Molecular symmetry operations

Axes of rotation are among the most common of molecular symmetry operations. A onefold axis is a rotation by a full turn, equivalent to the identity. A twofold rotation axis, as in the example of the water molecule, is sometimes called a dyad. Cyclopropane has a threefold axis perpendicular to the plane containing the carbon atoms it also has three twofold axes. Can you visualize them ... 

The symmetry elements (Section 1.18) of a molecule aid in locating its principal axes. Clearly, a molecular symmetry operation must carry the momental ellipsoid to an orientation indistinguishable from its original position. Consider a Cn rotation (n= M) such a rotation about any axis other than one of the three principal axes of the momental ellipsoid will send the ellipsoid to a nonequivalent spatial orientation. Hence a Cn symmetry axis of the molecule must coincide with one of its principal axes. (The converse is, of course, not true every molecule has three principal... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

Consider the symmetry behavior of the normal modes of H20 and COz. For water, the symmetry operations are E, C2 b), ov(ab), and av(bc), where the a and b principal axes lie in the molecular plane the bond angle in water is relatively obtuse, so that the C2 axis is the axis of intermediate moment of inertia. Each of these operations converts the displacement vectors of p, in Fig. 6.1 to an indistinguishable configuration the same is true for v2. Normal modes that are symmetric with respect to all the molecular symmetry operations are called totally symmetric. For H20, the modes and v2 are totally symmetric, but v3 is not. Figure 6.3 shows that v3 is antisymmetric with respect to C2(b). We can tabulate the behavior of the H20 normal modes with respect to the molecular symmetry operations ... 

The classification of molecular symmetry operations that we shall follow here is the conventional one (see, e.g., Tinkham [2]), involving rotations about a specified axis, denoted Cn for a counterclockwise rotation through reflections in a plane, de-... 

For a given molecule, Py is a physical quantity with a unique numerical value that must remain unchanged by any molecular symmetry operation such as rotation or inversion. Hence, tohaveanonzerovalue,F y,mustbetotallysymmetric thatis,r(i/fy) X T(if) X Titpj) = Xg, where denotes the symmetry species of etc. The dipole moment component... 

The analogous Schoenflies symbolism derived from molecular symmetry operations. 

The delocalized MOs [such as (15.20) for H2O] that satisfy (15.34) are called the canonical SCF MOs. The canonical MOs are eigenfunctions of the Fock operator, which commutes with the molecular symmetry operators, and hence the canonical MOs each transform according to one of the possible molecular symmetry species. A set of MOs like (15.33) satisfies (15.35) but not (15.34) and does not necessarily transform according to the molecular symmetry species. 

Table 7-1 The Contribution per Atom for Various Molecular Symmetry Operations to Determine Unshifted Atoms...

Furthermore, all these functions, which appear together in a given linear combination must transfonn exactly alike under the molecular symmetry operations. Thus the wave functions, of the anharmonic molecule belong to the same symmetry species as do those of the harmonic oscillator, and the symmetry species for a state of a real molecule can be identified with the species of the harmonic oscillator function which most closely approximates it and therefore contributes the largest term to the linear combination. 

For a minimal-basis-set MO treatment of H2O (Fig. 15.1 in Section 15.2), we start with the Oli, 02s, 02p 02py, and 02p inner-shell and valence oxygen AOs, and the Hi Is and H2IS valence AOs of the hydrogen atoms. Linear combinations of these seven basis AOs give LCAO approximations to the seven lowest MOs of water. As stated in Section 15.3, the MOs can be chosen so that upon application of the molecular symmetry operators each... 

As stated in Sections 15.3 and 15.5, the canonical MOs of a molecule can be chosen so that each one transforms according to one of the symmetry species of the molecule when the molecular symmetry operators are applied. The same is true of the normal-mode vibrations. For example, when each of the four symmetry operators of H2O is applied to the W) vibration in Fig. 15.16, the vibration displacement vectors end up looking unchanged. This normal mode therefore belongs to the totally symmetric symmetry species a,. Give the symmetry species of the other two normal modes in Fig. 15.16. Take the axes as in Fig. 15.1. 

The H-M symbolism derived from crystal symmetry operations. For image and movie representations of each point group, see the website http //cst-www.nrl.navy.mil/lattice/spcgrp/index.html The analogous Schoenflies symbolism derived from molecular symmetry operations. 

The set of all distinct symmetry operations that take a molecule into an equivalent arrangement constitutes a mathematical structure called a group. An area of mafhemafics called group theory helps organize information we collect about molecular symmetry operations, although group theory has much broader application. 

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