Antisymmetric

We separate the symmetric part and the antisymmetric part of the tensor ... 

The external reflection of infrared radiation can be used to characterize the thickness and orientation of adsorbates on metal surfaces. Buontempo and Rice [153-155] have recently extended this technique to molecules at dielectric surfaces, including Langmuir monolayers at the air-water interface. Analysis of the dichroic ratio, the ratio of reflectivity parallel to the plane of incidence (p-polarization) to that perpendicular to it (.r-polarization) allows evaluation of the molecular orientation in terms of a tilt angle and rotation around the backbone [153]. An example of the p-polarized reflection spectrum for stearyl alcohol is shown in Fig. IV-13. Unfortunately, quantitative analysis of the experimental measurements of the antisymmetric CH2 stretch for heneicosanol [153,155] stearly alcohol [154] and tetracosanoic [156] monolayers is made difflcult by the scatter in the IR peak heights. 

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.

The fact that allowed fennion states have to be antisymmetric, i.e., changed m sign by any odd pemuitation of the fennions, leads to an interesting result concerning the allowed states. Let us write a state wavefiinction for a system of n noninteracting fennions as... 

One of the consequences of this selection rule concerns forbidden electronic transitions. They caimot occur unless accompanied by a change in vibrational quantum number for some antisynnnetric vibration. Forbidden electronic transitions are not observed in diatomic molecules (unless by magnetic dipole or other interactions) because their only vibration is totally synnnetric they have no antisymmetric vibrations to make the transitions allowed. 

Table Bl.5.1 Independent non-vanishing elements of the nonlinear susceptibility, for an interface in the Ay-plane for various syimnetry classes. When mirror planes are present, at least one of them is perpendicular to they-axis. For SFIG, elements related by the pennutation of the last two elements are omitted. For SFG, these elements are generally distinct any syimnetry constraints are indicated in parentheses. The temis enclosed in parentheses are antisymmetric elements present only for SFG. (After [71])...

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

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. 

Since the total wave function must have the correct symmetry under the permutation of identical nuclei, we can determine the symmetiy of the rovi-bronic wave function from consideration of the corresponding symmetry of the nuclear spin function. We begin by looking at the case of a fermionic system for which the total wave function must be antisynmiebic under permutation of any two identical particles. If the nuclear spin function is symmetric then the rovibronic wave function must be antisymmetric conversely, if the nuclear spin function is antisymmebic, the rovibronic wave function must be symmetric under permutation of any two fermions. Similar considerations apply to bosonic systems The rovibronic wave function must be symmetric when the nuclear spin function is symmetric, and the rovibronic wave function must be antisymmetiic when the nuclear spin function is antisymmetric. This warrants... 

Next, we address some simple eases, begining with honronuclear diatomic molecules in E electronic states. The rotational wave functions are in this case the well-known spherical haimonics for even J values, Xr( ) symmetric under permutation of the identical nuclei for odd J values, Xr(R) is antisymmetric under the same pemrutation. A similar statement applies for any type molecule. 

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