Antisymmetric combination

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. 

Gleiter and Ginsburg found that 4-substituted-l,2,4-triazoline-3,5-dione reacted with the propellanes 36 and 37 at the syn face of the cyclohexadiene with respect to the hetero-ring. They ascribed the selectivity to the secondary orbital interaction between the orbitels (LUMO) of 36 and 37 with antisymmetrical combination of lone pair orbitals (HOMO ) of the triazolinediones (Scheme 24) [29]. 

A second, independent spectroscopic proof of the identity of 4 as rans-[Mo(N2)2(weso-prP4) was provided by vibrational spectroscopy. The comparison of the infrared and Raman spectrum (Fig. 7) shows the existence of two N-N vibrations, a symmetric combination at 2044 cm-1 and an antisymmetric combination at 1964 cm-1, indicating the coordination of two dinitrogen ligands. In the presence of a center of inversion the symmetric combination is Raman-allowed and the antisymmetric combination IR allowed. The intensities of vs and vaK as shown in Fig. 2 clearly reflect these selection rules. Moreover, these findings fully agree with results obtained in studies of other Mo(0) bis(dinitrogen)... 

Fig. 7. Comparison of the infrared (top) and Raman (bottom) spectrum of frans-[Mo(N2)2(nreso-prP4)] showing the symmetric N-N vibration at 2044 cm-1 and the antisymmetric combination at 1964 cm-1 (33).

The electronic states introduced by hydrogen in the band structure of Si are quite different depending upon the location of the impurity in the lattice. For H at the bond center we can, to a first approximation, treat the problem as involving only three states (Schaad, 1974 Fisch and Licciardello, 1978) the semiconductor bonding (b) and antibonding (a) states (which, in turn, are symmetric and antisymmetric combinations of hydrid orbitals on the two atoms) and the hydrogen Is orbital. The cor-... 

This seeming lack of stabilization is disconcerting when it is noticed that the antisymmetric combination of the two methylene C—H a bonds is of the right symmetry to mix in with the n system. As such, cyclopentadiene can be said to enjoy the possibility of 6n, and hence aromatic stabilization. 

Linear combination of the two lone-pair orbitals n and n.2) with reference to the relevant symmetry operation leads to a symmetric (n+) and an antisymmetric combination (n ). 

Coordinates and Potential Energy Surfaces. In any of the above formalisms one must identify the nuclear coordinates which contribute to the activation of the reactants. In the present case we confine our attention to two "active" modes — the solvent polarization mode ([Pg.262]

Typical potential energies associated with such a Hamiltonian are shown in Figure 4 as a function of the parameter 0 = x/2J. The coordinate is the antisymmetric combination. The symmetric mode clearly adds a term to the total energy independent of coupling. 

MeS, MeSe, Me2N, or cyclic systems containing these donor atoms. Two sharp bands are seen below 11 eV, of which that with the lowest IP is ascribed to ionization from the highest occupied it orbital, ir, which has a large contribution from the symmetric combination of the donor pt orbitals, and that with die second lowest to ionization from the antisymmetric combination of these orbitals (n ). Both IPs decrease with decreasing ionization potential of A (Table 21), that is, in the series S < Se < N. 

The energy factored force field for carbonyls contains interaction parameters as well as stretching parameters. Such parameters are invariably required to be positive, for CO groups attached to the same metal, by the experimental finding that symmetric combinations of the individual CO vibrations occur at higher frequencies than similar antisymmetric combinations. 

When is expressed as the antisymmetric combination given by the Slater determinant ... 

No discussion about strained-ring radical cations would be complete without the valence isomers quadricyclane (15 +) and norbornadiene, (16 +) 15 features two adjacent rigidly held cyclopropane rings, whereas 16 contains two ethene n systems well suited to probe through-space interactions.Molecular orbital considerations suggest the antisymmetric combination of the ethene n orbitals (16) or cyclopropane Walsh orbitals (15) as respective HOMOs of the two parent molecules. The radical ions have different state symmetries and their SOMOs have different orbital symmetries. 

Calculations and Experiments on the Stereomutation of Cyclopropane. In 1965, Hoffmann published a seminal paper on trimethylene, another name for propane-1,3-diyl (8). He used extended hiickel (EH) calculations and an orbital interaction diagram to show that hyperconjugative electron donation from the central methylene group destabilizes the symmetric combination of 2p-n AOs on the terminal carbons in the (0,0) conformation of this diradical. Hoffmann s calculations predicted that the resulting occupancy of the antisymmetric combination of 2p-n AOs in 8 should favor conrotatory opening of cyclopropane (7), as depicted in Figure 22.8. 

Alkyl substituents are not just stereochemical markers. At Cl and C3 of 8 they can hyperconjugatively donate electrons to both the symmetric and antisymmetric combinations of 2p-n AOs at these carbons. As a result, hyperconjugative destabilization of both these combinations of 2p-n AOs by electron donation from alkyl substituents at Cl and C3 should reduce the preference for the configuration in which the antisymmetric combination of 2p-n AOs is doubly occupied. In fact, our (2/2)CASSCF calculations on 8a and on its 1,3-dimethyl derivative indicated that the methyl substituents should wipe out almost completely the weak preference for conrotation over monorotation that we computed for the stereomutation of unsubstituted 7. ... 

Figure 6.3 The MO formed by interaction between the antisymmetric combination of H Is orbitals and the oxygen orbital (see also Figure 6.7). Bonding interactions are enhanced by mixing a small amount of O d character into the MO...

In the case of ET, the reaction path starts with the electron in LUMO of the donor. Along the reaction path, the orbital shifts character, from donor to acceptor orbital. At the transition state LUMO and LUMO+1 for the full system are symmetric and antisymmetric combinations of the two local LUMO s (fig.2). Since for any of the two orbitals, the negative of the orbital energy approximates the electron affinity according to Koopmans theorem, the following implication holds ... 

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