Position space electron density

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. 

It is apparent that only a trickle of work has been, and is currently being, done on momentum densities in comparison with the torrent of effort devoted to the position space electron density. Moreover, much of the early work on II( p) has suffered from an undue emphasis on linear molecules. Nevertheless, some useful insights into the electronic structure of molecules have been achieved by taking the electron momentum density viewpoint. The most recent phenomenal developments in computer hardware, quantum chemical methods and software for generating wavefunctions, and visualization software suggest that the time is ripe to mount a sustained effort to understand momentum densities from a chemical perspective. Readers of this chapter are urged to take part in this endeavor. 

Fig. 3. Contour plots (in the plane z = 1 bohr) of the position-space electron density for the occupied n-orbitals for trans-C2oH22- Red (/ iO and green (broken) contours denote regions in which the wavefunction has opposite phases. The labels mark the positions of the nuclei (projected onto the plane z = 1). Orbital rr, has the highest binding energy and orbital n o is the least strongly bound. Note also the orientation of the axes (the x-axis points along the chain and the z-axis perpendicular to the molecular plane), as well as the marked distances d and h to which reference is made in the text...

The momentum-space electron density, p p), is just as straightforward to evaluate as its analogue in position spac, p(r). The wavefiinction in momentum space, (/ ), is simply the Fourier transform of the r-space wavefunction (r) ... 

Symmetry is the invariance of an object (in chemistry, an object made of nuclei and electrons) with respect to a geometrical operation that transforms the object into itself without distortion. A particular aspect of symmetry is periodicity a system is said to have a structural periodicity when certain structural motifs or descriptors repeat themselves in space by translation with a given period. Symmetry and periodicity concern nuclear positions and electron density as well, and are extremely useful in describing the structure of crystalline systems. 

Static simulation a single structure is considered, and forces and energies are calculated with fixed nuclear positions and electron density, under the given potential. This technique samples a unique point in phase space, and therefore can be used only for periodic symmetric systems, ideally perfect crystals at zero temperature, for which static, vibration- and temperature-less lattice energies can be obtained, or harmonic vibrational frequencies can be guessed. 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

Many atomic and molecular properties depend on the electron density, and some depend on the gradient of the electron density evaluated at certain positions in space. 

Wave-like properties cause electrons to be smeared out rather than localized at an exact position. This smeared-out distribution can be described using the notion of electron density Where electrons are most likely to be found, there is high electron density. Low electron density correlates with regions where electrons are least likely to be found. Each electron, rather than being a point charge, is a three-dimensional particle-wave that is distributed over space in... 

Both objects are much less complicated than the total A -particle wavefunction itself, since they only depend on three spatial variables. The electron density is manifestly positive (or zero) everywhere in space while the spin-density can be positive or negative. If, by convention, there are more spin-up than spin-down electrons, the positive part of the spin-density will prevail and there will usually be only small regions of negative spin-density that arise from spin-polarization. This spin-polarization is physically important and is already included in the UHF method but not in the ROHF method that, by construction, can only describe the... 

Since two electrons of the same spin have a zero probability of occupying the same position in space simultaneously, and since t / is continuous, there is only a small probability of finding two electrons of the same spin close to each other in space, and an increasing probability of finding them an increasingly far apart. In other words the Pauli principle requires electrons with the same spin to keep apart. So the motions of two electrons of the same spin are not independent, but rather are correlated, a phenomenon known as Fermi correlation. Fermi correlation is not to be confused with the Coulombic correlation sometimes referred to without its qualifier simply as correlation . Coulombic correlation results from the Coulombic repulsion between any two electrons, regardless of spin, with the consequent loss of independence of their motion. The Fermi correlation is in most cases much more important than the Coulomb correlation in determining the electron density. 

Since the electrostatic potential is closely related to the electronic density, it may be useful to discuss how the information that can be obtained from V(r) differs from that provided by the p(r). Both are real physical properties, related by Eqs. (3.1) and (3.4). An important difference between V(r) and p(r) is that the electrostatic potential explicitly reflects the net effect of all of the nuclei and electrons at each point in space, whereas the electron density directly represents only the concentration of electrons at each point. A molecule s interactions with another chemical system is affected by its total charge distribution, both positive and negative, and thus can be better understood in terms of its electrostatic potential than its electronic density alone. Examples illustrating this point have been discussed elsewhere (Politzer and Daiker 1981 Politzer and Murray 1991). 

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