Cross-term, vibrational

Intensive use of cross-terms is important in force fields designed to predict vibrational spectra, whereas for the calculation of molecular structure only a limited set of cross-terms was found to be necessary. For the above-mentioned example, the coupling of bond-stretching (f and / and angle-bending (B) within a water molecule (see Figure 7-1.3, top left) can be calculated according to Eq. (30). 

Figure 7-13. Cross-terms combining internal vibrational modes such as bond stretch, angle bend, and bond torsion within a molecule.

The consistent force field (CFF) was developed to yield consistent accuracy of results for conformations, vibrational spectra, strain energy, and vibrational enthalpy of proteins. There are several variations on this, such as the Ure-Bradley version (UBCFF), a valence version (CVFF), and Lynghy CFF. The quantum mechanically parameterized force field (QMFF) was parameterized from ah initio results. CFF93 is a rescaling of QMFF to reproduce experimental results. These force fields use five to six valence terms, one of which is an electrostatic term, and four to six cross terms. 

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. 

The number and type of cross terms vary among different force fields. Thus, AMBER2 contains no cross terms, MM23 uses stretch-bend interactions only and MM34 uses stretch-bend, bend-bend and stretch-torsion interactions. Cross terms are essential for an accurate reproduction of vibrational spectra and for a good treatment of strained molecular systems, but have only a small effect on conformational energies. 

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

Of the three eigenvalue equations, the one of interest to us is the vibrational equation. It has a particularly simple form when normal coordinates are employed because the classical kinetic and potential energies then have no cross terms (see eqns (9-2.17) and (9-2.18)) and this fact leads to a simple form for their quantum mechanical analogues (the kinetic energy and potential energy operators). The vibrational equation is thus... 

The presence of these higher cross-terms tend to improve the ability of the force field to predict the properties of unusual systems (such as those which are highly strained) and also to enhance its ability to reproduce vibrational spectra. It must be noticed, however, that any of the cross terms listed above have been proven to be truly of the form in which they are written. No attempts have been reported to derive that or any other form of the coupling between different geometry distortions and to estimate the corresponding constants from some independent point of view. The class III force field will also take into account further features such as electronegativity and hyperconjugation. We shall turn to these problems later. 

As stated in Section 1.6, normal vibrations are completely independent of each other. This means that the potential and kinetic energies in terms of normal coordinates (Q) must be written without cross terms. Namely,... 

In the functional form of other force fields additional terms, such as the cross terms are included. These terms couple internal coordinates, e.g. changes of planar angle are coupled with adjacent bonds stretching. Cross terms are important in many cases, e.g. for a better reproduction of vibrational spectra of molecules. 

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 ... 

We have seen that the dependence of Bn(R) on the vibrational coordinate causes a mixing of the vibrational level of interest with neighbouring levels. This mixing results in centrifugal distortion corrections to all the various parameters Xn(R) in the perturbation Hamiltonian 3C when combined in a cross term. The operator has the same form as in the original term, for example, (2/3) /6T 0(S, S) for the spin spin dipolar term, multiplied by N2. The coefficient which qualifies this term has the general form... 

But consistent with the overall theme of this chapter, we need to ask ourselves whether we really have to conclude that the mechanism of vibrational energy relaxation is fundamentally electrostatic just because we find the overall relaxation rate to be sensitive to Coulombic forces. Let us attempt to get at this question through another mechanistic analysis of the INM vibrational influence spectrum, this time looking at the respective contributions of the electrostatic part of the solvent force on our vibrating bond, the nonelectrostatic part (in most simulations, the Lennard-Jones forces), and whatever cross terms there may be. 

The Buckingham Eq. (6-10) takes into account the fact that the infiuence of solvent dipolarity [characterized by /(cr) = (cr — l)/(2er- -1)] and solvent polarizability [characterized by f n ) = n — l)j 2n + 1)] on the solute IR vibrations are two independent effects. Based on the assumption that solute/solvent collision complexes are formed in solution, which should lead to a mutual correlation in dipolarity/ polarizability changes, Bekarek et al. have added a third cross-term f ef) f n ) to the two terms of Eq. (6-10) [379]. Indeed, using the modified three-term Eq. (6-12),... 

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