Linear molecules point groups

TABLE 6.4 Character tables for the linear molecule point groups and... 

You may recall a strange omission in the character tables of Chapter 6 with one exception, the symbols of the irreducible representations may be written using a capital letter (for overall symmetry) or a lower case letter (for the symmetry of a single-coordinate function, such as an orbital wavefunction). The exception is that the X representations of the linear molecule point groups are always written using capital letters, because X symmetry is not possible Irom a one-coordinate function. For example, tr molecular orbitals are always symmetric upon reflection through any vertical mirror plane, whereas X electronic states (which are antisymmetric under (7y)... 

Rigid linear molecules are a special case in which an extended MS group, rather than the MS group, is isomorphic to the point group of the equilibrium structure see chapter 17 of [1]. 

So, for any atom, the orbitals can be labeled by both 1 and m quantum numbers, which play the role that point group labels did for non-linear molecules and X did for linear molecules. Because (i) the kinetic energy operator in the electronic Hamiltonian explicitly contains L2/2mer2, (ii) the Hamiltonian does not contain additional Lz, Lx, or Ly factors. 

The point group is derived from by the inclusion of U , therefore, all linear molecules with a plane of symmetry perpendicular to the axis belong to D f. Acetylene... 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

Linear molecules belong to either the (with an inversion centre) or the (without an inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the (Table A. 3 7 in Appendix A) or (Table A. 16 in Appendix A) character table we can... 

Clearly, Eq. (E.12) shows that to a first approximation the electronic energy varies linearly with displacements in p, increasing for one component state while decreasing for the other. Thus, the potential minimum cannot be at p = 0. This is the statement of the Jahn-Teller theorem for a X3 molecule belonging to the D3h point group. 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

In summary, the molecular orbitals of a linear molecule can be labeled by their m quantum number, which plays the same role as the point group labels did for non-linear polyatomic molecules, and which gives the eigenvalue of the angular momentum of the orbital about the molecule s symmetry axis. Because the kinetic energy part of the... 

If the atom or molecule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear molecules and point group symmetry for nonlinear polyatomics), the trial wavefunctions should also conform to these spatial symmetries. This Chapter addresses those operators that commute with H, Pjj, S2, and Sz and among one another for atoms, linear, and non-linear molecules. 

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