Relationship between force constants and bond

Quantitative Relationships between Force Constants and Bond Distances... 

We observed in Section 2.4 that we expect there to be some relationship between force constants and bond distances, that is, bonds with greater force constants should be shorter. Furthermore, because the force constant (given our previous assumptions) is dominated by the repulsive part of the potential function, and the repulsions are primarily due to nonbonding core electrons which are more numerous row by row down the periodic table, we expect that the repulsive wall of the core is further from the nucleus in each successive row (i.e., increasing principal quantum number). So if a functional relationship between force constants and bond distances exists, we expect that the parameters of the relationship, whatever its functional form, will depend on which row(s) of the periodic table the bonded atoms occupy. 

Kurita E Matsuura H Ohno K (2004) Relationship between force constants and bond lengths forCX (X = C, Si, Ge, N, P, As, O, S, Se, F, Cl and Br) single and multiple bonds formulation of Badger s rule for universal use. Spectrochim Acta A 60 3013-3023 Murell JN (1960) The application of perturbation theory to the calculation of force constants. J Mol Spectr 4 446-456... 

For an ab initio calculation of Fn and Fgg, see [39]. Fn and Fig were used to establish a relationship between force constants and bond energies of bent XYg molecules [51]. 

The relationship between force constant and bond order can also be seen in the series of homonuclear diatomics N2, O2, and F2. For this series, the bond orders (approximate indicators of the bond strength) are 3, 2, and 1, respectively. The corresponding force constants are 2290, 1180, and 440Nm CO and NO, which also have bond orders of 3, have force constants of 1900Nm and... 

In conclusion, we have presented Raman results in different conjugated polymers for which the vibrational aiudysis has been successfuL A good assignment for IR and Raman modes has been performed, with a good detomination force constants which agree rather well with the nature of the associated chemical bonds. We have established a relationship between force constants and intemuclear distances which can be applied to the different conjugated polymers studied in our laboratory, but also to otho compounds. ... 

The relationships between bond length, stretching force constant, and bond dissociation energy are made clear by the potential energy curve for a diatomic molecule, the plot of the change in the internal energy AU of the molecule A2 as the internuclear separation is increased until the molecule dissociates into two A atoms ... 

R. M. Badger, /. Chem. Phys., 2, 2128 (1934). A Relationship Between Internuclear Distances and Bond Force Constants. 

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 most optimistic response to this situation is to claim that the force constant — t-bond order relationship is still valid, but that the reference points need to be changed V(CO)e itself is then a possible reference compound (76). The relationship can then only be quantified by using calculated orbital populations for the reference species, and can only be tested by more extended comparisons between calculated bond order and observed force constant. Precisely this test has been apphed to a whole group of substituted and unsubstituted octahedral carbonyls of groups VI and VII, the substituents in every case being hahde (77). The data used in fact were not force constants, but Cotton-Krainhanzel parameters this does not actually matter, since no reference molecules were used at all. Excellent agreement was found with an expression. 

To compensate for steric effects slight modification of ionization radii allows the derivation of observed dissociation energies, using the same general diatomic function for all covalent bonds, irrespective of bond order. Comparison of the modified characteristic radii and the fundamental ionization radii enables the calculation of harmonic force constants and defines a functional relationship between bonds of different order. 

Attempts have been made to calculate CO bond orders from CO-stretching frequencies and to use these parameters in discussing the electron distribution in the molecules. Such a procedure is doubtful, however, in view of certain limitations. First, force constants, calculated from the frequency data, are not necessarily rigorous. Second, a relationship between the bond order A (CO) and the CO-stretching force constant must be assumed. All such relationships that have been proposed are based on a linear dependence of the type N(CO) = ak + h where a, b are constants and k is the force constant and differ only in the values assumed for the constants a and b. In one approach, values of the CO-stretching force constants for free carbon monoxide, where JV (CO) = 3, and for the compound CH2=C=0, where N(CO) = 2, were used to define a and b (43, 280). Using this relation ip, the CO bond order of the compound Ni(CO)4 was calculated to be 2.75. On the basis of this result and from other considerations Bigorgne came to the conclusion tliat the NiC bond order was 0.25 (43). Similar calculations were made for the deriva-... 

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