Interaction force constants

The amonotonic order for Co and Ni, compared to the monotonic order for Cu, may best be rationalized in terms of a finite, positive kc=c,c=c interaction force constant for Co and Ni, rather than as the outcome of amonotonicity in the principal c=c force-constants. 

Force constants of have been calculated from the data in Table 2 using the general valence force field (GVFF) [148, 149] as well as the Urey-Bradley force field [80] (UBFF) although there is insufficient data to evaluate all the interaction constants since no isotopomers of Se have been measured by vibrational spectroscopy. The stretching and bond interaction force constants were reported as/r = 2.24 andf = 0.53 N cm, respectively [149]. However, because of the uncertainty regarding the Am mode of Se the published force constants [80, 148, 149] maybe unreliable. 

The effect shown in Fig. 9 is a result of the bond-bond interaction which is a characteristic feature for chains and rings of two-valent chalcogen atoms. It can also be recognized from the relatively large bond interaction force constants fir of such compounds. The stretching force constants /r(SS) of polysulfur compounds depend on the SS bond distances as shown in Fig. 10. The data used in this figure include several excited electronic states of the S2 molecule as well as the disulfide anion and a number of sulfur homocycles [77]. 

The interpretation of interaction parameters is far from stredghtforward. Calculations on model systems (2 c), and comparisons between parameters and force constants in real systems (4), show that the parameters contain several different terms. Thus in octahedral carbonyls, the cis interaction parameter, which is smaller than the trans parameter, originates mainly from the true cis interaction force constant. The trans interaction parameter, on the other hand, contains only a small direct contribution, while the main terms are indirect and involve MC,MC and MC.C O interaction constants. Calculations on substituted... 

With respect to nonionized organic solutes in aqueous solutions, the interfacial interaction force constants were expressed as constants that define the interfacial potential function as follows ... 

This way of expressing the overall modes for the pair of molecular units is only approximate, and it assumes that intramolecular coupling exceeds in-termolecular coupling. The frequency difference between the two antisymmetric modes arising in the pair of molecules jointly will depend on both the intra- and intermolecular interaction force constants. Obviously the algebraic details are a bit complicated, but the idea of intermolecular coupling subject to the symmetry restrictions based on the symmetry of the entire unit cell is a simple and powerful one. It is this symmetry-restricted intermolecular correlation of the molecular vibrational modes which causes the correlation field splittings. 

One conclusion is clear the dominant cubic and quartic interaction force constants are those associated with bond stretching, and these are not dissimilar to those of the corresponding diatomics. The same conclusion follows from a study of all other published data, and comparisons between bondstretching anharmonicity in related molecules are discussed further below (see Table 15). 

Cubic and quartic interaction force constants may be visualized in the following way, e.g. for C02 ... 

The two Ai stretchings of axial (ax) and equatorial (eq) CO groups are extensively coupled84, 87> 216. Approximate force-field calculations have been made which neglect coupling with other molecular vibrations and take into account only two CO stretching force constants (/ax,/eq) and two CO/CO stretch-stretch interaction force constants (/ax,e9 To avoid vagueness,... 

We can also consider cases in which the intrinsic barrier is altered. Two such effects are steric hindrance and contribution of charge-separated structures to the transition state. Steric hindrance raises the energy of the transition state compared to that of a similarly exothermic unhindered model. This can be accomodated by considering an increase in the intrinsic barrier, which therefore makes the isotope effect rise. In ref.11 this is alternatively interpreted in a quadratic representation of the surface as an increase in the interaction force constant, and thus also correlated with an increase in the tunnel correction. An example of such an enhancement is the large value of the isotope effect in the trityl radical mesitylenethiol reaction in Table 1. 

This formula related the bending force constant to the stretching force constants. Now the stretch-stretch interaction force constant 12 is usually smaller by a factor of 10 or so than tl and hence often can be neglected. Then from equation (137) ... 

NI3, in which each individual bond is weaker than that in M2 (as gauged by their relative force constants), the interaction force constant is almost certainly positive. Negative values of frr are encountered in KrF2, H2O and HCN, while unusually large values of frr are found in triatomlc anions such as HCl2, HF2, Cl3, ... 

The diagonal force constants fij with i = j describe the elasticity of a bond according to Hooke s law. The constants with i j, the interaction force constants however, describe the change of the elastic properties of one bond when another bond is deformed (Sec. 5.2). 

Such calculations are very approximate, especially because coupling with the bending vibrations of the chain and those of other groups (C H bonds), the interaction force constants, and the anharmonicity are ignored. This is discussed elsewhere in this book (see the following section as well as Secs. 4.2 and 5.2). However, these calculations allow insight into typical properties of vibrating molecules. 

From these frequencies and with the help of the corresponding G-Matrix elements (Wilson et al., 1955), the symmetry-adapted force constants (F) can be calculated directly. In the vibrations discussed here, F is a linear combination of stretching and interaction force constants / and f,r . 

If two different coordinates (r, and /y) occur in one potential energy term, interaction force constants are obtained. Their value differs from zero if distortions in one coordinate r, change the equilibrium position in r,. 

Force constant calculations are facilitated by applying symmetry concepts. Group theory is used to find the appropriate linear combination of internal coordinates to symmetry-adapted coordinates (symmetry coordinates). Based on these coordinates, the G matrix and the F matrix are factorized, which makes it possible to carry out separate calculations for each irreducible representation (c.f. Secs. 2.133 and 5.2). The main problem in calculating force constants is the choice of the potential function. Up until now, it has not been possible to apply a potential function in which the number of force constants corresponds to the number of frequencies. The number of remaining constants is only identical with the number of internal coordinates (simple valence force field SVFF) if the interaction force constants are neglected. If this force field is applied to symmetric molecules, there are often more frequencies than force constants. However, the values are not the same in different irreducible representations, a fact which demonstrates the deficiencies of this force field (Becher, 1968). 

Obviously, the addition of the values of all interaction force constants which are involved in F(lattice) simplify the result F(lattice) w /. Analogous relations between F(lattice) and the observed frequencies have been discussed for solid NaCl and Cap2 (Becher,... 

Changes in mn.cn) interaction force constant, have little effect on i qn and almost none on and i mn-... 

Therefore, it can be reasonably assumed that the correct assignment of the CO-stretching frequencies for these simple systems, containing all equivalent CO groups, corresponds to positive Cotton-Kraihanzel interaction force constants. 

Force fields which predicted the observed frequency data most satisfactorily have been those for which certain approximations were made about the valence interaction constants F. . Jones related a number of such interaction constants by the use of simple valence theory ( resonance interaction valence force field ) and obtained a satisfactory set of force constants for the compounds M(CO)6 (M = Cr, Mo, or W) (190, 191, 192, 193) and NiC(0)4 (189). However, the secular equations for governing the vibrations of these systems have most usually been solved by neglecting certain interaction force constants For instance,... 

Both Equations 5.1 and 5.2 can be solved with appropriate boundary conditions and interfacial interaction force constants, generated from liquid chromatography experiments. 

Table 11 records force constants and calculated bond orders in several azides. (/(N Nb/NtN ) is the bond-bond interaction force constant.)... 

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