Electron configuration

Electron atomic and molecular orbitals

A simple electron shell diagram oflithium

In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) inatomic or molecular orbitals.^[1] For example, the electron configuration of the neon atom is 1s² 2s² 2p⁶.

Electronic configurations describe electrons as each moving independently in an orbital, in an average field created by all other orbitals. Mathematically, configurations are described by Slater determinants or configuration state functions.

According to the laws of quantum mechanics, for systems with only one electron, an energy is associated with each electron configuration and, upon certain conditions, electrons are able to move from one configuration to another by emission or absorption of a quantum of energy, in the form of a photon.

For atoms or molecules with more than one electron, the motion of electrons are correlated and such picture is no longer exact. An infinite number of electronic configurations are needed to exactly describe any multi-electron system, and no energy can be associated with one single configuration. However, the electronic wave function is usually dominated by a very small number of configurations and therefore the notion of electronic configuration remains essential for multi-electron systems.

Electronic configuration of polyatomic molecules can change without absorption or emission of photon through vibronic couplings.

Knowledge of the electron configuration of different atoms is useful in understanding the structure of the periodic table of elements. The concept is also useful for describing the chemical bonds that hold atoms together. In bulk materials this same idea helps explain the peculiar properties of lasers and semiconductors.

Shells and subshells

	s (ℓ=0)	p (ℓ=1)
	m=0	m=0	m=±1
	s	pz	px	py
n=1
n=2

Electron configuration was first conceived of under the Bohr model of the atom, and it is still common to speak of shells and subshells despite the advances in understanding of thequantum-mechanical nature of electrons.

An electron shell is the set of allowed states electrons may occupy which share the same principal quantum number, n (the number before the letter in the orbital label). An atom's nth electron shell can accommodate 2n² electrons, e.g. the first shell can accommodate 2 electrons, the second shell 8 electrons, and the third shell 18 electrons. The factor of two arises because the allowed states are doubled due to electron spin—each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +1/2 (usually noted by an up-arrow) and one with a spin −1/2 (with a down-arrow).

A subshell is the set of states defined by a common azimuthal quantum number, ℓ, within a shell. The values ℓ = 0, 1, 2, 3 correspond to the s, p, d, and f labels, respectively. The maximum number of electrons which 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, ten electrons in a d subshell and fourteen electrons in an f subshell.

The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics,^[2] in particular the Pauli exclusion principle, which states that no two electrons in the same atom can have the same values of the four quantum numbers.^[3]

Notation

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 atomic orbital labels (e.g. for phosphorus the sequence 1s, 2s, 2p, 3s, 3p) with the number of electrons assigned to each orbital (or set of orbitals sharing the same label) 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, since all but the last few subshells are identical to those of one or another of the noble gases. Phosphorus, for instance, differs from neon (1s² 2s² 2p⁶) only by the presence of a third shell. Thus, the electron configuration of neon is pulled out, and phosphorus is written as follows: [Ne] 3s² 3p³. This convention is useful as it is the electrons in the outermost shell which most determine the chemistry of the element.

The order of writing the orbitals is not completely fixed: some sources group all orbitals with the same value of n together, while other sources (as here) follow the order given by Madelung's rule. Hence the electron configuration of iron can be written as [Ar] 3d⁶ 4s² (keeping the 3d-electrons with the 3s- and 3p-electrons which are implied by the configuration of argon) or as [Ar] 4s² 3d⁶ (following the Aufbau principle, see below).

The superscript 1 for a singly occupied orbital is not compulsory. It is quite common to see the letters of the orbital labels (s, p, d, f) written in an italic or slanting typeface, although the International 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 categorizing spectral lines as "sharp", "principal", "diffuse" and "fundamental" (or "fine"), based on their observed fine structure: their modern usage indicates orbitals with an azimuthal 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.^[4][5]

The electron configurations of molecules are written in a similar way, except that molecular orbital labels are used instead of atomic orbital labels (see below).

Energy — ground state and excited states

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 the ground state. Any other configuration is an excited state.

As an example, the ground state configuration of the sodium atom is 1s²2s²2p⁶3s, as deduced from the Aufbau principle (see below). The first excited state is obtained by promoting a 3s electron to the 3p orbital, to obtain the 1s²2s²2p⁶3p configuration, abbreviated as the 3p level. Atoms can move from one configuration to another by absorbing or emitting energy. In a sodium-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 of valence electrons (such as 3s for sodium) involves energies corresponding to photons of visible or ultraviolet light. The excitation of core electrons is possible, but requires much higher energies generally corresponding to x-ray photons. This would be the case for example to excite a 2p electron to the 3s level and form the excited 1s²2s²2p⁵3s² configuration.

The remainder of this article deals only with the ground-state configuration, often referred to as "the" configuration of an atom or molecule.

History

Niels Bohr was the first to propose (1923) that the periodicity in the properties of the elements might be explained by the electronic structure of the atom.^[6] His proposals were based on the then current Bohr 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).

The following year, E. C. Stoner incorporated Sommerfeld's third quantum number into the description of electron shells, and correctly predicted the shell structure of sulfur to be 2.8.6.^[7] However neither Bohr's system nor Stoner's could correctly describe the changes in atomic spectra in a magnetic field (the Zeeman effect).

Bohr was well aware of this shortcoming (and others), and had written to his friend Wolfgang Pauli to ask for his help in saving quantum theory (the system now known as "old quantum theory"). Pauli realized that the Zeeman effect must be due only to the outermost electrons of the atom, and was able to reproduce Stoner's shell structure, but with the correct structure of subshells, by his inclusion of a fourth quantum number and his exclusion principle (1925):^[8]

It should be forbidden for more than one electron with the same value of the main quantum number n to have the same value for the other three quantum numbers k [l], j [m_l] and m [m_s].

The Schrödinger equation, published in 1926, gave three of the four quantum numbers as a direct consequence of its solution for the hydrogen atom:^[2] this solution yields the atomic orbitals which 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),^[9] see below) for the order in which atomic orbitals are filled with electrons.

Aufbau principle and Madelung rule

The Aufbau principle (from the German Aufbau, "building up, construction") was an important part of Bohr's original concept of electron configuration. It may be stated as:^[10]

a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals 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 orbitals outside the range of the diagram, starting with 8s.)

The principle works very well (for the ground states of the atoms) for the first 18 elements, then decreasingly well for the following 100 elements. The modern form of the Aufbau principle describes an order of orbital energies given by Madelung's rule (or Klechkowski's rule). This rule was first stated by Charles Janet in 1929, rediscovered by Erwin Madelung in 1936,^[9] and later given a theoretical justification by V.M. Klechkowski^[11]

1. Orbitals are filled in the order of increasing n+l;
2. Where two orbitals have the same value of n+l, they are filled in order of increasing n.

This gives the following order for filling the orbitals:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, (8s, 5g, 6f, 7d, 8p, and 9s)

In this list the orbitals in parentheses are not occupied in the ground state of the heaviest atom now known (Uuo, Z = 118).

The Aufbau principle can be applied, in a modified form, to the protons and neutrons in the atomic nucleus, as in the shell model of nuclear physics and nuclear chemistry.

Periodic table

Electron configuration table

The form of the periodic table is closely related to the electron configuration of the atoms of the elements. For example, all the elements of group 2 have an electron configuration of [E] ns² (where [E] is an inert gas configuration), and have notable similarities in their chemical properties. In general, the periodicity of the periodic table in terms of periodic table blocks is clearly due to the number of electrons (2, 6, 10, 14...) needed to fill s, p, d, and f subshells.

The outermost electron shell is often referred to as the "valence shell" and (to a first approximation) determines the chemical properties. It should be remembered that the similarities in the chemical properties were remarked more than a century before the idea of electron configuration.^[12] It is not clear how far Madelung's rule explains(rather than simply describes) the periodic table,^[13] although some properties (such as the common +2 oxidation state in the first row of the transition metals) would obviously be different with a different order of orbital filling.

Shortcomings of the Aufbau principle

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; neither of these is true (although they are approximately true enough for the principle to be useful). 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 which cannot be calculated exactly^[14] (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, the energy levels are slightly split by the magnetic field of the nucleus, and by the quantum electrodynamic effects of the Lamb shift).

Ionization of the transition metals

The naive application of the Aufbau principle leads to a well-known paradox (or apparent paradox) in the basic chemistry of the transition metals. Potassium and calcium 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 has n+l  = 4 (n = 4, l = 0) while the 3d-orbital has n+l  = 5 (n = 3, l = 2). However, chromium and copper 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".

The apparent paradox arises when electrons are removed from the transition metal atoms to form ions. 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. The same is true when chemical compounds are formed. Chromium hexacarbonyl can be described as a chromium atom (not ion, it is in the oxidation state 0) surrounded by six carbon monoxide ligands: it isdiamagnetic, and the electron configuration of the central chromium atom is described as 3d⁶, i.e. the electron which was in the 4s-orbital in the free atom has passed into a 3d-orbital on forming the compound. This interchange of electrons between 4s and 3d is universal among the first series of the transition metals.^[15]

The phenomenon is only paradoxical if it is assumed that the energies of atomic orbitals are fixed and unaffected 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 doesn't. There is no special reason why the Fe²⁺ ion should have the same electron configuration as the chromium atom, given that iron has two more protons in its nucleus than chromium and that the chemistry of the two species is very different. When care is taken to compare "like with like", the paradox disappears.^[16]

Other exceptions to Madelung's rule

There are several more exceptions to Madelung's rule among the heavier elements, and it is more and more difficult to resort to simple explanations such as the stability of half-filled subshells. It is possible to predict most of the exceptions by Hartree–Fock calculations,^[17] which are an approximate method for taking account of the effect of the other electrons on orbital energies. 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 the speed of light. In general, these relativistic effects^[18] tend to decrease the energy of the s-orbitals in relation to the other atomic orbitals.^[19]