Symmetry consideration

Symmetry considerations go a long way in predicting, inter alia, properties of wavefunctions, electronic states and transitions between molecular states. The mathematical treatment of symmetry operations is part of a special branch of mathematics known as group theory. Only a brief outline of some of its most important applications is given here. 

The character table of a point group defines the symmetry properties of a (wave) function as either 1 for symmetric or —1 for antisymmetric with respect to each symmetry operation/ The first row lists all the symmetry operations of the point group and the first column lists the Mulliken symbols of all possible irreducible representations, the symmetry transformation properties that are allowed for wavefunctions. As an example, the character table for the D2h point group is given in Table 4.1. The character tables of all relevant point groups are given in many textbooks.134,273-275 The last column shows the transformation properties of the axes x, y and z, which are used to determine electronic dipole and transition moments (Section 4.5). 

Character tables of point groups of high symmetry (n 2) have entries other than 1, but we need not deal with such groups here. 

We now return to first-order perturbation theory for degenerate orbitals (Section 4.3). As any linear combination of two degenerate orbitals i/q and xj/j is equally valid, we set up a trial wavefunction function xj/jc = akixl/i-F akj t//7 and we have to solve the secular Equation 4.18. The eigenvalues of the unperturbed system will be equal for all linear combinations, = = (0). 

Equation 4.18 First order perturbation of two degenerate orbitals 

In general, it is advantageous to use the symmetry elements of a molecule in dealing with the molecular orbitals. For example, consider the symmetry properties of a homonuclear diatomic molecule 

A reflection in the plane which contains A-B also makes the molecule go into itself. An inversion of A and B through the center of the connection line takes A into B and B into A, and again (since A = B) there would not in a physical sense be any difference in the molecule after the symmetry operation. 

The symmetry operation of rotation by 180° around the x axis is called a jrotation around a twofold axis and is written C2( = Cggo/j g). 

The two wave functions change phase during certain symmetry operations. We call an arbitrary symmetry operation S. Operating on a wave function ip, we have S and we call X the eigen- 

We see immediately that rotating ipg by an angle 90° aroimd the line AB Will transform into ipg. Thus, if we choose (p in C to be 90° such that x goes to -y, we have, using matrix language and matrix multiplication, 

The theory of optical activity would be understood in terms of symmetry considerations at the first stage. The elements of symmetry are the geometric elements in relation to which the symmetry operations are carried out, and are classified in the following  

A molecule A is chiral, when its mirror image B is not superimposable on A. Only the groups C, Cn, n are compatible with chirality. Especially the molecules belonging to the C, group are called asymmetric. The molecules having Cn or Dn symmetry are also chiral and have an axial symmetry. 

When the symmetry of a given molecule has been determined, electronic transitions fall into the following three categories i a. electric allowed — magnetic forbidden b. magnetic allowed — electric forbidden [Groups C, Cnh, Dnh (n 2), S2n (n, odd), Oh, Td] 

The wavefunction (1), antisymmetric under exchange of space-spin variables of the two electrons, was conveniently factorized into a product of space and spin factors anti-parallel coupling of the spins (antisymmetric spin factor) then implied symmetry of the spatial factor and a consequent enhancement of electron density in the bond region. Such a factorization is unique to a 2-electron system for an 7V-electron system, a Pauli-compatible function must have the form 

The set of spatial functions d K, which may be either exact degenerate eigenfunctions of the Hamiltonian operator H or approximations to them, must then provide the associate representation D in which (using ep for the parity factor, 1 for P even or odd) 

The matrices D(P) with elements D(P) K form a matrix representation of the group of permutations and this representation is characterized by the labels S, Ms which indicate the total spin (S) and its z-component (Ms)- From (19) it follows that, when an orthonormal set of spin eigenfunctions is available, the matrix elements D(P) K may be expressed as 

In principle, it is easy to calculate the electronic energy for a wavefunction of type (18). With the assumptions that the spin functions are orthonormal and that H is spin-free, the energy expectation value is 

For future reference we note that the operators defined in (25) possess the properties 

In general, we represent the (p s as a column vector l i. We have then, S ijl = A hjj, where A is the so-called transformation matrix. 

The trace of the transformation matrix is equal to 2 cos p. For a rotation cp = tt/2, of course, the trace is equal to 0, as found before. Since ip5 and ipe are mixed under C, we continue the investigation of symmetry properties using matrix notation  

Real liquid crystal molecules are much more complex than the multilevel molecule studied in the preceding section. In general, they are anisotropic and possess permanent dipoles (i.e., the molecules themselves are polar). Nevertheless, in bulk form, these liquid crystalline molecules tend to align themselves such that their collective dipole moment is of vanishing value. 

Put another way, most phases of liquid crystals are characterized by centrosym-metry, due to the equivalence of the -n and n directions. This holds in the nematic phase (which belongs to the Deo symmetry group), the smectic-A phase (D symmetry), and the smectic-C phase (C2 , symmetry). As a result of such centrosyimne-try, the macroscopic second-order polarizability Xijk=0. 

The most well-known and extensively studied noncentrosymmetric liquid crystalline phase is the smectic—C phase. The helically modulated smectic C system, with a spatially varying spontaneous polarization, is locally characterized by a C2 symmetry the average polarization of the bulk is still vanishing. By unwinding such a helical stmeture, the system behaves as a crystal with C2 point symmetry. Such unwound SmC phases possess sizable second-order nonlinear polarizabilities,with (2) ranging from 8 X 10 m/V for DOBAMBC to about 0.2 X 10 mA for o-nitroalkoxyphenyl-biphenyl-carboxylate.  

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Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

Symmetry considerations forbid any nonzero off-diagonal matrix elements in Eq. (68) when f(x) is even in x, but they can be nonzero if f x) is odd, for example,/(x) = x. (Note that x itself hansforms as B2 [284].) Figure 3 shows the outcome for the phase by the continuous phase tracing method for cycling... 

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

Symmetry considerations have long been known to be of fundamental importance for an understanding of molecular spectra, and generally molecular dynamics [28-30]. Since electrons and nuclei have distinct statistical properties, the total molecular wave function must satisfy appropriate symmehy... 

The origin of a torsional barrier can be studied best in simple cases like ethane. Here, rotation about the central carbon-carbon bond results in three staggered and three eclipsed stationary points on the potential energy surface, at least when symmetry considerations are not taken into account. Quantum mechanically, the barrier of rotation is explained by anti-bonding interactions between the hydrogens attached to different carbon atoms. These interactions are small when the conformation of ethane is staggered, and reach a maximum value when the molecule approaches an eclipsed geometry. 

It should be stressed that although these symmetry considerations may allow one to anticipate barriers on reaction potential energy surfaces, they have nothing to do with the thermodynamic energy differences of such reactions. Symmetry says whether there will be symmetry-imposed barriers above and beyond any thermodynamic energy differences. The enthalpies of formation of reactants and products contain the information about the reaction s overall energy balance. 

Only one exception to the clean production of two monomer molecules from the pyrolysis of dimer has been noted. When a-hydroxydi-Zvxyljlene (9) is subjected to the Gorham process, no polymer is formed, and the 16-carbon aldehyde (10) is the principal product in its stead, isolated in greater than 90% yield. This transformation indicates that, at least in this case, the cleavage of dimer proceeds in stepwise fashion rather than by a concerted process in which both methylene—methylene bonds are broken at the same time. This is consistent with the predictions of Woodward and Hoffmann from orbital symmetry considerations for such [6 + 6] cycloreversion reactions in the ground state (5). 

Only certain types of crystalline materials can exhibit second harmonic generation (61). Because of symmetry considerations, the coefficient must be identically equal to zero in any material having a center of symmetry. Thus the only candidates for second harmonic generation are materials that lack a center of symmetry. Some common materials which are used in nonlinear optics include barium sodium niobate [12323-03-4] Ba2NaNb O lithium niobate [12031 -63-9] LiNbO potassium titanyl phosphate [12690-20-9], KTiOPO beta-barium borate [13701 -59-2], p-BaB204 and lithium triborate... 

A bonding interaction can be maintained only in the antarafacial mode. The 1,3-suprafacial shift of hydrogen is therefore forbidden by orbital symmetry considerations. The allowed... 

SECTION 13.2. ORBITAL SYMMETRY CONSIDERATIONS RELATED TO PHOTOCHEMICAL REACTIONS... 

Orbital Symmetry Considerations Related to Photochemical Reactions... 

The complementary relationship between thermal and photochemical reactions can be illustrated by considering some of the same reaction types discussed in Chapter 11 and applying orbital symmetry considerations to the photochemical mode of reaction. The case of [2ti + 2ti] cycloaddition of two alkenes can serve as an example. This reaction was classified as a forbidden thermal reaction (Section 11.3) The correlation diagram for cycloaddition of two ethylene molecules (Fig. 13.2) shows that the ground-state molecules would lead to an excited state of cyclobutane and that the cycloaddition would therefore involve a prohibitive thermal activation energy. 

A striking illustration of the relationship between orbital symmetry considerations and the outcome of photochemical reactions can be found in the stereochemistry of electrocyclic reactions. In Chapter 11, the distinction between the conrotatory and the disrotatory mode of reaction as a function of the number of electrons in the system was... 

Is the reaction concerted As was emphasized in Chapter 11, orbital symmetry considerations apply only to concerted reactions. The possible involvement of triplet excited states and, as a result, a nonconcerted process is much more common in photochemical reactions than in the thermal processes. A concerted mechanism must be established before the orbital symmetry rules can be applied. 

Scheme 13.1 lists some example of photochemical cycloaddition and electrocyclic reactions of the type that are consistent with the predictions of orbital symmetry considerations. We will discuss other examples in Section 13.4. 

As was mentioned in Section 13.2, the [27t + 27i] photocycloaddition of alkenes is an allowed reaction according to orbital symmetry considerations. Among the most useful reactions in this categoty, from a synthetic point of view, are intramolecular [27t + 2ti] cycloadditions of dienes and intermolecular [2ti + 2ti] cycloadditions of alkenes with cyclic a, -unsaturated carbonyl compounds. These reactions will be discussed in more detail in Section 6.4 of Part B. 

An intramolecular rearrangement of the conjugate acid of the triazene compound to form the oc-complex without an additional molecule of amine would correspond to a thermal [l,3]-sigmatropic rearrangement. However, such a mechanism can be ruled out on the grounds of the antarafacial pathway required from orbital symmetry considerations (Woodward-Hoffmann rules). 

These arguments go hand in hand with Extended Hiickel Theory (EHT), both being based on overlap (symmetry) considerations. In fact, an EHT calculation will provide almost exactly the same results as a skilful use of the qualitative MO building scheme we have provided in this section. 

For octacovalence a different equation is needed. From symmetry considerations we see that the OC—M—CO bond angle for M(CO)3 with three double bonds is the tetrahedral angle 109.47°. The upper curve in Fig. 1 has been drawn as a straight line passing through the points for n = 1 and n = 2 ... 

Symmetry considerations have also been advanced to explain predominant endo addition. In the case of 2 + 4 addition of butadiene to itself, the approach can be exo or endo. It can be seen (Fig. 15.10) that whether the HOMO of the diene overlaps with the LUMO of the alkene or vice versa, the... 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

The rearrangment of nitromethane to aei-nitromethane via the postulated 1,3-intramolecular hydrogen shift is a high barrier reaction (barrier height of 310 kJ/mol), in agreement with the predietion based on the higher tension of four-membered ring and orbital symmetry considerations. 

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