Fractional bonding

Some force fields, such as MMX, have atom types designated as transition-structure atoms. When these are used, the user may have to define a fractional bond order, thus defining the transition structure to exist where there is a... 

The resonating-valence-bond theory of metals discussed in this paper differs from the older theory in making use of all nine stable outer orbitals of the transition metals, for occupancy by unshared electrons and for use in bond formation the number of valency electrons is consequently considered to be much larger for these metals than has been hitherto accepted. The metallic orbital, an extra orbital necessary for unsynchronized resonance of valence bonds, is considered to be the characteristic structural feature of a metal. It has been found possible to develop a system of metallic radii that permits a detailed discussion to be given of the observed interatomic distances of a metal in terms of its electronic structure. Some peculiar metallic structures can be understood by use of the postulate that the most simple fractional bond orders correspond to the most stable modes of resonance of bonds. The existence of Brillouin zones is compatible with the resonating-valence-bond theory, and the new metallic valencies for metals and alloys with filled-zone properties can be correlated with the electron numbers for important Brillouin polyhedra. 

For example, tin, with v = 2-5, crystallizes with a unique atomic arrangement, in which each atom has six ligates, four at 3-016 A and two at 3-175 A. These distances have been used (1947) in assigning the bond numbers 0-48 and 0-26 to these bonds. It is clear that these bond numbers can be taken as and and that the choice of the structure and the value of its axial ratio (which determines the relative lengths of the two kinds of bonds) are the result of the effort of the tin atom to use its valency 2-5 in the formation of stable bonds with simple fractional bond numbers. 

The properties of some sulfide minerals can be discussed in a reasonably satisfactory way by assigning one or another of the structures of Table I to the sulfur atoms. For some minerals the ligancy of the sulfur atom is equal to its covalence, and for others it is greater, the bonds then having fractional bond numbers. It is often necessary to assign a hybrid structure to the sulfur atoms. I shall assume that the covalent bonds indicated in Table I always have the normal amount of covalent character, as given by the... 

This specific problem was addressed by Burgi and Dunitz (1987), using a Morse function modified to treat fractional bonds. The observed ground state structure of a given molecular fragment, e.g. the O-C-O fragment of the acetals [96], was then treated as a distorted version of a standard structure of known molecular dimensions. Using experimental values of the... 

Some molecules exist where the bonding electrons cannot be assigned to atom pairs, but belong to more than two cores, e.g. in the polyboranes. In these cases the model concept of covalently bound atom pairs as a rep-resention basis for chemical constitution using binary relations can be sustained by the assignment of fractional bond orders. 

Some years ago, in the course of a discussion of interatomic distances in metals, an equation for interatomic distances of fractional bonds, bonds with bond number less than 1, was proposed.55 This equation is... 

In a few molecules and crystals it is convenient to describe the interactions between the atoms in terms of the one-electron bond and the three-electron bond. Each of these bbnds is about half as strong as a shared-electron-pair bond each might be described as a half-bond.1 There are also many other molecules and crystals with structures that may be described as involving fractional bonds that result, from the resonance of bonds between two or more positions. Moat of these molecules and crystals have a smaller number of valence electrons than of stable bond orbitals. Substances of this type are called electron-deficient substances. The principal types of electron-deficient substances are discussed in the following sections (and in the next chapter, on metals). 

The principal innovations that have been made in the discussion of the theory of the chemical bond in this edition are the wide application of the electroneutrality principle and the use of an empirical equation (Sec. 7-10) for the evaluation of the bond numbers of fractional bonds from the observed bond lengths. A new theory of the structure of electron-deficient substances, the resonating-valence-bond theory, is described and used in the discussion of the boranes, ferrocene, and other substances. A detailed discussion of the valence-bond theory of the electronic structure of metals and intermetallic compounds is also presented. 

It is appropriate now to re-emphasize just what reasonably quantitative data one can derive from the proton and C13 spectra. In this early work we assume that atoms other than carbon and hydrogen are present in only negligible amounts. The atomic hydrogen to carbon ratios can be evaluated from the elemental analyses of the materials and can be used to normalize the hydrogen and carbon NMR measurements so that they add to unity. The proton spectrum gives, after this minor modification, three items of information hnr, the fraction of total atoms in the material which is present as hydrogen atoms directly bonded to aromatic carbons hay the fraction bonded to carbons situated a to aromatic rings and hp, the fraction bonded to other nonaromatic carbons. The carbon spectrum in conjunction with the aromaticity calibration curve yields c.r, the fraction of total atoms in the material... 

The second class of hexanuclear clusters also contains an octahedron of metal atoms, but they are coordinated by twelve halide ligands along the edges (Fig. 16.64b). Niobium and tantalum form clusters of this type. Here the bonding situation is somewhat more complicated The metal atoms are surrounded by a very distorted square prism of (bur metal and four halogen atoms. Furthermore, these compounds are electron deficient in the same sense as the boranes—there are fewer pairs of electrons than orbitals to receive them and so fractional bond orders of are obtained. 

In the Re3 cluster just described, we have an example of double bonds between M atoms. Triple and even quadruple M-M bonds (as well as bonds having fractional bond order) also occur. The best-documented triple bonds are found in the molecular species M2X6 (M = Mo, W X = OR, NR2, CR3). The bonding can be very simply described. Each M has six valence electrons, of which three are used to form M-X single bonds, leaving three to form a triple M-M bond, X3M MX3. The M-M bond lengths are typically about 40 pm shorter than single bonds. (See also Section 8.2.)... 

The bond order in a diatomic molecule is defined as one-half the difference between the number of electrons in bonding orbitals and the number of antibonding orbitals. The factor one-half preserves the concept of the electron pair and makes the bond order correspond to the multiplicity in the valence-bond formulation one for a single bond, two for a double bond, and three for a triple bond. Fractional bond orders are allowed, but are not within the scope of this discussion. 

To make this precise, we define a cut in a molecule to be the conceptual dissection of a covalent connectivity a cut of a covalent bond of formal order n is counted as being n cuts. In some molecules the formal order of certain covalent bonds is not an integer. In such cases one can either approximate the bond order by an integer, or use a fractional bond order consistently for counting the cuts the genus for such molecules may therefore be not an integer in the latter case. With these concepts, we can make the precise... 

If in the structure single bonds only occur, then the chemical valency is equal to the coordination number. For fractional bonds (as occur e.g. in PbS) with bond number n the relations are... 

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