According to the laws ofquantum mechanics, alevel of energy is associated with each electron configuration. In certain conditions, electrons are able to move from one configuration to another by the emission or absorption of aquantum of energy, in the form of aphoton.
Anelectron shell is theset ofallowed states that share the sameprincipal quantum number,n, that electrons may occupy. In eachterm of an electron configuration,n is thepositive integer that precedes eachorbital letter (e.g.helium's electron configuration is 1s2, thereforen = 1, and the orbital contains two electrons). An atom'snth electron shell can accommodate 2n2 electrons. For example, the first shell can accommodate two electrons, the second shell eight electrons, the third shell eighteen, and so on. The factor of two arises because the number of allowed states doubles with each successive shell due toelectron spin—each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +1⁄2 (usually denoted by an up-arrow) and one with a spin of −1⁄2 (with a down-arrow).
Asubshell is the set of states defined by a commonazimuthal quantum number,l, within a shell. The value ofl is in the range from 0 ton − 1. The valuesl = 0, 1, 2, 3 correspond to the s, p, d, and f labels, respectively. For example, the 3d subshell hasn = 3 andl = 2. The maximum number of electrons that can be placed in a subshell is given by 2(2l + 1). This gives two electrons in an s subshell, six electrons in a p subshell, and ten electrons in a d subshell.
The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics,[a] in particular thePauli exclusion principle, which states that no two electrons in the same atom can have the same values of the fourquantum numbers.[2]
Exhaustive technical details about the complete quantum mechanical theory of atomic spectra and structure can be found and studied in the basic book of Robert D. Cowan.[3]
Physicists and chemists use a standard notation to indicate the electron configurations of atoms and molecules. For atoms, the notation consists of a sequence of atomicsubshell labels (e.g. forphosphorus the sequence 1s, 2s, 2p, 3s, 3p) with the number of electrons assigned to each subshell placed as a superscript. For example,hydrogen has one electron in the s-orbital of the first shell, so its configuration is written 1s1.Lithium has two electrons in the 1s-subshell and one in the (higher-energy) 2s-subshell, so its configuration is written 1s2 2s1 (pronounced "one-s-two, two-s-one").Phosphorus (atomic number 15) is as follows: 1s2 2s2 2p6 3s2 3p3.
For atoms with many electrons, this notation can become lengthy and so an abbreviated notation is used. The electron configuration can be visualized as thecore electrons, equivalent to thenoble gas of the precedingperiod, and thevalence electrons: each element in a period differs only by the last few subshells. Phosphorus, for instance, is in the third period. It differs from the second-periodneon, whose configuration is 1s2 2s2 2p6, only by the presence of a third shell. The portion of its configuration that is equivalent to neon is abbreviated as [Ne], allowing the configuration of phosphorus to be written as [Ne] 3s2 3p3 rather than writing out the details of the configuration of neon explicitly. This convention is useful as it is the electrons in the outermost shell that most determine the chemistry of the element.
For a given configuration, the order of writing the orbitals is not completely fixed since only the orbital occupancies have physical significance. For example, the electron configuration of thetitanium ground state can be written as either [Ar] 4s2 3d2 or [Ar] 3d2 4s2. The first notation follows the order based on theMadelung rule for the configurations of neutral atoms; 4s is filled before 3d in the sequence Ar, K, Ca, Sc, Ti. The second notation groups all orbitals with the same value ofn together, corresponding to the "spectroscopic" order of orbital energies that is the reverse of the order in which electrons are removed from a given atom to form positive ions; 3d is filled before 4s in the sequence Ti4+, Ti3+, Ti2+, Ti+, Ti.
The superscript 1 for a singly occupied subshell is not compulsory; for examplealuminium may be written as either [Ne] 3s2 3p1 or [Ne] 3s2 3p. In atoms where a subshell is unoccupied despite higher subshells being occupied (as is the case in some ions, as well as certain neutral atoms shown to deviate from theMadelung rule), the empty subshell is either denoted with a superscript 0 or left out altogether. For example, neutralpalladium may be written as either[Kr] 4d10 5s0 or simply[Kr] 4d10, and thelanthanum(III) ion may be written as either[Xe] 4f0 or simply [Xe].[4]
It is quite common to see the letters of the orbital labels (s, p, d, f) written in an italic or slanting typeface, although theInternational Union of Pure and Applied Chemistry (IUPAC) recommends a normal typeface (as used here). The choice of letters originates from a now-obsolete system of categorizingspectral lines as "sharp", "principal", "diffuse" and "fundamental" (or "fine"), based on their observedfine structure: their modern usage indicates orbitals with anazimuthal quantum number,l, of 0, 1, 2 or 3 respectively. After f, the sequence continues alphabetically g, h, i... (l = 4, 5, 6...), skipping j, although orbitals of these types are rarely required.[5][6]
The electron configurations of molecules are written in a similar way, except thatmolecular orbital labels are used instead of atomic orbital labels (see below).
The energy associated to an electron is that of its orbital. The energy of a configuration is often approximated as the sum of the energy of each electron, neglecting the electron-electron interactions. The configuration that corresponds to the lowest electronic energy is called theground state. Any other configuration is anexcited state.
As an example, the ground state configuration of thesodium atom is 1s2 2s2 2p6 3s1, as deduced from the Aufbau principle (see below). The first excited state is obtained by promoting a 3s electron to the 3p subshell, to obtain the1s2 2s2 2p6 3p1 configuration, abbreviated as the 3p level. Atoms can move from one configuration to another by absorbing or emitting energy. In asodium-vapor lamp for example, sodium atoms are excited to the 3p level by an electrical discharge, and return to the ground state by emitting yellow light of wavelength 589 nm.
Usually, the excitation ofvalence electrons (such as 3s for sodium) involves energies corresponding tophotons of visible orultraviolet light. The excitation ofcore electrons is possible, but requires much higher energies, generally corresponding toX-ray photons. This would be the case for example to excite a 2p electron of sodium to the 3s level and form the excited 1s2 2s2 2p5 3s2 configuration.
The remainder of this article deals only with the ground-state configuration, often referred to as "the" configuration of an atom or molecule.
Irving Langmuir was the first to propose in his 1919 article "The Arrangement of Electrons in Atoms and Molecules" in which, building onGilbert N. Lewis'scubical atom theory andWalther Kossel's chemical bonding theory, he outlined his "concentric theory of atomic structure".[7] Langmuir had developed his work on electron atomic structure from other chemists as is shown in the development of theHistory of the periodic table and theOctet rule.
Niels Bohr (1923) incorporated Langmuir's model that theperiodicity in the properties of the elements might be explained by the electronic structure of the atom.[8] His proposals were based on the then currentBohr model of the atom, in which the electron shells were orbits at a fixed distance from the nucleus. Bohr's original configurations would seem strange to a present-day chemist:sulfur was given as 2.4.4.6 instead of 1s2 2s2 2p6 3s2 3p4 (2.8.6). Bohr used 4 and 6 followingAlfred Werner's 1893 paper. In fact, the chemists accepted the concept of atoms long before the physicists. Langmuir began his paper referenced above by saying,
«…The problem of the structure of atoms has been attacked mainly by physicists who have given little consideration to the chemical properties which must ultimately be explained by a theory of atomic structure. The vast store of knowledge of chemical properties and relationships, such as is summarized by the Periodic Table, should serve as a better foundation for a theory of atomic structure than the relatively meager experimental data along purely physical lines... These electrons arrange themselves in a series of concentric shells, the first shell containing two electrons, while all other shells tend tohold eight.…»
The valence electrons in the atom were described byRichard Abegg in 1904.[9]
In 1924,E. C. Stoner incorporatedSommerfeld's third quantum number into the description of electron shells, and correctly predicted the shell structure of sulfur to be 2.8.6.[10] However neither Bohr's system nor Stoner's could correctly describe the changes inatomic spectra in amagnetic field (theZeeman effect).
Bohr was well aware of this shortcoming (and others), and had written to his friendWolfgang Pauli in 1923 to ask for his help in saving quantum theory (the system now known as "old quantum theory"). Pauli hypothesized successfully that the Zeeman effect can be explained as depending only on the response of the outermost (i.e., valence) electrons of the atom. Pauli was able to reproduce Stoner's shell structure, but with the correct structure of subshells, by his inclusion of a fourth quantum number and hisexclusion principle (1925):[11]
It should be forbidden for more than one electron with the same value of the main quantum numbern to have the same value for the other three quantum numbersk [l],j [ml] andm [ms].
TheSchrödinger equation, published in 1926, gave three of the four quantum numbers as a direct consequence of its solution for the hydrogen atom:[a] this solution yields the atomic orbitals that are shown today in textbooks of chemistry (and above). The examination of atomic spectra allowed the electron configurations of atoms to be determined experimentally, and led to an empirical rule (known as Madelung's rule (1936),[12] see below) for the order in which atomic orbitals are filled with electrons.
Theaufbau principle (from theGermanAufbau, "building up, construction") was an important part ofBohr's original concept of electron configuration. It may be stated as:[13]
a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy subshells are filled before electrons are placed in higher-energy orbitals.
The approximate order of filling of atomic orbitals, following the arrows from 1s to 7p. (After 7p the order includes subshells outside the range of the diagram, starting with 8s.)
The form of theperiodic table is closely related to the atomic electron configuration for each element. For example, all the elements ofgroup 2 (the table's second column) have an electron configuration of [E] ns2 (where [E] is anoble gas configuration), and have notable similarities in their chemical properties. The periodicity of the periodic table in terms ofperiodic table blocks is due to the number of electrons (2, 6, 10, and 14) needed to fill s, p, d, and f subshells. These blocks appear as the rectangular sections of the periodic table. The single exception ishelium, which despite being an s-block atom is conventionally placed with the othernoble gasses in the p-block due to its chemical inertness, a consequence of its full outer shell (though there is discussion in the contemporary literature on whether this exception should be retained).
The electrons in thevalence (outermost) shell largely determine each element'schemical properties. The similarities in the chemical properties were remarked on more than a century before the idea of electron configuration.[b]
The aufbau principle rests on a fundamental postulate that the order of orbital energies is fixed, both for a given element and between different elements; in both cases this is only approximately true. It considers atomic orbitals as "boxes" of fixed energy into which can be placed two electrons and no more. However, the energy of an electron "in" an atomic orbital depends on the energies of all the other electrons of the atom (or ion, or molecule, etc.). There are no "one-electron solutions" for systems of more than one electron, only a set of many-electron solutions that cannot be calculated exactly[c] (although there are mathematical approximations available, such as theHartree–Fock method).
The fact that the aufbau principle is based on an approximation can be seen from the fact that there is an almost-fixed filling order at all, that, within a given shell, the s-orbital is always filled before the p-orbitals. In ahydrogen-like atom, which only has one electron, the s-orbital and the p-orbitals of the same shell have exactly the same energy, to a very good approximation in the absence of external electromagnetic fields. (However, in a real hydrogen atom, theenergy levels are slightly split by the magnetic field of the nucleus, and by thequantum electrodynamic effects of theLamb shift.)
The naïve application of the aufbau principle leads to a well-knownparadox (or apparent paradox) in the basic chemistry of thetransition metals.Potassium andcalcium appear in the periodic table before the transition metals, and have electron configurations [Ar] 4s1 and [Ar] 4s2 respectively, i.e. the 4s-orbital is filled before the 3d-orbital. This is in line with Madelung's rule, as the 4s-orbital hasn + l = 4 (n = 4,l = 0) while the 3d-orbital hasn + l = 5 (n = 3,l = 2). After calcium, most neutral atoms in the first series of transition metals (scandium throughzinc) have configurations with two 4s electrons, but there are two exceptions.Chromium andcopper have electron configurations [Ar] 3d5 4s1 and [Ar] 3d10 4s1 respectively, i.e. one electron has passed from the 4s-orbital to a 3d-orbital to generate a half-filled or filled subshell. In this case, the usual explanation is that "half-filled or completely filled subshells are particularly stable arrangements of electrons". However, this is not supported by the facts, astungsten (W) has a Madelung-following d4 s2 configuration and not d5 s1, andniobium (Nb) has an anomalous d4 s1 configuration that does not give it a half-filled or completely filled subshell.[15]
The apparent paradox arises when electrons areremoved from the transition metal atoms to formions. The first electrons to be ionized come not from the 3d-orbital, as one would expect if it were "higher in energy", but from the 4s-orbital. This interchange of electrons between 4s and 3d is found for all atoms of the first series of transition metals.[d] The configurations of the neutral atoms (K, Ca, Sc, Ti, V, Cr, ...) usually follow the order 1s, 2s, 2p, 3s, 3p, 4s, 3d, ...; however the successive stages of ionization of a given atom (such as Fe4+, Fe3+, Fe2+, Fe+, Fe) usually follow the order 1s, 2s, 2p, 3s, 3p, 3d, 4s, ...
This phenomenon is only paradoxical if it is assumed that the energy order of atomic orbitals is fixed and unaffected by the nuclear charge or by the presence of electrons in other orbitals. If that were the case, the 3d-orbital would have the same energy as the 3p-orbital, as it does in hydrogen, yet it clearly does not. There is no special reason why the Fe2+ ion should have the same electron configuration as the chromium atom, given thatiron has two more protons in its nucleus than chromium, and that the chemistry of the two species is very different. Melrose andEric Scerri have analyzed the changes of orbital energy with orbital occupations in terms of the two-electron repulsion integrals of theHartree–Fock method of atomic structure calculation.[16] More recently Scerri has argued that contrary to what is stated in the vast majority of sources including the title of his previous article on the subject, 3d orbitals rather than 4s are in fact preferentially occupied.[17]
In chemical environments, configurations can change even more: Th3+ as a bare ion has a configuration of [Rn] 5f1, yet in most ThIII compounds the thorium atom has a 6d1 configuration instead.[18][19] Mostly, what is present is rather a superposition of various configurations.[15] For instance, copper metal is poorly described by either an [Ar] 3d10 4s1 or an [Ar] 3d9 4s2 configuration, but is rather well described as a 90% contribution of the first and a 10% contribution of the second. Indeed, visible light is already enough to excite electrons in most transition metals, and they often continuously "flow" through different configurations when that happens (copper and its group are an exception).[20]
Similar ion-like 3dx 4s0 configurations occur intransition metal complexes as described by the simplecrystal field theory, even if the metal hasoxidation state 0. For example,chromium hexacarbonyl can be described as a chromium atom (not ion) surrounded by sixcarbon monoxideligands. The electron configuration of the central chromium atom is described as 3d6 with the six electrons filling the three lower-energy d orbitals between the ligands. The other two d orbitals are at higher energy due to the crystal field of the ligands. This picture is consistent with the experimental fact that the complex isdiamagnetic, meaning that it has no unpaired electrons. However, in a more accurate description usingmolecular orbital theory, the d-like orbitals occupied by the six electrons are no longer identical with the d orbitals of the free atom.
There are several more exceptions toMadelung's rule among the heavier elements, and as atomic number increases it becomes more and more difficult to find simple explanations such as the stability of half-filled subshells. It is possible to predict most of the exceptions by Hartree–Fock calculations,[21] which are an approximate method for taking account of the effect of the other electrons on orbital energies. Qualitatively, for example, the 4d elements have the greatest concentration of Madelung anomalies, because the 4d–5s gap is larger than the 3d–4s and 5d–6s gaps.[22]
For the heavier elements, it is also necessary to take account of theeffects of special relativity on the energies of the atomic orbitals, as the inner-shell electrons are moving at speeds approaching thespeed of light. In general, these relativistic effects[23] tend to decrease the energy of the s-orbitals in relation to the other atomic orbitals.[24] This is the reason why the 6d elements are predicted to have no Madelung anomalies apart from lawrencium (for which relativistic effects stabilise the p1/2 orbital as well and cause its occupancy in the ground state), as relativity intervenes to make the 7s orbitals lower in energy than the 6d ones.
The table below shows the configurations of the f-block (green) and d-block (blue) atoms. It shows the ground state configuration in terms of orbital occupancy, but it does not show the ground state in terms of the sequence of orbital energies as determined spectroscopically. For example, in the transition metals, the 4s orbital is of a higher energy than the 3d orbitals; and in the lanthanides, the 6s is higher than the 4f and 5d. The ground states can be seen in theElectron configurations of the elements (data page). However this also depends on the charge: acalcium atom has 4s lower in energy than 3d, but a Ca2+ cation has 3d lower in energy than 4s. In practice the configurations predicted by the Madelung rule are at least close to the ground state even in these anomalous cases.[25] The empty f orbitals in lanthanum, actinium, and thorium contribute to chemical bonding,[26][27] as do the empty p orbitals in transition metals.[28]
Vacant s, d, and f orbitals have been shown explicitly, as is occasionally done,[29] to emphasise the filling order and to clarify that even orbitals unoccupied in the ground state (e.g.lanthanum 4f orpalladium 5s) may be occupied and bonding in chemical compounds. (The same is also true for the p-orbitals, which are not explicitly shown because they are only actually occupied for lawrencium in gas-phase ground states.)
Electron shells filled in violation of Madelung's rule[30] (red) Predictions for elements 109–112[31]
The various anomalies describe the free atoms and do not necessarily predict chemical behavior. Thus for example neodymium typically forms the +3 oxidation state, despite its configuration[Xe] 4f4 5d0 6s2 that if interpreted naïvely would suggest a more stable +2 oxidation state corresponding to losing only the 6s electrons. Contrariwise, uranium as[Rn] 5f3 6d1 7s2 is not very stable in the +3 oxidation state either, preferring +4 and +6.[33]
The electron-shell configuration of elements beyondhassium has not yet been empirically verified, but they are expected to follow Madelung's rule without exceptions untilelement 120.Element 121 should have the anomalous configuration[Og] 8s25g0 6f0 7d08p1, having a p rather than a g electron. Electron configurations beyond this are tentative and predictions differ between models,[34] but Madelung's rule is expected to break down due to the closeness in energy of the 5g, 6f, 7d, and 8p1/2 orbitals.[31] That said, the filling sequence 8s, 5g, 6f, 7d, 8p is predicted to hold approximately, with perturbations due to the huge spin-orbit splitting of the 8p and 9p shells, and the huge relativistic stabilisation of the 9s shell.[35]
Every system has the tendency to acquire the state of stability or a state of minimum energy, and so chemical elements take part in chemical reactions to acquire a stable electronic configuration similar to that of its nearestnoble gas. An example of this tendency is twohydrogen (H) atoms reacting with oneoxygen (O) atom to formwater (H2O). Neutral atomic hydrogen has one electron in itsvalence shell, and on formation of water it acquires a share of a second electron coming from oxygen, so that its configuration is similar to that of its nearest noble gashelium (He) with two electrons in its valence shell. Similarly, neutral atomic oxygen has six electrons in its valence shell, and acquires a share of two electrons from the two hydrogen atoms, so that its configuration is similar to that of its nearest noble gasneon with eight electrons in its valence shell.
In asolid, the electron states become very numerous. They cease to be discrete, and effectively blend into continuous ranges of possible states (anelectron band). The notion of electron configuration ceases to be relevant, and yields toband theory.
The most widespread application of electron configurations is in the rationalization ofchemical properties, in bothinorganic andorganic chemistry. In effect, electron configurations, along with some simplified forms ofmolecular orbital theory, have become the modern equivalent of thevalence concept, describing the number and type ofchemical bonds that anatom can be expected to form.
Foratoms ormolecules with more than oneelectron, the motion of electrons arecorrelated and such a picture is no longer exact. A very large number of electronic configurations are needed to exactly describe any multi-electron system, and precisely associating a certain energy level with any single configuration is not possible. However, the electronicwave function is usually dominated by a very small number of configurations and therefore the notion of electronic configuration remains essential for multi-electron systems.
A fundamental application of electron configurations is in the interpretation ofatomic spectra. In this case, it is necessary to supplement the electron configuration with one or moreterm symbols, which describe the differentenergy levels available to an atom. Term symbols can be calculated for any electron configuration, not just theground-state configuration listed in tables, although not all the energy levels are observed in practice. It is through the analysis of atomic spectra that the ground-state electron configurations of the elements were experimentally determined.
^Electrons areidentical particles, a fact that is sometimes referred to as "indistinguishability of electrons". A one-electron solution to a many-electron system would imply that the electrons could be distinguished from one another, and there is strong experimental evidence that they can't be. The exact solution of a many-electron system is an-body problem withn ≥ 3 (the nucleus counts as one of the "bodies"): such problems have evadedanalytical solution since at least the time ofEuler.
^There are some cases in the second and third series where the electron remains in an s-orbital.
^The labels are written in lowercase to indicate that they correspond to one-electron functions. They are numbered consecutively for each symmetry type (irreducible representation in thecharacter table of thepoint group for the molecule), starting from the orbital of lowest energy for that type.
^Xu, Wei; Ji, Wen-Xin; Qiu, Yi-Xiang; Schwarz, W. H. Eugen; Wang, Shu-Guang (2013). "On structure and bonding of lanthanoid trifluorides LnF3 (Ln = La to Lu)".Physical Chemistry Chemical Physics.2013 (15):7839–47.Bibcode:2013PCCP...15.7839X.doi:10.1039/C3CP50717C.PMID23598823.
^Miessler, G. L.; Tarr, D. A. (1999).Inorganic Chemistry (2nd ed.). Prentice-Hall. p. 38.
^abHoffman, Darleane C.; Lee, Diana M.; Pershina, Valeria (2006). "Transactinides and the future elements". In Morss; Edelstein, Norman M.; Fuger, Jean (eds.).The Chemistry of the Actinide and Transactinide Elements (3rd ed.). Dordrecht, The Netherlands:Springer Science+Business Media.ISBN978-1-4020-3555-5.
^Jørgensen, Christian K. (1988). "Influence of rare earths on chemical understanding and classification".Handbook on the Physics and Chemistry of Rare Earths. Vol. 11. pp. 197–292.doi:10.1016/S0168-1273(88)11007-6.ISBN978-0-444-87080-3.
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