Counting schemes

Soon after this discovery the harnessing of the technique to the measurement of all the U isotopes and all the Th isotopes with great precision immediately opened up the entire field of uranium and thorium decay chain studies. This area of study was formerly the poaching ground for radioactive measurements alone but now became part of the wonderful world of mass spectrometric measurements. (The same transformation took place for radiocarbon from the various radioactive counting schemes to accelerator mass spectrometry.)... 

The many higher boranes such as B5H9 and BgH 2 are similarly electron deficient and cannot be described by a single Lewis structure. They can often be described in terms of a combination of two- and three-center bonds. Alternatively, their structures can be rationalized by electron-counting schemes such as those proposed by Wade. Analysis of the electron density of these molecules by the AIM method shows that there are bond paths between all adjacent pairs of atoms. So from the point of view of the AIM theory there are bonds between each adjacent pair of atoms, but these cannot all be regarded as Lewis two-center, two-electron bonds as is the case in B2H6. 

When going beyond the lowest-order term in derivatives, we need a counting scheme. For theories with only one relevant scale (such as QCD at zero chemical potential), each derivative is suppressed by a factor of This is not the case for theories with multiple scales. In the CFL phase, we have both and the gap, A, and the general form of the chiral expansion is [31] ... 

The alloy /J NiAl is a solid with a CsCl-type structure in which one atom is located at the corners, and the second atom at the center of the unit cell. The valence-electron to atom ratio is often quoted as 1.5, using a counting scheme in which the transition metal has zero valence and the A1 is considered as trivalent. 

The electron-counting scheme which allows to understand this situation and which also works for the simpler clusters was developed from the MO theory for boron hydride polyhedra 389-393). It rests on the fact that a regular triangular faced polyhedron of n boron atoms in the molecule requires n + 1 bonding... 

In order to make transition metal clusters comparable to borane polyhedra, the following counting scheme is adopted (1) each metal atom uses one orbital for each... 

A transition element has 5 additional valence orbitals, the 5d orbitals, and therefore 10 additional electrons are required per atom to fill the valence shell of each metal atom. A closo cluster consisting only of transition metal atoms should have a total of 14/i + 2 valence electrons. A capped cluster should have 14n, a nido cluster 14/i + 4, and an arachno cluster 14n+6. The combined formula 4/i+2 + 10m would represent the total electron count for a closo cluster, A mMm, of n atoms that contains m transition metal atoms and n -m main group atoms.Table 8.2 summarizes the main rules, and the following examples show how the total electron counting scheme is applied. 

In both [Fe4X4(N0)4](X = S or Se) and [Fe4S2(NO)4(NCMe3)2] the total valence electron count is 60. This is the number characteristic of tetrahedral tetranuclear metal clusters, such as [Ir4(CO)12], in the Wade and Mingos skeletal-electron counting schemes (76, 77) and, furthermore, each iron atom in these clusters obeys the 18-electron rule, provided that it forms single Fe-Fe bonds to each of the other iron atoms in the tetrahedron. 

A variety of irradiation, decay, and counting schemes can be employed by NAA in order to enhance the sensitivities of different subsets of elements. For example, short irradiation times followed short decay times are used to measure the short-lived elements, and long irradiation times followed by long decay times are used to measure the long-lived elements. For obsidian, the short-lived... 

A simplified electron-counting scheme (10) is also introduced in this module, providing a guide to likely products according to the 16- to 18-electron... 

Electron-counting scheme for macropolyhedral boranes mno rule... 

A generalized electron-counting scheme, known as the mno rule, is applicable to a wide range of polycondensed polyhedral boranes and heteroboranes, metal -laboranes, metallocenes, and any of their combinations. According to this mno rule, the number of electron pairs N necessary for a macropolyhedral system to be stable is... 

Allyl ligands have features common to metal-alkyl and metal-olefin complexes, and can act as three-electron donor ligands according to the electron counting scheme where total covalency is assumed. The metal ion in these complexes interacts with all three carbon atoms of the allyl functionality in an equivalent manner. 

For polyhedral clusters (sometimes called deltahedral, because the faces are all triangles resembling the Greek letter delta) the ancestor of all electron counting schemes is the correlation proposed by Wade between borane (or carborane) cages and metal carbonyl cages. Wade first drew attention to the similarity of a M(CO)3 unit and a BH (or CH) unit, a relationship that we would now call isolobality (Section 1-6). He then proposed that the 2n + 2 rule for closo boranes (Chapter 5) would also apply to closo metal cluster species such as [Os CO) ]2, and that 2n + 4 and 2n + 6 electron counts would, similarly, be appropriate for stable M clusters with nido and arachno structures. Hydrogen atoms are assumed to contribute one electron each, an interstitial carbon atom four electrons, and so on. 

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