Orbital amplitude

Orbital amplitudes, 54 Orbital energies, 51, 53, 287 Orbital interactions general rules, 10, 38 Orbital labels group theoretical, 46, 51 localized types, 51 Orbital occupancies, 57 Oxygen, 88 Ozone, 151... 

Keywords Chemical orbital theory. Electron delocalization. Frontier orbital. Orbital amplitude, Orbital energy, Orbital interaction. Orbital mixing rule, Orbital phase, Orbital phase continuity, Orbital phase environment. Orbital synunetry, Reactivity, Selectivity... 

From Electron Density to Frontier Orbital Amplitude... 

Keywords Cycloadditions, Chemical orbital theory. Donor-acceptor interaction. Electron delocalization band. Electron transfer band, Erontier orbital. Mechanistic spectrum, NAD(P)H reactions. Orbital amplitude. Orbital interaction. Orbital phase. Pseudoexcitation band. Quasi-intermediate, Reactivity, Selectivity, Singlet oxygen. Surface reactions... 

Keywords Orbital mixing. Orbital amplitude. Orbital phase. Orbital polarization. Orbital deformation, Regioselectivity, Stereoselectivity, n Facial selectivity... 

The n orbital amplitudes of ethene are identical on both carbons. Unsymmetrical substitutions polarize the n orbital. Electron acceptors or electrophiles attack the carbon with the larger r amplitude. The polarization of frontier orbitals is important for regioselectivities of reactions. Here, mechanism of the n orbital polarization of ethene by methyl substitution [4] is described (Scheme 5). 

According to the frontier orbital theory, a bond preferentially forms between the atoms with the largest frontier orbital amplitudes (Sect. 3.4 in the Chapter Elements of a Chemical Orbital Theory by Inagaki in this volume). This is applicable for the regioselectivities of Diels-Alder reactions [15]. The orbital mixing rules are shown here to be useful to understand and design the regioselectivities. 

The regioselectivities of Diels-Alder reactions are also understood in terms of the orbital phase continuity [29]. The selectivity is also explained by the frontier orbital amplitude [30]. 

Scheme 1 From electron density to orbital amplitude...

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

Two general differences are apparent. TheFock-matrix element general reduction of a-orbital amplitude at each center (by the normalization factor 2-1/2) as well as the more directed p-rich character of n compared with a. For the present comparison this results in a difference... 

Figure 10.1. The first SCVB orbital for the allyl radical. The orbital amplitude is given in a plane parallel to the radical and 0.5 A distant.

In Fig. 10.1 we show an altitude drawing of the orbital amplitude of the first of the SCVB orbitals of the allyl n system. The third can be obtained by merely reflecting this one in the y-z plane of the molecule. It is seen to be concentrated at... 

Figure 10.6. The first SCVB orbital for the BeH molecule and associated with the Be nucleus. This has the general appearance of an 5-p hybrid pointed toward the H atom, and we denote it the inner hybrid, hi. The orbital amplitude is given in the x-z plane, which contains the nuclei.

Figure 10.9. The three SCVB orbitals for the Li atom. The orbital amplitudes times the radial coordinate are shown.

For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular, localization of excess spin must be well predicted. ROHF methods leave something to be desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially identical, they make no contribution to P only singly occupied orbitals contribute. As discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance of spin polarization compared to tire simple analysis of orbital amplitude for singly occupied molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998). 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

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