Nuclear symmetry

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]. 

Transition structures can be dehned by nuclear symmetry. For example, a symmetric Spj2 reaction will have a transition structure that has a higher symmetry than that of the reactants or products. Furthermore, the transition structure is the lowest-energy structure that obeys the constraints of higher symmetry. Thus, the transition structure can be calculated by forcing the molecule to have a particular symmetry and using a geometry optimization technique. 

The aim of this contribution is to review the present understanding of the hadronic EoS, and the nuclear symmetry energy (SE) in particular. In the past the latter quantity has been computed mostly in terms of the Brueckner-Hartree-Fock (BHF) approach. In order to get an idea about the accuracy of the BHF result we present a recent calculation [1] in terms of the self-consistent... 

Symmetry-induced kinetic isotope effects cause changes in the rate of a chemical process involving identical nuclei when the nuclear symmetry of the system is altered... 

A number of studies have measured the isotopic distribution in atmospheric ozone. There are three naturally occurring isotopes of oxygen, lftO, l70, and lxO, which might initially be expected to be represented statistically in atmospheric ozone. However, both stratospheric and tropospheric ozone have been measured to be enriched in the heavier isotopes over what one would expect statistically (e.g., see Mauersberger, 1981 Mauersberger et al., 1993 Krankowsky et al., 1995). A variety of explanations of this fractionation have been put forth, including nuclear symmetry restrictions on the 02 + O reaction that forms 03 (Hellene, 1996), the preferential dissociation of heavy ozone to form vibrationally excited 02 (u > 26) that then... 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

To understand these distributions, one needs to consider the fission transition nucleus. Figure 11.22 shows a coordinate system for describing this nucleus in terms of its quantum numbers J, the total angular momentum M, the projection of J upon a space-fixed axis, usually taken to be the direction of motion of the fissioning system, and K, the projection of J upon the nuclear symmetry axis. 

Then K h the root-mean-square projection of 7 upon the nuclear symmetry axis, can be calculated as ... 

Figure 20. Energy dependence of total cross section for He (2 5) + He calculated from potentials of Fig. 14 and Table 111. Oscillations at low energies attributable to nuclear-symmetry. Glory effect is amplified in curve I, in which difference between cross sections for identical and distinguishable particles is plotted on an expanded scale. 

The approach to even-even and odd-even nuclei based on nuclear symmetries in EBM/IEEM has received much attention in recent years [ARI76, ARI78, ARI79,BAL8l,BAL83,IAC79,IAC80]. 

A.l.Maergoiz, E.E.Nikitin, and J.Troe, Adiabatic channel study of the capture of nitrogen and oxygen molecules by an ion Effect of nuclear symmetry and spin-spin interaction. Z. Phys D- Atoms, Molecules and Clusters, 36, 339 (1996)... 

First, we recast in our notation a calculation originally done by Coulson (1961). With reference to Figures 2.16 and 2.17, we choose the molecule to lie in the yz plane. Nuclear symmetry (Magnasco, 2007,2009a) shows that H2O has Civ symmetry, with two symmetry planes (yz and zx) whose intersection determines a C2 binary axis directed along z. 2d is the interbond (valence) angle. 

Unfortunately, if spatial symmetrization (in contrast to spin symmetrization considered above) is used, discontinuities will arise when spatial symmetrybreaking nuclear motion is considered. Thus broken spatial symmetry is best treated via the methods discussed in the next section that unfortunately do not yield states of pure spatial symmetry. (The alternative energy-projection method of Noodleman when self-consistently applied also becomes discontinuous during nuclear symmetry breaking.)... 

Comparison of the results of the previous effort, where it was assumed that excitation and absorption spectrum coincided,7 with the present results shows that there is really not much of a difference. Because of the higher sensitivity, MEs belonging to P(l) are detected over a range of almost 8 GHz instead of 3.5 GHz as found previously, and in accordance with that their number has increased. We now have 35 triplet states over 7.6 GHz, which leads to a density of 140 states per cm-1. The calculated density of pyrazine triplet vibronic states around the singlet energy appears to be around 100 cm-1,13-15 but if we want a comparison, we should take into account that they are triplets (times 3) and that nuclear symmetry permits interaction only between equal symmetry species. Table II gives the nuclear spin symmetry species of pyrazine and their statistical weight. The J = 0, K = 0 has the symmetry At and therefore can only interact with Ae triplet rovibronic states, which constitute th of the triplet manifold. We therefore expect about 3 x 17/48 x 100 a 106 triplet vibronic states per cm-1 to be available for the interaction, which compares favorably to the density of 140 per cm-1 as found from the density of the ME spectrum. 

Very recently, de Lange et al.30 carefully studied the intensities of the low J manifold of pyrazine with a nanosecond laser, which averages reasonably well over the quantum yields of the MEs. Taking a clue from the beautiful results obtained on benzene by Riedle et al.,31 they fitted the intensities to a quantum yield determined by Coriolis coupling of S, to S0. They assumed a Boltzmann distribution in the ground state, except for the J" = 0, K" = 0 state, which for nuclear symmetry reasons can only be reached by a A J" = — 2 transition. 

Analysis of symmetry also leads to this conclusion. From Table II it can be noted that the lowest J, K states are mostly of different nuclear symmetry and therefore cannot mix.30 This symmetry selection rule is considerably relaxed at higher J, there being many more K states of the same symmetry. 

The nuclear symmetry character of the electronic wave function is represented in the term symbol by introducing the superscript + or — after taking cognizance of the presence of the subscript g or u discussed in Section 48a, the combinations... 

The principal use of the nuclear symmetry character is in determining the allowed values of the rotational quantum number K of the molecule. The complete wave functions for a molecule (including the nuclear-spin function) must be either symmetric or antisymmetric in the nuclei, depending on the nature of the nuclei involved. If the nuclei have no spins, then the existent functions are of one or the other of the types listed below. 

The nuclear symmetry of the complete wave function (including rotation of the molecule but not nuclear spin). Selection rule Symmetric-antisymmetric transitions not allowed. 

In contrast to the vibrational effect, the rotational effect on hydrogen dissociation on Cu is much less understood, until very recently. Most 3D quantum calculations have used the plane rotor model, which is not appropriate for studying rotational effects. The studies of Refs. 113, 114, and 117 using the spherical rotor treatment have obtained important results on the effect of rotational orientation and the nuclear symmetry. The rotational orientational effect is clearly shown in Fig. 15, where reaction probabilities for different initial rotational orientation states are plotted as a function of kinetic energy. Significant enhancement of reaction probability is seen for the state with j = m ( helicopter mode) while the m = o ( cartwheel mode) is least effective for dissociation. 

Another effect found in the flat-surface study of Refs. 113, 114 and 117 is the nuclear symmetry effect which states that for initial rotation states satisfying j + m =... 

Fig. 1.22a. Crossing of energy surfaces of electron states of different symmetry for the formaldehyde + methane reaction with the symmetry plane retained b avoided crossing when nuclear symmetry is distorted [127]. (Reproduced with permission from the American Chemical Society)...

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