Bond length deformation

Bonding in a diatomic molecule may be described by the curve given in Fig. 2.5, which represents the potential energy (K(r)) as a function of the structure (r). The bonding force constant, k, is given by the second derivative of the potential with respect to the structural parameter r, which corresponds to the curvature of the potential energy function. The anharmonicity can be described by higher-order derivatives. 

Computationally, a Morse function may be mimicked by a Taylor expansion, where the first term (quadratic) describes a harmonic potential and successive higher-order terms are included as anharmonic corrections (Eq. 2.14). 

A harmonic potential is a good approximation of the bond stretching function near the energy minimum (Fig. 2.7). Therefore, most programs use this approximation (see Eq. 2.6) however the limits of the simplification have to be kept in mind, in those cases where the anharmonicity becomes important. Apart from the possibility of including cubic terms to model anharmonicity fsee the second term in Eq. 2.14), which is done in the programs MM2 and MM3[1,2,2 241, the selective inclusion of 1,3-nonbonded interactions can also be used to add anharmonicity to the total potential energy function. 

In spectroscopy, force constants are obtained from normal coordinate analyses of vibrational spectra1251. There, harmonic potentials are usually satisfactory because the motions are small116,251. Spectroscopic parameters are usually good first guesses for an empirical force field. 

Bonding of ligands to alkali, alkaline earth and rare earth metal ions is mainly electrostatic. Consequently, these bonds can be described as a combination of electrostatic and van der Waals terms (see Chapter 14). 

The experimentally observed parameters re, De and ke are not directly related to the corresponding parameters r0, D0 and kQ, describing the theoretical curve 

In molecular mechanics bond stretching is sometimes modeled using a Morse function (Eq. 3.13)[7() where a describes the curvature and D the depth of the potential function (Fig. 3.6). 

This treats the bond as a mechanical spring whose force constant is strong for small and weak for large interatomic distances. The disadvantage of using a Morse function in empirical force field calculations is that an exponential in addition to the square function and three parameters are involved, increasing the time requirement for the minimization process and the complexity of the force field parameterization. 

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The simple cycloalkanes (CH2)n with n = 5 to 12 are the compounds most frequently studied by force field calculations (8, 9, 11, 12,17, 21). This preference results from their simple structure, from the abundant available experimental material (structural (46), thermo-chemical (47) and vibrational spectroscopic (27, 48, 49) data), and from the fact that, apart from bond length deformations, all other strain factors (angle deformations, unfavourable torsion angles, strongly repulsive nonbonded interactions) are important for the calculation of their properties. The cycloalkanes are thus good candidates for testing force fields. For a more detailed discussion we choose cyclodecane, a so-called medium-ring compound. 

The repulsion increases exponentially, and it is steeper than the bond length deformation potential. The attractive force is usually modeled by a 1/r6 term, while various possibilities exist for the repulsion. The functions used in modem programs include, apart from the Morse potential (Eq. 2.13), the Lennard-Jones potential (Eq. 2.25)[401 (e.g., AMBER1411), the Buckingham potential (Eq. 2.26)[421 (e.g., MOMEC[81), or a modification thereof, the Hill potential (Eq. 2.27) (e.g., MM2, MM3[1,2,231). 

The repulsion increases exponentially, and it is steeper than the bond length deformation potential. The attractive force is usually modeled by a Hr6 term while... 

More accurate force constants for a number of transition metal complexes with ammine ligands have been derived by normal-coordinate analyses of infrared spectra[130, 31l The fundamental difference between spectroscopic and molecular mechanics force constants (see Section 3.4) leads to the expectation that some empirical adjustment of the force constants may be necessary even when these force constants have been derived by full normal-coordinate analyses of the infrared data. This is even more important for force constants associated with valence angle deformation (see below). It is unusual for bond-length deformation terms to be altered substantially from the spectroscopically derived values. 

Dynamic processes such as conformational interconversion or bond length deformation associated with changes in electronic or oxidation states have energy barriers associated with them. It is sometimes possible to obtain measures of these barriers, either directly or indirectly, but there are no experimental methods for determining the mechanisms by which these changes occur. Also, if the barriers are low it can be almost impossible to obtain experimental measures of them. Molecular mechanics calculations can be used to obtain theoretically based estimates of the barriers, irrespective of their height, and can also give mechanistic information. 

By molecular mechanics we mean a method by which we calculate the total energy of a molecule in a particular geometry with reference to a hypothetical molecule with no bond-angle or bond-length deformations, no torsional strain and no steric repulsion and with a given number of single and multiple bonds. The energy difference is obtained as the sum of six components ... 

R = Raman, IR = infrared, p = polarized, d - depolarized. Bond length deformation. 

Equation [3] looks complicated at first sight, but is just Eq, [2] to which a cubic term has been added. The cubic constant has the value - 2.00 times the quadratic constant. The factor of 143.88 converts the units to kcal/mol. Judicious. selection of the force constant parameter for this cubic expression allowed for accurate treatments of bond length deformations in a wide variety of molecules. 

Thus there appears to be no average bond length deformation beyond experimental error (2-0 esd). Only two of the bonds deviate from their counterparts by more than 3 esd (4-6 esd in both cases). The conclusion from this example is that crystal packing does not influence bond lengths in hydrocarbons much, if at all, at current levels of accuracy of measurement. 

Dynamic processes such as conformational interconversion or bond length deformation associated with changes in electronic or oxidation states have energy barriers... 

Thus, the theoretical modulus of elasticity depends on the two force constants for the bond length deformation and bond angle expansion, as well as on bond length, bond angle, and molecular cross-sectional area. Thus, polymers with about the same cross-sectional area can have very different moduli of elasticity (Table 11-3). 

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