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: :
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 principle works very well (for the ground states of the atoms) for the known 118 elements, although it is sometimes slightly wrong. 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, •
Subshells are filled in the order of increasing
n + . • Where two subshells have the same value of
n + , 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, , 6f, 7d, 8p, and 9s) In this list the subshells in parentheses are not occupied in the ground state of the heaviest atom now known (
Og,
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 . The form of the
periodic table is closely related to the atomic electron configuration for each element. For example, all the elements of
group 2 (the table's second column) have an electron configuration of [E]
ns (where [E] is a
noble gas configuration), and have notable similarities in their chemical properties. The periodicity of the periodic table in terms of
periodic 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 is
helium, which despite being an s-block atom is conventionally placed with the other
noble 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 the
valence (outermost) shell largely determine each element's
chemical properties. The similarities in the chemical properties were remarked on more than a century before the idea of electron configuration.
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; 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 (although there are mathematical approximations available, such as the
Hartree–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 a
hydrogen-like atom, which only has one electron, calculations indicate that the s-orbital and the p-orbitals of the same shell have exactly the same energy, which in reality is 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 naïve 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] 4s and [Ar] 4s 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 + = 4 (
n = 4, = 0) while the 3d-orbital has
n + = 5 (
n = 3, = 2). After calcium, most neutral atoms in the first series of transition metals (
scandium through
zinc) have configurations with two 4s electrons, but there are two exceptions.
Chromium and
copper have electron configurations [Ar] 3d 4s and [Ar] 3d 4s 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, as
tungsten (W) has a Madelung-following d s configuration and not d s, and
niobium (Nb) has an anomalous d s configuration that does not give it a half-filled or completely filled subshell. 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. This interchange of electrons between 4s and 3d is found for all atoms of the first series of transition metals. 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 Fe 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. Melrose and
Eric Scerri have analyzed the changes of orbital energy with orbital occupations in terms of the two-electron repulsion integrals of the
Hartree–Fock method of atomic structure calculation. 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. 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. Mostly, what is present is rather a superposition of various configurations. Similar ion-like 3d 4s configurations occur in
transition metal complexes as described by the simple
crystal field theory, even if the metal has
oxidation state 0. For example,
chromium hexacarbonyl can be described as a chromium atom (not ion) surrounded by six
carbon monoxide ligands. The electron configuration of the central chromium atom is described as 3d 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 is
diamagnetic, meaning that it has no unpaired electrons. However, in a more accurate description using
molecular orbital theory, the d-like orbitals occupied by the six electrons are no longer identical with the d orbitals of the free atom.
Other exceptions to Madelung's rule There are several more exceptions to
Madelung'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, 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. For the heavier elements, it is also necessary to take account of the
effects 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 tend to decrease the energy of the s-orbitals in relation to the other atomic orbitals. 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 the
Electron configurations of the elements (data page). However this also depends on the charge: a
calcium 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. The empty f orbitals in lanthanum, actinium, and thorium contribute to chemical bonding, as do the empty p orbitals in transition metals. Vacant s, d, and f orbitals have been shown explicitly, as is occasionally done, to emphasise the filling order and to clarify that even orbitals unoccupied in the ground state (e.g.
lanthanum 4f or
palladium 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.) 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 that if interpreted naïvely would suggest a more stable +2 oxidation state corresponding to losing only the 6s electrons. Contrariwise, uranium as is not very stable in the +3 oxidation state either, preferring +4 and +6. The electron-shell configuration of elements beyond
hassium has not yet been empirically verified, but they are expected to follow Madelung's rule without exceptions until
element 120.
Element 121 should have the anomalous configuration , having a p rather than a g electron. Electron configurations beyond this are tentative and predictions differ between models, but Madelung's rule is expected to break down due to the closeness in energy of the , 6f, 7d, and 8p1/2 orbitals. That said, the filling sequence 8s, , 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. == Open and closed shells ==