The spin density

The unpaired electron has the complication that it is not localized on a single point but, in general, is delocalized on the entire molecule. So, in every point of space where the molecular orbital (MO) containing the unpaired electron has a non-zero value, the average electron magnetic moment sensed by the nucleus is different from zero and is proportional to (Sz) times the fraction of unpaired electron present at that point. Such a fraction is called spin density p, which for a single electron is given by the square of its wavefunction at that point. 

The situation is further complicated by the fact that the presence of an unpaired electron in a MO polarizes the paired electrons in the core. This is called spin polarization. The MO containing the paired electrons is modified by the presence of the unpaired electron in another MO in such a way that the electron with the spin aligned with the unpaired electron will have a slight preference to occupy the region of space of its MO which is closer to the unpaired electron itself (this is a manifestation of Hund s rule). Conversely, the other electron with spin antiparallel to the unpaired electron will have a slight preference to occupy regions of its MO far from the unpaired electron. This is illustrated in Fig. 2.1 for the case of two atomic orbitals. 

At this point we recall that the unpaired electron may have some probability to sit just on the nucleus. Type s orbitals have maximal electron density on the nucleus, as their wavefunction is of the type exp(—r), where r is the electron-nucleus distance. Therefore, if the unpaired electron occupies an s orbital, or an MO containing an s orbital, there will be a finite probability that the electron resides on that nucleus. The amount of unpaired electrons residing on the nucleus is the spin density p at the nucleus. The spin density at the nucleus or in 

The magnetic nucleus-unpaired electron interaction the hyperfine shift 31 

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An alternative fomuilation of the nearest-neighbour Ising model is to consider the number of up f T land down [i] spins, the numbers of nearest-neighbour pairs of spins IT 11- U fl- IT Hand their distribution over the lattice sites. Not all of the spin densities are independent since... 

Making use of the relations between the spin densities, the energy of a given spm configuration can be written in tenns of the numbers of down spins [4] and nearest-neighbour down spins [44] ... 

In equation (bl. 15.24), r is the vector coimecting the electron spin with the nuclear spin, r is the length of this vector and g and are the g-factor and the Boln- magneton of the nucleus, respectively. The dipolar coupling is purely anisotropic, arising from the spin density of the impaired electron in an orbital of non-... 

The CIDNP spectrum is shown in figure B 1.16.1 from the introduction, top trace, while a dark spectrum is shown for comparison in figure B 1.16.1 bottom trace. Because the sign and magnitude of the hyperfine coupling constant can be a measure of the spin density on a carbon, Roth et aJ [10] were able to use the... 

Spin den sitieshelp to predict the observed coupling con slants in electron spin rcsonan ce (HSR) spectroscopy. From spin density plots you can predict a direct relalitin sh ip between the spin density on a carbon atom an d th c couplin g con stan t assti-ciated with ati adjacent hydrogen. 

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

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 ... 

Gunnarsson O and B I Lundqvist 1976. Exchange and Correlation in Atoms, Molecules, and Solids by the Spin-density-functional Formalism. Physical Review B13.-4274-4298. 

FIGURE 10 3 (a) The spin density (yellow) in allyl radical is equally divided between the two allylic carbons There is a much smaller spin density at C 2 hydrogen (b) The odd electron is in an orbital that is part of the allylic tt system... 

The unpaired electron in benzyl radical is shared by the benzylic carbon and by the nng carbons that are ortho and para to it as shown by the spin density surface in Figure 119 Delocalization of the unpaired electron from the benzylic carbon to the ortho and para positions can be explained on the basis of resonance contributions from the fol lowing structures... 

Once you have calculated an ab initio or a semi-empirical wave function via a single point calculation, geometry optimization, molecular dynamics or vibrations, you can plot the electrostatic potential surrounding the molecule, the total electronic density, the spin density, one or more molecular orbitals /i, and the electron densities of individual orbitals You can examine orbital energies and select orbitals for plotting from an orbital energy level diagram. 

The spin density defines the excess probability of finding spin-up over spin-down electrons at a point in space and is zero everywhere for closed-shell RHF situations. The spin density at the position of a nucleus is a prime determinant of electron spin resonance (ESR) spectra. 

The EPR spectra of semidione radical anions can provide information on the spin density at the individual atoms. "The semidione derived from butane-2,3-dione, for example, has a spin density of 0.22 at each oxygen and 0.23 at each carbonyl carbon. The small amount of remaining spin density is associated with the methyl groups. This extensive delocalization is consistent with the resonance picture of the semidione radical anion. 

Analyze the hyperfine coupling in the spectrum of the butadiene radical anion given in Fig. 12.PI I. What is the spin density at each carbon atom according to the McConnell equation ... 

In the aUyl radical, the spin density is divided between the two terminal carbon atoms In the four other compounds, the C2 carbon atom retains an unpaired electron. Foi the Mg and Be cases, both the C2 carbon atom and the substituent have an unpairec electron. In the Be compound, the spin density is localized mostly on the substitueni atoms, while for the Mg compound, a bit more of the density remains near the C carbon. 

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 second set of illustrations show the spin density plotted on the electron density isosurface the spin density provides the shading for the isodensity surface dark areas indicate positive (excess a) spin density and light areas indicate negative (excess P) spin density. For example, in the allyl radical, the spin density is concentrated around the two terminal carbons (and away from the central carbon). In the Be form, it is concentrated around the substituent, and in acetyl radical, it is centered around the C2 carbon atom. 

Hohenberg and Kohn demonstrated that is determined entirely by the (is a functional of) the electron density. In practice, E is usually approximated as an integral involving only the spin densities and possibly their gradients ... 

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. 

Repeat your analysis for localized and delocalized allyl radical and allyl anion. Focus on location of the spin density in the former and on the negative charge in the latter. 

Repeat your analysis fox phenoxy radical. Instead of charge, focus on the spin density. Calculate the delocalization energy using phenoxy radical at phenol geometry. Is it of the same order of magnitude as that for phenoxy anion Explain. 

Examine the geometry of the most stable radical. Is the bonding in the aromatic ring fuUy delocalized (compare to model alpha-tocopherol), or is it localized Also, examine the spin density surface of the most stable radical. Is the unpaired electron localized on the carbon (oxygen) where bond cleavage occurred, or is it delocalized Draw all of the resonance contributors necessary for a full description of the radical s geometry and electronic structure. 

The process is exothermic, suggesting that the phenoxy radical is particularly stable. Display the spin density surface for phenoxy radical. Is the unpaired electron localized or delocalized over several centers Is the unpaired electron in the a or 7t system Draw appropriate Lewis structures that account for your data. 

Examine the spin density surface for BHT radical. Is the unpaired electron localized or delocalized Examine BHT radical as a space-filling model. What effect do the bulky tert-butyl groups have on the chemistry of the species (Hint BHT radical does not readily add to alkenes or abstract hydrogens from other molecules.)... 

Compare the spin density surface for vitamin E radical to those of phenoxy and BHT radicals (see also Chapter 16, Problem 2). Are there significant differences among the three If so, elaborate. What is the function of the long alkyl chain in vitamin E Examine an electrostatic potential map for vitamin E radical. Do you expect it to be soluble in aqueous (polar) or non-aqueous (non-polar) environments, or both ... 

First, try to draw resonance contributors for both ground state and triplet anthrone. Then display a spin density surface for the triplet state of anthrone. (Note that the spin density surface shows the location of both unpaired electrons, one of which may be in a 7t orbital and one of which may be in a o orbital.) Where are the two unpaired electrons Are they localized or delocalized Given that spin delocalization generally leads to stabilization, would you expect the triplet state of anthrone to be stable ... 

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