Spectroscopic force fields

Force constants derived from normal coordinate analyses of infrared spectra have formed the basis for the parameterization of many molecular mechanics force fields. However, empirical adjustment of these molecular mechanics force fields has led to appreciable differences between the spectroscopic and molecular mechanics force constants. It is important to understand why this should be so and to appreciate the difference between spectroscopic and molecular mechanics force fields. 

Let us consider the description of a simple three-atom molecule by the three most frequently used spectroscopic parameterization schemes, the general central 

While the GCF (Eq. 3.36) uses only interatomic distances (Ar, Aq), in the GVF the nonbonded interactions are replaced by valence angles (Eq. 3.37), and in the UBF (Eq. 3.38) both the interatomic distances and the valence angles are used (re and qe in Eqs. 3.36-3.38, correspond to ideal distances)17 1. 

In each model the coupling of vibrations is taken into account by the addition of cross terms. Two important facts arise from this and a general appreciation of Eq. 3.36-3.38  

A molecule-independent, generalized force field for predictive calculations can be obtained by the inclusion of additional terms such as van der Waals and torsional angle interactions. This adds an additional anharmonic part to the potential (see below) but, more importantly, also leads to changes in the whole force field thus the force constants used in molecular mechanics force fields are not directly related to parameters obtained and used in spectroscopy. It is easy to understand this dissimilarity since in spectroscopy the bonding and angle bending potentials describe relatively small vibrations around an equilibrium geometry that, at least 

Let us consider the description of a simple three-atom molecule by the three most frequently used spectroscopic parameterization schemes, the general central force field (GCF), the general valence force field (GVF) and the Urey-Bradley force field (UBF Fig. 2.14). 

9 An adaption of the GCF to molecular mechanics1 1 has been discussed in Section 2.1. 

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The influence of bilinear cross terms of this type in force field caculations has been studied systematically only once so far (79). They are standard for vibrational-spectroscopic force field expressions (20), and accordingly vibrational frequencies depend considerably more sensitively on cross terms than e.g. conformational parameters. An example for the significant influence of cross terms also with respect to the latter is described in Section 6.1.3. 

We have seen that for our calculations essentially two types of force fields have to be considered VFF- and UBFF-expressions. The main difference with repect to spectroscopic force fields consists in the superposition of nonbonded interactions. The force fields used so far for our purposes are almost exclusively simple valence force fields without cross terms, and a veriety of UB-force fields. Only recently could experiences be gathered with a valence force field that includes a number of important cross terms (79). Vibrational spectroscopic force fields of both types have been derived and tested with an overwhelming amount of experimental data. The comprehensive investigations of alkanes by Schachtschneider and Snyder (26) may be mentioned out of numerous examples. The insights gained from this voluminous spectroscopic work are important also when searching for suitable potentials for our force-field calculations. 

A force field is considered transferable from an arbitrary molecule A to another molecule B if the agreement of properties calculated fori (geometrical, vibrational, thermochemical, and other properties) with the respective experimental values is as good as for A. In our calculations we are dealing with force fields which describe entire families of molecules. Within these families, properties of only a fraction of their members are known experimentally while it is our aim to predict the others computationally. It is therefore clear that the problem of transferability is of decisive importance for force field calculations of unknown systems or those with unknown properties or. Traditional vibrational spectroscopic force fields in most cases reproduce well the frequencies of a single molecule or a family of closely related molecules however, they are not transferable to molecules of different strain. Subsequently we comment on this point in somewhat more detail. 

The molecular mechanics technique has been called by many different names, including Westheimer method, strain-energy method, conformational energy calculations, empirical potential energy calculations, atom-atom pair potential method, and force field calculations. Empirical force field is widely used, but somewhat long, and many authors omit empirical, leading to confusion with spectroscopic force field calculations. Molecular mechanics (11) now appears to be favored (10a) and is used (abbreviated as MM) throu out this chapter. 

Fig. 2.14. Parameters used for spectroscopic force field calculations of H20.

The force constants may not be transferred from one spectroscopic force field into another one, i.e., they are not transferable physical quantities. 

The force constants may not be transferred from one spectroscopic force field... 

The fact that the second derivative, d U/db in Eq. [24], contains a slight contamination from nonbonded interactions and third-order terms is an example of how parameter correlation can arise because it is not a pure bond stretch. If this derivative were simply used as the bond-stretching force constant, as in spectroscopic force fields, it would not be transferable to other molecules where the coupling or nonbonded interaction may differ. This problem is a general one and can be quite serious. As already discussed in previous sections, one possible resolution of this problem lies in the use of many molecular environments to determine all contributing terms. If we simultaneously fit many different alkanes, i.e., ethane, propane, butane, etc., with the full force field and assume... 

The second element in the procedure is the choice of a spectroscopic force field, i.e., a set of Fy. This could be an empirical force field, but since we wish to provide as complete a description as possible at this stage, a scaled ab initio force field is the one of choice. The scaling process also connects the force constants to experimental frequencies and band assignments, and avoids the problems of having incorrect eigenvectors because of using inappropriate basis sets [26]. 

One item that does seem to be of perhaps more importance concerns the vibrational spectra. MM4 has a root-mean-square (rms) error of about 25 cm in overall vibrational calculations for a test set of alkanes. (This corresponds to only 0.07 kcal/ mol.) Spectroscopists commonly claim that they can measure vibrational frequencies down to 1 cm without difficulty (in fairly small molecules), and numerous spectroscopic force fields in the literature indicate that they can often calculate these frequencies with average errors of 10cm or less. We have not worked on this problem very seriously. The MM4 force field is designed fo calculate a great many different quantities... 

The per chain modulus of this pol3nner is about equal to that of diamond in the [110] direction. A polyethylene fiber with the same per chain mechanical properties would have an ultimate tensile strength in excess of one million psi. The theoretical modulus calculated for a defect free polydiacetylene chain using a spectroscopic force field is within 10% of the observed modulus. This contrasts with the case for conventional polymers, where the bulk tensile modulus is typically much less than 50% of the theoretical (spectroscopic) modulus. 

Molecular mechanics force field developers have ba.sed their parameters on a variety of data types. However, there is often a bias towards data from isolated molecules, as obtained from gas-phase electron diffraction or ah initio quantum mechanics, because it can be assumed that there is no environmentally induced distortion. At least one spectroscopic force field for carbohydrates is under development, with the high precision of vibrational frequencies making attractive calibration data. However, purely spectroscopic force fields... 

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