Mulliken spin densities

As shown in Table 1, the calculated Mulliken spin densities of Fe and 02 moieties (ppe and p02) confirm that the calculated systems have indeed the closed-shell singlet, open-shell singlet and triplet spin states. Based on the calculated total electronic energies, the Weiss-type open-shell singlet (2) is most stable, followed... 

Table 1 Mulliken spin densities, charges, and energies of optimized Mb02 models...

Fig. 5 Results of two Ehrenfest simulations on benzene cation with fixed nuclei left side) and with nuclei moving right side). The top figures plot the evolution of the Mulliken spin densities as the function of time. The electron and nuclear motions are represented on the bottom moat diagrams (the nuclear geometry in blue and the electronic character in pink). With nuclei fixed, the electronic character of the system between a set of quinoid/antiquinoid structures. With moving nuclei, the oscillations in the electronic character seem damped until the nuclear geometry slowly catches the electronic character...

Table 14.2 Mulliken spin densities on the various atoms of the series a monoradicals for the RDFT and UDFT solutions...

Table 14.7 Mulliken spin densities on the triplet RDFT and UDFT solutions of series...

Figure 1 ROHF and UHF valence MOs of the methyl radical. The numbers under each MO denote the ratio of the AO coefficients on carbon and on the hydrogens (from an STO-3G calculation). Note how this ratio differs in the paired UHF MOs, illustrating the tendency of the P electrons to avoid the carbon atom where the a electron in the SOMO is localized. Note also that the ratio of AO coefficients in the ROHF MOs is about the average of those in each of the corresponding pairs for UHF MOs. The numbers at the bottom denote Mulliken spin densities p on carbon and on each hydrogen attached to it.

The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half-electron calculation for the triplet. Koopman s theorem is not followed for half-electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable. 

Considering a five-point spin density distribution (central ion and four nitrogens) for the determination of the dipolar proton hfs tensors in Ag(TPP) (5.5), the computed ADD principal values are found to be close to the experimental results. It should be noted that in Ag(TPP) the Mulliken population, UN, on the nitrogen nearest to the pyrrole proton provides a larger contribution to ADD along the Ag-H direction than the population UAg. 

Fig.12. Qualitative MO scheme for complex 1 (St = 0, BS(1,1)) (top) and spin density plot with Mulliken spin populations (bottom).

The spin density calculation, which analyzes the difference between each atom s Mulliken population (see Section 9.1.3.2) of a and ft electrons, indicated the unpaired electron to be highly delocalized, with populations on the ortho and para carbons of the phenyl... 

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])...

For all models, the configuration of the ligands coordinated to the Ni atom is tetrahedral in the high-spin state and square planar in the low-spin state. The Mulliken atomic spin densities and charges of each state are shown in Table 9-3. In the high-spin state, the SI, S3 and Ni atoms have 0.5-0.8 spin densities, while the SI and Ni atoms in la have hardly any spin. In addition, each of the SI, S3 and Ni atoms in 4b, 4b, and 4b" have 0.4-0.9 spin densities except the atoms abstracting the first H radical. The second H radical is captured easily on atoms with large spin densit. 

Table 9-3. Mulliken atomic spin densities and charge densities in all complexes...

Table 9-4. Mulliken atomic spin densities and charge densities in anion complexes (2a, 2h, 3a-, 3b-, TSa, TSh ) and dianion complexes (2a2-, 2b2-, 3a2, 3b2-, TSa2-, TSb2 )...

Fig. 5. Sample trajectory leading to the electron-transfer product, (a) Electronic energy, (b) atomic distances, (c) Mulliken group charges, and (d) spin density. Reprinted with permission from Yamataka et al. (1999).

The net charges and net spin densities obtained from Mulliken population analysis are listed in Table 3. The net charges of An are all ca. +2.9 except for Pa, for which the charge is +3.5. For all of the complexes, the charges on the C atoms are all about -0.2, and the H atoms are essentially neutral, consistent with the picture that the charge transfer occurs primarily between the Cpn orbitals to the An orbitals, as expected. The net spin density increases from Th to Am, consistent with the increase in the number of metal-localized electrons in the series of actinocenes. 

Table 3 Mulliken Net Charges and Spin Densities for the An, C, and H Atoms in An(Cot>2...

Fig. 2 Optimized geometries, calculated Mulliken charges (spin densities) of the true S (n=3-7) minima at the B3PW91/6-311+G level of theory...

Analyzed together with spin densities and Mulliken spin populations, the results of Table V allow us to determine the character of each type of free radical (Leroy, 1983b). To illustrate this, we consider two typical examples HCO and HO0. The properties of these species are shown in Figs. 12 and 13, respectively. 

Spin density calculations analyze the difference between each atom s population of a- and [3-electrons using various techniques. Mulliken population analysis140 is still often used for determining spin densities despite its deficiencies experienced sometimes when large basis sets are used. The Lowdin population analysis Scheme141 tries to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme however it is somewhat still basis set dependent. The natural bond orbital (NBO) methodology142 is a whole set of analysis techniques, including natural population analysis (NPA), which is less basis set dependent than the Mulliken scheme and often considered one of the most reliable methods for population analysis. 

Symmetry and stability analysis. The semi-empirical unrestricted Hartree-Fock (UHF) method was used for symmetry and stability analysis of chemical reactions at early stage of our theoretical studies.1,2 The BS MOs for CT diradicals are also expanded in terms of composite donor and acceptor MOs to obtain the Mulliken CT theoretical explanations of their electronic structures. Instability in chemical bonds followed by the BS ab initio calculations is one of the useful approaches for elucidating electronic structures of active reaction intermediates and transition structures.2 The concept is also useful to characterize chemical reaction mechanisms in combination with the Woodward-Hoffman (WH) orbital symmetry criterion,3 as illustrated in Figure 1. According to the Woodward-Hoffmann rule,3 there are two types of organic reactions orbital-symmetry allowed and forbidden. On the other hand, the orbital instability condition is the other criterion for distinguishing between nonradical and diradical cases.2 The combination of the two criteria provides four different cases (i) allowed nonradical (AN), (ii) allowed radical (AR), (iii) forbidden nonradical (FN), and (iv) forbidden radical (FR). The charge and spin density populations obtained by the ab initio BS MO calculations are responsible for the above classifications as shown in Fig. 1. 

Another property which is interesting for a radical is the spin-density. We can compute the spin populations on the carbon atoms using a Mulliken analysis. The CASSCF wave function is used. The spin-density will come entirely from the tt system because we have used a closed shell for the a electrons. 

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