Atomic orbitals nodal surfaces

CT and the antibondmg orbital ct ( sigma star ) The bonding orbital is characterized by a region of high electron probability between the two atoms while the antibondmg orbital has a nodal surface between them... 

Front-side approach is disfavored both because the density of the a orbital is less in the region between the carbon and the leaving group and because front-side approach would involve both a bonding and an antibonding interaction with the tr orbital since it has a nodal surface between the atoms. 

It can be shown, from wave-mechanical calculations, that the Is orbital (quantum numbers n = 1, Z = 0, m = 0, corresponding to the classical K shell) is spherically symmetrical about the nucleus of the atom, and that the 2s orbital (quantum numbers n = 2, Z = 0, m = 0) is similarly spherically symmetrical, but at a greater distance from the nucleus there is a region between the two latter orbitals where the probability of finding an electron approaches zero (a spherical nodal surface). 

Molecular orbitals will be very irregular three-dimensional functions with maxima near the nuclei since the electrons are most likely to be found there and falling off toward zero as the distance from the nuclei increases. There will also be many zeros defining nodal surfaces that separate phase changes. These requirements are satisfied by a linear combination of atom-centered basis functions. The basis functions we choose should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei since, when the electron is close to one atom and far from the others, its distribution will resemble an AO of that atom. And yet they should be simple enough that mathematical operations required in the solution of the Fock equations can actually be carried out efficiently. The first requirement is easily satisfied by choosing hydrogenic AOs as a basis... 

A nodal plane or surface is the locus of all points at which a wave-function has zero amplitude as a result of its changing sign on passing from one side of the surface to the other the probability of finding an electron on such a surface is zero. o-Bonds and o-orbitaJs are defined as those having no nodal surface which contains the bond axis such bonds and orbitals will be symmetric about the bond axis. In this section we consider which AOs of a central atom A (which is bonded to a set of other atoms) can be combined to form a hybrid orbital which is symmetric about the bond axis and therefore capable of cr-bonding. 

In contrast to o -orbitals and o-bonds, a tt-orbital or w-bond is defined as one which has one nodal surfaoe or plane containing the bond axis. (The reader might note that, though we will not deal with them, d-orbitals and d-bonds have two nodal surfaces which intersect on the bond axis.) If a w-bond is to be formed in MO fashion by combining two orbitals, one on each of the two bonded atoms, it is obviously necessary that each orbital have 7r-character with respect to the bond axis and that their two nodal planes coincide. The formation of a w-type bonding MO from two ir-type AOs is shown in Fig. 11-4.1, where the plus and minus signs refer to the sign of the wavefunction. 

It is conventional to label the internuclear axis in a diatomic molecule as z. Thus the three 2p(F) orbitals can be labelled 2p, 2py and 2p2 2px and 2py have their lobes directed perpendicular to the internuclear axis, and have nodal surfaces containing that axis, while 2pr clearly overlaps in o fashion with ls(H). (The reader may wonder whether this orientation of 2p, 2py and 2pz is obligatory, or whether it is chosen for convenience. For a spherically-symmetric atom, there are no constraints in choosing a set of three Cartesian axes. Any set of orthogonal p orbitals can be transformed into another equally acceptable set, by a simple rotation which does not change the electron density distribution of the atom. The overlap integral between a hydrogen Is orbital and the set of three 2p(F) orbitals is the... 

We now consider the possibility of n bonding. If the ligands have a full set of orbitals for this purpose - i.e. if there are two suitable orbitals on each L atom, p or d orbitals each with a nodal surface containing the M-L axis - it is easy to show that the orbitals can now engage in n... 

Figure 18 Contour plots of the ns and np atomic orbitals of C, Si, and Sn (contour values 0.0, 0.02, 0.05, 0.10, 0.2, 0.5 bohr 3/2 nodal surfaces dash-dotted). Dots indicate the positions of the nuclei in the corresponding AH radical. Ge 4s and 4p are not shown they are only slightly larger than Si 3s and 3p.

The V2 factor is for normalization, just as in Figure 6.3. One orbital points more to the right in Figure 6.8, so it can overlap well with the hydrogen atom to the right the other orbital overlaps well with the hydrogen atom to the left. Notice that the nodal surface (the surface between the positive and negative lobes where f = 0) for these orbitals is neither a plane (as it is for the 2p orbitals) nor a sphere (as it is for the 2s orbital). 

There is no quantum-mechanical evidence for the localization of electron pairs between atomic nuclei, and atomic orbitals, in so far as they correspond to spherical surface harmonics, have their nodal curves in the surface of the density sphere. Sets of real hybrid orbitals are physically undefined. To understand intramolecular interactions as a quantum phenomenon it is necessary to approach the problem with the minimum of assumptions and to state all essential assumptions clearly and precisely at the outset. 

Figure 6. Coordinate system (a) and nodal surfaces (b) and (c) for a saturated ketone R—CO—R. The nodal plane (XZ) of the n orbital (b) bisects the R—C—R angle and is perpendicular to the plane of the ketone. The nodal surfaces of the x orbital (c) where the plane of the carbonyl group is a nodal plane (YZ, horizontal) and there is another nodal surface (A, vertical), not necessarily a plane, perpendicular to the C=0 axis and intersecting it between the carbon and oxygen atoms, [(b) and (c) Redrawn and modified from J.B. Lambert, H.F. Shurvell, D. Lightner and R.G. Cooks, Introduction to Organic Spectroscopy, Macmillan, New York, 1987.]...

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