Diatomic molecules symmetry axis

Rather than giving the general expression for the Hellmann-Feynman theorem, we focus on the equation for a general diatomic molecule, because from it we can leam how p influences the stability of a bond. We take the intemuclear axis as the z axis. By symmetry, the x and y components of the forces on the two nuclei in a diatomic are zero. The force on a nucleus a therefore reduces to the z component only, Fz A, which is given by... 

Problem 10-5. In a homonuclear diatomic molecule, taking the molecular axis as z, the pair of LCAO-MO s tpi = 2p A + PxB and tp2 = 2 PyA + 2 PyB forms a basis for a degenerate irreducible representation of D h, as does the pair 3 = 2pxA PxB and 4 = PxA — PxB Identify the symmetry species of these wave functions. Write down the four-by-four matrices for the direct product representation by examining the effect of the group elements on the products 0i 03, 0i 04, V 2 03) and 02 04- Verify that the characters of the direct product representation are the products of the characters of the individual representations. 

For molecules of axial symmetry such as diatomic molecules, if the molecular axis is aligned with the z-axis, we have Q20 = qi, the quadrupole strength all other ( 2m vanish. 

Next we consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia. To achieve inversion of all particles with respect to space-fixed axes, we first rotate all the electrons and nuclei by 180° about the c axis (which is perpendicular to the molecular plane) we then reflect all the electrons in the molecular ab plane. The net effect of these two transformations is the desired space-fixed inversion of all particles. (Compare the corresponding discussion for diatomic molecules in Section 4.7.) The first step rotates the electrons and nuclei together and therefore has no effect on the molecule-fixed coordinates of either the electrons or the nuclei. (The abc axes rotate with the nuclei.) Thus the first step has no effect on tpel. The second step is a reflection of electronic spatial coordinates in the molecular plane this is a symmetry plane and the corresponding operator Oa has the possible eigenvalues +1 and — 1 (since its square is the unit operator). The electronic wave functions of a planar molecule can thus be classified as having... 

For a symmetric top, the selection rules are such that we can determine only B0 [see (5.85)]. Knowledge of Ib°, the moment of inertia about a principal axis perpendicular to the symmetry axis, is not sufficient to determine the molecular structure, except for a diatomic molecule. To get added information, the microwave spectra of isotopically substituted spe-... 

A parallel band of a linear molecule has no (7-branch lines. (The single vibration of a diatomic molecule is a parallel mode.) See Fig. 6.6. In C02, we see from Fig. 6.2 that v3 changes the dipole-moment component along the symmetry axis hence the v3 fundamental band is a parallel band in contrast, the v2 fundamental is a perpendicular band. 

The symmetry operations E, C, and av (reflection in a plane that contains the axis A-B) are present. All molecules that possess these symmetry properties have the point-group symmetry Coov The orbitals are characterized by symbols similar to those used for a homonuclear diatomic molecule, such as a, n, etc. The character table for CMV is given in Table 2-2. 

In diatomic molecules, or more generally in localized two-center bonds, orbitals of two atoms that may be combined to form an MO are those that have the same symmetry about the axis between the two atomic centers in the bond. This rule does not extend to constructing MOs that extend over three or more atoms more complicated rules would be necessary. 

Free atoms are spherically symmetrical, which implies conservation of their angular momenta. Quantum-mechanically this means that both Lz and L2 are constants of the motion when V = V(r). The special direction, denoted Z, only becomes meaningful in an orienting field. During a chemical reaction such as the formation of a homonuclear diatomic molecule, which occurs on collisional activation, a local held is induced along the axis of approach. Polarization also happens in reactions between radicals, in which case it is directed along the principal symmetry axes of the activated reactants. When two radicals interact they do so by anti-parallel line-up of their symmetry axes, which ensures that any residual angular momentum is optimally quenched. The proposed sequence of events is conveniently demonstrated by consideration of the interactions between simple hydrocarbon molecules. 

Consider then the case of diatomic molecules in E electronic states. H< and e electronic states have cylindrical symmetry about the Z axis, and the tr dipole matrix element de e, is parallel to the molecular (Z) axis. Hence orilv the d... 

We note the interesting result that the transformation properties of functions of coordinates which are defined in the molecule-fixed axis system (that is, electronic or vibrational) are the same under E as they are under reflection symmetry of electronic or vibrational states of diatomic molecules. 

Heteronuclear diatomic molecules belong to the symmetry group Cxv. States which are invariant under rotation about the symmetry axis are called E states. These are the states for which A = 0. II states are those for which A = 1, A states those for which A = 2, and so on. In addition, the S states are further characterized by the designation + and — according to whether they remain invariant or change sign when subjected to the operation ov (reflection in a plane in which the symmetry axis lies). 

Heteronuclear diatomic molecules, as well as non symmetrical linear molecules, such as HCN, are classified as Coot), since cylindrical symmetry can be regarded as a rotational axis of infinite order. The groups C /, contain a horizontal plane of symmetry oh in addition to the -fold rotation axis. Many of these molecules are planar, such as rrani-dichloroethylene and boric acid ... 

In Chapters 4 and 6 we have distinguished between a molecular orbitals and 7T molecular orbitals in diatomic molecules on the basis of the symmetry of the m.o.s with respect to rotation around the intemuclear axis whereas a a orbital has cylindrical symmetry, a tt orbital changes sign upon a rotation of 180°. In Chapters 7 and 8 we have extended the notion of a orbitals to polyatomic molecules by referring to the local symmetry with respect to each X-Y intemuclear axis. We will now study systems of tt m.o.s in polyatomic species. 

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