Theoretical bonding

Quantum mechanical calculation of molecular dynamics trajectories can sim ulate bon d breakin g and frtrm ation.. Although you dt) n ot see th e appearance or disappearan ce ofhonds, you can plot the distan ce between two bonded atom s.. A distan ce excccdi n g a theoretical bond length suggests bond breaking. 

Table 1 Molecular parameters of the diatomic oxides and sulfides of carbon and silicon derived experimentally (force constant f and bond energy BE) and theoretically (bond distance d, charge Q, and Shared Electron Number SEN).

The recently determined X-ray crystal structure of cyclopentacorannulene 25 provides an additional opportunity to test the performance of theory in predicting the minimum energy structures of nonplanar conjugated systems. Table 4 presents the C-C bond lengths in the crystal and a comparison with ab initio values calculated at HF/3-21G and HF/6-3IG levels. The agreement between experimental and theoretical bond lengths is very satisfactory with root mean square devia-... 

Figure 12.2. A comparison between reorganizational energies and theoretical bond energies of alkane CH bonds (kcal/mol). The numbering refers to that shown in Table 12.3 [233].

Angle of a tetrahedron is 109.5° this exists only if there are no unshared electrons. 1 point for theoretical bond angle of a tetrahedron. 

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3. ... 

In many compounds, the experimental bond valences, S, and the theoretical bond valences, s, are both found to be equal to the bond fluxes, <1>, within the limits of experimental uncertainty. This is an empirical observation that is not required by any theory. For this reason, and because there are occasions when the differences between them are significant and contain important information about the crystal chemistry, it is convenient to retain a different name for each of these three quantities to indicate the ways in which they have been determined. The bond flux is determined from the calculation of the Madelung field, the theoretical bond valence is calculated from the network equations (3.3) and (3.4), and the experimental bond valence is determined from the observed bond lengths using eqn (3.1) or (3.2). 

Table 3.1 A comparison of the bond fluxes, theoretical bond valences, s, and experimental bond valences, S, and the corresponding bond lengths, R, in CaCrF5(10286)...

In cases where the experimental and theoretical bond valences are different, the bond capacitances do not cancel, but the experimental bond valences continue to give a good estimate of the bond flux (Preiser et al. 1999). In these cases, discussed in Chapters 8 and 12, the theoretical bond valences can be used to determine a reference bond length against which the sizes of the strains in the observed bond lengths can be measured. 

Rules 3.3 and 3.4, through the corresponding network equations (3.3) and (3.4), can be solved (see Appendix 3) to give theoretical bond valences which, for unstrained structures, are equal to the bond fluxes and experimental bond valences. Taken together. Rules 3.3 and 3.4 are equivalent to the statement ... 

The theoretical bond valence, which is calculated using the equal valence rule (3.4), clearly gives a very poor estimate of both the flux and the experimental bond valence since in these compounds the valence is not equally distributed between the bonds. Nevertheless, it provides a useful reference against which to measure the strain introduced by the electronic asymmetry. ... 

The bond strain in this chapter, as in most other places in this book, is defined as the difference between the observed bond lengths and the bond lengths calculated from the theoretical bond valences. A bond strain index that measures this strain is defined in eqn (12.1). 

Fig. 8.9. Bond graphs of (a) ZnSb206 (30409) and (b) ZnV206 (30880) showing the theoretical bond valences.

Fig. 8.9(b), ZnV20g is able to crystallize with the brannerite structure whose theoretical bond valences, calculated from the network equations ((3.3) and (3.4)) and shown in Fig. 8.9(b), already predict an out-of-centre distortion for the ion. ZnV20g thus adopts a bond graph that supports the electronically induced distortion. In this case the adoption of a lower symmetry bond graph is favoured because it is able to reduce the bond strain. 

The bond valence model may also be used to refine the structure since it is based on the same assumptions as the two-body potential method. The network equations (3.3) and (3.4), can be used to predict the theoretical bond valences as soon as the bond graph is known. From these one can determine the expected bond... 

Typically lattice-induced strain results in the bonds around one cation being stretched and the bonds around another cation being compressed as found in BaRuOs (10253) by Santoro et al. (1999, 2000). When this happens, the valence sum rule will be violated around the cations in question but the valence still distributes itself as uniformly as possible among the bonds, so that the experimental bond valences determined from the bond lengths remain as close as possible to the theoretical bond valences. For this reason the BSI is typically smaller than the GII for lattice-induced strains, though the opposite is true for compounds with electronically induced strain where the valence sum rule remains well obeyed. 

In principle the bond valence parameters could be obtained by comparing the experimental bond valences, S, determined using an initial set of bond valence parameters, against the theoretical bond valences, s, calculated using the network eqns (3.3) and (3.4). These initial values could then be refined to minimize the differences given by the expression (Al.l) ... 

Figure 3.11 Potential energy diagram of a diatomic molecule. E is the potential energy and r the internuclear distance. The theoretical bond dissociation energy is in the ground state and E in the associative excited state Si, while T] is a dissociative excited state...

Experimental and Theoretical Bond Lengths (A) in Chrysene (Burns and Iball, 1960)... 

Table 1 Selected theoretical bond lengths and bond angles...

J. P. Malrieu, in Models of Theoretical Bonding, Z. B. Maksic, Ed., Springer Verlag, New York, 1990, pp. 108-136. The Magnetic Description of Conjugated Hydrocarbons. 

More precise comparisons may be made on symmetrical structures for which a theoretical bond order may be calculated by averaging the possible canonical forms with, for S5 compounds, the additional assumption that all canonical forms written with divalent sulfur atoms are equivalent. 

Finally, the sum of the above total (diatomic) entropy-covalency and information-ionicity indices determines the overall information-theoretic bond multiplicity in the molecular fragment in question ... 

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