Ungerade symmetry

For the homonuclear example, the and CSFs undergo Cl coupling to form a pair of states of symmetry (the CSF cannot partake in this Cl mixing because it is of ungerade symmetry the states can not mix because they are of triplet spin symmetry). The Cl mixing of the and 1 CSFs is described in terms of a 2x2 secular problem... 

E s are the unperturbed energies of the electronic and vibrational states, respectively, and Bm is a constant energy factor which depends on the M excited state. It appears from Eq. (6.5) that ungerade symmetry of inverse energy mismatch between the relevant levels. 

Within the approximation that the valence electronic states can be described adequately as combinations of the above valence CSFs, the three JE, JE, and CSFs must be combined to form the three lowest energy valence electronic states of E symmetry. For the homonuclear case, the E CSF does not couple with the other two because it is of ungerade symmetry, while the other CSFs JE and1E have gerade symmetry and do combine. 

Ultraviolet (UV) radiation, 244 Ungerade symmetry, 48. 160 Ungerade orbitals, 17-18, 160, 438... 

Figure 13.3 depicts the lowest four eigenfunctions of the ungerade symmetry (multiplied with the X — A transition dipole function). They are anti-symmetric with respect to the interchange of i i and R2 and therefore they have a node on the symmetry line i i = R2. Some examples for gerade states will be shown in Figure 14.4. The assignment Imn ) reflects the leading term in expansion (13.7). For example, 21 ) means that the function dominates the expansion while the coefficients for the other basis functions are considerably smaller. The corresponding wavefunction is approximately given by...

Dealing with a case of nearly degenerate 20g and 2ct 3(12b 1 Jrg) configurations one may ask what kind of spacial correlations are involved. In Sects. 7.1 and 7.2 we discussed the breaking of the gerade-ungerade symmetry of the molecule and dynamic localization of the hole, i.e. hopping of a localized hole between the nuclei. That discussion was based... 

Fig. 2.9 Angular wave functions of s, p, d, and / orbitals illustrating gerade and ungerade symmetry (a) s orbital, gerade (b) p orbital, ungerade-, (c) pictorial representation of symmetry of p orbital (d) orbital, gerade-, (e) pictorial representation of symmetry of d orbital (f) d.i orbital, gerade-, (g) / j orbital, ungerade.

The potential energy surfaces (LEPS A from Refs. (8.10)). the effective potentials U (n=0 gerade and ungerade symmetry) and the bound states of the collinear IHI and lOI are shown in Fig. 4. As argued for the Cl+HCl system the reaction is very adiabatic, and thus the effective potentials and the adiabatic eigen-energies e (r) coincide within 10 eV. It should be noticed that isotopic suBstitution of the hydrogen atom by its heavier isotope deuterium tends to destabilize the vibrational bonded molecule. 

For states with ungerade symmetry, the agreement between the transition energies calculated with active spaces C1-C2 and D is in most cases acceptable, with a maximum difference of 0.2 eV. An exception is the first transition. In this case the difference between the... 

Crystal fields of uneven (ungerade) symmetry are able to lift the parity selec- ion rule. The reader should not underestimate the importance of such a, seemingly-Kademic, statement there would be no color television, and no energy-.saving lumi-lescent lamps without uneven-symmetry crystal fields, as we will see later. 

Finally, we can simplify the expansion further by taking into account the spatial symmetry of the system. Both To> and 22> are of gerade symmetry while is of ungerade symmetry because it contains one orbital with gerade and one with ungerade symmetry. Therefore, this single excitation will not mix with q) 22>. Thus we can write the Cl expansion,... 

The pluses and minuses indicate the sign of the atomic basis functions in the LCAO description of the four orbitals. Note that a and r are of gerade symmetry while b and s are of ungerade symmetry. The HF wave function... 

Photoassociation scenarios of the nonpolar Rb2 and polar KRb molecules differ due to the presence of the gerade-ungerade symmetry the dipole transition from the... 

The excited molecule has two half-filled MOs, for which the electron spins can be opposite (singlet) or the same (triplet). The sum of electron orbital momenta wiU remain zero (giving S if the electron is excited from wj / = — 1 to — 1 or from -I-1 to +1, but will change to +2 or —2 (giving A) if changed from —1 to +1 or from - -1 to — 1. In all cases, the excited state has an odd number of electrons in the MOs of ungerade symmetry, so all terms will have a u subscript. The term symbols, then, are, respectively Su, Eu, Au, Au. ... 

For homonuclear diatomics, the positive combination has ungerade symmetry and the negative has gerade symmetry. Labeling the positive and negative combinations (for... 

If an object has ungerade symmetry with respect to inversion, then it will have a negative character in the i column and the operations generated from combinations of the rotational subgroup members and i will have the opposite sign to that of the simple rotation involved. These are the representations labelled u . 

Now we will consider each class of operations to narrow down the possible standard irreducible representations that can be present until we arrive at only one option. The 3 under the E class reminds us that there are three orbitals being represented, and so our combination must consist of either three E-type representations or one E and one doubly degenerate representation. Under the inversion centre i the total character is —3, and so all three orbitals must be reversed by the inversion. This means that any irreducible representation present must have ungerade symmetry if we assigned a gerade representation, then it would contribute positively under i. This eliminates all gerade representations from further consideration. 

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