Period 6 element

From Wikipedia, the free encyclopedia

A period 6 element is one of the chemical elements in the sixth row (or period) of the periodic table of the chemical elements, including the lanthanides. The periodic table is laid out in rows to illustrate recurring (periodic) trends in the chemical behaviour of the elements as their atomic number increases: a new row is begun when chemical behaviour begins to repeat, meaning that elements with similar behaviour fall into the same vertical columns. The sixth period contains 32 elements, tied for the most with period 7, beginning with caesium and ending with radon. Lead is currently the last stable element; all subsequent elements are radioactive. For bismuth, however, its only primordial isotope, ²⁰⁹Bi, has a half-life of more than 10¹⁹ years, over a billion times longer than the current age of the universe. As a rule, period 6 elements fill their 6s shells first, then their 4f, 5d, and 6p shells, in that order; however, there are exceptions, such as gold.

Properties

This period contains the lanthanides, also known as the rare earths. Many lanthanides are known for their magnetic properties, such as neodymium. Many period 6 transition metals are very valuable, such as gold, however many period 6 other metals are incredibly toxic, such as thallium. Period 6 contains the last stable element, lead. All subsequent elements in the periodic table are radioactive. After bismuth, which has a half-life or more than 10¹⁹ years, polonium, astatine, and radon are some of the shortest-lived and rarest elements known; less than a gram of astatine is estimated to exist on earth at any given time.

Atomic characteristics

Chemical element			Block	Electron configuration
55	Cs	Caesium	s-block	[Xe] 6s1
56	Ba	Barium	s-block	[Xe] 6s2
57	La	Lanthanum	f-block	[Xe] 5d1 6s2
58	Ce	Cerium	f-block	[Xe] 4f1 5d1 6s2
59	Pr	Praseodymium	f-block	[Xe] 4f3 6s2
60	Nd	Neodymium	f-block	[Xe] 4f4 6s2
61	Pm	Promethium	f-block	[Xe] 4f5 6s2
62	Sm	Samarium	f-block	[Xe] 4f6 6s2
63	Eu	Europium	f-block	[Xe] 4f7 6s2
64	Gd	Gadolinium	f-block	[Xe] 4f7 5d1 6s2
65	Tb	Terbium	f-block	[Xe] 4f9 6s2
66	Dy	Dysprosium	f-block	[Xe] 4f10 6s2
67	Ho	Holmium	f-block	[Xe] 4f11 6s2
68	Er	Erbium	f-block	[Xe] 4f12 6s2
69	Tm	Thulium	f-block	[Xe] 4f13 6s2
70	Yb	Ytterbium	f-block	[Xe] 4f14 6s2
71	Lu	Lutetium	d-block	[Xe] 4f14 5d1 6s2
72	Hf	Hafnium	d-block	[Xe] 4f14 5d2 6s2
73	Ta	Tantalum	d-block	[Xe] 4f14 5d3 6s2
74	W	Tungsten	d-block	[Xe] 4f14 5d4 6s2
75	Re	Rhenium	d-block	[Xe] 4f14 5d5 6s2
76	Os	Osmium	d-block	[Xe] 4f14 5d6 6s2
77	Ir	Iridium	d-block	[Xe] 4f14 5d7 6s2
78	Pt	Platinum	d-block	[Xe] 4f14 5d9 6s1
79	Au	Gold	d-block	[Xe] 4f14 5d10 6s1
80	Hg	Mercury	d-block	[Xe] 4f14 5d10 6s2
81	Tl	Thallium	p-block	[Xe] 4f14 5d10 6s2 6p1
82	Pb	Lead	p-block	[Xe] 4f14 5d10 6s2 6p2
83	Bi	Bismuth	p-block	[Xe] 4f14 5d10 6s2 6p3
84	Po	Polonium	p-block	[Xe] 4f14 5d10 6s2 6p4
85	At	Astatine	p-block	[Xe] 4f14 5d10 6s2 6p5
86	Rn	Radon	p-block	[Xe] 4f14 5d10 6s2 6p6

• In many periodic tables, the f-block is erroneously shifted one element to the right, so that lanthanum and actinium become d-block elements, and Ce–Lu and Th–Lr form the f-block tearing the d-block into two very uneven portions. This is a holdover from early erroneous measurements of electron configurations. Lev Landau and Evgeny Lifshitz pointed out in 1948 that lutetium is not an f-block element, and since then physical, chemical, and electronic evidence has overwhelmingly supported that the f-block contains the elements La–Yb and Ac–No, as shown here and as supported by International Union of Pure and Applied Chemistry reports dating from 1988 and 2021.

s-block elements

Caesium

Main article: Caesium

Caesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-gold alkali metal with a melting point of 28 °C (82 °F), which makes it one of only five elemental metals that are liquid at (or near) room temperature. Caesium is an alkali metal and has physical and chemical properties similar to those of rubidium and potassium. The metal is extremely reactive and pyrophoric, reacting with water even at−116 °C (−177 °F). It is the least electronegative element having a stable isotope, caesium-133. Caesium is mined mostly from pollucite, while the radioisotopes, especially caesium-137, a fission product, are extracted from waste produced by nuclear reactors.

Two German chemists, Robert Bunsen and Gustav Kirchhoff, discovered caesium in 1860 by the newly developed method of flame spectroscopy. The first small-scale applications for caesium have been as a "getter" in vacuum tubes and in photoelectric cells. In 1967, a specific frequency from the emission spectrum of caesium-133 was chosen to be used in the definition of the second by the International System of Units. Since then, caesium has been widely used in atomic clocks.

Since the 1990s, the largest application of the element has been as caesium formate for drilling fluids. It has a range of applications in the production of electricity, in electronics, and in chemistry. The radioactive isotope caesium-137 has a half-life of about 30 years and is used in medical applications, industrial gauges, and hydrology. Although the element is only mildly toxic, it is a hazardous material as a metal and its radioisotopes present a high health risk in case of radioactivity releases.

Barium

Main article: Barium

Barium is a chemical element with the symbol Ba and atomic number 56. It is the fifth element in Group 2, a soft silvery metallic alkaline earth metal. Barium is never found in nature in its pure form due to its reactivity with air. Its oxide is historically known as baryta but it reacts with water and carbon dioxide and is not found as a mineral. The most common naturally occurring minerals are the very insoluble barium sulfate, BaSO₄ (barite), and barium carbonate, BaCO₃(witherite). Barium's name originates from Greek barys (βαρύς), meaning "heavy", describing the high density of some common barium-containing ores.

Barium has few industrial applications, but the metal has been historically used to scavenge air in vacuum tubes. Barium compounds impart a green color to flames and have been used in fireworks. Barium sulfate is used for its density, insolubility, and X-ray opacity. It is used as an insoluble heavy additive to oil well drilling mud, and in purer form, as an X-ray radiocontrast agent for imaging the human gastrointestinal tract. Soluble barium compounds are poisonous due to release of the soluble barium ion, and have been used as rodenticides. New uses for barium continue to be sought. It is a component of some "high temperature" YBCOsuperconductors, and electroceramics.

f-block elements (lanthanides)

Main article: Lanthanides

The lanthanide or lanthanoid (IUPAC nomenclature) series comprises the fifteen metallic chemical elements with atomic numbers 57 through 71, from lanthanum through lutetium. These fifteen elements, along with the chemically similar elements scandium and yttrium, are often collectively known as the rare-earth elements.

The informal chemical symbol Ln is used in general discussions of lanthanide chemistry. All but one of the lanthanides are f-block elements, corresponding to the filling of the 4f electron shell; lanthanum, a d-block element, is also generally considered to be a lanthanide due to its chemical similarities with the other fourteen. All lanthanide elements form trivalent cations, Ln³⁺, whose chemistry is largely determined by the ionic radius, which decreases steadily from lanthanum to lutetium.

The lanthanide elements are the group of elements with atomic number increasing from 57 (lanthanum) to 71 (lutetium). They are termed lanthanide because the lighter elements in the series are chemically similar to lanthanum. Strictly speaking, both lanthanum and lutetium have been labeled as group 3 elements, because they both have a single valence electron in the d shell. However, both elements are often included in any general discussion of the chemistry of the lanthanide elements.

In presentations of the periodic table, the lanthanides and the actinides are customarily shown as two additional rows below the main body of the table, with placeholders or else a selected single element of each series (either lanthanum or lutetium, and either actinium or lawrencium, respectively) shown in a single cell of the main table, between barium and hafnium, and radium and rutherfordium, respectively. This convention is entirely a matter of aesthetics and formatting practicality; a rarely used wide-formatted periodic table inserts the lanthanide and actinide series in their proper places, as parts of the table's sixth and seventh rows (periods).

d-block elements

Lutetium

Main article: Lutetium

Lutetium (/ljuːˈtiːʃiəm/ lew-TEE-shee-əm) is a chemical element with the symbol Lu and atomic number 71. It is the last element in the lanthanide series, which, along with the lanthanide contraction, explains several important properties of lutetium, such as it having the highest hardness or density among lanthanides. Unlike other lanthanides, which lie in the f-block of the periodic table, this element lies in the d-block; however, lanthanum is sometimes placed on the d-block lanthanide position. Chemically, lutetium is a typical lanthanide: its only common oxidation state is +3, seen in its oxide, halides and other compounds. In an aqueous solution, like compounds of other late lanthanides, soluble lutetium compounds form a complex with nine water molecules.

Lutetium was independently discovered in 1907 by French scientist Georges Urbain, Austrian mineralogist Baron Carl Auer von Welsbach, and American chemist Charles James. All of these men found lutetium as an impurity in the mineral ytterbia, which was previously thought to consist entirely of ytterbium. The dispute on the priority of the discovery occurred shortly after, with Urbain and von Welsbach accusing each other of publishing results influenced by the published research of the other; the naming honor went to Urbain as he published his results earlier. He chose the name lutecium for the new element but in 1949 the spelling of element 71 was changed to lutetium. In 1909, the priority was finally granted to Urbain and his names were adopted as official ones; however, the name cassiopeium (or later cassiopium) for element 71 proposed by von Welsbach was used by many German scientists until the 1950s. Like other lanthanides, lutetium is one of the elements that traditionally were included in the classification "rare earths."

Lutetium is rare and expensive; consequently, it has few specific uses. For example, a radioactive isotope lutetium-176 is used in nuclear technology to determine the age of meteorites. Lutetium usually occurs in association with the element yttrium and is sometimes used in metal alloys and as a catalyst in various chemical reactions. ¹⁷⁷Lu-DOTA-TATE is used for radionuclide therapy (see Nuclear medicine) on neuroendocrine tumours.

Hafnium

Main article: Hafnium

Hafnium is a chemical element with the symbol Hf and atomic number 72. A lustrous, silvery gray, tetravalent transition metal, hafnium chemically resembles zirconium and is found in zirconium minerals. Its existence was predicted by Dmitri Mendeleev in 1869. Hafnium was the penultimate stable isotope element to be discovered (rhenium was identified two years later). Hafnium is named for Hafnia, the Latin name for "Copenhagen", where it was discovered.

Hafnium is used in filaments and electrodes. Some semiconductor fabrication processes use its oxide for integrated circuits at 45 nm and smaller feature lengths. Some superalloys used for special applications contain hafnium in combination with niobium, titanium, or tungsten.

Hafnium's large neutron capture cross-section makes it a good material for neutron absorption in control rods in nuclear power plants, but at the same time requires that it be removed from the neutron-transparent corrosion-resistant zirconium alloys used in nuclear reactors.

Tantalum

Main article: Tantalum

Tantalum is a chemical element with the symbol Ta and atomic number 73. Previously known as tantalium, the name comes from Tantalus, a character from Greek mythology. Tantalum is a rare, hard, blue-gray, lustrous transition metal that is highly corrosion resistant. It is part of the refractory metals group, which are widely used as minor component in alloys. The chemical inertness of tantalum makes it a valuable substance for laboratory equipment and a substitute for platinum, but its main use today is in tantalum capacitors in electronic equipment such as mobile phones, DVD players, video game systems and computers. Tantalum, always together with the chemically similar niobium, occurs in the minerals tantalite, columbite and coltan (a mix of columbite and tantalite).

Tungsten

Main article: Tungsten

Tungsten, also known as wolfram, is a chemical element with the chemical symbol W and atomic number 74. The word tungsten comes from the Swedish language tung sten directly translatable to heavy stone, though the name is volfram in Swedish to distinguish it from Scheelite, in Swedish alternatively named tungsten.

A hard, rare metal under standard conditions when uncombined, tungsten is found naturally on Earth only in chemical compounds. It was identified as a new element in 1781, and first isolated as a metal in 1783. Its important ores include wolframite and scheelite. The free element is remarkable for its robustness, especially the fact that it has the highest melting point of all the non-alloyed metals and the second highest of all the elements after carbon. Also remarkable is its high density of 19.3 times that of water, comparable to that of uranium and gold, and much higher (about 1.7 times) than that of lead. Tungsten with minor amounts of impurities is often brittle and hard, making it difficult to work. However, very pure tungsten, though still hard, is more ductile, and can be cut with a hard-steel hacksaw.

The unalloyed elemental form is used mainly in electrical applications. Tungsten's many alloys have numerous applications, most notably in incandescent light bulb filaments, X-ray tubes (as both the filament and target), electrodes in TIG welding, and superalloys. Tungsten's hardness and high density give it military applications in penetrating projectiles. Tungsten compounds are most often used industrially as catalysts.

Tungsten is the only metal from the third transition series that is known to occur in biomolecules, where it is used in a few species of bacteria. It is the heaviest element known to be used by any living organism. Tungsten interferes with molybdenum and copper metabolism, and is somewhat toxic to animal life.

Rhenium

Main article: Rhenium

Rhenium is a chemical element with the symbol Re and atomic number 75. It is a silvery-white, heavy, third-row transition metal in group 7 of the periodic table. With an estimated average concentration of 1 part per billion (ppb), rhenium is one of the rarest elements in the Earth's crust. The free element has the third-highest melting point and highest boiling point of any element. Rhenium resembles manganese chemically and is obtained as a by-product of molybdenum and copper ore's extraction and refinement. Rhenium shows in its compounds a wide variety of oxidation states ranging from −1 to +7.

Discovered in 1925, rhenium was the last stable element to be discovered. It was named after the river Rhine in Europe.

Nickel-based superalloys of rhenium are used in the combustion chambers, turbine blades, and exhaust nozzles of jet engines, these alloys contain up to 6% rhenium, making jet engine construction the largest single use for the element, with the chemical industry's catalytic uses being next-most important. Because of the low availability relative to demand, rhenium is among the most expensive of metals, with an average price of approximately US$4,575 per kilogram (US$142.30 per troy ounce) as of August 2011; it is also of critical strategic military importance, for its use in high performance military jet and rocket engines.

Osmium

Main article: Osmium

Osmium is a chemical element with the symbol Os and atomic number 76. It is a hard, brittle, blue-gray or blue-black transition metal in the platinum family and is the densest naturally occurring element, with a density of 22.59 g/cm³ (slightly greater than that of iridium and twice that of lead). It is found in nature as an alloy, mostly in platinum ores; its alloys with platinum, iridium, and other platinum group metals are employed in fountain pen tips, electrical contacts, and other applications where extreme durability and hardness are needed.

Iridium

Main article: Iridium

Iridium is the chemical element with atomic number 77, and is represented by the symbol Ir. A very hard, brittle, silvery-white transition metal of the platinum family, iridium is the second-densest element (after osmium) and is the most corrosion-resistant metal, even at temperatures as high as 2000 °C. Although only certain molten salts and halogens are corrosive to solid iridium, finely divided iridium dust is much more reactive and can be flammable.

Iridium was discovered in 1803 among insoluble impurities in natural platinum. Smithson Tennant, the primary discoverer, named the iridium for the goddess Iris, personification of the rainbow, because of the striking and diverse colors of its salts. Iridium is one of the rarest elements in the Earth's crust, with annual production and consumption of only three tonnes. ¹⁹¹
Ir and ¹⁹³
Ir are the only two naturally occurring isotopes of iridium as well as the only stable isotopes; the latter is the more abundant of the two.

The most important iridium compounds in use are the salts and acids it forms with chlorine, though iridium also forms a number of organometallic compounds used in industrial catalysis, and in research. Iridium metal is employed when high corrosion resistance at high temperatures is needed, as in high-end spark plugs, crucibles for recrystallization of semiconductors at high temperatures, and electrodes for the production of chlorine in the chloralkali process. Iridium radioisotopes are used in some radioisotope thermoelectric generators.

Iridium is found in meteorites with an abundance much higher than its average abundance in the Earth's crust. For this reason the unusually high abundance of iridium in the clay layer at the Cretaceous–Paleogene boundary gave rise to the Alvarez hypothesis that the impact of a massive extraterrestrial object caused the extinction of dinosaurs and many other species 66 million years ago. It is thought that the total amount of iridium in the planet Earth is much higher than that observed in crustal rocks, but as with other platinum group metals, the high density and tendency of iridium to bond with iron caused most iridium to descend below the crust when the planet was young and still molten.

Platinum

Main article: Platinum

Platinum is a chemical element with the chemical symbol Pt and an atomic number of 78.

Its name is derived from the Spanish term platina, which is literally translated into "little silver". It is a dense, malleable, ductile, precious, gray-white transition metal.

Platinum has six naturally occurring isotopes. It is one of the rarest elements in the Earth's crust and has an average abundance of approximately 5 μg/kg. It is the least reactive metal. It occurs in some nickel and copper ores along with some native deposits, mostly in South Africa, which accounts for 80% of the world production.

As a member of the platinum group of elements, as well as of the group 10 of the periodic table of elements, platinum is generally non-reactive. It exhibits a remarkable resistance to corrosion, even at high temperatures, and as such is considered a noble metal. As a result, platinum is often found chemically uncombined as native platinum. Because it occurs naturally in the alluvial sands of various rivers, it was first used by pre-Columbian South American natives to produce artifacts. It was referenced in European writings as early as 16th century, but it was not until Antonio de Ulloa published a report on a new metal of Colombian origin in 1748 that it became investigated by scientists.

Platinum is used in catalytic converters, laboratory equipment, electrical contacts and electrodes, platinum-resistance thermometers, dentistry equipment, and jewelry. Because only a few hundred tonnes are produced annually, it is a scarce material, and is highly valuable. Being a heavy metal, it leads to health issues upon exposure to its salts, but due to its corrosion resistance, it is not as toxic as some metals. Its compounds, most notably cisplatin, are applied in chemotherapy against certain types of cancer.

Gold

Main article: Gold

Gold is a dense, soft, shiny, malleable and ductile metal. It is a chemical element with the symbol Au and atomic number 79.

Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a transition metal and a group 11 element. It is one of the least reactive chemical elements solid under standard conditions. The metal therefore occurs often in free elemental (native) form, as nuggets or grains in rocks, in veins and in alluvial deposits. Less commonly, it occurs in minerals as gold compounds, usually with tellurium.

Gold resists attacks by individual acids, but it can be dissolved by the aqua regia (nitro-hydrochloric acid), so named because it dissolves gold. Gold also dissolves in alkaline solutions of cyanide, which have been used in mining. Gold dissolves in mercury, forming amalgam alloys. Gold is insoluble in nitric acid, which dissolves silver and base metals, a property that has long been used to confirm the presence of gold in items, giving rise to the term the acid test.

Gold has been a valuable and highly sought-after precious metal for coinage, jewelry, and other arts since long before the beginning of recorded history. Gold standards have been a common basis for monetary policies throughout human history, later being supplanted by fiat currency starting in the 1930s. The last gold certificate and gold coin currencies were issued in the U.S. in 1932. In Europe, most countries left the gold standard with the start of World War I in 1914 and, with huge war debts, failed to return to gold as a medium of exchange.

A total of 165,000 tonnes of gold have been mined in human history, as of 2009. This is roughly equivalent to 5.3 billion troy ounces or, in terms of volume, about 8500 m³, or a cube 20.4 m on a side. The world consumption of new gold produced is about 50% in jewelry, 40% in investments, and 10% in industry.

Besides its widespread monetary and symbolic functions, gold has many practical uses in dentistry, electronics, and other fields. Its high malleability, ductility, resistance to corrosion and most other chemical reactions, and conductivity of electricity led to many uses of gold, including electric wiring, colored-glass production and even gold leaf eating.

It has been claimed that most of the Earth's gold lies at its core, the metal's high density having made it sink there in the planet's youth. Virtually all of the gold that mankind has discovered is considered to have been deposited later by meteorites which contained the element. This supposedly explains why, in prehistory, gold appeared as nuggets on the earth's surface.

Mercury

Main article: Mercury (element)

Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum ( < Greek "hydr-" water and "argyros" silver). A heavy, silvery d-block element, mercury is the only metal that is liquid at standard conditions for temperature and pressure; the only other element that is liquid under these conditions is bromine, though metals such as caesium, francium, gallium, and rubidium melt just above room temperature. With a freezing point of −38.83 °C and boiling point of 356.73 °C, mercury has one of the narrowest ranges of its liquid state of any metal.

Mercury occurs in deposits throughout the world mostly as cinnabar (mercuric sulfide). The red pigment vermilion is mostly obtained by reduction from cinnabar. Cinnabar is highly toxic by ingestion or inhalation of the dust. Mercury poisoning can also result from exposure to water-soluble forms of mercury (such as mercuric chloride or methylmercury), inhalation of mercury vapor, or eating seafood contaminated with mercury.

Mercury is used in thermometers, barometers, manometers, sphygmomanometers, float valves, mercury switches, and other devices though concerns about the element's toxicity have led to mercury thermometers and sphygmomanometers being largely phased out in clinical environments in favor of alcohol-filled, galinstan-filled, digital, or thermistor-based instruments. It remains in use in scientific research applications and in amalgam material for dental restoration. It is used in lighting: electricity passed through mercury vapor in a phosphor tube produces short-wave ultraviolet light which then causes the phosphor to fluoresce, making visible light.

p-block elements

Thallium

Main article: Thallium

Thallium is a chemical element with the symbol Tl and atomic number 81. This soft gray other metal resembles tin but discolors when exposed to air. The two chemists William Crookes and Claude-Auguste Lamy discovered thallium independently in 1861 by the newly developed method of flame spectroscopy. Both discovered the new element in residues of sulfuric acid production.

Approximately 60–70% of thallium production is used in the electronics industry, and the remainder is used in the pharmaceutical industry and in glass manufacturing. It is also used in infrared detectors. Thallium is highly toxic and was used in rat poisons and insecticides. Its use has been reduced or eliminated in many countries because of its nonselective toxicity. Because of its use for murder, thallium has gained the nicknames "The Poisoner's Poison" and "Inheritance Powder" (alongside arsenic).

Lead

Main article: Lead

Lead is a main-group element in the carbon group with the symbol Pb (from Latin: plumbum) and atomic number 82. Lead is a soft, malleable other metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed to air. Lead has a shiny chrome-silver luster when it is melted into a liquid.

Lead is used in building construction, lead-acid batteries, bullets and shots, weights, as part of solders, pewters, fusible alloys and as a radiation shield. Lead has the highest atomic number of all of the stable elements, although the next higher element, bismuth, has a half-life that is so long (much longer than the age of the universe) that it can be considered stable. Its four stable isotopes have 82 protons, a magic number in the nuclear shell model of atomic nuclei.

Lead, at certain exposure levels, is a poisonous substance to animals as well as for human beings. It damages the nervous system and causes brain disorders. Excessive lead also causes blood disorders in mammals. Like the element mercury, another heavy metal, lead is a neurotoxin that accumulates both in soft tissues and the bones. Lead poisoning has been documented from ancient Rome, ancient Greece, and ancient China.

Bismuth

Main article: Bismuth

Bismuth is a chemical element with symbol Bi and atomic number 83. Bismuth, a trivalent other metal, chemically resembles arsenic and antimony. Elemental bismuth may occur naturally uncombined, although its sulfide and oxide form important commercial ores. The free element is 86% as dense as lead. It is a brittle metal with a silvery white color when newly made, but often seen in air with a pink tinge owing to the surface oxide. Bismuth metal has been known from ancient times, although until the 18th century it was often confused with lead and tin, which each have some of bismuth's bulk physical properties. The etymology is uncertain but possibly comes from Arabic bi ismid meaning having the properties of antimony or German words weisse masse or wismuth meaning "white mass".

Bismuth is the most naturally diamagnetic of all metals, and only mercury has a lower thermal conductivity.

Bismuth has classically been considered to be the heaviest naturally occurring stable element, in terms of atomic mass. Recently, however, it has been found to be very slightly radioactive: its only primordial isotope bismuth-209 decays via alpha decay into thallium-205 with a half-life of more than a billion times the estimated age of the universe.

Bismuth compounds (accounting for about half the production of bismuth) are used in cosmetics, pigments, and a few pharmaceuticals. Bismuth has unusually low toxicity for a heavy metal. As the toxicity of lead has become more apparent in recent years, alloy uses for bismuth metal (presently about a third of bismuth production), as a replacement for lead, have become an increasing part of bismuth's commercial importance.

Polonium

Main article: Polonium

Polonium is a chemical element with the symbol Po and atomic number 84, discovered in 1898 by Marie Skłodowska-Curie and Pierre Curie. A rare and highly radioactive element, polonium is chemically similar to bismuth and tellurium, and it occurs in uraniumores. Polonium has been studied for possible use in heating spacecraft. As it is unstable, all isotopes of polonium are radioactive. There is disagreement as to whether polonium is a post-transition metal or metalloid.

Astatine

Main article: Astatine

Astatine is a radioactive chemical element with the symbol At and atomic number 85. It occurs on the Earth only as the result of decay of heavier elements, and decays away rapidly, so much less is known about this element than its upper neighbors in the periodic table. Earlier studies have shown this element follows periodic trends, being the heaviest known halogen, with melting and boiling points being higher than those of lighter halogens.

Until recently most of the chemical characteristics of astatine were inferred from comparison with other elements; however, important studies have already been done. The main difference between astatine and iodine is that the HAt molecule is chemically a hydride rather than a halide; however, in a fashion similar to the lighter halogens, it is known to form ionic astatides with metals. Bonds to nonmetals result in positive oxidation states, with +1 best portrayed by monohalides and their derivatives, while the higher are characterized by bond to oxygen and carbon. Attempts to synthesize astatine fluoride have been met with failure. The second longest-living astatine-211 is the only one to find a commercial use, being useful as an alpha emitter in medicine; however, only extremely small quantities are used, and in larger ones it is very hazardous, as it is intensely radioactive.

Astatine was first produced by Dale R. Corson, Kenneth Ross MacKenzie, and Emilio Segrè in the University of California, Berkeley in 1940. Three years later, it was found in nature; however, with an estimated amount of less than 28 grams (1 oz) at given time, astatine is the least abundant element in Earth's crust among non-transuranium elements. Among astatine isotopes, four (with mass numbers 215, 217, 218 and 219) are present in nature as the result of decay of heavier elements; however, the most stable astatine-210 and the industrially used astatine-211 are not.

Radon

Main article: Radon

Radon is a chemical element with symbol Rn and atomic number 86. It is a radioactive, colorless, odorless, tasteless noble gas, occurring naturally as the decay product of uranium or thorium. Its most stable isotope, ²²²Rn, has a half-life of 3.8 days. Radon is one of the densest substances that remains a gas under normal conditions. It is also the only gas that is radioactive under normal conditions, and is considered a health hazard due to its radioactivity. Intense radioactivity also hindered chemical studies of radon and only a few compounds are known.

Radon is formed as part of the normal radioactive decay chain of uranium and thorium. Uranium and thorium have been around since the earth was formed and their most common isotope has a very long half-life (14.05 billion years). Uranium and thorium, radium, and thus radon, will continue to occur for millions of years at about the same concentrations as they do now. As the radioactive gas of radon decays, it produces new radioactive elements called radon daughters or decay products. Radon daughters are solids and stick to surfaces such as dust particles in the air. If contaminated dust is inhaled, these particles can stick to the airways of the lung and increase the risk of developing lung cancer.

Radon is responsible for the majority of the public exposure to ionizing radiation. It is often the single largest contributor to an individual's background radiation dose, and is the most variable from location to location. Radon gas from natural sources can accumulate in buildings, especially in confined areas such as attics and basements. It can also be found in some spring waters and hot springs.

Epidemiological studies have shown a clear link between breathing high concentrations of radon and incidence of lung cancer. Thus, radon is considered a significant contaminant that affects indoor air quality worldwide. According to the United States Environmental Protection Agency, radon is the second most frequent cause of lung cancer, after cigarette smoking, causing 21,000 lung cancer deaths per year in the United States. About 2,900 of these deaths occur among people who have never smoked. While radon is the second most frequent cause of lung cancer, it is the number one cause among non-smokers, according to EPA estimates.

Biological role

Of the period 6 elements, only tungsten is known to have any biological role in organisms. However, gold, platinum, mercury, and some lanthanides such as gadolinium have applications as drugs.

Toxicity

Most of the period 6 elements are toxic (for instance lead) and produce heavy-element poisoning. Promethium, polonium, astatine and radon are radioactive, and therefore present radioactive hazards.

Molecular orbital

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Molecular_orbital

 

Complete acetylene (H–C≡C–H) molecular orbital set. The left column shows MO's which are occupied in the ground state, with the lowest-energy orbital at the top. The white and grey line visible in some MO's is the molecular axis passing through the nuclei. The orbital wave functions are positive in the red regions and negative in the blue. The right column shows virtual MO's which are empty in the ground state, but may be occupied in excited states.

In chemistry, a molecular orbital (/ɒrbədl/) is a mathematical function describing the location and 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 terms atomic orbital and molecular orbital were introduced by Robert S. Mulliken in 1932 to mean one-electron orbital wave functions. At an elementary level, they are used to describe the region of space in which a function has a significant amplitude.

In an isolated atom, the orbital electrons' location is determined by functions called atomic orbitals. When multiple atoms combine chemically into a molecule, the electrons' locations are determined by the molecule as a whole, so the atomic orbitals combine to form molecular orbitals. The electrons from the constituent atoms occupy the molecular orbitals. Mathematically, molecular orbitals are an approximate solution to the Schrödinger equation for the electrons in the field of the molecule's atomic nuclei. They are usually 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–Fock or self-consistent field (SCF) methods.

Molecular orbitals are of three types: bonding orbitals which have an energy lower than the energy of the atomic orbitals which formed them, and thus promote the chemical bonds which hold the molecule together; antibonding orbitals which have an energy higher than the energy of their constituent atomic orbitals, and so oppose the bonding of the molecule, and non-bonding orbitals which have the same energy as their constituent atomic orbitals and thus have no effect on the bonding of the molecule.

Overview

A molecular orbital (MO) can be used to represent the regions in a molecule where an electron occupying that orbital is likely to be found. Molecular orbitals are approximate solutions to the Schrödinger equation for the electrons in the electric field of the molecule's atomic nuclei. However calculating the orbitals directly from this equation is far too intractable a problem. Instead they 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 (or one pair of) electron(s). Most commonly a MO is represented as a linear combination of atomic orbitals (the LCAO-MO method), especially in qualitative or very 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. Necessarily this is an approximation, and highly accurate descriptions of the molecular electronic wave function do not have orbitals (see configuration interaction).

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, the application of a symmetry operation S (e.g., a reflection, rotation, or inversion) 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 (sigma) or antisymmetric (pi) with respect to reflection in the molecular plane. If molecules with degenerate orbital energies are also 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 (and complementary to) 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 closely 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. (The discussion in the rest of this article will focus on canonical molecular orbitals. For further discussions on localized molecular orbitals, see: natural bond orbital and sigma-pi and equivalent-orbital models.)

Formation of molecular orbitals

Molecular orbitals arise from allowed interactions between atomic orbitals, which are allowed if the symmetries (determined from group theory) of the atomic orbitals are compatible with each other. Efficiency of atomic orbital interactions is determined from the overlap (a measure of how well two orbitals constructively interact with one another) between two atomic orbitals, which is significant if the atomic orbitals are close in energy. Finally, the number of molecular orbitals formed must be equal to the number of atomic orbitals in the atoms being combined to form the molecule.

Qualitative discussion

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.

Linear combinations of atomic orbitals (LCAO)

Main article: Linear combination 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 (LCAO) 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

${\displaystyle \mathrm{\Psi }=c_a{\psi }_a+c_b{\psi }_b}$

${\displaystyle {\mathrm{\Psi }}^{\ast }=c_a{\psi }_a-c_b{\psi }_b}$

where ${\displaystyle \mathrm{\Psi }}$ and ${\displaystyle {\mathrm{\Psi }}^{\ast }}$ are the molecular wavefunctions for the bonding and antibonding molecular orbitals, respectively, ${\displaystyle {\psi }_a}$ and ${\displaystyle {\psi }_b}$ are the atomic wavefunctions from atoms a and b, respectively, and ${\displaystyle c_a}$ and ${\displaystyle c_b}$ are adjustable coefficients. These coefficients can be positive or negative, depending on the energies and symmetries of the individual atomic orbitals. As the two atoms become closer together, their atomic orbitals overlap to produce areas of high electron density, and, as a consequence, molecular orbitals are formed between the two atoms. The atoms are held together by the electrostatic attraction between the positively charged nuclei and the negatively charged electrons occupying bonding molecular orbitals.

Bonding, antibonding, and nonbonding MOs

When atomic orbitals interact, the resulting molecular orbital can be of three types: bonding, antibonding, or nonbonding.

Bonding MOs:

• Bonding interactions between atomic orbitals are constructive (in-phase) interactions.
• Bonding MOs are lower in energy than the atomic orbitals that combine to produce them.

Antibonding MOs:

• Antibonding interactions between atomic orbitals are destructive (out-of-phase) interactions, with a nodal plane where the wavefunction of the antibonding orbital is zero between the two interacting atoms
• Antibonding MOs are higher in energy than the atomic orbitals that combine to produce them.

Nonbonding MOs:

• Nonbonding MOs are the result of no interaction between atomic orbitals because of lack of compatible symmetries.
• Nonbonding MOs will have the same energy as the atomic orbitals of one of the atoms in the molecule.

Sigma and pi labels for MOs

The type of interaction between atomic orbitals can be further categorized by the molecular-orbital symmetry labels σ (sigma), π (pi), δ (delta), φ (phi), γ (gamma) etc. These are the Greek letters corresponding to the atomic orbitals s, p, d, f and g respectively. The number of nodal planes containing the internuclear axis between the atoms concerned is zero for σ MOs, one for π, two for δ, three for φ and four for γ.

σ symmetry

Further information: Sigma bond

A MO with σ symmetry results from the interaction of either two atomic s-orbitals or two atomic p_z-orbitals. An MO will have σ-symmetry if the orbital is symmetric with respect to the axis joining the two nuclear centers, the internuclear axis. This means that rotation of the MO about the internuclear axis does not result in a phase change. A σ* orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ* orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis.

π symmetry

Further information: Pi bond

A MO with π symmetry results from the interaction of either two atomic p_x orbitals or p_y orbitals. An MO will have π symmetry if the orbital is asymmetric with respect to rotation about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change. There is one nodal plane containing the internuclear axis, if real orbitals are considered.

A π* orbital, pi antibonding orbital, will also produce a phase change when rotated about the internuclear axis. The π* orbital also has a second nodal plane between the nuclei.

δ symmetry

Further information: Delta bond

A MO with δ symmetry results from the interaction of two atomic d_xy or d_x²-y² orbitals. Because these molecular orbitals involve low-energy d atomic orbitals, they are seen in transition-metal complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, and a δ* antibonding orbital also has a third nodal plane between the nuclei.

φ symmetry

Further information: Phi bond

Suitably aligned f atomic orbitals overlap to form phi molecular orbital (a phi bond)

Theoretical chemists have conjectured that higher-order bonds, such as phi bonds corresponding to overlap of f atomic orbitals, are possible. There is no known example of a molecule purported to contain a phi bond.

Gerade and ungerade symmetry

For molecules that possess a center of inversion (centrosymmetric molecules) there are additional labels of symmetry that can be applied to molecular orbitals. Centrosymmetric molecules include:

• Homonuclear diatomics, X₂
• Octahedral, EX₆
• Square planar, EX₄.

Non-centrosymmetric molecules include:

• Heteronuclear diatomics, XY
• Tetrahedral, EX₄.

If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade (g) symmetry, from the German word for even. If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade (u) symmetry, from the German word for odd. For a bonding MO with σ-symmetry, the orbital is σ_g (s' + s'' is symmetric), while an antibonding MO with σ-symmetry the orbital is σ_u, because inversion of s' – s'' is antisymmetric. For a bonding MO with π-symmetry the orbital is π_u because inversion through the center of symmetry for would produce a sign change (the two p atomic orbitals are in phase with each other but the two lobes have opposite signs), while an antibonding MO with π-symmetry is π_g because inversion through the center of symmetry for would not produce a sign change (the two p orbitals are antisymmetric by phase).

MO diagrams

Main article: Molecular orbital diagram

The qualitative approach of MO analysis uses a molecular orbital diagram to visualize bonding interactions in a molecule. In this type of diagram, the molecular orbitals are represented by horizontal lines; the higher a line the higher the energy of the orbital, and degenerate orbitals are placed on the same level with a space between them. Then, the electrons to be placed in the molecular orbitals are slotted in one by one, keeping in mind the Pauli exclusion principle and Hund's rule of maximum multiplicity (only 2 electrons, having opposite spins, per orbital; place as many unpaired electrons on one energy level as possible before starting to pair them). For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (although is still necessary for a quantitative approach). Some properties:

• A basis set of orbitals includes those atomic orbitals that are available for molecular orbital interactions, which may be bonding or antibonding
• The number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion or the basis set
• If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called symmetry-adapted atomic orbitals (SO)), which belong to the representation of the symmetry group, so the wave functions that describe the group are known as symmetry-adapted linear combinations (SALC).
• The number of molecular orbitals belonging to one group representation is equal to the number of symmetry-adapted atomic orbitals belonging to this representation
• Within a particular representation, the symmetry-adapted atomic orbitals mix more if their atomic energy levels are closer.

The general procedure for constructing a molecular orbital diagram for a reasonably simple molecule can be summarized as follows:

1. Assign a point group to the molecule.

2. Look up the shapes of the SALCs.

3. Arrange the SALCs of each molecular fragment in order of energy, noting first whether they stem from s, p, or d orbitals (and put them in the order s < p < d), and then their number of internuclear nodes.

4. Combine SALCs of the same symmetry type from the two fragments, and from N SALCs form N molecular orbitals.

5. Estimate the relative energies of the molecular orbitals from considerations of overlap and relative energies of the parent orbitals, and draw the levels on a molecular orbital energy level diagram (showing the origin of the orbitals).

6. Confirm, correct, and revise this qualitative order by carrying out a molecular orbital calculation by using commercial software.

Bonding in molecular orbitals

Orbital degeneracy

Main article: Degenerate orbital

Molecular orbitals are said to be degenerate if they have the same energy. For example, in the homonuclear diatomic molecules of the first ten elements, the molecular orbitals derived from the p_x and the p_y atomic orbitals result in two degenerate bonding orbitals (of low energy) and two degenerate antibonding orbitals (of high energy).

Ionic bonds

Main article: Ionic bond

When the energy difference between the atomic orbitals of two atoms is quite large, one atom's orbitals contribute almost entirely to the bonding orbitals, and the other atom's orbitals contribute almost entirely to the antibonding orbitals. Thus, the situation is effectively that one or more electrons have been transferred from one atom to the other. This is called an (mostly) ionic bond.

Bond order

Main article: Bond order

The bond order, or number of bonds, of a molecule can be determined by combining the number of electrons in bonding and antibonding molecular orbitals. A pair of electrons in a bonding orbital creates a bond, whereas a pair of electrons in an antibonding orbital negates a bond. For example, N₂, with eight electrons in bonding orbitals and two electrons in antibonding orbitals, has a bond order of three, which constitutes a triple bond.

Bond strength is proportional to bond order—a greater amount of bonding produces a more stable bond—and bond length is inversely proportional to it—a stronger bond is shorter.

There are rare exceptions to the requirement of molecule having a positive bond order. Although Be₂ has a bond order of 0 according to MO analysis, there is experimental evidence of a highly unstable Be₂ molecule having a bond length of 245 pm and bond energy of 10 kJ/mol.

HOMO and LUMO

Main article: HOMO and LUMO

The highest occupied molecular orbital and lowest unoccupied molecular orbital are often referred to as the HOMO and LUMO, respectively. The difference of the energies of the HOMO and LUMO is called the HOMO-LUMO gap. This notion is often the matter of confusion in literature and should be considered with caution. Its value is usually located between the fundamental gap (difference between ionization potential and electron affinity) and the optical gap. In addition, HOMO-LUMO gap can be related to a bulk material band gap or transport gap, which is usually much smaller than fundamental gap.

Examples

Homonuclear diatomics

Homonuclear diatomic MOs contain equal contributions from each atomic orbital in the basis set. This is shown in the homonuclear diatomic MO diagrams for H₂, He₂, and Li₂, all of which containing symmetric orbitals.

H₂

Electron wavefunctions for the 1s orbital of a lone hydrogen atom (left and right) and the corresponding bonding (bottom) and antibonding (top) molecular orbitals of the H₂ molecule. The real part of the wavefunction is the blue curve, and the imaginary part is the red curve. The red dots mark the locations of the nuclei. The electron wavefunction oscillates according to the Schrödinger wave equation, and orbitals are its standing waves. The standing wave frequency is proportional to the orbital's kinetic energy. (This plot is a one-dimensional slice through the three-dimensional system.)

As a simple MO example, consider the electrons in a hydrogen molecule, H₂ (see molecular orbital diagram), with the two atoms labelled H' and H". The lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule. However, the following symmetry adapted atomic orbitals do:

1s' – 1s"	Antisymmetric combination: negated by reflection, unchanged by other operations
1s' + 1s"	Symmetric combination: unchanged by all symmetry operations

The symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. Because the H₂ molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (hence more stable) than two free hydrogen atoms. This is called a covalent bond. The bond order is equal to the number of bonding electrons minus the number of antibonding electrons, divided by 2. In this example, there are 2 electrons in the bonding orbital and none in the antibonding orbital; the bond order is 1, and there is a single bond between the two hydrogen atoms.

He₂

On the other hand, consider the hypothetical molecule of He₂ with the atoms labeled He' and He". As with H₂, the lowest energy atomic orbitals are the 1s' and 1s", and do not transform according to the symmetries of the molecule, while the symmetry adapted atomic orbitals do. The symmetric combination—the bonding orbital—is lower in energy than the basis orbitals, and the antisymmetric combination—the antibonding orbital—is higher. Unlike H₂, with two valence electrons, He₂ has four in its neutral ground state. Two electrons fill the lower-energy bonding orbital, σ_g(1s), while the remaining two fill the higher-energy antibonding orbital, σ_u*(1s). Thus, the resulting electron density around the molecule does not support the formation of a bond between the two atoms; without a stable bond holding the atoms together, the molecule would not be expected to exist. Another way of looking at it is that there are two bonding electrons and two antibonding electrons; therefore, the bond order is 0 and no bond exists (the molecule has one bound state supported by the Van der Waals potential).

Li₂

Dilithium Li₂ is formed from the overlap of the 1s and 2s atomic orbitals (the basis set) of two Li atoms. Each Li atom contributes three electrons for bonding interactions, and the six electrons fill the three MOs of lowest energy, σ_g(1s), σ_u*(1s), and σ_g(2s). Using the equation for bond order, it is found that dilithium has a bond order of one, a single bond.

Noble gases

Considering a hypothetical molecule of He₂, since the basis set of atomic orbitals is the same as in the case of H₂, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H₂ + 2 He, so the molecule is very unstable and exists only briefly before decomposing into hydrogen and helium. In general, we find that atoms such as He that have full energy shells rarely bond with other atoms. Except for short-lived Van der Waals complexes, there are very few noble gas compounds known.

Heteronuclear diatomics

While MOs for homonuclear diatomic molecules contain equal contributions from each interacting atomic orbital, MOs for heteronuclear diatomics contain different atomic orbital contributions. Orbital interactions to produce bonding or antibonding orbitals in heteronuclear diatomics occur if there is sufficient overlap between atomic orbitals as determined by their symmetries and similarity in orbital energies.

HF

In hydrogen fluoride HF overlap between the H 1s and F 2s orbitals is allowed by symmetry but the difference in energy between the two atomic orbitals prevents them from interacting to create a molecular orbital. Overlap between the H 1s and F 2p_z orbitals is also symmetry allowed, and these two atomic orbitals have a small energy separation. Thus, they interact, leading to creation of σ and σ* MOs and a molecule with a bond order of 1. Since HF is a non-centrosymmetric molecule, the symmetry labels g and u do not apply to its molecular orbitals.

Quantitative approach

To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals that are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree–Fock method, which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of Gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations, which are in fact a particular representation of the Hartree–Fock equation. There are a number of programs in which quantum chemical calculations of MOs can be performed, including Spartan.

Simple accounts often suggest that experimental molecular orbital energies can be obtained by the methods of ultra-violet photoelectron spectroscopy for valence orbitals and X-ray photoelectron spectroscopy for core orbitals. This, however, is incorrect as these experiments measure the ionization energy, the difference in energy between the molecule and one of the ions resulting from the removal of one electron. Ionization energies are linked approximately to orbital energies by Koopmans' theorem. While the agreement between these two values can be close for some molecules, it can be very poor in other cases.