Spins density

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. 

It was noted in Section 5.3 that when the 1-electron density matrix is written in the form (5.3.12) the difference between the a and /3 components allows us to define a resultant spin density, essentially as the excess density of up-spin electrons compared with down-spin. In recent years, with the development of magnetic resonance techniques, this quantity has acquired great importance. To indicate its origin we note tiiat fhe expectation value of the z component of spin angular momentum may be written, using for clarity the explicit form (5.2.12), 

Now for any spin eigenstate, p may be expressed in terms of its components according to (5.3.12), and the spin integration may then be 

For ri = ri, the integrand represents the contribution to the expectation value arising from the volume element dri in ordinary 3-dimensional space we therefore define (cf. (5.3.15)) 

for a spin eigenstate, the expectation-value expression yields 

The spin density defined above in (5.9.1) is in fact just one component of a vector density Q = (Q, Qy, Q, the other two components may be calculated by generalizing (5.9.2) and writing 

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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] ... 

As before, we note that the resonance frequency of a nucleus at position r is directly proportional to the combined applied static and gradient fields at that location. In a gradient G=G u, orthogonal to the slice selection gradient, the nuclei precess (in the usual frame rotating at coq) at a frequency ciD=y The observed signal therefore contains a component at this frequency witli an amplitude proportional to the local spin density. The total signal is of the fomi... 

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-... 

Hence, a measurement of hyperfme coupling constants provides infonnation on spin densities at certain positions in the molecule and thus renders a map of tlie electronic wavefiinction. 

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... 

Besides molecular orbitals, other molecular properties, such as electrostatic potentials or spin density, can be represented by isovalue surfaces. Normally, these scalar properties are mapped onto different surfaces see above). This type of high-dimensional visualization permits fast and easy identification of the relevant molecular regions. 

To display properties on molecular surfaces, two different approaches are applied. One method assigns color codes to each grid point of the surface. The grid points are connected to lines chicken-wire) or to surfaces (solid sphere) and then the color values are interpolated onto a color gradient [200]. The second method projects colored textures onto the surface [202, 203] and is mostly used to display such properties as electrostatic potentials, polarizability, hydrophobidty, and spin density. 

Above all, spin density is most significant for radicals. Their unpaired electrons can be localized rapidly, by visualizing this property on the molecule. 

You can also plot ihe electrostatic polenlial. the total charge density. or the total spin density determined during a semi-enipincal or ah initio calculation. This information is useful in determining reactivity and correlating calculalional results with experimental data. Th ese examples illustrate uses of lb ese plots ... 

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 ... 

Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

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. 

This simple model allows one to estimate spin densities at eaeh earbon eenter and provides insight into whieh eenters should be most amenable to eleetrophilie or nueleophilie attaek. For example, radieal attaek at the C5 earbon of the nine-atom system deseribed earlier would be more faeile for the ground state F than for either F or F. In the former, the unpaired spin density resides in /5, whieh has non-zero amplitude at the C5 site x=L/2 in F and F, the unpaired density is in /4 and /6, respeetively, both of whieh have zero density at C5. These densities refleet the values (2/L)F2 sin(n7ikRcc/L) of the amplitudes for this ease in whieh L = 8 x Rcc for n = 5, 4, and 6, respeetively. 

The CNDO and CNDO/S methods apply the ZDO approximation to all integrals, regardless of whether the orbitals are loeated on the same atom or not. In the INDO method, whieh was designed to improve the treatment of spin densities at nuelear eenters and to handle singlet-triplet energy differenees for open-shell speeies, exehange integrals... 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

Chipman DZP+diffuse Available for H(6.vl/i) through l0s6p2d). Optimized to reproduce high-accuracy spin density results. 

Many functions, such as electron density, spin density, or the electrostatic potential of a molecule, have three coordinate dimensions and one data dimension. These functions are often plotted as the surface associated with a particular data value, called an isosurface plot (Figure 13.5). This is the three-dimensional analog of a contour plot. 

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