The Methane Molecule

The four (unnormalized) bonding MOs are then appropriately written as  

The calculation of the matrix elements follows as usual, giving  

The lowest roots of the secular equations will give the Hiickel energy of the eight valence electrons as a function of angle 6  

The resulting MOs are symmetry MOs, delocalized over the entire molecule. As it will be shown for H20, a description more adherent to the chemical picture of four localized C—H bonds can be obtained in terms of the orthogonal transformation16 connecting occupied MOs, the first relation being  

We now apply our model to investigate the formation of four C-H bonds in the methane molecule CH4 of symmetry Tj (Magnasco, 2004a). For this molecule, tetrahedral sp hybridization is completely determined by molecular symmetry. We use the usual notation for the eight valence AOs, calling s, x, y, z the 2s and 2p orbitals on C, and, 4 the Is orbitals 

---

In general, each nomial mode in a molecule has its own frequency, which is detemiined in the nonnal mode analysis [24]- Flowever, this is subject to the constraints imposed by molecular synmietry [18, 25, 26]. For example, in the methane molecule CFI, four of the nonnal modes can essentially be designated as nonnal stretch modes, i.e. consisting primarily of collective motions built from the four C-FI bond displacements. The molecule has tetrahedral synmietry, and this constrains the stretch nonnal mode frequencies. One mode is the totally symmetric stretch, with its own characteristic frequency. The other tliree stretch nonnal modes are all constrained by synmietry to have the same frequency, and are refened to as being triply-degenerate. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Here we present some results from studies on systems with N 2,3, and 4. (N=1 would correspond to the methane molecule). The molecules are shown in Figure 8. 

Measuring and Using Numbers Compare the measured bond angle in the methane molecule to the accepted bond angle, 109.5°. Account for any differences in the two values. 

The dissociation energy for C-H bond in methane (E = 436 kj/mol) is one of the highest among all organic compounds. Its electronic structure (i.e., the lack of n- and n-electrons), lack of polarity, and any functional group makes it extremely difficult to thermally decompose the methane molecule into its constituent elements. 

The mechanism of thermal decomposition (pyrolysis) of methane has been extensively studied [90,91]. Because C-H bonds in the methane molecule are significantly stronger than C-H and C-C bonds of the products, the secondary and tertiary reactions contribute at the very early stages of the reaction, which obscure the initial processes. According to Holmen et al. [92], the overall methane thermal decomposition reaction at high temperatures can be described as a stepwise dehydrogenation as follows ... 

When a covalent bond breaks to produce radicals, i.e. one electron of the bond pair goes to each atom, homolytic fission has occurred. These highly reactive chlorine radicals attack the methane molecules. 

In many gaseous state reactions of technological importance, short-lived intermediate molecules which are formed by the decomposition of reacting species play a significant role in the reaction kinetics. Thus reactions involving the methane molecule, CH4, show the presence of a well-defined dissociation product, CH3, the methyl radical, which has a finite lifetime as a separate entity and which plays an important part in a sequence or chain of chemical reactions. 

Figure 19. Interactions between the jt MOs of the two double bonds and the pseudo-jt orbitals of the methano group in 20 and the methane molecule in 21.

For example, in the methane molecule (CH4), the four sp3 hybrid orbitals of the carbon atom overlap end to end with one Is orbital from each hydrogen atom to form four C — H bonds. Those bonds are all o bonds. 

This is the type of hybridization of the carbon atom in the methane molecule. 

However, if you link the hydroxyl group with the methane molecule rather than the ethane, you get the potentially toxic chemical called methyl alcohol, or wood spirit. Similarly, if you add what s called an aldehyde group (-CHO) instead of the hydroxyl group, you will get one of a variety of chemicals called aldehydes, of which a common one is the gas formaldehyde (HCHO), widely used in the manufacture of plastics and glues. This gas can be an irritant and potentially dangerous if inhaled. 

The distinction between these two terms may be more evident if described in terms of a simple example, the C-H bond. The enthalpy of dissociation of the C-H bond depends on the nature of the molecular species from which the H atom is being separated. For example, in the methane molecule... 

A molecule of methane contains just five atoms one of carbon and four of hydrogen. In chemical representations of molecules, each element is identified by a symbol. Carbon is represented by the symbol C hydrogen is represented by the symbol H. Thus, the molecular formula for methane is CH4. This representation, or model, tells us just one simple fact the methane molecule contains one carbon and four hydrogen atoms. ... 

A great deal of experience informs us that carbon atoms frequently form molecules in which they are simultaneously linked to four other atoms, hi contrast, hydrogen atoms are usually linked to only one other atom. It follows that the methane molecule is structured with a central carbon atom to which are bonded the four hydrogen atoms. In such a structure, only carbon-hydrogen (C-H) bonds exist. We can represent methane in the following way, in which the solid lines are symbols for chemical bonds ... 

The structure of the methane molecule is that of a regular tetrahedron, with the carbon atom in the center and the four hydrogen atoms at the four comers. 

The methane molecule is a very important molecule in organic chemistry, the geometry around the tetravalent carbon atom being basic to the understanding of the structure, isomerism and optical activity of a very large number of compounds. It is a tetrahedral molecule belonging to the tetrahedral point group, Td. 

The carbon two-valent state is converted to its four-valent state by unpairing the 2s electrons and promoting one of them to the third 2p orbital. The electrons from the four hydrogen atoms (red) make up four pairs of or bonding electrons, which repel each other to a position of minimum repulsion to give the tetrahedral shape for the methane molecule. 

Table 6.2 Symmetries of the valence orbitals of the carbon atom and the group orbitals of the hydrogen atoms of the methane molecule...

Figure 3.5 Chemists use the structural formula to show a visual layout of the elements and bonds present in a compound. The molecular model shown here provides a three-dimensional view of the arrangement of atoms in a compound as expressed by the compound s chemical formula. For example, the molecular formula of the methane molecule above illustrates the chemical formula CH. ...

Substituting a diird chlorine on the methane molecule results in the compound whose proper name is trichloromediane (tri- for three chloro- for chlorine and methane, the hydrocarbon s name for die one-carbon chain). It is more commonly known as chloroform. Its molecular formula is CHC13. Chlorofonn is a heavy, colorless, volatile liquid with a sweet taste and characteristic odor. It is classified as non-flammable, but it will bum if exposed to high temperatures for long periods of time. It is narcotic by inhalation and toxic in high concentrations. It is an insecticide and a fumigant and is very useful in the manufacture of refrigerants. 

The mechanism of cage-to-cage diffusion was investigated by plotting the distance between the center of mass of the methane molecule and the center of the parent supercage versus the distance to the center of the daughter supercage (as has been described for Xe diffusion in NaY zeolite). 

It was found that the diffusion of methane from one cage to another takes place by the surface-mediated route, whereby the methane molecule skates along the inner surface of the cages. At 300 K, the residence times of a methane molecule within a given cage were found to be of the order of a few picoseconds. This is substantially shorter than the residence time of Xe (14), notwithstanding their similar sizes. The reason for this difference lies in the very different masses of the two sorbates. 

Bandyopadhyay and Yashonath (31), in an extension of their work on MD studies of noble gas diffusion, presented MD results for methane diffusion in NaY and NaCaA zeolites. The zeolite models were the same as those used in the noble gas simulations (13, 15, 17, 18, 20, 28, 29) and the zeolite lattice was held rigid. The methane molecule was approximated as a single interaction center and the guest-host potential parameters were calculated from data of Bezus et al. (49) (for the dispersive term) and by setting the force on a pair of atoms equal to zero at the sum of their van der Waals radii (for the repulsive term). Simulations were run for 600 ps with a time step of 10 fs. 

---