Shell orbital combination

When we discussed sp3 hybrid orbitals in Section 1.6, we said that the four valence-shell atomic orbitals of carbon combine to form four equivalent sp3 hybrids. Imagine instead that the 2s orbital combines with only two of the three available 2p orbitals. Three sp2 hybrid orbitals result, and one 2p orbital remains unchanged- The three sp2 orbitals lie in a plane at angles of 120° to one another, with the remaining p orbital perpendicular to the sp2 plane, as shown in Figure 1.13. 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

The HOMO, 4ai (only valence shell orbitals are numbered), represents a combination of halogen p AOs destabilized by an antibonding interaction with a central atom sp hybrid orbital. Thus this MO is antibonding E—Hal MO in nature and has quite a large contribution from a central atom valence s orbital. This explains the quite low IEs observed for ionizations from 4a 1 MO, incompatible with a lone-pair orbital mainly localized on the central atom. This MO determines the Lewis base properties of the central atom in EHal2. 

As an example, consider methane. If the carbon atom L-shell orbitals are arranged as tetrahedral hybrids, we can take the tat, t td configuration and combine this with an 3aSbScSd configuration of the four hydrogen atoms. Table 1 shows some numbers of states associated with these orbitals. It is... 

But how do we account for the bond angles in water (104°) and ammonia (107°) when the only atomic orbitals are at 90° to each other All the covalent compounds of elements in the row Li to Ne raise this difficulty. Water (H2O) and ammonia (NH3) have angles between their bonds that are roughly tetrahedral and methane (CH4) is exactly tetrahedral but how can the atomic orbitals combine to rationalize this shape The carbon atom has electrons only in the first and second shells, and the Is orbital is too low in energy to contribute to any molecular orbitals, which leaves only the 2s and 2p orbitals. The problem is that the 2p orbitals are at right angles to each other and methane does not have any 90° bonds. (So don t draw any either Remember Chapter 2.). Let us consider exactly where the atoms are in methane and see if we can combine the AOs in such a way as to make satisfactory molecular orbitals. 

The total wave function for open-shell systems [8] is, in general, a sum of several antisymmetrized products, each of which contains a closed-shell core and a partially occupied open shell. The combinal set of orbitals is defined by... 

Particular geometries (spatial orientations of atoms in a molecule) can be related to particular bonding patterns in molecules. These bonding patterns led to the concept of hybridization, which was derived from a mathematical model of bonding. In that model, mathematical functions (wave functions) for the s and p orbitals in the outermost electron shell are combined in various ways (hybridized) to produce geometries close to those deduced from experiment. 

We learned in Chapter 5 that an isolated atom has its electrons arranged in orbitals in the way that leads to the lowest total energy for the atom. Usually, however, these pure atomic orbitals do not have the correct energies or orientations to describe where the electrons are when an atom is bonded to other atoms. When other atoms are nearby as in a molecule or ion, an atom can combine its valence shell orbitals to form a new set of orbitals that is at a lower total energy in the presence of the other atoms than the pure atomic orbitals would be. This process is called hybridization, and the new orbitals that are formed are called hybrid orbitals. These hybrid orbitals can overlap with orbitals on other atoms to share electrons and form bonds. Such hybrid orbitals usually give an improved description of the experimentally observed geometry of the molecule or ion. 

For benzene, the molecular orbital theory states that the six p-orbitals combine to give six molecular orbitals. The three lower-energy molecular orbitals are bonding molecular orbitals, and these are completely filled by the six electrons (which are spin-paired). There are no electrons in the (higher-energy) antibonding orbitals, and hence benzene has a closed bonding shell of delocalised Jt-electrons. 

The strongest pieces of evidence complex atoms provide in favour of independent electron modes and simple Bohr-Sommerfeld quantisation are (i) the existence of Rydberg series and (ii) the regularity of the periodic table of the elements. As a corollary, we should look for quantum chaos (if it occurs) in atoms for which there is some breakdown in the quality of the shell structure, combined with prolific and heavily perturbed overlapping series of interacting levels. These conditions are most readily met, as will be shown below, in the spectra of the alkaline-earth elements, as a result of d-orbital collapse. 

For the E = S substate of a A > 0 state, construct Slater determinants by combining the spin functions specified in Table 3.4 with the open-shell orbital products from step 1 above, always in the same order. Each spin... 

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