Methane, symmetry point group

A symmetry analysis of the basis orbitals for methane (T point group) identifies the representations for the central... 

Linearly polarized light can be assigned a specific irreducible representation of a symmetry point group. If the electronic ground state of methane has A,g symmetry and x-polarized light has the label T, what are the symmetry label(s) of allowed excited electronic states Use the character table in Appendix 3. 

For methane, the vectors point at the comers of a tetrahedron or at alternate corners of a cube. Using the point group, we can use these four vectors as the basis of a reducible representation. As usual, the character for each vector is 1 if it remains unchanged by the symmetry operation, and 0 if it changes position in any other way (reversing direction is not an option for hybrids). The reducible representation for these four vectors is then r = Aj + T. ... 

We consider first groups lacking a C axis. A molecule with no elements of symmetry (other than the identity E) belongs to the point group C. The identity element is equivalent to Ci, a rotation by In/l radians or 360°. Very few small molecules are this unsymmetrical—one example is the trisubstituted methane ... 

As a straightforward example we can calculate the ( ieff)H of any hydrogen atom of methane. The axis system is most conveniently oriented through the carbon atom along the three S4 axes of improper rotation that characterise the point group symmetry of this Td molecule. In this axis system each hydrogen atom is equidistant from each axis, r, and the rotational inertia about each axis is equivalent, 1= 4/mh Then... 

To take a concrete example, any MO calculation of the electronic structure of the ethene (ethylene) molecule wiU generate two lowest MOs (almost degenerate) which are just the in-phase and out-of-phase linear combinations of the basis functions used to describe the Is shells of the carbon atoms. The fact that they occur as molecular orbitals rather than remaining actually unchanged as atomic orbitals is simply an artifact of the symmetry of ethene the MOs are computed as symmetric or antisymmetric with respect to the operations of the point group which in this case includes reflection in a plane perpendicular to the C—C axis. If a calculation is carried through on the isoelectronic methanal (formaldehyde) molecule the oxygen Is AO and the carbon Is AO survive the calculation almost unscathed as the lowest MOs . 

Methane. The AOs for a minimal-basis-set SCF calculation of CH4 are the carbon Is, 2s, 2p 2py, and 2p AOs and a Is AO on each hydrogen atom. The point group of CH4 is S rf. Group theory (see Cotton or Schonland) gives the possible symmetry... 

As a simple illustration we choose methane CH4, which belongs to the point group Td, so that the equivalence of the four CH EBOs Xi, X2, X and X4, is strictly enforced by symmetry (drawing 18). If we assign a dot to each of the EBOs X, and if we link... 

Because of the high symmetry of methane, the corresponding CMOs cpj are completely predetermined, i.e. they are independent of the explicit values of the parameters A and B. One of the many possible, real representations of the methane CMOs in terms of the EBOs 2 is given by equations 20. These CMOs, which belong to the irreducible representations A and (triply degenerate) T2 of the point group are graphically... 

The PE spectra of methane and ethane are rather special cases, because of their high symmetry. The point groups of methane and of ethane, the latter in its staggered conformation, are and D d respectively, with the consequence that some of their valence shell orbitals are degenerate. They are indicated in bold type in the following list ... 

A loss of the orientational order of a molecular crystal due to free rotation of molecules around x, y and z-axes with the positional order remained results in plastic crystals. The point group symmetry increases to that characteristic of crystals with spherical atoms. However, such crystals are much softer. An example is sohd methane CH4 at low temperature. 

The last statement of the above example mentioned a set of orbitals. A similar procedure can be applied to a combination of atomic orbitals, just as the four sp hybrid orbitals above were treated in combination. For example, in methane the four Is atomic orbitals of hydrogen, which the symmetry operations in move onto each other, represent such a set of orbitals. Perhaps not so surprisingly, the set of four Is atomic orbitals can also be labeled as A 0 T2 in the point group. 

Point group. The classic example of this point group is methane shown in Fig. 3.4. Methane has four threefold axes, three mutually perpendicular twofold axes (coincident with 4 axes), and two mutually perpendicular planes of symmetry through each twofold axis, or a total of six planes of symmetry. 

Fig. 3.4. Symmetry elements of methane (point group, T. TABLE 3.2...

Point Group T. The symmetry of methane (and all molecules of tetrahedral symmetry) is that of point group T. There are three mutually perpendicular twofold axes and four threefold axes. Each pair of threefold axes is associated with a mirror plane. It is of interest that the group has no center of symmetry. 

Tetrahedral methane belongs to the T point group. Replacement of one H by F yields CHjF which now belongs to the C3, point group. The question is What is the effective symmetry of the four bonds connecting carbon to HjF We will argue that the effective symmetry is not but T, that is to say, the four bond orbitals of the... 

The molecular structure of tetra(silyl)methane has been determined in the gas phase by electron diffraction. The molecule has the standard point group T symmetry (neopentane type) with a Si—C—Si—H torsion angle of 20.04° [Si—C 1.8751(7), Si—H 1.486(4) A, C—Si—H 108.5(6)°]. Crystals are cubic with space group F43m and Z = 4. Owing to the extremely small number of observed reflections, a detailed refinement of the crystal structure was not possible. 

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