Nuclear selection rules

Cordonnier M, Uy D, Dickson R M, Kew K E, Zhang Y and Oka T 2000 Selection rules for nuclear spin modifications in ion-neutral reactions involving Hg" J. Chem. Phys. 113 3181-93... 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

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. 

Thus in the lowest order approximation the angle x is eliminated from the off-diagonal matrix elements of [second and third of Eqs. (60)] it solely determines the selection rules for matrix elements of Hg with respect to nuclear basis functions. 

As in atoms, the selection rule breaks down as the nuclear charge increases. For example, triplet-singlet transitions are strictly forbidden in FI2 but in CO the a U — transition is observed weakly. 

Thus the change in the direction of the spin angular momentum of the electron effectively imparts some singlet character to a triplet state and, conversely, triplet character to a singlet state. This relaxes the spin selection rule since J S St dr is no longer strictly zero. The greater the nuclear charge,... 

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 allowed transitions occur between levels Ex and E3, and between levels Ei and E4 as indicated by the dashed lines. These transitions are governed by selection rules which require that the electron spin changes by one unit while the nuclear spin remains unchanged. Under certain rather restricted conditions these selection rules no longer apply and forbidden transitions occur. 

Infrared, Raman, microwave, and double resonance techniques turn out to offer nicely complementary tools, which usually can and have to be complemented by quantum chemical calculations. In both experiment and theory, progress over the last 10 years has been enormous. The relationship between theory and experiment is symbiotic, as the elementary systems represent benchmarks for rigorous quantum treatments of clear-cut observables. Even the simplest cases such as methanol dimer still present challenges, which can only be met by high-level electron correlation and nuclear motion approaches in many dimensions. On the experimental side, infrared spectroscopy is most powerful for the O—H stretching dynamics, whereas double resonance techniques offer selectivity and Raman scattering profits from other selection rules. A few challenges for accurate theoretical treatments in this field are listed in Table I. 

For Figure 3.27, note that lines 1, 3, 4, and 6, obey the selection rule IAm/l = l for the allowed y transitions between the nuclear sublevels, whereas lines 2 and 5 obey the Am/=0 selection rule. For a isotropic (gx=gy=gz=g) sample in which the effective magnetic field is parallel to the observed y rays, the intensity of the Am/=0 lines vanishes so that only four lines are seen in the spectrum (Figure 3a of reference 34). The same lines that are missing in the isotropic case will be maximized when the effective magnetic field is perpendicular to the y rays (Figure 3b of reference 34). For a uniaxial case (gx=gy=0 and 5 0, Figure 3c of reference 34) or the extreme anisotropic case (gz gx,gy), the intensities of the absorption lines are independent of the... 

The selection rules for radiationless transitions are just the opposite of those for radiative transitions. The nuclear kinetic operator is symmetric. The symmetric aromatic molecules normally have symmetrical ground state and antisymmetrical excited state. Therefore, allowed transitions are ... 

Further, we will find in this chapter that wavefunctions (nuclear or electronic) must be functions which form bases for the irreducible representations of the point group to which the molecule belongs. With this knowledge we are able to determine which integrals over molecular wavefunctions are necessarily zero and this in turn (next chapter) leads to well known spectroscopic selection rules. 

In this section we derive certain rules which will determine whether or not an integral over given electronic or nuclear wavefunctions vanishes from such rules we can deduce spectroscopic selection rules. Consider the integral... 

An alternative but not so general selection rule (it is restricted to the harmonic oscillator approximation) is that jVoI> / v5i dr is zero if dfiJdQ (evaluated for the equilibrium nuclear configuration) is zero, i.e. if there is no linear dependence of the dipole moment on the normal coordinate Q . 

The selection rule (4.138) differs from previously discussed selection rules in that it holds well for nonradiative transitions, as well as for radiative transitions. In deriving (4.138), we made no reference to the operator d, beyond the statement that it did not involve the nuclear spin coordinates. For any time-dependent perturbation that does not involve nuclear spin, the selection rule (4.138) will hold. Thus molecular collisions will not cause nonradiative transitions between symmetric and antisymmetric rotational levels of a homonuclear diatomic molecule. If we somehow start with all the molecules in symmetric levels, the collisions will not populate the antisymmetric levels. 

Now suppose we cool H2 to a very low temperature. Because of the selection rule (4.138), molecules originally in levels with odd values of J will all fall to the /= l,t>=0 level, whereas molecules originally in levels with even values of J will all fall to the /=0,t =0 level. Actually, (4.138) is not absolutely inviolable. There are very weak interactions between the nuclear spins and the electrons as a result, the separation of the molecular... 

For C22H2, the nuclear spin of C12 is zero and contributes a factor of 1 to the nuclear statistical weights. The statistical weights are therefore the same as in H2. For the ground vibronic state, the even J levels are s and have nuclear statistical weight 1, corresponding to the one possible ns the odd J levels are a and have nuclear statistical weight 3. The usual selection rule (4.138) holds for collisions as well as radiative transitions, and we have ortho and para acetylene. The two forms have not been separated. 

The preceding discussion has not taken into account a selection rule that results from the existence of nuclear spin. We now consider the effect of nuclear spin on selection rules. 

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