In the thematic review dedicated to polyoxometalate (POM) chemistry published in Chemical Reviews in 1998, no contribution was devoted to theory. This is not surprising because computational modelling of molecular metal-oxide clusters was in its infancy at that time. Nowadays, the situation has completely changed and modern computational methods have been successfully applied to study the structure, electronic properties, spectroscopy and reactivity of POM clusters. Indeed, the progress achieved during the past decade has been spectacular and herein we critically review the most important papers to provide the reader with an almost complete perspective of the field.

A series of systematic DFT calculations were conducted on Keggin [SiW(9)M(3)O(40)](n-), M = Mo, V, and Nb; and Wells-Dawson anions [P(2)M(18)O(62)],(6-) M = W and Mo; [P(2)M(15)M(3)'O(62)](m-), M = W and Mo, M' = W, Mo, and V to analyze the redox properties and the basicity of the external oxygen sites in polyoxometalates with nonequivalent addenda metals. The energy and composition of the lowest unoccupied orbitals, formally delocalized over the addenda atoms, determine the redox properties of a polyoxometalate. When a Mo(6+) substitutes one W(6+) in the 1:12 tungstate, the energy of the LUMO decreases and the cluster is more easily reduced. The tungstoniobates behave differently because the niobium orbitals insert into the tungsten band and the reduction of [SiW(9)Nb(3)O(40)](7-) yields the blue species SiW(9)Nb(3) 1e and not the cluster SiW(9)Nb(2)Nb(IV). In Wells-Dawson structures, the polar and equatorial sites have different electron affinities and the reduction preferentially occurs in the equatorial sites. Inserting ions with larger electron affinities into the polar sites can modify this traditional conduct. Hence, the trisubstituted [P(2)W(15)V(3)O(62)](9-) anion is reduced in the vanadium polar sites. By means of molecular electrostatic potential maps and the relative energy of the various protonated forms of [SiW(9)V(3)O(40)](7-) and [SiW(9)Mo(3)O(40)](4-), we established the basicity scale: OV(2) > OMo(2) > OW(2) > OV > OW > OMo. Finally, a continuum model for the solvent enabled us to compare anions with different total charges.

In this review we summarise the quantum chemistry studies carried out by several groups over the last ten years on polyoxometalates, or polyoxoanions. This is an immense family of compounds made up of transition metal ions in their highest oxidation state and oxo ligands. The continuous progress of computers in general, and quantum chemistry software in particular, has enabled a number of topics in polyoxometalate chemistry to be studied from the electronic structure of the most representative polyoxometalate, the so-called Keggin anion, to the factors governing the inclusion complexes and the magnetism in reduced complexes.

Keggin heteropolyanions [XM(12)O(40)](n-) have various isomeric structures, alpha and beta being the most common. Conventionally, the alpha structure appears to be the most stable, but calculations carried out at the DFT level for X = P(V), Si(IV), Al(III), As(V), Ge(IV), and Ga(III) and M = W(VI) and Mo(VI) show that this stability depends on several factors, particularly on the nature of the heteroatom (X) and the total charge of the cluster. In this paper, we apply the clathrate model to the Keggin molecule to carry out a fragment-interaction study to elucidate when and why the traditional relative stability of various isomers can be inverted. The fully oxidized anions that have inverted the traditional stability trend in this series are [AlW(12)O(40)](5-) and [GaW(12)O(40)](5-), both of which contain a third-group heteroatom and an overall charge of -5. beta-isomers are always more easily reduced than alpha-isomers. This experimental observation suggests that reduction favors the stability of beta-isomers and one of the most important results of this study is that the alpha/beta inversion is achieved in most cases after the second reduction. The alpha- and beta-isomers may have different properties because the energy of the LUMO, a symmetry-adapted d(xy)-metal orbital, is different.

Calculations based on density functional theory (DFT) have been carried out to investigate the electronic and magnetic properties of the alpha-Keggin anions mentioned in the title. The atomic populations and the distribution of the electron density computed for the studied clusters support the hypothesis that an oxidized Keggin anion is an XO(4)(n-) clathrate inside a neutral M(12)O(36) cage. The energy gap between the band of occupied orbitals, formally delocalized over the oxo ligands, and the unoccupied d-metal orbitals, delocalized over the addenda, has been found to be independent of the central ion. However, substitution of a W or a Mo by V modifies the relative energy of the LUMO and then induces important changes in the redox properties of the cluster. In agreement with the most recent X-ray determination of [Co(III)W(12)O(40)](5-) and with the simplicity of the (183)W NMR and (17)O NMR spectra observed for this anion the calculations suggest that [Co(III)W(12)O(40)](5-) has a slightly distorted T(d ) geometry. For the parent cluster [CoW(12)O(40)](6-) the quadruplet corresponding to the anion encapsulating a Co(II) was found to be approximately 1 eV more stable than the species formed by a Co(III) and 1 e delocalized over the sphere of tungstens. The one-electron reduction of [Co(II)W(12)O(40)](6-) and [Fe(III)W(12)O(40)](5-) leads to the formation of the 1 e blue species [Co(II)W(12)O(40)](7-) and [Fe(III)W(12)O(40)](6-). The blue-iron cluster is considerably antiferromagnetic, and in full agreement with this behavior the low-spin state computed via a Broken Symmetry approach is 196 cm(-1) lower than the high-spin solution. In contrast, the cobalt blue anion has a low ferromagnetic coupling with an S-T energy gap of +20 cm(-1). This blue species is more stable than the alternative reduction product [Co(I)W(12)O(40)](7-) by more than 0.7 eV.