Atomic orbital coefficients

Experimentally determined maximum absolute ionization cross sections for the inert gases and a range of small molecules are compared with the predictions of DM, BEB, and EM calculations in Table 1. Atomic orbital coefficients for the DM calculations were determined at the Hartree-Fock level and the EM cross sections are volume averaged for calculations carried out at the HF/6-31G level. Hie same data are plotted in Figure 5 with the calculated values on the ordinate and the experimental result on the abscissa. The heavy line represents a direct correspondence between experiment and theory. Although the ab initio EM method performs well for the calculation of qm and Em,T,17 the DM and BEB methods allow for the calculation of the cross section as a function of the electron energy, i.e. the ionization... 

The dipolar parts of the analyzed hfs tensors have been compared with calculated values obtained from first order expressions of the electron-nuclear dipole interaction (5.3)57. The coefficients of the atomic orbitals used in this computation, which considers all two- and three-center contributions, are obtained from an extended Huckel calculation (ethyl groups replaced by protons). It has been found that almost 100% of the unpaired electron is located on the CuS4 fragment so that the replacement of the ethyl groups by protons is of minor importance for the calculation of the atomic orbital coefficients. The experimental and theoretical hfs data, summarized in Table 8, are found to... 

Figure 3. MO diagram for the ai and a2u orbitals of porphine. The atomic orbital coefficients are proportional to the size of the circles solid or dashed circles indicate sign.

When vj/j is a normalized wavefunction, the atomic orbital coefficients fulfill the condition... 

It is therefore somewhat surprising that in the biperylenyl anion radical Baumgarten et al. (1993) observed a localization of the unpaired electron in one moiety only, although the steric situation is anticipated to be similar to that in the binaphthyl anion radical. This is explained by the fact that the atomic orbital coefficients of the single-occupied molecular orbital for the bridgehead positions are larger for the binaphthyl anion radical than for the biperynyl anion radical. Therefore, the interaction of the separate entities is more decoupled in the biperynyl case. 

Fig. 4 SOMO of C60 ion and Mulliken charge spin density (italics). The magnitude of 7r atomic orbital coefficients of SOMO is represented by the size of circles and its sign by open (+) and filled (-) circles. (Reproduced from [20])...

Fig. VI-2. Ultraviolet transition ir4 — it for 2-aminothiazole atomic orbital coefficients in...

Heteronuclear diatomic molecules are naturally somewhat more complicated than the homonuclear comprehensive comparisons with homonuclear molecules were given by Mulliken [15]. The atomic orbital coefficients in the molecular orbitals ofheteronu-clear diatomic molecules are no longer determined by symmetry alone, and the electrons in the molecular orbitals may be shared equally between atoms, or may be almost localised on one atom. The molecular orbitals can still be classified as a or n, but in the absence of a centre-of-symmetry the g/u classification naturally disappears. Some heteronuclear molecules contain atoms which are sufficiently similar that the molecular orbitals resemble those shown in figure 6.7. In many other cases, however, the atoms are very different. This is particularly the case for hydride systems, like the HC1 molecule,... 

Two features of the results presented in the table are of particular interest. The first is the very large Fermi contact interaction constant, given by ly = b + (c/3), of 1.947 GHz. The calculated value for an impaired electron in a 4s atomic orbital of Sc is 2.823 GHz, suggesting that the orbital of the unpaired electron in ScO has 69% 4s character, or an atomic orbital coefficient of 0.83. Ab initio calculations by Bauschlicher and Langhoff [68] indicate that the primary electron configuration for the ground state of ScO is... 

Here, C and C n are the sulfur atomic orbital coefficients in v ia and ib, rL is the position vector, and (%n % n) is an overlap integral. The upshot of this is that the same type of dithiolene MO (e.g., (ground-state and excited-state wave functions for enhanced CT transition intensity, and this intensity is a direct consequence of the dmetai — dithiolene covalency. 

The hyperfine coupling constants of protons attached to a re-center can be determined by EPR/ENDOR spectroscopic measurements [15, 16] these coupling constants are proportional to the local spin densities at the re-centers as predicted by the corresponding atomic orbital coefficients of the... 

Once the dominant FMO interaction has been identified, then a choice as to which regiochemical outcome is preferred is based upon a comparison of the sizes of the coefficients at the atomic orbital centers involved in the formation of the new bonds. Consequently, at least a qualitative knowledge of the form of the FMOs is useful in assessing the relative merits of one over the other regiochemical outcome. If, as is usually but not always (vide irfra) the case, the coefficients in both the HOMO and the LUMO differ in size, then the new overlap can occur in either of two ways. It can readily be shown, however, that overlap between the atomic orbitals with the larger coefficients (and, by default, the two centers with the smaller atomic orbital coefficients overlap at the other site of bond formation) is preferred over the alternative. - ... 

In frontier oibital terms, the regiochemistry is governed largely by the atomic orbital coefficients at the termini of the reaction pailners, which are tered by the substituents. Hence, in normal Diels-Alder reactions a diene substituent at C-1 has the tendency to direct the addition of a carbonyl-conjugated al-kene towards the ortho product (X), whereas a substituent at C-2 favors the para product (XI). The... 

Also kno vn as the donor superdelocalizability, it is defined as the sum over all the unoccupied molecular orbitals (Nmo Nocc)> Nmo being the total number of molecular orbitals, and over the atomic orbitals i of the ath atom of the square atomic orbital coefficient c, divided by the energy of the corresponding molecular orbital, that is. 

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