The Selection Rule for

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

The matrix elements (60) represent effective operators that still have to act on the functions of nuclear coordinates. The factors exp( 2iAx) determine the selection rules for the matrix elements involving the nuclear basis functions. 

For a symmetric rotor molecule the selection rules for the rotational Raman spectmm are... 

Strategy Use the selection rules for the four quantum numbers to find the sets that could not occur. For the valid sets, identify the principal level and sublevel... 

Figure 1. Stereofacial selectivity rule for the Sharpless asymmetric epoxidation.

Stability or instability of a scheme from the primary family depends only on selection rules for the operator R. From the point of view of stability theory the arbitrariness in the choice of the operator R is restricted by the following requirements ... 

The only difference between the methods we have mentioned above lies in the selection rules for the parameter In the minimal residual... 

Verify the selection rules for the hydrogen atom as given in the last paragraph of Section 12.3.3. 

The general relation which must be satisfied in order to bring about an appreciable stabilization energy in the chemical interaction has been given by Eq. (3.20) and Eq. (3.25 b). Such relations frequently provide a selection rule for the occurrence of stereoselective reactions. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, S, P, Ur, It, t>c, and lc-The terms of H involving the parameters aj, ac, and bo are diagonal with respect to both the lT and lc quantum numbers, while the hi term connects with one another the basis functions with l T = lT 2, l c = Zc T 2. The c terms couple with each other the electronic species —A and A. The selection rules for the vibrational quantum numbers are v Tjc = vT/c, t)j/c 2, vT/c 4. 

Upon entering the interaction frame of the rf irradiation for the CNvn or RN n sequences [cf. (14)] and taking the first-order effective Hamiltonians [cf. (17a) and (18a)], it is possible to establish the following selection rules for the averaging (and conversely recoupling) of the various interactions described in (45) as... 

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... 

Equation (7.9). By analyzing this matrix element, we can establish the selection rules for the transition. 

From the properties of the two 6-j symbols in Eq. (20) the specific selection rules for the 7c / coupling scheme are derived as ... 

The selection rules for the QM harmonic oscillator pennit transitions only for An = 1 (see Section 14.5). As Eq. (9.47) indicates diat the energy separation between any two adjacent levels is always hm, the predicted frequency for die = 0 to n = 1 absorption (or indeed any allowed absorption) is simply v = o). So, in order to predict die stretching frequency within the harmonic oscillator equation, all diat is needed is the second derivative of the energy with respect to bond stretching computed at die equilibrium geometry, i.e., k. The importance of k has led to considerable effort to derive analytical expressions for second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF and select other levels of theory, although they can be quite expensive at some of the more highly correlated levels of theoiy. 

Selection rules for the electronic energy transfer by dipole-dipole interactions are the same as those for corresponding electric dipole transitions in the isolated molecules. The spin selection rule requires that the total multiplicity of the donor arid the acceptor, prior to and after the act of transfer, must be preserved. This implies that M0. - Mq and MA — MA where M s denote the multiplicity of the states (Section 2.5.1). 

The mechanism of the reaction that occurs at a platinum anode in CH3CN-Et4NC104 has been elucidated by showing that iV,N,AMriphenyl-o-phenylenediamine (50) is oxidized quantitatively at + 1.3 V versus SCE to give the dication of (49), which then can be reduced to 49 at —0.7 V versus SCE. Selection rules for the conversion of diphenylamines to dihydrophen-azines are given.109... 

Evaluation of this integral (Problem 3.3) shows that it vanishes for m — n even, and is nonzero for m — n odd. The selection rule for the particle in a box is the following electric-dipole transitions are allowed if and only if the change in quantum number is odd. 

Mclver and Stanton used group theory to derive selection rules for the symmetry of transition states. For example, they proved that the exchange reaction H2 + D2- 2HD cannot have a tetrahedral transition state. See J. W. Mclver and R. E. Stanton, J. Am. Chem. Soc., 94, 8618 (1972) J. W. Mclver, Accounts Chem. Res., 7, 72 (1974). 

The selection rules for the Raman spectrum turn out to depend not on the matrix elements of the electric dipole moment, but on the matrix elements of the molecular polarizability, which we now define. The application of an electric field E to a molecule gives rise to an induced electric dipole moment djnd (which is in addition to the permanent dipole moment d). If E= "> 1 + yl+ >zk, then the induced dipole moment has the components... 

A spherical top is a special case of a symmetric top the rotational energy depends only on 7, and the J selection rule for the vibration-rotation transitions is the same as for symmetric tops ... 

The selection rules for the fundamentals of these modes are obtained immediately from the right columns of the character table and are... 

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