Nonbonding molecular orbital coefficients

In any nonbonding molecular orbital (NBMO), the sum of the coefficients of the atoms s adjacent to a given atom r is zero. Hence the NBMO coefficients can be calculated very easily. The benzyl radical provides a nice example. All of the non-starred atoms have coefficients of zero. We give the para atom an arbitrary coefficient a. The sum of the coefficients of the atoms adjacent to the meta carbons must be zero, so the ortho coefficients are —a. For the sum of the coefficients around the ipso carbon to cancel, the benzylic carbon coefficient must be 2a. The value for a is given by the normalization condition ... 

A similar study for H2O (ref. 111), now including both bonding and nonbonding m.o.s, leads to two equivalent molecular orbitals localized on O (which are approximately sp hybrids) and to two equivalent molecular orbitals localized at each 0-H bond. The latter are, approximately, the result of the superposition of the Is orbital of H with an sp hybrid orbital of O (along with a small contribution from the Is orbital of the other H atom). For an sp orbital, the square of the coefficient of 2p is four times that for 2s. 

Spin densities (p) are theoretical quantities, defined as the sum of the squared atomic orbital coefficients in the nonbonding semi-occupied molecular orbital (SOMO) of the radical species (Hiickel theory). For monoradical species, the spin density is connected to the experimental EPR hyperfine coupling constant a through the McConnell equation [38]. This relation provides the opportunity to test the spin density dependence of the D parameter [Eq. (8)] for the cyclopentane-1,3-diyl triplet diradicals 10 by comparing them with the known experimental hyperfine coupling constants (ap) of the corresponding substituted cumyl radicals 14 [39]. The good semiquadratic correlation (Fig. 9) between these two EPR spectral quantities demonstrates unequivocally that the localized triplet 1,3-diradicals 9-11 constitute an excellent model system to assess electronic substituent effects on the spin density in cumyl-type monoradicals. 

Fig. 5.1 Correlating the atomic orbitals (AOs) of xenon and of two fluorines with the molecular orbitals (MOs) ofXep2. The two fluorines are bonded to the xenon by a three-center, two-electron bond corresponding to the molecular orbital (MO). This axial MO has electron density on all three atoms as shown by the fact that the AO coefficients in this MO are all nonzero. The other two axial electrons reside in the MO xl/i, this is nonbonding as shown by the feet that the xenon coefficient in this MO is zero, i.e. 2 contributes electron density only to the two fluorines. The antibonding MO is empty. The octet rule is clearly not violated. The situation is analogous to that in NF5 (Chapter 4, Fig. 4.5)...

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