Fractional bond orders

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. 

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. 

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... 

The inclusion of bonds with fractional bond orders extends the scope of the mathematical model of the constitutional chemistry from classical organic chemistry also to modern inorganic and organometallic chemistry. 

In all cases with p, = p2 = 1 (also if V, = V2), V is linearly dependent on n. Vasa function of n has a maximum only if either p, or p2, or both, are larger than 1. Since < 1, the decrease of energy with partial formation of one of the bonds, Virtf1, is smaller than the amount proportional to the fractional bond order (V nj) if p, > 1. The larger p, the smaller is the energy set free in the fractional bond forming process. 

Two main definitions of bond order indices are reported below. Moreover, the term fractional bond order was suggested to refer to the inverse of any bond order index. Fractional bond order permits individual treatment of a and it molecular systems a bonds give simple graphs, while jt bonds introduce a weighted molecular framework with weights less than one [Randic et al, 1980]. 

An alternative ID number is calculated in the same way using - fractional bond order instead of conventional bond order to accomplish a gradual attenuation of the role of paths of longer lengths. 

Fractional bond orders are possible, as in the cases of 02+ and 02, when the number of antibonding electrons is odd. 

The equations for k(n) and D n) derived in this way have the same form as the Pauling definition of fractional bond order [24], Herschbach and Laurie s relationship between stretching force constant and bond distance [193], and Johnston and Parr s expressions for the bond-energy and bond-order relationships [25]. With reasonable values of p 2 A Dq 100 kcal mol and n = 1, dD(/"el/dre is about 400kcalmol thus reproducing the order of magnitude of the experimental slope for acetal hydrolysis. 

The major issues raised in these studies were (I) the question of local symmetries of the species along the reaction path for a nucleophilic addition to a trigonal center and (II) the correlations between the fractional bond order, the pyramidalization of the trigonal center, and the bond distance during the breaking and forming of bonds. 

In regard to the second problem, correlation of the fractional bond order and pyramidalization, the newer crystallographic and computational evidence consistently supports the original model, albeit a modification of its quantitative aspects is clearly necessary. Similarly, the simple relations invoking bond-order conservation during bond breaking and forming proved useful also in the more recent studies. 

We have already described several examples of bonding pictures that involve the delocalization of electrons. In cases such as BF3 and SFg, this leads to fractional bond orders. We now consider two linear XY2 species in which there is only one occupied MO with Y—X—Y bonding character. This leads to the formation of a three-centre two-electron (3c-2e) bonding interaction. 

Usually the bond order corresponds to the number of bonds described by the valence bond theory. Fractional bond orders exist in species that contain an odd number of electrons, such as the nitrogen oxide molecule, NO (15 electrons) and the superoxide ion, 02 (17 electrons). 

Partial bonding, such as that occurring in resonance hybrids, often leads to fractional bond orders. For O3, we have... 

There is a variety of chemical species whose constitution cannot be adequately described in terms of electron pair bonds between pairs of cores. Examples are the boron hydrides, the transition metal complexes of hydrocarbons and their anions, such as Tt-allyl nickel or ferrocene, or the hypothetical intermediate of the limiting Sjy mechanism 88). In these cases the molecular structure can be adequately represented by the model concepts of imdlicenter bonds and fractional bond orders. 

EM with fractional bond orders can be represented by fe-matrices having rational entries. The off-diagonal entries are... 

The rational de-matrices B can be converted into equivalent nX > matrices with integral entries by multiplication with the smallest common denominator of the fractional bond orders. By this the mathematical properties of the integral de-matrices and of their reaction matrices can also be used for chemical systems with fractional bond orders. 

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