H Bond Length versus Vibrational Frequency

Over the next several pages we will discnss some properties of the C-H bond and how they are related, and what they can teach us about cheuiistry. But keep this in mind The C-H bond is a typical covalent bond, which means that whatever we can learn about that particular bond is likely to be widely applicable to other covalent bonds. 

It was mentioned earlier (Chapter 2) that the vibrational frequency of a C-H bond depends upon the bond length. Other things being equal, the shorter bond is the stronger bond. Since it is very easy to accurately measure these vibrational fiequencies in most cases, and very difficnlt to measure the corresponding C-H bond lengths by neutron diffraction, or other methods, this relationship has potential uses. 

The relationship for C-H bonds based on McKean s equation has an advantage over Badger s and Herschbach s equations in that McKean recognized the different ways in which the bond lengths can be described (Chapter 2) and always used the same physical quantity, Tq. Combining this equation with the well-known relationship between the stretching force constant (k), vibrational frequency (v), velocity of light (c), and C-H reduced mass (/r) in the harmonic oscillator approximation we find [Eq. (6.2)]  

McKean s Eq. (6.1) is expected to (and does) work well in the absence of serious differences in the steric and electrostatic effects exerted upon the bonds in question. By substituting the molecular mechanics parameter (k) for the actual bond length (ro), Eq. (6.3) can be used in a general way with a molecular mechanics force field. To this end, one also needs to use Eq. (6.3) to determine the force parameter (k,) for the bond under examination, and that replaces the ordinary force constant. The rest of the force field will take care of any unusual steric or electrostatic effects. 

Since the C-H bond is a typical covalent bond, one would expect that this relationship between bond length and vibrational frequency should hold not only for the C-H bond but for covalent bonds in general. It was easier to see the existence of this relationship with the C-H bond because that particular bond shows vibrational frequencies that are well removed from the other bonds in the molecule (because of the small mass of the hydrogen), and the coupling of the particular bond in question with other bonds is normally negligible. 

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