Spin Density Description

Electron Correlation, Including Spin Density Description.—In this Appendix we discuss ... 

Spin Density Description.—To conclude this Appendix, we shall briefly summarize the idea behind the generalization of the density description to allow for non-zero spin density m(r) defined in equation (176). This is discussed in the work of Stoddart and March151 and by a number of other workers later.152-153... 

Bernholc, Z., and N. A. W. Holzwarth (1983). Local spin-density description of multiple metal-metal bonding Moj and Ctj. Phys. Rev. Lett. 50, 1451-54. 

A UHF wave function may also be a necessary description when the effects of spin polarization are required. As discussed in Differences Between INDO and UNDO, a Restricted Hartree-Fock description will not properly describe a situation such as the methyl radical. The unpaired electron in this molecule occupies a p-orbital with a node in the plane of the molecule. When an RHF description is used (all the s orbitals have paired electrons), then no spin density exists anywhere in the s system. With a UHF description, however, the spin-up electron in the p-orbital interacts differently with spin-up and spin-down electrons in the s system and the s-orbitals become spatially separate for spin-up and spin-down electrons with resultant spin density in the s system. 

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 electron- and spin-densities are the only building blocks of a much more powerful theory the theory of reduced density matrices. Such one-particle, two-particle,. .. electron- and spin-density matrices can be defined for any type of wavefunction, no matter whether it is of the HF type, another approximation, or even the exact wavefunction. A detailed description here would be inappropriate... 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

The phenomenon of electron pairing is a consequence of the Pauli exclusion principle. The physical consequences of this principle are made manifest through the spatial properties of the density of the Fermi hole. The Fermi hole has a simple physical interpretation - it provides a description of how the density of an electron of given spin, called the reference electron, is spread out from any given point, into the space of another same-spin electron, thereby excluding the presence of an identical amount of same-spin density. If the Fermi hole is maximally localized in some region of space all other same-spin electrons are excluded from this region and the electron is localized. For a closed-shell molecule the same result is obtained for electrons of p spin and the result is a localized a,p pair [46]. 

This chapter introduces and illustrates isosurface displays of molecular orbitals, electron and spin densities, electrostatic potentials and local ionization potentials, as well as maps of the lowest-unoccupied molecular orbital, the electrostatic and local ionization potentials and the spin density (on top of electron density surfaces). Applications of these models to the description of molecular properties and chemical reactivity and selectivity are provided in Chapter 19 of this guide. 

As far as an NMR experiment is concerned the state of the products of reaction a is unambiguously determined, within the so-called sudden approximation, (40) by the state of the substrates involved just prior to the reaction as well as by the reaction mechanism. This problem is considered in detail in Section III.C. Needless to say, the states involved are understood as the corresponding spin density matrices. In a description of the dynamic equilibrium considered, we may take into account all the stereochemically feasible reactions or we may divide the entire set of the latter into subsets of NMR-nondifferentiable reactions and then consider only one reaction from each of the subsets. (41, 42) In the latter method, the probability n a, which refers to subset a, is the sum of probability given by ... 

To understand chemical processes, it is useful to have information besides total energies. Electron localization methods provide insight on the behavior of electrons in molecules. Properties, such as electron density, spin density, and the electron pair localization function (EPLF) [33], can routinely be computed by post-processing. The EPLF provides a quantitative description of electron pairing in molecular systems and has similarities to the electron localization function (ELF) of Becke and Edgecombe [34]. The QMC method is a particularly well-suited approach for obtaining such information because the simple and general definition of EPLF is easily evaluated in QMC. 

The spin density distribution in the 2A2 excited state requires the derivation of all the contributing determinants as done for allyl radical. A full treatment is given in Exercise 8.5, while here we provide an approximate description. Already at the outset one can recall that the coefficient of the QC determinant in the excited state s wave function is zero, and we therefore expect very different spin density distribution than in the ground state. To proceed, we first express the resonance structures as products of the bonds and the odd electron. Thus... 

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