Atomic number lobes

Fig. 8.34. The isolobal analogy between molecular fragments (symbolized by a double arrow with a loop). All species shown are electrically neutral and are derived conceptually from the first row that contains the generic" compounds CH4, CrLg, Mn(CO)3(C5H4), where L are ligands that offer an electron pair each. The second row is a result of an alchemical transmutation we remove an electron pair with a ligand L, but compensate for it by increasing the atomic number of the central atom by 1 (its additional electron enters the empty lobe). In such a way, all species in the second row have an orbital lobe carrying a single electron and each pair of them is isolobal. Similarly, in the third row, all species share the same two-lobe structure (all are mutually isolobal), while in the fourth row, we have the same three-lobe isolobal structures.

Be careful not to confuse the atomic number Z with the coordinate z.) The exponential term in Eq. (4-24) is spherically symmetric, resembling a diffuse Is orbital. The function z vanishes in the xy plane and becomes increasingly positive or negative as we move away from the plane in either direction. As a result, looks as sketched in Fig. 4-6. It has nearly spherical lobes, one with positive phase and one with negative phase. When i/r2p is squared, the contour lines remain unchanged in relative position, but the function becomes everywhere positive in sign. 

As Increases, the detailed shapes of the p orbitals become more complicated (the number of nodes increases, just as for s orbitals). Nevertheless, the directionality of the orbitals does not change. Each p orbital is perpendicular to the other two in its set, and each p orbital has its lobes along its preferred axis, where electron density is high. To an approaching atom, therefore, an electron in a 3p orbital presents the same characteristics as one in a 2p orbital, except that the 3p orbital is bigger. Consequently, the shapes and relative orientations of the 2p orbitals in Figure 7-22 represent the prominent spatial features of all p orbitals. 

The orbital angular-momentum quantum number, , defines the shape of the atomic orbital (for example, s-orbitals have a spherical boundary surface, while p-orbitals are represented by a two-lobed shaped boundary surface). can have integral values from 0 to (n - 1) for each value of n. The value of for a particular orbital is designated by the letters s, p, d and f, corresponding to values of 0, 1, 2 and 3 respectively (Table 1.2). 

The principal quantum number n is the most important determinant of the radius and energy of the electron atomic orbital. The orbital shape quantum number I determines the shape of the atomic orbital. When / = 1, the atomic orbital is called an s orbital there are two s orbitals for each value of n, and they are spherically symmetric in space around the nucleus. When I = 2, the orbitals are called the p orbitals there are six p orbitals, and they have a dumbbell shape of two lobes that are diametrically opposed. When I = 3 and 4, we have 10 d orbitals and 14 f orbitals. The orbital orientation quantum number m controls the orientation of the orbitals. For the simplest system of a single electron in a hydrogen atom, the most stable wave function Is has the following form ... 

The trick is to make two equivalent orbitals in Be out of the atomic orbitals so that each hydrogen will see essentially the same electronic environment. We can accomplish this by mixing the 2s orbital and one of the empty 2p orbitals (say, the 2p ) to form two equivalent orbitals we call sp" hybrids, since they have both s and p characteristics. As with molecular orbital theory, we have to end up with the same number of orbitals we started with. The bonding lobes on the new spa and spb orbitals on Be are 180° apart, just as we need to form BeH2. In this manner, we can mix any type of orbitals we wish to come up with specific bond angles and numbers of equivalent orbitals. The most common combinations are sp, sp, and sp hybrids. In sp hybrids, one and one p orbital are mixed to get two sp orbitals, both of which... 

Combining four atomic orbitals on the same atom gives the same total number of hybrid orbitals. Each of these has one-quarter s character and three-quarters p character. The sp3 orbital has a planar node through the nucleus like a p orbital but one lobe is larger than the other because of the extra contribution of the 2s orbital, which adds to one lobe but subtracts from the other. 

Fig. 4. Ribbon diagram of human diferric lactoferrin, showing the organization of the molecule, with the N-lobe above and C-lobe below. The four domains (Nl, N2, Cl, C2), the interlobe connecting peptide (H), and the C-terminal helix (C) are indicated. The glycosylation sites in various transferrins are shown by triangles and numbered (1, human transferrin 2, rabbit transferrin 3, human lactoferrin 4, bovine lactoferrin 5 chicken ovotransferrin). The interdomain backbone strands in each lobe can be seen behind the iron atoms. Adapted from Baker et al. (82), with permission.

Fig. 18.4. The 1,3-diene-cyclobutene interconversion. The orbitals shown are not molecular orbitals, but a basis set of p-atomic orbitals, (a) Disrotatory ring closure gives zero sign inversion, (b) Conrotatory ring closure gives one sign inversion. We could have chosen to show any other basis set (e.g., another basis set would have two plus lobes above the plane and two below, etc.). This would change the number of sign inversion, but the disrotatory mode would stiU have an even number of sign inversions, and the conrotatory mode an odd number, whichever basis set was chosen.

A general equation can be derived that describes the variation in direction of the valence electron density about the nucleus. The distortion from sphericity caused by valence electrons and lone-pair electrons is approximated by this equation, which includes a population parameter, a radial size function, and a spherical harmonic function, equivalent to various lobes (multipoles). In the analysis the core electron density of each atom is assigned a fixed quantity. For example, carbon has 2 core electrons and 4 valence electrons. Hydrogen has no core electrons but 1 valence electron. Experimental X-ray diffraction data are used to deri e the parameters that correspond to this function. The model is now more complicated, but gives a better representation of the true electron density (or so we would like to think). This method is useful for showing lone pair directionalities, and bent bonds in strained molecules. Since a larger number of diffraction data are included, the geometry of the molecular structure is probably better determined. 

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