Catalysis
Catalysis is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst, not consumed in the catalyzed reaction and can continue to act repeatedly. Because of this, only small amounts of catalyst are required to alter the reaction rate in principle. In general, chemical reactions occur faster in the presence of a catalyst because the catalyst provides an alternative reaction pathway with a lower activation energy than the non-catalyzed mechanism. In catalyzed mechanisms, the catalyst reacts to form a temporary intermediate, which regenerates the original catalyst in a cyclic process. A substance which provides a mechanism with a higher activation energy does not decrease the rate because the reaction can still occur by the non-catalyzed route. An added substance which does reduce the reaction rate is not considered a catalyst but a reaction inhibitor. Catalysts may be classified as either heterogeneous. A homogeneous catalyst is one whose molecules are dispersed in the same phase as the reactant's molecules.
A heterogeneous catalyst is one whose molecules are not in the same phase as the reactant's, which are gases or liquids that are adsorbed onto the surface of the solid catalyst. Enzymes and other biocatalysts are considered as a third category. In the presence of a catalyst, less free energy is required to reach the transition state, but the total free energy from reactants to products does not change. A catalyst may participate in multiple chemical transformations; the effect of a catalyst may vary due to the presence of other substances known as inhibitors or poisons or promoters. Catalyzed reactions have a lower activation energy than the corresponding uncatalyzed reaction, resulting in a higher reaction rate at the same temperature and for the same reactant concentrations. However, the detailed mechanics of catalysis is complex. Catalysts may bind to the reagents to polarize bonds, e.g. acid catalysts for reactions of carbonyl compounds, or form specific intermediates that are not produced such as osmate esters in osmium tetroxide-catalyzed dihydroxylation of alkenes, or cause dissociation of reagents to reactive forms, such as chemisorbed hydrogen in catalytic hydrogenation.
Kinetically, catalytic reactions are typical chemical reactions. The catalyst participates in this slowest step, rates are limited by amount of catalyst and its "activity". In heterogeneous catalysis, the diffusion of reagents to the surface and diffusion of products from the surface can be rate determining. A nanomaterial-based catalyst is an example of a heterogeneous catalyst. Analogous events associated with substrate binding and product dissociation apply to homogeneous catalysts. Although catalysts are not consumed by the reaction itself, they may be inhibited, deactivated, or destroyed by secondary processes. In heterogeneous catalysis, typical secondary processes include coking where the catalyst becomes covered by polymeric side products. Additionally, heterogeneous catalysts can dissolve into the solution in a solid–liquid system or sublimate in a solid–gas system; the production of most industrially important chemicals involves catalysis. Most biochemically significant processes are catalysed.
Research into catalysis is a major field in applied science and involves many areas of chemistry, notably organometallic chemistry and materials science. Catalysis is relevant to many aspects of environmental science, e.g. the catalytic converter in automobiles and the dynamics of the ozone hole. Catalytic reactions are preferred in environmentally friendly green chemistry due to the reduced amount of waste generated, as opposed to stoichiometric reactions in which all reactants are consumed and more side products are formed. Many transition metals and transition metal complexes are used in catalysis as well. Catalysts called. A catalyst works by providing an alternative reaction pathway to the reaction product; the rate of the reaction is increased as this alternative route has a lower activation energy than the reaction route not mediated by the catalyst. The disproportionation of hydrogen peroxide creates oxygen, as shown below. 2 H2O2 → 2 H2O + O2This reaction is preferable in the sense that the reaction products are more stable than the starting material, though the uncatalysed reaction is slow.
In fact, the decomposition of hydrogen peroxide is so slow that hydrogen peroxide solutions are commercially available. This reaction is affected by catalysts such as manganese dioxide, or the enzyme peroxidase in organisms. Upon the addition of a small amount of manganese dioxide, the hydrogen peroxide reacts rapidly; this effect is seen by the effervescence of oxygen. The manganese dioxide is not consumed in the reaction, thus may be recovered unchanged, re-used indefinitely. Accordingly, manganese dioxide catalyses this reaction. Catalytic activity is denoted by the symbol z and measured in mol/s, a unit, called katal and defined the SI unit for catalytic activity since 1999. Catalytic activity is not a kind of reaction rate, but a property of the catalyst under certain conditions, in relation to a specific chemical reaction. Catalytic activity of one katal of a catalyst means one mole of that catalyst will catalyse 1 mole of the reactant to product in one second. A catalyst may and will have different catalytic activity for di
Ligand
In coordination chemistry, a ligand is an ion or molecule that binds to a central metal atom to form a coordination complex. The bonding with the metal involves formal donation of one or more of the ligand's electron pairs; the nature of metal–ligand bonding can range from covalent to ionic. Furthermore, the metal–ligand bond order can range from one to three. Ligands are viewed as Lewis bases, although rare cases are known to involve Lewis acidic "ligands". Metals and metalloids are bound to ligands in all circumstances, although gaseous "naked" metal ions can be generated in a high vacuum. Ligands in a complex dictate the reactivity of the central atom, including ligand substitution rates, the reactivity of the ligands themselves, redox. Ligand selection is a critical consideration in many practical areas, including bioinorganic and medicinal chemistry, homogeneous catalysis, environmental chemistry. Ligands are classified in many ways, including: charge, the identity of the coordinating atom, the number of electrons donated to the metal.
The size of a ligand is indicated by its cone angle. The composition of coordination complexes have been known since the early 1800s, such as Prussian blue and copper vitriol; the key breakthrough occurred when Alfred Werner reconciled isomers. He showed, among other things, that the formulas of many cobalt and chromium compounds can be understood if the metal has six ligands in an octahedral geometry; the first to use the term "ligand" were Alfred Stock and Carl Somiesky, in relation to silicon chemistry. The theory allows one to understand the difference between coordinated and ionic chloride in the cobalt ammine chlorides and to explain many of the inexplicable isomers, he resolved the first coordination complex called hexol into optical isomers, overthrowing the theory that chirality was associated with carbon compounds. In general, ligands are viewed as the metals as electron acceptors; this is because the ligand and central metal are bonded to one another, the ligand is providing both electrons to the bond instead of the metal and ligand each providing one electron.
Bonding is described using the formalisms of molecular orbital theory. The HOMO can be of ligands or metal character. Ligands and metal ions can be ordered in many ways. Metal ions preferentially bind certain ligands. In general,'hard' metal ions prefer weak field ligands, whereas'soft' metal ions prefer strong field ligands. According to the molecular orbital theory, the HOMO of the ligand should have an energy that overlaps with the LUMO of the metal preferential. Metal ions bound to strong-field ligands follow the Aufbau principle, whereas complexes bound to weak-field ligands follow Hund's rule. Binding of the metal with the ligands results in a set of molecular orbitals, where the metal can be identified with a new HOMO and LUMO and a certain ordering of the 5 d-orbitals. In an octahedral environment, the 5 otherwise degenerate d-orbitals split in sets of 2 and 3 orbitals. 3 orbitals of low energy: dxy and dyz 2 of high energy: dz2 and dx2−y2The energy difference between these 2 sets of d-orbitals is called the splitting parameter, Δo.
The magnitude of Δo is determined by the field-strength of the ligand: strong field ligands, by definition, increase Δo more than weak field ligands. Ligands can now be sorted according to the magnitude of Δo; this ordering of ligands is invariable for all metal ions and is called spectrochemical series. For complexes with a tetrahedral surrounding, the d-orbitals again split into two sets, but this time in reverse order. 2 orbitals of low energy: dz2 and dx2−y2 3 orbitals of high energy: dxy and dyzThe energy difference between these 2 sets of d-orbitals is now called Δt. The magnitude of Δt is smaller than for Δo, because in a tetrahedral complex only 4 ligands influence the d-orbitals, whereas in an octahedral complex the d-orbitals are influenced by 6 ligands; when the coordination number is neither octahedral nor tetrahedral, the splitting becomes correspondingly more complex. For the purposes of ranking ligands, the properties of the octahedral complexes and the resulting Δo has been of primary interest.
The arrangement of the d-orbitals on the central atom, has a strong effect on all the properties of the resulting complexes. E.g. the energy differences in the d-orbitals has a strong effect in the optical absorption spectra of metal complexes. It turns out that valence electrons occupying orbitals with significant 3 d-orbital character absorb in the 400–800 nm region of the spectrum; the absorption of light by these electrons can be correlated to the ground state of the metal complex, which reflects the bonding properties of the ligands. The relative change in energy of the d-orbitals as a function of the field-strength of the ligands is described in Tanabe–Sugano diagrams. In cases where the ligand has low energy LUMO, such orbitals participate in the bonding; the metal–ligand bond can be further stabilised by a formal donation of electron density back to the ligand in a process known as back-bonding. In this case a filled, c
Noble gas
The noble gases make up a group of chemical elements with similar properties. The six noble gases that occur are helium, argon, krypton and the radioactive radon. Oganesson is variously predicted to be a noble gas as well or to break the trend due to relativistic effects. For the first six periods of the periodic table, the noble gases are the members of group 18. Noble gases are highly unreactive except when under particular extreme conditions; the inertness of noble gases makes them suitable in applications where reactions are not wanted. For example, argon is used in incandescent lamps to prevent the hot tungsten filament from oxidizing; the properties of the noble gases can be well explained by modern theories of atomic structure: their outer shell of valence electrons is considered to be "full", giving them little tendency to participate in chemical reactions, it has been possible to prepare only a few hundred noble gas compounds. The melting and boiling points for a given noble gas are close together, differing by less than 10 °C.
Neon, argon and xenon are obtained from air in an air separation unit using the methods of liquefaction of gases and fractional distillation. Helium is sourced from natural gas fields that have high concentrations of helium in the natural gas, using cryogenic gas separation techniques, radon is isolated from the radioactive decay of dissolved radium, thorium, or uranium compounds. Noble gases have several important applications in industries such as lighting and space exploration. A helium-oxygen breathing gas is used by deep-sea divers at depths of seawater over 55 m. After the risks caused by the flammability of hydrogen became apparent, it was replaced with helium in blimps and balloons. Noble gas is translated from the German noun Edelgas, first used in 1898 by Hugo Erdmann to indicate their low level of reactivity; the name makes an analogy to the term "noble metals", which have low reactivity. The noble gases have been referred to as inert gases, but this label is deprecated as many noble gas compounds are now known.
Rare gases is another term, used, but this is inaccurate because argon forms a considerable part of the Earth's atmosphere due to decay of radioactive potassium-40. Pierre Janssen and Joseph Norman Lockyer discovered a new element on August 18, 1868 while looking at the chromosphere of the Sun, named it helium after the Greek word for the Sun, ἥλιος. No chemical analysis was possible at the time, but helium was found to be a noble gas. Before them, in 1784, the English chemist and physicist Henry Cavendish had discovered that air contains a small proportion of a substance less reactive than nitrogen. A century in 1895, Lord Rayleigh discovered that samples of nitrogen from the air were of a different density than nitrogen resulting from chemical reactions. Along with Scottish scientist William Ramsay at University College, Lord Rayleigh theorized that the nitrogen extracted from air was mixed with another gas, leading to an experiment that isolated a new element, from the Greek word ἀργός. With this discovery, they realized.
During his search for argon, Ramsay managed to isolate helium for the first time while heating cleveite, a mineral. In 1902, having accepted the evidence for the elements helium and argon, Dmitri Mendeleev included these noble gases as group 0 in his arrangement of the elements, which would become the periodic table. Ramsay continued his search for these gases using the method of fractional distillation to separate liquid air into several components. In 1898, he discovered the elements krypton and xenon, named them after the Greek words κρυπτός, νέος, ξένος, respectively. Radon was first identified in 1898 by Friedrich Ernst Dorn, was named radium emanation, but was not considered a noble gas until 1904 when its characteristics were found to be similar to those of other noble gases. Rayleigh and Ramsay received the 1904 Nobel Prizes in Physics and in Chemistry for their discovery of the noble gases; the discovery of the noble gases aided in the development of a general understanding of atomic structure.
In 1895, French chemist Henri Moissan attempted to form a reaction between fluorine, the most electronegative element, argon, one of the noble gases, but failed. Scientists were unable to prepare compounds of argon until the end of the 20th century, but these attempts helped to develop new theories of atomic structure. Learning from these experiments, Danish physicist Niels Bohr proposed in 1913 that the electrons in atoms are arranged in shells surrounding the nucleus, that for all noble gases except helium the outermost shell always contains eight electrons. In 1916, Gilbert N. Lewis formulated the octet rule, which conc
Transition metal
In chemistry, the term transition metal has three possible meanings: The IUPAC definition defines a transition metal as "an element whose atom has a filled d sub-shell, or which can give rise to cations with an incomplete d sub-shell". Many scientists describe a "transition metal" as any element in the d-block of the periodic table, which includes groups 3 to 12 on the periodic table. In actual practice, the f-block lanthanide and actinide series are considered transition metals and are called "inner transition metals". Cotton and Wilkinson expand the brief IUPAC definition by specifying; as well as the elements of groups 4 to 11, they add scandium and yttrium in group 3, which have a filled d subshell in the metallic state. Lanthanum and actinium in group 3 are, classified as lanthanides and actinides respectively. English chemist Charles Bury first used the word transition in this context in 1921, when he referred to a transition series of elements during the change of an inner layer of electrons from a stable group of 8 to one of 18, or from 18 to 32.
These elements are now known as the d-block. In the d-block the atoms of the elements have between 10 d electrons; the elements of groups 4–11 are recognized as transition metals, justified by their typical chemistry, i.e. a large range of complex ions in various oxidation states, colored complexes, catalytic properties either as the element or as ions. Sc and Y in group 3 are generally recognized as transition metals. However, the elements La–Lu and Ac–Lr and group 12 attract different definitions from different authors. Many chemistry textbooks and printed periodic tables classify La and Ac as group 3 elements and transition metals, since their atomic ground-state configurations are s2d1 like Sc and Y; the elements Ce–Lu are considered as the "lanthanide" series and Th–Lr as the "actinide" series.</ref> The two series together are classified as f-block elements, or as "inner transition elements". Some inorganic chemistry textbooks include Ac with the actinides; this classification is based on similarities in chemical behaviour and defines 15 elements in each of the two series though they correspond to the filling of an f subshell, which can only contain 14 electrons.
A third classification defines the f-block elements as La–Yb and Ac–No, while placing Lu and Lr in group 3. This is based on the Aufbau principle for filling electron subshells, in which 4f is filled before 5d, so that the f subshell is full at Yb, while Lu has an s2f14d1 configuration; however La and Ac are exceptions to the Aufbau principle with electron configuration s2d1, so it is not clear from atomic electron configurations whether La or Lu should be considered as transition metals. Zinc and mercury are excluded from the transition metals, as they have the electronic configuration d10s2, with no incomplete d shell. In the oxidation state +2 the ions have the electronic configuration …d10. However, these elements can exist in other oxidation states, including the +1 oxidation state, as in the diatomic ion Hg2+2; the group 12 elements Zn, Cd and Hg may therefore, under certain criteria, be classed as post-transition metals in this case. However, it is convenient to include these elements in a discussion of the transition elements.
For example, when discussing the crystal field stabilization energy of first-row transition elements, it is convenient to include the elements calcium and zinc, as both Ca2+ and Zn2+ have a value of zero, against which the value for other transition metal ions may be compared. Another example occurs in the Irving–Williams series of stability constants of complexes; the recent synthesis of mercury fluoride has been taken by some to reinforce the view that the group 12 elements should be considered transition metals, but some authors still consider this compound to be exceptional. Although meitnerium and roentgenium are within the d-block and are expected to behave as transition metals analogous to their lighter congeners iridium and gold, this has not yet been experimentally confirmed. Early transition metals are on the left side of the periodic table from group 3 to group 7. Late transition metals are on the right side of the d-block, from group 8 to 11; the general electronic configuration of the d-block elements is d1–10n s0–2.
The period 6 and 7 transition metals add f0–14 electrons, which are omitted from the tables below. The Madelung rule predicts that the typical electronic structure of transition metal atoms can be written as ns2dm where the inner d orbital is predicted to be filled after the valence-shell's s orbital is filled; this rule is however only approximate – it only holds for some of the transition elements, only in their neutral ground state. The d-sub-shell is denoted as d - sub-shell; the number of s electrons in the outermost s sub-shell is one or two except palladium, with no electron in that s-sub shell in its ground state. The s-sub-shell in the valence shell is represented as e.g. 4s. In the periodic table, the transition metals are present in eight groups, with some authors including some elements in groups 3 o
Transition metal imido complex
In coordination chemistry and organometallic chemistry, transition metal imido complexes is a coordination compound containing an imido ligand. Imido ligands can be terminal or bridging ligands; the parent imido ligand has the formula NH, but most imido ligands are have alkyl or aryl groups in place of H. The imido ligand is viewed as a dianion, akin to oxide. In some terminal imido complexes, the M=N-C angle is 180º but the angle is decidedly bent. Complexes of the type M=NH are assumed to be intermediates in nitrogen fixation by synthetic catalysts. Imido ligands are observed as doubly and, less triply bridging ligands. Metal-imido complexes are generated from metal oxo complexes, they arise by condensation of amines and metal oxides and metal halides: LnMO + H2NR → LnMNR + H2OThis approach is illustrated by the conversion of MoO2Cl2 to the diimido derivative MoCl22, precursors to the Schrock carbenes of the type Mo2. LnMCl2 + 3 H2NR → LnMNR + 2 RNH3ClAryl isocyanates react with metal oxides concomitant with decarboxylation: LnMO + O=C=NR → LnMNR + CO2 Some are generated from the reaction of low-valence metal complexes with azides: LnM + N3R → LnMNR + N2A few imido complexes have been generated by the alkylation of metal nitride complexes: LnMN- + RX → LnMNR + X- Metal imido complexes are of academic interest.
They are however assumed to be intermediates in ammoxidation catalysis, in the Sharpless oxyamination, in nitrogen fixation
Atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus; the term atomic orbital may refer to the physical region or space where the electron can be calculated to be present, as defined by the particular mathematical form of the orbital. Each orbital in an atom is characterized by a unique set of values of the three quantum numbers n, ℓ, m, which correspond to the electron's energy, angular momentum, an angular momentum vector component; each such orbital can be occupied by a maximum of two electrons, each with its own spin quantum number s. The simple names s orbital, p orbital, d orbital and f orbital refer to orbitals with angular momentum quantum number ℓ = 0, 1, 2 and 3 respectively; these names, together with the value of n, are used to describe the electron configurations of atoms.
They are derived from the description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp, principal and fundamental. Orbitals for ℓ > 3 continue alphabetically, omitting j because some languages do not distinguish between the letters "i" and "j". Atomic orbitals are the basic building blocks of the atomic orbital model, a modern framework for visualizing the submicroscopic behavior of electrons in matter. In this model the electron cloud of a multi-electron atom may be seen as being built up in an electron configuration, a product of simpler hydrogen-like atomic orbitals; the repeating periodicity of the blocks of 2, 6, 10, 14 elements within sections of the periodic table arises from the total number of electrons that occupy a complete set of s, p, d and f atomic orbitals although for higher values of the quantum number n when the atom in question bears a positive charge, the energies of certain sub-shells become similar and so the order in which they are said to be populated by electrons can only be rationalized somewhat arbitrarily.
With the development of quantum mechanics and experimental findings, it was found that the orbiting electrons around a nucleus could not be described as particles, but needed to be explained by the wave-particle duality. In this sense, the electrons have the following properties: Wave-like properties: The electrons do not orbit the nucleus in the manner of a planet orbiting the sun, but instead exist as standing waves, thus the lowest possible energy an electron can take is similar to the fundamental frequency of a wave on a string. Higher energy states are similar to harmonics of that fundamental frequency; the electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron. The charge on the electron acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's wave function. Particle-like properties: The number of electrons orbiting the nucleus can only be an integer.
Electrons jump between orbitals like particles. For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon; the electrons retain particle-like properties such as: each wave state has the same electrical charge as its electron particle. Each wave state has a single discrete spin depending on its superposition. Thus, despite the popular analogy to planets revolving around the Sun, electrons cannot be described as solid particles. In addition, atomic orbitals do not resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and oddly shaped "atmosphere", distributed around a tiny planet. Atomic orbitals describe the shape of this "atmosphere" only when a single electron is present in an atom; when more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection tends toward a spherical zone of probability describing the electron's location, because of the uncertainty principle.
Atomic orbitals may be defined more in formal quantum mechanical language. In quantum mechanics, the state of an atom, i.e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion into linear combinations of anti-symmetrized products of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. A state is a function of the coordinates of all the electrons, so that their motion is correlated, but this is approximated by this independent-particle model of products of single electron wave functions. In atomic physics, the atomic spectral lines correspond to transitions between quantum states of an atom; these states are labeled by a set of quantum numbers summarized in the term symbol and associated with particular electron configurations, i.e. by occupation schemes of atomic orbitals (for example, 1s2
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region; the term orbital was introduced by Robert S. Mulliken in 1932 as an abbreviation for one-electron orbital wave function. At an elementary level, it is used to describe the region of space in which the function has a significant amplitude. Molecular orbitals are constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms, they can be quantitatively calculated using the Hartree -- self-consistent field methods. A molecular orbital can be used to represent the regions in a molecule where an electron occupying that orbital is to be found. Molecular orbitals are obtained from the combination of atomic orbitals, which predict the location of an electron in an atom.
A molecular orbital can specify the electron configuration of a molecule: the spatial distribution and energy of one electron. Most a MO is represented as a linear combination of atomic orbitals in qualitative or approximate usage, they are invaluable in providing a simple model of bonding in molecules, understood through molecular orbital theory. Most present-day methods in computational chemistry begin by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin; this is an approximation, accurate descriptions of the molecular electronic wave function do not have orbitals. Molecular orbitals are, in general, delocalized throughout the entire molecule. Moreover, if the molecule has symmetry elements, its nondegenerate molecular orbitals are either symmetric or antisymmetric with respect to any of these symmetries.
In other words, application of a symmetry operation S to molecular orbital ψ results in the molecular orbital being unchanged or reversing its mathematical sign: Sψ = ±ψ. In planar molecules, for example, molecular orbitals are either symmetric or antisymmetric with respect to reflection in the molecular plane. If molecules with degenerate orbital energies are considered, a more general statement that molecular orbitals form bases for the irreducible representations of the molecule's symmetry group holds; the symmetry properties of molecular orbitals means that delocalization is an inherent feature of molecular orbital theory and makes it fundamentally different from valence bond theory, in which bonds are viewed as localized electron pairs, with allowance for resonance to account for delocalization. In contrast to these symmetry-adapted canonical molecular orbitals, localized molecular orbitals can be formed by applying certain mathematical transformations to the canonical orbitals; the advantage of this approach is that the orbitals will correspond more to the "bonds" of a molecule as depicted by a Lewis structure.
As a disadvantage, the energy levels of these localized orbitals no longer have physical meaning. Molecular orbitals arise from allowed interactions between atomic orbitals, which are allowed if the symmetries of the atomic orbitals are compatible with each other. Efficiency of atomic orbital interactions is determined from the overlap between two atomic orbitals, significant if the atomic orbitals are close in energy; the number of molecular orbitals formed must be equal to the number of atomic orbitals in the atoms being combined to form the molecule. For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the "Linear combination of atomic orbitals molecular orbital method" ansatz. Here, the molecular orbitals are expressed as linear combinations of atomic orbitals. Molecular orbitals were first introduced by Friedrich Hund and Robert S. Mulliken in 1927 and 1928; the linear combination of atomic orbitals or "LCAO" approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones.
His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This qualitative approach to molecular orbital theory is part of the start of modern quantum chemistry. Linear combinations of atomic orbitals can be used to estimate the molecular orbitals that are formed upon bonding between the molecule's constituent atoms. Similar to an atomic orbital, a Schrödinger equation, which describes the behavior of an electron, can be constructed for a molecular orbital as well. Linear combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to the Hartree–Fock equations which correspond to the independent-particle approximation of the molecular Schrödinger equation. For simple diatomic molecules, the wavefunctions obtained are represented mathematically by the equations Ψ = c a ψ a + c b ψ b {\displaystyle \Psi
