Reference Bonds

Equation (10.2) is our starting point. Applying the Hellmann-Feynman theorem, we get 

Direct applications of Eq. (10.12) are generally difficult to handle—this is why the more efficient charge-dependent energy formulas were developed in the first place. Most thorough tests were made for selected carbon-carbon bonds [13,14,44,108] (Table 1.1). 

But there are ways to circumvent the difficulty of solving Eq. (10.12), at least in an approximate manner. Consider the Hellmann-Feynman derivative (10.4) for a hydrogen atom bonded to atom I 

A small change of Ski, written Aski, can be evaluated in an approximate manner by replacing the first term of Eq. (10.12) with a modified one, leaving the rest of Eq. (10.12) unchanged, so that (with y = 2) 

with the help of Eq. (10.13), for hydrogen hnked to atom /, we have 

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For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

Table 4.1 Force constants and reference bond lengths for selected bonds [Allinger 1977],...

The coefficients C are chosen to ensure that the function has a minimum at the appropriate reference bond angle. For linear, trigonal, square planar and octahedral coordination, Fourier series with just two terms are used with a Cq term and a term for n = 1, 2, 3 or 4, respectively ... 

The energy parameters used for the reference polyene by Hess and Schaad were developed on a strictly empirical basis. Subsequendy, Moyano and Paniagua developed an alternative set of reference bond energies on a theoretical basis. These values are shown... 

The 6th, 7th, and 8th terms of expression (9) are sometimes neglected or is related to (24,25). The potential constants of UBFF s, including the reference parameters, often deviate markedly from the corresponding VFF-constants. Especially the reference bond lengths b may assume values which hardly agree with intuitive ideas about strain-free bond lengths. 

The Bond function describes a central atom of reference and the atoms bonded to it. B211 states that there is a central atom of reference bonded to one atom by a double bond (2) and to two other atoms by single bonds (1). The order of the Bond function arguments corresponds to this Bond function notation. These arguments are not simple atomic symbols, but Atom functions that can relate considerable information about the atom. In this Bond function, Atom(C,1,0,0,0) is the central atom. The next three arguments are atoms that are bonded to this central atom the first, Atom(0,3,0,0,0) by a double bond the next two, Atom(C,2,0,0,0) and Atom(H,4,0,0,0), by single bonds. 

Fortunately, we have something better in store. Eirst we calculate a few reference bond energies sh, using Eq. (10.12), and subsequently modify these bond energies as the electron densities p are varied. 

This means that the reference bond energies, corresponding to a reference electron density p°(r) have been modified by the change A(V f + 2V ) to give the energies sj i corresponding to p(r). We also define... 

Equation (10.34) indicates how the intrinsic energy of a chemical bond linking atoms k and I depends on the electronic charges carried by the bond-forming atoms Eki is for a reference bond with net charges qk and (fi at atoms k and I, respectively, whereas Ski corresponds to modified charges qk = qi + Aqk and qi = qf + Aqi. Fki follows from Eq. (10.26). These formulas are the basics of this theory. From here on, we focus on simplifications and application. 

It turns to our advantage to consider eh and Fj i jointly. The idea is best explained by an example. Suppose that the CC and CH bond of ethane were selected as reference bonds, with energies ecc and ecH. respectively, and F =0. New references are required for olefins, specifically tailored for C(sp )—C(sp ) and C(sp )—H bonds the reference for C(sp )-C(sp ) bonds is deduced from that representing C(sp )—C(sp ) while C(sp )—H is derived from C(ip )—H by incorporating the appropriate parts of F into the new reference energies ... 

While most convenient in computations, the su reference energy does not embrace an actual physical situation. The true physical reference bond, as it is found in an appropriately selected reference molecule, is su, with reference atomic charges qh and qf at the bond-forming atoms k and I, respectively efi is amenable to direct calculations efl is not. 

Any application of Eqs. (10.37)-(10.40) requires a solid knowledge of the appropriate set of reference bond energies sa, of the bond energy parameters a/ i and, finally, of the appropriate atomic charges. 

Calculation of Reference Bond Energies [Eq. (10.39)]. The parameters indicated in Table 10.4 are ready for use in the bond energy formula, Eq. (10.39). The following examples, in part based on detailed results given in Chapters 15 and 16 for nitrogen- and oxygen-containing molecules, illustrate the procedure and report the input data. 

TABLE 10.4. Reference Bond Energies (kcal/mol) and a/a Parameters... 

The modification of reference charges that involve readjustments of reference bond energies, evaluated by the function atiAq ... 

Accordingly, 10 new reference bond energies must be dehned. This is done with respect to the well-known bonds of ethane. In addition, seven additional parameters are to be evaluated by means of our standard formulas, Eqs. (10.41) and (11.16), with proper consideration of a- and rr-electron populations. All these transformations, including those dictated by changes of internuclear distances, are straightforward but require attention. 

If this simple expression is sufficient to account for the discrepancies observed in calculations of olefins based on alkane reference bonds [i.e., calculations using Eq. (10.33) with F = 0], we can claim that the principle (11.8) is, indeed, satisfied by ethylenic hydrocarbons. 

Despite the marked differences in both geometric parameters and the SCF Ar values between the molecules involved in this comparison, there are striking regularities the F value calculated for propene is, for all practical reasons, th that of ethylene (which takes care of 3 CH bonds) plus th that of tetramethylethylene (for the CC bond). Capitalizing on this idea, we may well consider transferable bond contributions modeled after Eq. (11.12) and use them to generate new reference bond energies satisfying Eq. (10.36). 

This considerably simplified approach works extremely well in molecular calculations. The bottom line is that we can safely proceed with Eq. (10.37) as long as we use the appropriate reference bond energies sy, because these reference energies enjoy a high degree of transferability. 

Here we start off with the CC and CH reference bond energies of ethane and use them to get the CC and CH bonds occurring in unsamrated hydrocarbons by considering both the contribution of F [Eq. (11.12)] and the change of charge given by au,Aqi [Eq. (10.37)]. The new reference bonds thus obtained are indicated in Table 11.2. 

TABLE 11.2. Selected Reference Bonds and Their Energies (kcal/mol)... 

Eq. (10.33). It must be included in the definition of the relevant reference bond energy. This formulation, which is the simplest possible one, is adequate because it turns out that, for all practical purposes, the amount of conjugation associated with a given bond can be treated as a constant (representing a rough but acceptable approximation) unless, of course, changes of molecular structure force a dismption... 

A new reference bond energy must be introduced at this point for the CO single bond of diethylether, selected as reference ... 

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. 

Bond order Bond order n n(A B) = exp a [p (p) - fi] where a and fi are determined from two suitable reference bonds AB... 

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