And electron spin density

Reduction by electron transfer yields superoxide ion (02 -)> which has its negative charge and electronic spin density delocalized between the two oxygens. As such, it has limited radical character (H-OO bond energy AGbf, 72kcalmoU ) and is a weak Bronsted base in water... 

State average values such as dipole and quadrupole moments and electron spin densities are usually evaluated as expectation values of their corresponding quantum-mechanical operators. For example, the electronic contribution to the dipole moment operator is... 

Fig. 6.16 Partial geometry and schematic sketch middle) and electron spin density (SD) maps (top) of the unpaired electron of the model for (a) substitutional and (b) interstitial N-doping in an anatase Ti02 crystal lattice. In the middle the N atom is represented by a black sphere, O atoms are represented by yellow spheres, and Ti atoms are represented by small grey spheres. The SD plot is in the plane perpendicular to that containing the three Ti atoms bound to the N species. The figure is adapted from [104] with permission from the American Institute of Physics...

Fig. 5.2 Electron density and electron spin density transferability as viewed through the source function, in n-alkyl radicals. The electron (a) and electron spin (b) densities at a terminal C-H bcp are reported along with their total SF contributions from the various CH2 and CH3 groups in each system, c is the same as (b), but for a (3, +1) -V p non bonded charge concentration (NBCC) reference point associated to the unpaired electron. All values are given in atomic units (au)...

Dipolar coupling. Within a point-dipole approximation, the through-space dipolar interaction between a nuclear spin and electron spin density outside the nucleus, 77°, is expressed in tensor form as... 

Local spin density functional theory (LSDFT) is an extension of regular DFT in the same way that restricted and unrestricted Hartree-Fock extensions were developed to deal with systems containing unpaired electrons. In this theory both the electron density and the spin density are fundamental quantities with the net spin density being the difference between the density of up-spin and down-spin electrons ... 

Local exchange and correlation functionals involve only the values of the electron spin densities. Slater and Xa are well-known local exchange functionals, and the local spin density treatment of Vosko, Wilk and Nusair (VWN) is a widely-used local correlation functional. 

Both the oxygen and sulfur atoms have two lone pairs while the C/ carbon has ar unpaired electron, and in both cases the double bond shifts from the two carbor atoms to the carbon and the substituent. In acetyl radical, the electron density i centered primarily on the C2 carbon, and the spin density is drawn toward the lattei more than toward the former. In contrast, the density is more balanced between thf two terminal heavy atoms with the sulfur substituent (similar to that in allyl radical with a slight bias toward the sulfur atom. These trends can be easily related to th< varying electronegativity of the heavy atom in the substituent. 

The spin density surface is a tool which helps us find the unpaired electrons in these unusual molecules. Spin density is defined as the difference between the spin up and spin down electron clouds, and a spin density surface is constructed by connecting together points in the electron cloud where the spin density has an arbitrarily chosen value. 

As mentioned in the start of Chapter 4, the correlation between electrons of parallel spin is different from that between electrons of opposite spin. The exchange energy is by definition given as a sum of contributions from the a and /3 spin densities, as exchange energy only involves electrons of the same spin. The kinetic energy, the nuclear-electron attraction and Coulomb terms are trivially separable. 

In Fig. 3 the number of possible classical structures arising from the spin being localised at each carbon atom (top half) is compared to the UHFAA spin density results (lower half). Note that the number of classical structures when the unpaired electron is at sites 2, 5 and 6 is the same as for the double bonds involving atoms 1 and 2, 1 and 5 or 1 and 6 in Ceo- The correlation between the number of classical structures and the spin density is excellent. With only one exception all centres with the number of classical structures larger than 2200 show positive spin density and all those less than 2200 show negative spin density. This anticipated correlation can be further quantified. 

The magnetic field seen by the probe neutron is solely due to the magnetic dipole moment density of the unpaired electrons. In other words, the magnetisation density is simply related to the electron spin density by a multiplicative factor, and there is no ambiguity in its definition. 

When ESR spectra were obtained for the benzene anion radical, [C6II6] and the methyl radical, CH3, the proton hyperfine coupling constants were found to be 3.75 and 23.0 G, respectively, i.e. they differ by about a factor of 6. Since the carbon atom of CH3 has a spin density corresponding to one unpaired electron and the benzene anion carries an electron spin density of 1/6, the two results suggest that the proton coupling to an electron in a n-orbital is proportional to the spin density on the adjacent carbon atom ... 

P has been computed using Hartree-Fock atomic orbital wavefunctions and can be found in several published tabulations14 17 and in Appendix 1. Because of the (r 3) dependence of P, dipolar coupling of a nuclear spin with electron spin density on another atom is usually negligible. 

The muon and 29Si hyperfine parameters provide compelling evidence in support of the BC model. In the simple molecular-orbital model proposed by Cox and Symons (1986) the muon is located at the center of a Si—Si bond near a node in the unpaired electron spin density, which is... 

---