Hybrid orbitals INDEX

We say that two hybrid orbitals are equivalent if they have the same hybridization index m. Recalling that the squares of the coefficients of a given orbital summed over the whole hybridized set must equal unity, we can easily see that... 

To confirm that hi, hi, and hs are equivalent to each other, we can calculate their hybridization indices and see that they are identical. The hybridization index n of a hybrid orbital is defined as... 

The introduction of hybrid-based MO s into the theoretical treatment of paraffinic hydrocarbons 50> has made it possible to extend the applicability of Eqs. (3.21) and (3.24) to a wide variety of saturated compounds. The index has been called delocalizability" of an atomic orbital and is defined by the following formulae 60 68> ... 

Stepped surfaces withstand cyclic oxidation-reduction treatments (146) like [111] and some other low-index planes. Steps have either [311] or [110] structures. They are claimed to be the only places where orbital hybridization does not take place (136). No wonder that such platinum (138) and iridium (147) surfaces have enhanced activity in Cg dehydrocyclization of n-heptane. 

Del Re and coworkers [131] were concerned with the relation of s character in hybrids to bond angles and have considered hybridization as described by local orbitals, determined by requiring that hybrids on different atoms have minimal overlap unless they participate in the same bond. Alternate approaches are provided by the bond index of Wiberg [132] and by the Trindle-Sinanoglu procedure [133] for the application of the physical criterion of Lennard-Jones and Pople [134,135], requiring that an electron in a localized orbital interact maximally with the electron sharing that orbital. 

Reed and Allen, using their bond polarity index, have assigned values of 0.000, 0.027, and 0.050, respectively (compared to H —0.032 and F 0.189) [108]. Without attempting to be too quantitative, convenient values of the core energies of hybrid atomic orbitals, in units, recognizing that changes in coordination number also occur, are approximately... 

The reason for expressing the relative contributions ofj and p in terms of the awkward-looking factors a/ 1/(1 + m) and Vm/(l + m) is that this form guarantees that the generalized hybrid A 1.9 will be normalized. (See Problem 1.7.) We have thus built in automatically one of the restraints on our hybrids. The quantity m is the hybridization index, and is the number that appears as the superscript in the standard designation of hybrid type. Thus an sp3 orbital, m = 3, always has the form... 

It is an easy matter to write the correct expressions for a pair of equivalent hybrids with a given angle a between them. Hybridization index m is found immediately from Equation A1.17. A direction (0ly i) must be chosen for the first hybrid, and a direction (02, < 2) for the second found such that the angle between them will be a. The orientation of the hybrids with respect to the coordinate system is arbitrary it will be easiest to set up the orbitals if they are oriented so that the first points along one of the axes (say x) and the second lies in one of the coordinate planes (say x, y), or if the two are placed in one of the coordinate planes with a coordinate axis bisecting the angle a. 

It can be easily checked that the hybrids given in either eq. (3.4.35) or eq. (3.4.36) are sp3 hybrids i.e., their hybridization index (population ratio of p orbitals and s orbital) is 3. The four sp3 hybrids of C are directed toward the corners of a tetrahedron and suitable for forming four localized bonding orbitals with four hydrogen Is orbitals. Such a bonding picture for CH4 is familiar to all chemistry students. 

To make this bonding description quantitative, it is useful to describe an sp orbital in terms of its hybridization index, n, and its hybridization parameter, 2, defined as X = n. Now the fractional s character of the ith orbital is given by the expression 1 /(I + 2 ). For example, die fraction of... 

The view of variable hybridization has some experimental support. The magnitude of C- H NMR coupling constants is expected to be proportional to the amount of carbon s character in the bond, because only s orbitals have density at the carbon nucleus and can affect neighboring nuclear spin states. In several systems, a clear correlation has been observed between NMR coupling constants and percent s character, as predicted from the geometry and the associated hybridization index. 

The overlap integrals between the carbon valence orbitals and the hydrogen Is orbital decreases in the order of HCsCH, CH2=CH2, and CH4 (with the hybridization index for the carbon atom equal to 1, 2, and 3, respectively). Assuming that the C-H bond strength increases with the overlap integral, we can conclude that the overlap integrals determined in (i) correlate well with the given spectral... 

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