Nuclear exchange symmetry

As with homonuclear diatomic molecules, the letters s and a are used to give the nuclear-exchange symmetry of / ns for polyatomic molecules. 

We can extend the method to heteronuclear diatomics with relative ease. Since the heteronuclear molecule does not have the nuclear exchange symmetry the g and u character is lost, but the rest of the notation remains the same. The correlation diagram becomes more complicated because of the difference in energies of the electron on the two different separated atoms. It is worth noting that isoelectronic species such as N=N, C=0,... 

Same as hydrogen, under normal circumstances, deuterium (normal-D2) consists of ortho-D2 and para-D2. However, because the D atom nuclear spin quantum number / = 1 is boson, nuclear exchange symmetry of the total wave function is symmetrical. Para-D2 corresponds to the energy levels at even-number rotational quantum states (7 = 0, 2,4,...) ortho-D2 corresponds to odd-number states 0 =1 3, 5,...). The ratio of ortho-D2 to para-D2 is 2 1. We prepared ortho-D2 using the same method as para-H2. 

Prior to digressing on the subject of nuclear exchange symmetry, we mentioned that a new symmetry element besides (molecule-fixed) was required to classify electronic-rotational states in homonuclear diatomics. A logical choice is i (molecule-fixed), an operation which belongs to but not It may be shown that z (molecule-fixed) is equivalent to Xj (space-fixed), and so the procedures worked out in the foregoing discussion may be used to classify ji/ZeiZrot) as either (s) or (a) under in lieu of determining their behavior under molecule-fixed inversion. The dipole moment operator ft in homonuclear molecules is (s) under Xj [11]. This leads to the conclusion that only states ij/e Xrot > with like symmetry under Xff can be connected by El transitions in electronic band spectra ... 

Until very recently, however, the same could not be said for reactive systems, which we define to be systems in which the nuclear wave function satisfies scattering boundary conditions. It was understood that, as in a bound system, the nuclear wave function of a reactive system must encircle the Cl if nontrivial GP effects are to appear in any observables [6]. Mead showed how to predict such effects in the special case that the encirclement is produced by the requirements of particle-exchange symmetry [14]. However, little was known about the effect of the GP when the encirclement is produced by reaction paths that loop around the CL... 

The correct permutational symmetry was implemented into the wave functions by projection onto irreducible representations of the total symmetry group for heteronuclear species and for homonuclear species, where e refers to electron exchange and H refers to nuclear exchange. The irreducible representations chosen were singlets in all cases. 

For CH2=CH2, there are four symmetry modifications, which might be called ss, sa, as, and aa. The letters refer to the symmetry of V / ns f°r nuclear exchanges brought about by rotations of the framework about two of the C2 axes. There are three C2 axes, but since C2(z) = C2(x)C2(y), the symmetry behavior with respect to two of the C2 rotations determines the behavior with respect to the third C2 rotation. 

The symmetry of pcl for linear polyatomic molecules follows from the same arguments as for diatomic molecules thus pcl is symmetric with respect to nuclear exchange for 2, 2 ,... functions and antisymmetric for e2, e2, ... functions. 

Ptj, the symmetry operation involving interchange of identical particles (nuclei or electrons). All particles axe either Bosons or Fermions, and the total wavefunction must, respectively, be even or odd upon interchange of any pair of identical particles. The total wavefunction of a homonuclear molecule, exclusive of the nuclear spin part, is classified s or a according to whether it is even or odd with respect to nuclear exchange. Since electrons are Fermions, the total molecular wavefunction must be odd with respect to permutation of any two electrons. This requirement is satisfied by the determinantal form of the electronic wavefunction (see Section 3.2.4). 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

Cyanuric acid is a titrable weak acid (pffai — 6.88, pifa2 — H-40, pffas — 13.5) (10). The pH of a saturated aqueous solution of pure CA at room temperature is - 4.8. Thermodynamic properties of CA are given ia Table 1. Spectroscopic data are available (1 3). Proton nmr is of limited usefulness because of proton exchange and CA s symmetry and low solubiUty. Nuclear quadmpole resonance measurements ( " N) have been reported (12). 

To treat the reagents and products as indistinguishable, one must make the total (electronic + nuclear) wave function symmetric under a cyclic exchange of nuclei, which is equivalent to making it symmetric under rotations 2n/3, 4n/3. about the threefold axis of symmetry. Mead showed that, because the electronic wave function >]> is antisymmetric under 2ti/3> must be symmetrized... 

As noted above at higher densities the EoS is sensitive to 3NF contributions. Whereas the 3NF for low densities seems now well understood its contribution to nuclear matter densities remains unsettled. In practice in calculations of the symmetry energy in the BHF approach two types of 3NF have been used in calculations in ref.[4] the microscopic 3NF based upon meson exchange by Grange et al. was used, and in ref. [15] as well in most VCS calculations the Urbana interaction. The latter has in addition to an attractive microscopic two-pion exchange part a repulsive phenomenological part constructed in such a way that the empirical saturation point for SNM is reproduced. Also in practice in the BHF approach to simplify the computational efforts the 3NF is reduced to a density dependent two-body force by averaging over the position of the third particle. 

NMR and EPR techniques provide unique information on the microscopic properties of solids, such as symmetry of atomic sites, covalent character of bonds, strength of exchange interactions, and rates of atomic and molecular motion. The recent developments of nuclear double resonance, the Overhauser effect, and ENDOR will allow further elucidation of these properties. Since the catalytic characteristics of solids are presumably related to the detailed electronic and geometric structure of solids, a correlation between the results of magnetic resonance studies and cata lytic properties can occur. The limitation of NMR lies in the fact that only certain nuclei are suitable for study in polycrystalline or amorphous solids while EPR is limited in that only paramagnetic species may be observed. These limitations, however, are counter-balanced by the wealth of information that can be obtained when the techniques are applicable. 

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