Symmetry selection rule

The selection rules for infrared and Raman activity of molecular vibrations in symmetry terms are as follows. 

We must therefore consider first how to determine the symmetry of molecular vibrations and, secondly, what the consequences of these symmetry selection rules might be. If you are not already familiar with the use of symmetry in molecules, please read Section 2.3 before continuing. 

As shown in Section 1.7, IR and Raman activities for small molecules can be determined by inspection of their normal modes. Clearly, it is difficult to apply such an approach to large and complex molecules. This problem can be solved by using the group theoretical consideration described next. 

According to quantum mechanics (18, 19), the selection rule for the IR spectrum is determined by the integrals 

ny and /rz are the x, y and z components of the dipole moment at the electronic ground state, respectively, i/v and l/v, are vibrational wavefunctions where v and v are the vibrational quantum numbers before and after the transition, respectively. Qa is the normal coordinate of the normal vibration, a. If one of these integrals is nonzero, this vibration is infrared-active. If all three integrals are zero, it is infrared-inactive. 

Using (1-57) as an example, let us determine whether such an integral is zero or nonzero. For this purpose, we first expand fix in terms of the normal coordinate, Qa  

The integral in the first term vanishes because of the orthogonality of ij/v and (except for v = v , no transition). In order for the second term to be nonzero. 

As stated before, a transition between different vibrational states may take place if the respective coefficients in the dipole moment expansion as a power series to normal coordinates are not equal to zero. For a fundamental transition at least one of the components dp dQk, 5py/9Qi and Sp /SQk must not be zero. For overtone transitions a non-zero value is required for at least one of the derivatives 52pz/aOk2 

A group theoretical analysis may determine the uffiared active transitions by considering flie symmetry properties of the vibrational wave functions of tire interacting states and of molecular dipole moment. In most general terms, a transition between vibrational states V and V will take place if at least one of the component matrix elements differs from zero 

If all three dipole matrix elements are zero the transition will be symmetry forbidden or infrared inactive. 

The symmetiy selection rales aie derived by studying the effect of synunetry operations on die matrix elements. The rales stem from the property of dipole moment matrix elements to be invariant with respect to a symmetiy operation (R) 

Since the transition dipole is a physical observable, it is evident that its value should be independent under symmetiy operations. In other words, die intramolecular change distribution and fluctuations are invariant with respect to symmetry operations. 

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Quack M 1977 Detailed symmetry selection rules for reactive collisions Mol. Phys. 34 477-504... 

Often it is possible to resolve vibrational structure of electronic transitions. In this section we will briefly review the symmetry selection rules and other factors controlling the intensity of individual vibronic bands. 

The right-hand side of Eq. (96) is of course the weighted direct sum of the irreducible representations. By convention the totally symmetric irreducible representation corresponds to t = 1. Thus, if n(1> = 0, the integral in Eq. (95) vanishes. The transitions m -> nandm n are then forbidden by the symmetry selection rules. Thte principle can be illustrated by the following example. 

With its substitution in Eq. (99) it becomes evident from the orthogonality of the Hermite polynomials, that all matrix elements are equal to zero, with the exception of v = v — 1 and vf = u +1. Thus, the selection rule for vibrational transitions (in the harmonic approximation) is An — 1. It is not necessary to evaluate the matrix elements unless there is an interest in calculating the intensities of spectral features resulting from vibrational transitions (see problem 18). It should be evident that transitions such as Av - 3 are forbidden under this more restrictive selection rule, although they are permitted under the symmetry selection rule developed in the previous paragraphs. 

A. Jorio, A.G.S. Filho, V.W. Brar, A.K. Swan, M.S. Unlu, B.B. Goldberg, A. Righi, J.H. Hafner, C.M. Lieber, R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Polarized resonant Raman study of isolated single-wall carbon nanotubes symmetry selection rules, dipolar and multipolar antenna effects. Phys. Rev. B 65, 121402.1-121402.4 (2002). 

Hence, according to the symmetry selection rule, n —> n transitions are allowed but n —> ti transitions are forbidden. However, in practice the n —> it transition is weakly allowed due to coupling of vibrational and electronic motions in the molecule (vibronic coupling). Vibronic coupling is a result of the breakdown of the Born-Oppenheimer approximation. 

Recent theoretical and spectroscopic studies indicate that in aliphatic dienes and trienes, excitation to the spectroscopic l1 state usually results in facile twisting about the termini in the stereochemical sense dictated by orbital symmetry selection rules for the appropriate electrocyclic ring closure, motions which are often accompanied by some degree of planarization of the carbon framework. In general, relatively minor distortions... 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

If we start with states of tt-symmetry (dashed lines) we find three distinct peaks in the XES spectra reflecting the occupied states. The 1 a2u and lelgTr-like orbitals are essentially intact from the gas phase, while the third state, labeled e2u, is not seen for the free molecule. Based on symmetry-selection rules, it can be shown that this state is derived from the lowest unoccupied molecular orbital (LUMO) e2utt -orbital that becomes slightly occupied upon adsorption. We anticipate a similar bonding mechanism as discussed in the previous section for adsorbed ethylene with the exception of a weaker rehybridization due to the extra stability in the -system from the aromatic character. 

If the analogy that is drawn between the Si=Si dimer on the Si(100)-2 x 1 surface and an alkene group is reasonable, then certain parallels might be expected to exist between cycloaddition reactions in organic chemistry and reactions that occur between alkenes or dienes and the silicon surface. In other words, cycloaddition products should be observed on the Si(100)-2 x 1 surface. Indeed, this prediction has been borne out in a number of studies of cycloaddition reactions on Si(100)-2x1 [14], as well as on the related surfaces of Ge(100)-2 x 1 (see Section 6.2.1) and C(100)-2 x 1 [192-195]. On the other hand, because the double-bonded description is only an approximation, deviations from the simple picture are expected. A number of studies have shown that the behavior differs from that of a double bond, and the asymmetric character of the dimer will be seen to play an important role. For example, departures from the symmetry selection rules developed for organic reactions are observed at the surface. Several review articles address cycloaddition and related chemistry at the Si(100)-2 x 1 surface the reader is referred to Refs. [10-18] for additional detail. 

The fact that there are many electronic transitions possible, however, does not mean that they can or will occur. There are complex selection rules based on the symmetry of the ground and excited states of the molecule under examination. Basically, electronic transitions are allowed if the orientation of the electron spin does not change during the transition and if the symmetry of the initial and final functions is different these are called the spin and symmetry selection rules, respectively. However, the so-called forbidden transitions can still occur, but give rise to weak absorptions. 

Energy state Electronic configuration Molecular state symmetry Selection rule for polarization of transition dipole... 

The 196 cm"1 mode is antisymmetric and thereby optically inactive and does not appear in the Raman spectrum. Since a direct optical excitation of the mode is excluded by symmetry selection rules we conclude that it is solely excited by the single proton transfer which breaks the symmetry. This demonstrates for the first time that the coherent excitation of a vibrational mode results exclusively from an ultrafast reactive process. 

Using i/ orbitai and orbital in this form, the integrals implicit in equation (45) are likely to be of the order of the values of C, or C and the band oscillator strengths of the order of their squares. The orbital symmetry selection rule discussed at the end of Section 6.5.1 will have some effect in determining band intensities, but even orbitally forbidden transitions can be expected to appear with finite intensity. 

Both transitions would be weak, partly because forbidden by the Franck-Condon principle and partly because both are related to the forbidden transition (1). In addition, (1A) is forbidden by the symmetry-selection rules and has therefore an extra reason to be weak compared with (IB). Similarly, for bent upper states, (2) and (3) have to be replaced... 

A) should be very weak because it is forbidden by the symmetry-selection rules and is related to the forbidden transition (2). Provided, however, that the upper state is not strongly bent, (3A) should be strong since it is allowed by the selection rules and is related to the intense transition (3). 

The contributions of optically forbidden electronic states to the x(3) of centrosymmetric structures are of particular interest. (18) Each of the terms in a sum-over-states calculation of x(3) involves the product of transition moments between a sequence of four states. There are symmetry selection rules that govern which states which can contribute to the individual terms. In a centrosymmetric molecule the symmetry of the contributing states must be in a sequence g -> u --> g --> u --> g.(19) This means that all the non-zero terms in the summation which determines the hyperpolarizibility must include an excited electronic state of g symmetry (or the ground state) as an intermediate state. The tetrakis(cumylphenoxy)phthalocyanines are approximately centrosymmetric and many of the new electronic states in a metal phthalocyanine will be of g symmetry. Such states may well contribute to the dependence of the hyperpolarizibility on metal substitution. 

R. Wallace, Chem. Phys., 37, 285 (1979). Vibronic State Symmetry, Selection Rules and Transition Probabilities for a Molecular Rearrangement Process. The Butadiene-Cyclobutene Rearrangement. 

AMg = 0, where S, Mg characterize the dimer states. The single-ion mechanism also allows transitions with AS = 1 and AMg = 1. In addition, both mechanisms lead to orbital, i.e. symmetry selection rules. In order to make full use of the latter in the assignment of dimer states it is essential to have single crystal data. 

Hamiltonian for the X2Eig — E2B2u electronic manifold of Bz+, involving extended group theoretical considerations, is given in Ref. [23]. The relevant submatrices for the cases treated below will be quoted in subsequent sections. Here we only note a general symmetry selection rule for the mode / in order to linearly couple the states i and j ... 

Mode assignment and symmetry selection rules. Equations (5) and (8) contain the matrix element (fi V v). When this matrix element is zero, the conductance at the vibration threshold will be zero. Hence, the symmetry of the electronic states p, and z/, and of the vibration can determine when this matrix element will be zero. 

In addition to the symmetry selection rule, there is also another selection rule on spin AS = 0, i.e., only the transitions between states with the same spin are allowed. The fact that we sometimes do observe spin-forbidden d-d transitions is mainly due to spin-orbit coupling, which will be discussed later in this section. 

This form of the wave function fixes (among other things) the number of electrons in the 7r-system. In variance with the general theory, the one-electron transfers between the subsystems (w- and general theory) are vanishing due to the symmetry selection rules ... 

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