8938-546: The regularity of repeating unit-cellular-structure that characterizes salts, covalent crystals, and metals. Molecules are generally held together by covalent bonding . Several non-metallic elements exist only as molecules in the environment either in compounds or as homonuclear molecules, not as free atoms: for example, hydrogen. While some people say a metallic crystal can be considered a single giant molecule held together by metallic bonding , others point out that metals behave very differently than molecules. A covalent bond

9047-456: The results of previous Electron Paramagnetic Resonance (EPR) experiments to be explained. In general, the result of an orbital operator acting on vibronic states can be replaced by an effective orbital operator acting on purely electronic states. In first order, the effective orbital operator equals the actual orbital operator multiplied by a constant, whose value is less than one, known as a first-order (Ham) reduction factor. For example, within

9156-405: The same atoms in different arrangements are called isomers . Also carbohydrates, for example, have the same ratio (carbon:hydrogen:oxygen= 1:2:1) (and thus the same empirical formula) but different total numbers of atoms in the molecule. The molecular formula reflects the exact number of atoms that compose the molecule and so characterizes different molecules. However different isomers can have

9265-404: The same atomic composition while being different molecules. The empirical formula is often the same as the molecular formula but not always. For example, the molecule acetylene has molecular formula C 2 H 2 , but the simplest integer ratio of elements is CH. The molecular mass can be calculated from the chemical formula and is expressed in conventional atomic mass units equal to 1/12 of

9374-414: The same energy minimum. The "transition" between them is thus more oscillatory than one would normally expect, and their time-averaged populations are close to 1/2. For a more typical scenario a more general conical intersection is "required". Molecule A molecule is a group of two or more atoms that are held together by attractive forces known as chemical bonds ; depending on context,

9483-405: The same length and strength, which yields a molecular structure as shown below: The science of molecules is called molecular chemistry or molecular physics , depending on whether the focus is on chemistry or physics. Molecular chemistry deals with the laws governing the interaction between molecules that results in the formation and breakage of chemical bonds, while molecular physics deals with

9592-572: The same time different biochemical activities. Molecular spectroscopy deals with the response ( spectrum ) of molecules interacting with probing signals of known energy (or frequency , according to the Planck relation ). Molecules have quantized energy levels that can be analyzed by detecting the molecule's energy exchange through absorbance or emission . Spectroscopy does not generally refer to diffraction studies where particles such as neutrons , electrons, or high energy X-rays interact with

9701-506: The shape of the pertinent potential energy surfaces (PES) and the energy levels for the nuclear motion on the JT-split PES. These energy levels are not vibrational energy levels in the traditional sense because of the intricate coupling to the electronic motion that occurs, and are better termed vibronic energy levels. The new field of ‘ vibronic coupling ’ or ‘vibronic coupling theory’ was born. A further breakthrough occurred upon

9810-497: The significance of the fit. Instead, well-founded theoretical investigations became possible which greatly improved the insight into the phenomena at hand and into the details of the underlying mechanisms. While recognizing the JTE distortion as a concrete example of the general spontaneous symmetry breaking mechanism, the exact degeneracy of the involved electronic state was identified as a non-essential ingredient for this symmetry breaking in polyatomic systems. Even systems that in

9919-517: The smallest particles of pure chemical substances that still retain their composition and chemical properties. This definition often breaks down since many substances in ordinary experience, such as rocks , salts , and metals , are composed of large crystalline networks of chemically bonded atoms or ions , but are not made of discrete molecules. The modern concept of molecules can be traced back towards pre-scientific and Greek philosophers such as Leucippus and Democritus who argued that all

10028-484: The so-called autocorrelation function C ( t ) {\displaystyle C(t)} reflects the motion of the wavepacket after an optical (= vertical) transition to the APES of the final electronic state. Typically it will move on the timescale of a vibrational period which is (for small molecules) of the order of 5–50 fs, i.e. ultrafast. Besides a nearly periodic motion, mode–mode interactions with very irregular (also chaotic) behaviour and spreading of

10137-404: The strength of the JT coupling which varies from 1 in zero coupling to 0 in very strong coupling. Furthermore, when second-order perturbation corrections are included, additional terms are introduced involving additional numerical factors, known as second-order (Ham) reduction factors. These factors are zero when there is no JT coupling but can dominate over first-order terms in strong coupling, when

10246-616: The system is unstable against asymmetric distortions, which leads to a symmetry lowering. In this particular case there are infinitely many isoenergetic JT distortions. The Q i {\displaystyle Q_{i}} giving these distortions are arranged in a circle, as shown by the red curve in the figure. Quadratic coupling or cubic elastic terms lead to a warping along this "minimum energy path", replacing this infinite manifold by three equivalent potential minima and three equivalent saddle points. In other JT systems, linear coupling results in discrete minima. The high symmetry of

10355-454: The system parameters such as coupling constants etc. Fits of these parameters to experimental data were often doubtful and inconclusive. The situation changed in the 1980s when efficient ab initio methods were developed and computational resources became powerful enough to allow for a reliable determination of these parameters from first principles. Apart from wave function -based techniques (which are sometimes considered genuinely ab initio in

10464-410: The term may or may not include ions that satisfy this criterion. In quantum physics , organic chemistry , and biochemistry , the distinction from ions is dropped and molecule is often used when referring to polyatomic ions . A molecule may be homonuclear , that is, it consists of atoms of one chemical element , e.g. two atoms in the oxygen molecule (O 2 ); or it may be heteronuclear ,

10573-593: The trivial group containing only the identity element). There are only two one-dimensional point groups, the identity group and the reflection group. Point groups in two dimensions , sometimes called rosette groups . They come in two infinite families: Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups. The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of

10682-498: The two components of an e mode are usually labelled Q θ {\displaystyle Q_{\theta }} and Q ϵ {\displaystyle Q_{\epsilon }} , which in octahedral symmetry transform as 3 z 2 − r 2 {\displaystyle 3z^{2}-r^{2}} and x 2 − y 2 {\displaystyle x^{2}-y^{2}} respectively. Eigenvalues of

10791-428: The two degenerate e g orbitals, leading to a doubly degenerate electronic ground state. Such complexes distort along one of the molecular fourfold axes (always labelled the z axis), which has the effect of removing the orbital and electronic degeneracies and lowering the overall energy. The distortion normally takes the form of elongating the bonds to the ligands lying along the z axis, but occasionally occurs as

10900-668: The ultimate particle of oxygen and the ultimate particle of nitrogen were 6, then the strength of the force would be divided accordingly, and similarly for the other combinations of ultimate particles. Amedeo Avogadro created the word "molecule". His 1811 paper "Essay on Determining the Relative Masses of the Elementary Molecules of Bodies", he essentially states, i.e. according to Partington 's A Short History of Chemistry , that: The smallest particles of gases are not necessarily simple atoms, but are made up of

11009-447: The undistorted symmetric configuration present electronic states which are near in energy but not precisely degenerate, can show a similar tendency to distort. The distortions of these systems can be treated within the related theory of the pseudo Jahn–Teller effect (in the literature often referred to as "second-order JTE"). This mechanism is associated to the vibronic couplings between adiabatic PES separated by nonzero energy gaps across

11118-409: The universe is composed of atoms and voids . Circa 450 BC Empedocles imagined fundamental elements ( fire ( [REDACTED] ), earth ( [REDACTED] ), air ( [REDACTED] ), and water ( [REDACTED] )) and "forces" of attraction and repulsion allowing the elements to interact. A fifth element, the incorruptible quintessence aether , was considered to be the fundamental building block of

11227-411: The unpaired electron produces a E g state, which is Jahn–Teller active. The third electron can occupy either of the orbitals comprising the e g shell: the mainly 3 z 2 − r 2 {\displaystyle 3z^{2}-r^{2}} orbital or the mainly x 2 − y 2 {\displaystyle x^{2}-y^{2}} orbital. If

11336-432: The vaporization/sublimation of such materials does produce separate molecules where electrons are still transferred fully enough for the bonds to be considered ionic rather than covalent. Most molecules are far too small to be seen with the naked eye, although molecules of many polymers can reach macroscopic sizes, including biopolymers such as DNA . Molecules commonly used as building blocks for organic synthesis have

11445-518: The very nature of the degeneracy, the Hamiltonian takes the form of a matrix referring to the degenerate wave function components. A matrix element between states Ψ a {\displaystyle \Psi _{a}} and Ψ b {\displaystyle \Psi _{b}} generally reads as: The expansion can be truncated after terms linear in the Q i {\displaystyle Q_{i}} , or extended to include terms quadratic (or higher) in

11554-418: The wavepacket may also occur. Near a conical intersection this will be accompanied/complemented by nonradiative transitions (termed internal conversion) to other APESs occurring on the same ultrafast time scale. For the JT case the situation is somewhat special, as compared to a general conical intersection, because the different JT potential sheets are symmetry-related to each other and have (exactly or nearly)

11663-531: The whole spectral intensity distribution P ( E ) {\displaystyle P(E)} of an electronic transition more information is encoded. It has been used to decide on the presence (or absence) of the geometric phase which is accumulated during the pseudorotational motion around the JT (or other type of) conical intersection. Prominent examples of either type are the ground (X) or an excited (B) state of Na 3 . The Fourier transform of P ( E ) {\displaystyle P(E)} ,

11772-432: Was an exception in the original treatment and was later treated separately. The formal mathematical proof of the Jahn–Teller theorem rests heavily on symmetry arguments, more specifically the theory of molecular point groups . The argument of Jahn and Teller assumes no details about the electronic structure of the system. Jahn and Teller made no statement about the strength of the effect, which may be so small that it

11881-526: Was learning the Dalton hook-and-eye bonding method , which was the mainstream description of bonds between atoms at the time. Pauling, however, was not satisfied with this method and looked to the newly emerging field of quantum physics for a new method. In 1926, French physicist Jean Perrin received the Nobel Prize in physics for proving, conclusively, the existence of molecules. He did this by calculating

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