Symmetric and antisymmetric

Figure Al.2.8. Typical energy level pattern of a sequence of levels with quantum numbers nj for the number of quanta in the symmetric and antisymmetric stretch. The bend quantum number is neglected and may be taken as fixed for the sequence. The total number of quanta (n + n = 6) is the polyad number, which...

Figure Al.2.10. Birth of local modes in a bifurcation. In (a), before the bifiircation there are stable anhamionic symmetric and antisymmetric stretch modes, as in figure Al.2.6. At a critical value of the energy and polyad number, one of the modes, in this example the symmetric stretch, becomes unstable and new stable local modes are bom in a bifurcation the system is shown shortly after the bifiircation in (b), where the new modes have moved away from the unstable syimnetric stretch. In (c), the new modes clearly have taken the character of the anliamionic local modes.

In summary, for a homonuclear diatomic molecule there are generally (2/ + 1) (7+1) symmetric and (27+1)7 antisymmetric nuclear spin functions. For example, from Eqs. (50) and (51), the statistical weights of the symmetric and antisymmetric nuclear spin functions of Li2 will be and respectively. This is also true when one considers Li2 Li and Li2 Li. For the former, the statistical weights of the symmetric and antisymmetiic nuclear spin functions are and, respectively for the latter, they are and in the same order. 

Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

Infrared The absorptions of interest m the IR spectra of amines are those associated with N—H vibrations Primary alkyl and arylammes exhibit two peaks m the range 3000-3500 cm which are due to symmetric and antisymmetric N—H stretching modes... 

The symmetric and antisymmetric stretching vibrations of methylamine can be viewed on Learning By Modeling... 

In order that i/ r A always antisymmetric for IT2 the antisymmetric ij/ are associated with even J states and the symmetric with odd J states, as shown in Figure 5.18. Interchange between states with symmetric and antisymmetric is forbidden so that IT2 can be regarded as consisting of two distinct forms ... 

The wave functions in Equations (7.26) and (7.27) are symmetric and antisymmetric, respectively, to electron exchange. 

Comparison of UV data for 3-aminoisothiazoles with those of reference compounds confirms that they exist in the 3-amino form. A more recent investigation of 4-aminoisothiazole (76MI41701) using deuterium exchange experiments of the type described in Section 4.01.5.2, and analysis of the symmetric and antisymmetric NH2 stretching frequencies in its IR spectrum, show that this compound also exists in the 4-amino form. 

The symmetric and antisymmetric parts of I are the tensors of stretching d and spin w, respectively. 

It is the velocity gradient when R = I, i.e., in a pure stretch, and will be called the unrotated velocity gradient. While U is symmetric, / in general fails to be so. The symmetric and antisymmetric parts of / will be called the unrotated stretching d and unrotated spin w... 

The second term is antisymmetric, as may be verified by differentiating QQ = /, so that decomposing into symmetric and antisymmetric parts... 

Fig. 6. Examples of (a) mirror symmetry and (b) non-symmetry with respect to the tube axis. The HO-LU band crossing in (a) changes into avoid crossing in (b). Notations S and A signify symmetric and antisymmetric with respect to the mirror symmetry, respectively, for instance.

The two intragap stales g (x) arc the symmetric and antisymmetric superpositions of the midgap stales localized near the soliton and the anlisoliton ... 

FIGURE 2 la The three normal vibrational modes of 11,0. Two of these modes are principally stretching motions of the bonds, but mode v2 is primarily bending, (b) The four normal vibrational modes of C02. The first two are symmetrical and antisymmetrical stretching motions, and the last two are perpendicular bending motions. 

In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

The subscripts S and A represent the words symmetric and antisymmetric, respectively (Sec. 29a) the wave function Ls... 

It is found in this way that crystalline hydrogen at temperatures somewhat below the melting point is a nearly perfect solid solution of symmetric and antisymmetric molecules, the latter retaining the quantum weight 3 for the state with j = 1 as well as the spin quantum weight 3. This leads to the expression... 

The possibility of the expression of the entropy of hydrogen as the sum of these terms was first noted by Giauque, who observed that it indicated the formation of nearly ideal solid solutions between symmetrical and antisymmetrical hydrogen and the retention of the quantum weight 9 for the latter. 

In crystals for which n0 is large, such as iodine, the lowest symmetric and the lowest antisymmetric state have practically the same energy and properties, and each corresponds to one eigenfunction only. As a result a mixture of symmetric and antisymmetric molecules at low temperatures will behave as a perfect solid solution, each molecule having just its spin quantum weight, and the entropy of the solid will be the translational entropy plus the same entropy of mixing and spin entropy as that of the gas. This has been verified for I2 by Giauque.17 Only at extremely low temperatures will these entropy quantities be lost. 

The wave functions for the two inner-core spherons can, of course, be described as the symmetric and antisymmetric combinations of l.t and Ip-functions. The Nilsson (19) treatment of neutron and proton orbitals in deformed nuclei is completely compatible with the foregoing discussion, which provides a structural interpretation of it. 

In a KI matrix the electronic absorption maximum of 82 - is observed at 400 nm, and the 88 stretching vibration by a Raman line at 594 cm k 83 shows a Raman line at 546 cm and an infrared absorption at 585 cm which were assigned to the symmetric and antisymmetric stretching vibrations, respectively. The bromides and iodides of Na, K, and Rb have also been used to trap 82 - but the wavenumbers of the 88 stretching vibration differ by as much as 18 cm- from the value in KI. The anion S3- has been trapped in the chlorides, bromides and iodides of Na, K, and Rb [120]. While the disulfide monoanion usually occupies a single anion vacancy [116, 122], the trisulfide radical anion prefers a trivacancy (one cation and two halide anions missing) [119]. 

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