Molecular orbitals of methane

Fig. 1.12 The molecular orbitals of methane from the interaction of the orbitals of tetrahedral H4 and a C atom...

Figure 1 Molecular orbitals of methane with l)4h symmetry...

Fig. 1.15 One contour of the wave function for the four filled molecular orbitals of methane...

Mixing of atomic orbitals on carbon with hydrogen Is orbitals to make molecular orbitals of methane. (Adapted from reference 138.)... 

Graphics with semiempirical MO calculations using a CAChe WorkSystem. For a more complete discussion of the molecular orbitals of methane, see Jorgensen, W. L. Salem, L. The Organic Chemist s Book of Orbitals Academic Press New York, 1973 p. 68. 

This orientation of the molecule reveals that methane possesses three twofold symmetry axes, one each along the x, y, and z axes. Because of this molecular symmetry, the proper molecular orbitals of methane must possess symmetry with respect to these same axes. There are two possibilities the orbital may be unchanged by 180° rotation about the axis (symmetric), or it may be transformed into an orbital of identical shape but opposite sign by the symmetry operation (antisymmetric). The carbon 2s-orbital is symmetric with respect to each axis, but the three 2p-orbitals are each antisymmetric to two of the axes and symmetric with respect to one. The combinations which give rise to molecular orbitals that meet these symmetry requirements are shown in Fig. 1.11. 

Figure A11.1 The radial functions used with s-type basis functions for C atoms in the 6-31C basis set. a) The six primitive Caussians (dashed lines) are shown scaled by their contraction coefficients (dp in equation Al 1.2). Their sum gives the contracted function (solid bold line) used for the core region, b) The three primitive Caussians (dashed lines) scaled by the contraction coefficients and the contracted function (solid bold line) used for the valence region, c) Example use of all three basis functions to form the C(2s) atomic orbital in the 2a, molecular orbital of methane. The three basis functions are shown as dashed lines scaled by the SCF coefficients given in the formula. The resulting summed radial function is shown as the bold solid line.

Such a transformation can be used for relocalizing a given set of delocalized molecular orbitals in conformity with the chemical formula. For instance, the occupied orbitals of methane can be transformed into orbitals very close to simple two-center MO s constructed from tetrahedral sp3 hybrid orbitals and Is hydrogen orbitals 24,25,26) a. unitary transformation can hardly modify the wave function, except for an immaterial phase factor therefore, it leads to a description which is as valid as that in terms of the canonical delocalized Hartree-Fock orbitals. Of course, the localization obtained in this way is not perfect, but it is usually much better than is often believed. In the case of methane, the best localized orbitals are uniquely determined by symmetry 27> for less symmetric molecules one needs a criterion for best localization 28 29>, a problem on which we shall not insist here. A careful inspection reveals that there are three classes of compounds ... 

The orbitals of methane, CH4, and those of the related fragments CH3, CH2, and CH can be described using the molecular orbital method, as we have done for all the systems studied so far in this book. But the valence-bond approach, introduced by L. Pauling, can also be used this is perhaps the simplest way to establish an initial relationship between the electronic structures of organic and inorganic fragments. 

FIGURE 1.14 The hypothetical formation of methane from an sp -hybrldlzed oarbon atom and four hydrogen atoms. In orbital hybridization we combine orbitals, not electrons. The electrons can then be placed In the hybrid orbitals as necessary for bond formation, but always In accordanoe with the Pauli principle of no more than two eleotrons (with opposite spin) in each orbital. In this illustration we have placed one electron in each of the hybrid carbon orbitals. In addition, we have shown only the bonding molecular orbital of each C—H bond because these are the orbitals that contain the electrons in the lowest energy state of the molecule. 

The bond angles at the carbon atoms of ethane, and of all alkanes, are also tetrahedral like those in methane. A satisfactory model for ethane can be provided by ry) -hybridized carbon atoms. Figure 1.19 shows how we might imagine the bonding molecular orbitals of an ethane molecule being constructed from two ry) -hybridized carbon atoms and six hydrogen atoms. 

Figure 3.6 shows the LCAO method for generating molecular orbitals of diatomic molecules such as H2. In real molecules, the atomic orbitals of elemental carbon are not really transformed into the molecular orbitals found in methane (CH4). Figure 3.6 represents a mathematical model that mixes atomic orbitals to predict molecular orbitals. Molecular orbitals exist in real molecules and the LCAO model attempts to use known atomic orbitals for atoms to predict the orbitals in the molecule. Molecular orbitals and atomic orbitals are very different in shape and energy, so it is not surprising that the model used for diatomic hydrogen fails for molecules containing other than s-orbitals. 

Hence we have two molecular orbitals, one along the line of centres, the other as two sausage-like clouds, called the n orbital or n bond (and the two electrons in it, the n electrons). The double bond is shorter than a single C—C bond because of the double overlap but the n electron cloud is easily attacked by other atoms, hence the reactivity of ethene compared with methane or ethane. 

For a molecule as simple as Fl2, it is hard to see much difference between the valence bond and molecular orbital methods. The most important differences appear- in molecules with more than two atoms. In those cases, the valence bond method continues to view a molecule as a collection of bonds between connected atoms. The molecular- orbital method, however, leads to a picture in which the sane electron can be associated with many, or even all, of the atoms in a molecule. We ll have more to say about the similarities and differences in valence bond and molecular- orbital theory as we continue to develop their principles, beginning with the simplest alkanes methane, ethane, and propane. 

The development of molecular orbital theory (MO theory) in the late 1920s overcame these difficulties. It explains why the electron pair is so important for bond formation and predicts that oxygen is paramagnetic. It accommodates electron-deficient compounds such as the boranes just as naturally as it deals with methane and water. Furthermore, molecular orbital theory can be extended to account for the structures and properties of metals and semiconductors. It can also be used to account for the electronic spectra of molecules, which arise when an electron makes a transition from an occupied molecular orbital to a vacant molecular orbital. 

FIGURE 3.37 The molecular orbital energy-level diagram for methane and the occupation of the orbitals by the eight valence electrons of the atoms. 

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