Nodal planes

Electrons m a tr bond are called it electrons The probability of finding a tt elec Iron IS highest m the region above and below the plane of the molecule The plane of the molecule corresponds to a nodal plane where the probability of finding a tt electron IS zero... 

Figure 114 also shows fhe orbifal overlaps and nodal properties of fhe benzene MOs Recall fhaf a wave funclion changes sign on passing fhrough a nodal plane and is... 

For all orbitals except s there are regions in space where 0, ) = 0 because either Yimt = 0 or = 0. In these regions the electron density is zero and we call them nodal surfaces or, simply, nodes. For example, the 2p orbital has a nodal plane, while each of the 3d orbitals has two nodal planes. In general, there are I such angular nodes where = 0. The 2s orbital has one spherical nodal plane, or radial node, as Figure 1.7 shows. In general, there are (n — 1) radial nodes for an ns orbital (or n if we count the one at infinity). 

The ordering in Figure 7.31 of the MOs in terms of energy follows the rules that decreased x character and increased number of nodal planes increase the energy, but some details rely on experimental data. 

Each axial mode has its own characteristic pahem of nodal planes and the frequency separation Av between modes is given by Equation (9.4). If the radiation in the cavity can be modulated at a frequency of cjld then the modes of the cavity are locked both in amplitude and phase since t, the time for the radiation to make one round-trip of the cavity (a distance 2d), is given by... 

The phenyl cation is an extremely unstable cation, as is reflected by the high hydride affinity shown in Table 5.2. In this case, the ring geometry opposes rehybridization so the vacant orbital retains sp character. Because the empty orbital is in the nodal plane of the ring, it receives no stabilization firom the n electrons. 

The HOMO-1 is localized on the central rings and has no contribution from the phenyl rings. There are nodal planes running perpendicular to the viewing plane every 45 degrees. There is another nodal plane that is the molecular plane. 

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

It is notable that pyridine is activated relative to benzene and quinoline is activated relative to naphthalene, but that the reactivities of anthracene, acridine, and phenazine decrease in that order. A small activation of pyridine and quinoline is reasonable on the basis of quantum-mechanical predictions of atom localization encrgies, " whereas the unexpected decrease in reactivity from anthracene to phenazine can be best interpreted on the basis of a model for the transition state of methylation suggested by Szwarc and Binks." The coulombic repulsion between the ir-electrons of the aromatic nucleus and the p-electron of the radical should be smaller if the radical approaches the aromatic system along the nodal plane rather than perpendicular to it. This approach to a nitrogen center would be very unfavorable, however, since the lone pair of electrons of the nitrogen lies in the nodal plane and since the methyl radical is... 

Node (Section 1.2) A surface of zero electron density within an orbital. For example, a p orbital has a nodal plane passing through the cenler of the nucleus, perpendicular to the axis of the orbital. 

The transition state for disproportionation requires overlap of the p C—H bond undergoing scission and the p-orbital containing the unpaired electron.18 This requirement rationalizes the specificity observed in disproportionation of radicals 29 (Section 1.4,2) and provides an explanation for the persistency of the triisopropylmcthyl radical (33) and related species (Section 1.4.3.2).166 In the case of 33, the P-bydrogens are constrained to lie in the nodal plane of the p-orbital due to stcric buttressing between the methyls of the adjacent isopropyls. 

All p-orbitals have boundary surfaces with similar shapes, including one nodal plane. Note that the orbital has opposite signs (as depicted by the depth of color) on each side of the nodal plane. 

We encounter a different type of bond in a nitrogen molecule, N2. There is a single electron in each of the three 2p-orbitals on each atom (33). However, when we try to pair them and form three bonds, only one of the three orbitals on each atom can overlap end to end to form a (T-bond (Fig. 3.10). Two of the 2/7-orbitals on each atom (2px and 2py) are perpendicular to the internuclear axis, and each one contains an unpaired electron (Fig. 3.11, top). When the electrons in one of these p-orbitals on each N atom pair, the orbitals can overlap only in a side-by-side arrangement. This overlap results in a TT-bond, a bond in which the two electrons lie in two lobes, one on each side of the internuclear axis (Fig. 3.11, bottom). More formally, a 7T-bond has a single nodal plane containing the internuclear axis. Although a TT-bond has electron density on each side of the internuclear axis, it is only one bond, with the electron cloud in the form of two lobes, just as a p-orbital is one orbital with two lobes. In a molecule with two Tr-bonds, such as N2, the... 

FIGURE 3.10 A (T-bond is formed by the pairing ol electron spins in two 2p7-orbitals on neighboring atoms. At this stage, we are ignoring the interactions of any 2p,-(and 2p -) orbitals that also contain unpaired electrons, because they cannot form electron pair may be found anywhere within the boundary surface shown in the bottom diagram. Notice that the nodal plane of each p7-orbital survives in the tr-bond. 

FIGURE 3.30 Two p-orbitals can overlap side by side to give bonding (below) and antibonding (above) u-orbitals. Note that the latter has a nodal plane between the two nuclei. Both orbitals have a nodal plane through the two nuclei and look like p-orbitals when viewed along the internuclear axis. 

The two atomic orbitals that contribute to the antibonding orbital in Eq. 2 are each proportional to e r a°, where r is the distance of the point from its parent nucleus. Confirm that there is a nodal plane lying halfway between the two nuclei. 

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