Symmetry 4, antisymmetric

Dominant spectral feature Vibrations destroying molecular symmetry antisymmetric stretching and deformation modes Vibrations preserving molecular symmetry symmetric stretching modes... 

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... 

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... 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32), Flectrons in the antisymmetric orbital r r have a 7ero probability of occurring at the node in u where U] = rj. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge... 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-eleetron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Zp Sp P eommute with total =Zi (i), as in the linear-moleeule ease. The additional symmetry present in the spherieal atom refleets itself in the faet that Lx, and Ly now also eommute with H and A. However, sinee does not eommute with Lx or Ly, new quantum... 

It is essential to realize that the energies (i H Oi> of the CSFs do n represent the energies of the true electronic states Ek the CSFs are simply spin- and spatial-symmetry adapted antisymmetric functions that form a basis in terms of which to expand the true electronic states. For R-values at which the CSF energies are separated widely, the true Ek are rather well approximated by individual (i H Oi> values such is the case near Rg. 

The first is the g or m symmetry property which indicates that ij/ is symmetric or antisymmetric respectively to inversion through the centre of the molecule (see Section 4.1.3). Since the molecule must have a centre of inversion for this property to apply, states are labelled g or m for homonuclear diatomics only. The property is indicated by a postsubscript, as in... 

The second symmetry property applies to all diatomics and concerns the symmetry of with respect to reflection across any (n ) plane containing the intemuclear axis. If is symmetric to (i.e. unchanged by) this reflection the state is labelled -I- and if it is antisymmetric to (i.e. changed in sign by) this reflection the state is labelled —as in or Ig. This symbolism is normally used only for I states. Although U, A, doubly degenerate state is... 

It was explained in Section 4.3.2 that the direct product of two identical degenerate symmetry species contains a symmetric part and an antisymmetric part. The antisymmetric part is an A (or 2") species and, where possible, not the totally symmetric species. Therefore, in the product in Equation (7.79), Ig is the antisymmetric and Ig + Ag the symmetric part. 

The CO2 laser is a near-infrared gas laser capable of very high power and with an efficiency of about 20 per cent. CO2 has three normal modes of vibration Vj, the symmetric stretch, V2, the bending vibration, and V3, the antisymmetric stretch, with symmetry species (t+, ti , and (7+, and fundamental vibration wavenumbers of 1354, 673, and 2396 cm, respectively. Figure 9.16 shows some of the vibrational levels, the numbering of which is explained in footnote 4 of Chapter 4 (page 93), which are involved in the laser action. This occurs principally in the 3q22 transition, at about 10.6 pm, but may also be induced in the 3oli transition, at about 9.6 pm. 

According to one classification (15,16), symmetrical dinuclear PMDs can be divided into two classes, A and B, with respect to the symmetry of the frontier molecular orbital (MO). Thus, the lowest unoccupied MO (LUMO) of class-A dyes is antisymmetrical and the highest occupied MO (HOMO) is symmetrical, and the TT-system contains an odd number of TT-electron pairs. On the other hand, the frontier MO symmetry of class-B dyes is the opposite, and the molecule has an even number of TT-electron pairs. 

Fig. 2. Simplified molecular orbital diagram for a low spia octahedral complex, such as [Co(NH3 )g, where A = energy difference a, e, and t may be antisymmetric (subscript ungerade) or centrosymmetric (subscript, gerade) symmetry orbitals. See text.

Consider for definiteness the antisymmetric case. We choose the origin of the coordinate system in one of the wells, and the center of symmetry has the coordinates Qt,Q-) (fig- 31). Inside the well the classical trajectories are Lissajous figures bordering on the rectangle formed by the lines Q = Q , and Q = —Q , where Q are the turning-point coordinates. 

This geometry possesses three important elements of symmetry, the molecular plane and two planes that bisect the molecule. All MOs must be either symmetric or antisymmetric with respect to each of these symmetry planes. With the axes defined as in the diagram above, the orbitals arising from carbon 2p have a node in the molecular plane. These are the familiar n and n orbitals. 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

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