Methane bond vectors

TABLE 10.1 Reduction of the C-H bond vector basis representation in methane into its irreducible components. 

Fig. 11-3.1. A set of vectors v1( va, V , and v4 representing the four o hybrid orbitals used by carbon to bond the four hydrogens in methane.

The dipole moments of molecules are often treated as being equal to the vector sum of the bond dipoles of the various bonds in the molecules. It is almost impossible to measure the dipole moment of an individual bond within a molecule. For example, molecules such as methane, carbon tetrachloride, and p-dichlorobenzene have no dipole moments, whereas molecules such as methylene chloride and m-dichlorobenzene do. The vector sum treatment could be made to agree quantitatively with all known dipole moments if the bond moments were treated as variables that depend on the nature of the particular molecule in which the bonds were located. 

The four linear combinations with Ai and T2 symmetries of the four vectors zi,. .., Z4 forming a bonds with orbitals on atom A may be easily obtained by the technique of projection operators. Therefore, only the results are given in Table 7.1.8, where all linear combinations of ligand orbitals will be listed. It is noted that the four combinations of the z vectors are identical to the combinations of hydrogen Is orbitals obtained for methane. 

Dipole moments are a conceptually simple, sometimes overinterpreted, reflection of molecular charge distributions. In the discussion that follows, it should be remembered that in molecules of high symmetry, the resultant of several bond moment vectors may be equal in magnitude to a single bond moment vector in the opposite direction. For example, methane has a molecular dipole moment of zero. Mathematically this is because the resultant of three C—H moments at mutual angles of 109° 48 is exactly equal and opposite to the fourth bond moment in the molecule. 

A molecular dipole moment is the vector sum of the individual bond dipole moments. Molecular dipole moments are not easy to predict because they depend on the bond angles and other factors that vary with the specific molecule. Table 6-1 lists the experimentally measured dipole moments of the halogenated methanes. Notice how the four symmetrically oriented polar bonds of the carbon tetrahalides cancel to give a molecular dipole moment of zero. 

Equations (3.5), (3.6), and (3.7) enable us to compute the dehydronic field, that is, the mechanical equivalent of the dehydration propensity of hydrogen bonds made along the folding process. This computation requires the evaluation of the gradient (R) = — Vr [4tt (R)]-1 qcf/t of the electrostatic energy with respect to the position vector R of the test hydrophobe (in the simulations we adopted methane as test hydrophobe). This analysis is motivated by the need to support the two-state kinetic picture outlined in Fig. 3.1. To normalize for the number of hydrogen bonds formed at any given time, we computed the dehydronic field per... 

In Section 1.3, we saw that for molecules with one covalent bond, the dipole moment of the bond is identical to the dipole moment of the molecule. For molecules that have more than one covalent bond, the geometry of the molecule must be taken into account because both the magnitude and the direction of the individual bond dipole moments (the vector sum) determine the overall dipole moment of the molecule. Symmetrical molecules, therefore, have no dipole moment. For example, let s look at the dipole moment of carbon dioxide (CO2). Because the carbon atom is bonded to two atoms, it uses sp orbitals to form the C—O a bonds. The remaining two p orbitals on carbon form the two C—O tt bonds. The individual carbon-oxygen bond dipole moments cancel each other— because sp orbitals form a bond angle of 180°—giving carbon dioxide a dipole moment of zero D. Another symmetrical molecule is carbon tetrachloride (CCI4). The four atoms bonded to the sp hybridized carbon atom are identical and project symmetrically out from the carbon atom. Thus, as with CO2, the symmetry of the molecule causes the bond dipole moments to cancel. Methane also has no dipole moment. 

Dipole moments of diatomic molecules can be calculated directly. In more complex molecules, vector addition of the individual bond dipole moments gives the net molecular dipole moment. However, it is usually not possible to calculate molecular dipoles directly from bond dipoles. Table 3.8 shows experimental and calculated dipole moments of chloro-methanes. The values calculated from vectors use C—H and C—Cl bond dipole moments of 1.3 X 10 ° and 4.9 X 10 C m, respectively, and tetrahedral bond angles. Clearly, calculating dipole moments is more complex than simply adding the vectors for individual bond moments. However, for many purposes, a qualitative approach is sufficient. 

Fig. 1.16. Geometries of transition state structures in the reaction of enantiotopomerization of tetrahedral methane as calculated by the MINDO/3 method [20]. Bond lengths are in A, arrows indicate the transition vector components...

Fig. 7.19 Streamlines of the current density vector fieid in methane CH4 on a piane through the C nucleus, normal to the uniform external magnetic fieid B paraiiei to the C3 symmetry axis along a CH bond. The (2, 0) stagnation points are identified via the stagnation graph of Fig. 7.18...

Yurchenko et al. [90] also reported ab initio DMSs of methane computed using the CCSD(T)-F12c/aug-cc-pVTZ-F12 level of theory and the finite field method, with a field of 0.005 a.u., as first derivatives with respect to the external electric field in the dipole approximation. In order to represent these ab initio DMSs ]l of CH4 analytically a symmetrized version of the MB representation (91) was used, where ]i is projected onto three symmetrized combinations of vectors associated with the four molecular bonds C-H, (i= 1, 2, 3, 4). This is analogous to the approach to represent the ab initio dipole moment of NH3, see Eqs. (61-63) from [79]. In order to define the reference axes the following three symmetrically independent reference vectors nr(F = p2x, F2y, F2z) spanning the irreducible representation F2 of d(M) [7] were introduced ... 

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