Symmetry transformation

The symmetry group to which an A(B, C,. . . ) molecule belongs is determined by the arrangement of the pendent atoms. The A atom, being unique, must lie on all planes and axes of symmetry. The orbitals that atom A uses in forming the A—(B, C,. . . ) bonds must therefore be discussed and classified in terms of the set of symmetry operations generated by these axes and planes—that is, in terms of the overall symmetry of the molecule. Thus, our first order of business is to examine the wave functions for AOs and consider their transformation (symmetry) properties under the various operations which constitute the point group of the A(B, C,. . . ) molecule. 

One can also transform symmetry operators whenever the coordinate system itself gets changed, as for instance in selecting an alternate setting for a space group. Then one must use a similarity transformation in the old system let the 4x4 symmetry operator be denoted by Q3, and in the new system as Qy let the coordinate system transformation be represented by the 4x4 matrix S, whose inverse matrix is S-1 then the similarity transformation yields ... 

In addition to utilizing molecular symmetry, it is also necessary to utilize transform symmetry where it exists. A transform possesses a symmetry element if the SUBSTRUCTURE possesses the element and it is preserved by the MANIPULATION and SCOPE/LIMITATIONS statements, e.g., the Diels Alder transform ... 

Table 2. The Signatures of g under Different Transformations Symmetry...

In the ground state, if the symmetry of MOs of the reactant matches that of the products that are nearest in energies, then reaction is thermally allowed. However, if the symmetry of MOs of the reactant matches that of the product in the first excited state but not in the ground state, then the reaction is photochemically allowed (Figure 1.12). When symmetries of the reactant and product molecular orbitals differ, the reaction does not occur in a concerted manner. It must be noted that a symmetry element becomes irrelevant when orbitals involved in the reaction are aU symmetric or antisymmetric. In conclusion, we can say that in pericyclic transformations, symmetry properties of the reactants and products remain conserved. 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

Electi ocyclic reactions are examples of cases where ic-electiDn bonds transform to sigma ones [32,49,55]. A prototype is the cyclization of butadiene to cyclobutene (Fig. 8, lower panel). In this four electron system, phase inversion occurs if no new nodes are fomred along the reaction coordinate. Therefore, when the ring closure is disrotatory, the system is Hiickel type, and the reaction a phase-inverting one. If, however, the motion is conrotatory, a new node is formed along the reaction coordinate just as in the HCl + H system. The reaction is now Mdbius type, and phase preserving. This result, which is in line with the Woodward-Hoffmann rules and with Zimmerman s Mdbius-Huckel model [20], was obtained without consideration of nuclear symmetry. This conclusion was previously reached by Goddard [22,39]. 

The system provides an opportunity to test our method for finding the conical intersection and the stabilized ground-state structures that are formed by the distortion. Recall that we focus on the distinction between spin-paired structures, rather than true minima. A natural choice for anchors are the two C2v stmctures having A2 and B, symmetry shown in Figures 21 and 22 In principle, each set can serve as the anchors. The reaction converting one type-I structirre to another is phase inverting, since it transforms one allyl structure to another (Fig. 12). 

The remaining combinations vanish for symmetry reasons [the operator transforms according to B (A") hreducible representation]. The nonvanishing of the off-diagonal matrix element fl+ is responsible for the coupling of the adiabatic electronic states. 

In (10), both 5i and 52 appear as independent constants. If, in addition, the dynamical system possesses some symmetry, then these numbers may satisfy a further relation. To illustrate this fact, let us consider the simplest case where we have a symmetry transformation k in the problem with k = id. Then one can show (see again [6]) ... 

Moreover, our Hamiltonian system possesses an additional symmetry — it is equivariant under the transformation (52,P2) —(92, 2). In other words each of these sets is a candidate for a set B mentioned in the assumptions of Corollary 4. Thus, by this result, both of these sets are almost invariant with... 

W, g potential functions, k 1, has been discussed in various papers (see, for example, [6, 11, 9, 16, 3]). It has been pointed out that, for step-sizes /j > e = 1/ /k, the midpoint method can become unstable due to resonances [9, 16], i.e., for specific values of k. However, generic instabilities arise if the step-size k is chosen such that is not small [3, 6, 18], For systems with a rotational symmetry this has been shown rigorously in [6j. This effect is generic for highly oscillatory Hamiltonian systems, as argued for in [3] in terms of decoupling transformations and proved for a linear time varying system without symmetry. 

A eonerete example will help elarify these eoneepts. In C3V symmetry, the n orbitals of the eyelopropenyl anion transform aeeording to ai and e symmetries... 

Let us eonsider the vibrational motions of benzene. To eonsider all of the vibrational modes of benzene we should attaeh a set of displaeement veetors in the x, y, and z direetions to eaeh atom in the moleeule (giving 36 veetors in all), and evaluate how these transform under the symmetry operations of D6h- For this problem, however, let s only inquire about the C-H stretehing vibrations. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

For a function to transform according to a specific irreducible representation means that the function, when operated upon by a point-group symmetry operator, yields a linear combination of the functions that transform according to that irreducible representation. For example, a 2pz orbital (z is the C3 axis of NH3) on the nitrogen atom... 

Using the orthogonality of eharaeters taken as veetors we ean reduee the above set of eharaeters to Ai + E. Henee, we say that our orbital set of three Ish orbitals forms a redueible representation eonsisting of the sum of and E IR s. This means that the three 1 sh orbitals ean be eombined to yield one orbital of Ai symmetry and a pair that transform aeeording to the E representation. 

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