Electron density notation

The electron density in both molecular orbitals is symmetrical about the axis between the two nuclei. This means that both of these are sigma orbitals. In MO notation, the Is bonding orbital is designated as eru. The antibonding orbital is given the symbol An asterisk designates an antibonding orbital... 

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

The expressions Eqs. (2), (4) are completely general. To address the aspects important for the TMCs modelling, i.e. the energies of the corresponding electronic states, we notice that the statement that the motion of electrons is correlated can be given an exact sense only with use of the two-electron density matrix Eq. (4). Generally, it looks like [35] (with subscripts and variables notations w omitted for brevity) ... 

In discussing the nature of various functionals, it is convenient to adopt some of the notation commonly used in the field. For instance, the functional dependence of Exc on the electron density is expressed as an interaction between the electron density and an energy density that is dependent on the electron density, viz. 

Wyckoff et al. (62) have provided a preliminary coordinate list of all nonhydrogen atoms in RNase-S. Along with the list is a series of notations on the quality of the map and the fit of the atomic model to the electron density contours. The following comments concerning group accessibilities are based on this coordinate list, but detailed interpretations must be made with caution in view of the uncertainties in many parts of the structure. 

For all pseudoazulenes an electron deficiency in the six-membered ring and an excess in electrons in the five-membered ring were calculated (Fig. 1 and Eq. 7), again in accordance with azulene.219 The heteroatom predominantly influences the electron density at the carbon atoms. As follows from calculations with identical parameter notations, the electronegativity of the heteroatom is essentially responsible57 (see Fig. 1). In pseudoazulenes of... 

The Laue and the Bragg condition give us information about the angular distribution of the diffraction peaks. To calculate the peak intensities, we have to know more about the scattering properties of the atoms or molecules in the crystal. In the case of X-rays and electrons the scattering probability is proportional to the electron density ne(r) within the crystal. Since n,(r) has to have the same periodicity as the crystal lattice, we can write it as a three-dimensional Fourier series (using the notation eikx = cos kx + i sin kx) ... 

In this book we are particularly interested in simple descriptions of structures that are easily visualized and providing information of the chemical environment of the ions and atoms involved. For metals, there is an obvious pattern of structures in the periodic table. The number of valence electrons and orbitals are important. These factors determine electron densities and compressibilities, and are essential for theoretical band calculations, etc. The first part of this book covers classical descriptions and notation for crystals, close packing, the PTOT system, and the structures of the elements. The latter and larger part of the book treats the structures of many crystals organized by the patterns of occupancies of close-packed layers in the PTOT system. 

The variations of the one-electron densities 6Pria with a = a, ,v,( and the polarity (/ — P[l) of the bond with m = 1 deserve some discussion. As it is seen from eqs. (3.86), (3.105) each bond incident to an atom contributes an increment to the quasitorque and to the pseudotorque acting upon its hybridization tetrahedron. In the equilibrium these increments separately sum up to zero. We can think that the equilibrium shape and orientation of the hybridization tetrahedron is obtained within a TATO DMM model applied to the entire system. Then, within such a model, there exists an atom corresponding to the left end of the bond with m = 1 having number Li according to our previous notation. The HOs obtained in this approximation provide an initial guess for HOs in the system including those of the atom Pi, which... 

It is clear that the entire electronic density in a molecule has the role of determining the nuclear distribution hence bonding, consequently, chemical bonding cannot be confined to lines in space. It is well understood that bond diagrams represent only an oversimplified, "short-hand" notation for the actual molecular structure, nevertheless, as most successful notations do, chemical bonds as formal lines have acquired an almost unquestioned reputation of their own as if they were truly responsible for holding molecules together. 

In most interactions between two reactants, local shape complementarity of functional groups is of importance. A local shape complementarity of molecular electron densities represented by FIDCOs implies complementary curvatures for complementary values of the charge density threshold parameters a. For various curvature domains of a FIDCO, we shall use the notations originally proposed for complete molecues [2], For example, the symbol D2(b),i(a, Fj) stands for the i-th locally convex domain of a FIDCO G(a) of functional group Fj, where local convexity, denoted by subscript 2(b), is interpreted relative to a reference curvature b. For locally saddle type and locally concave domains relative to curvature b, the analogous subscripts 1(b) and 0(b) are used, respectively. 

Two important tools of electron density modeling and shape analysis are the concepts of molecular isodensity contour, MIDCO G(K,a) and the associated density domain DD(K,a) [27], Here the nuclear arrangement (also called the nuclear configuration) is denoted by K, whereas the electron density threshold is denoted by a. Each MIDCO G(K,a) is the collection of all those points r of the three-dimensional space where the electron density p(K,r) of the molecule M of conformation K is equal to the threshold a. The density domain DD(K,a) is the collection of all points r where the electron density p(K,r) is greater than or equal to the threshold a. Using formal notations,... 

F RaI where the notation for P ° and P is analogous to that in en tin 671 anH R. electron density at the nucleus. The indirect spin-spin coupling between nuclei A and B, which is the one observed in ... 

A detailed review of the basic concepts of fuzzy sets can be found in other chapters of this volume. Here only the specific notations and the fuzzy set concepts most relevant to the molecular shape problem are reviewed, followed by a simple proof for a special fuzzy set generalization of the Hausdorff distance, motivated by the quantum chemical properties of fuzzy electronic densities of molecules. 

The concept of density domains can be introduced in the context of isodensity contour surfaces. A molecular isodensity contour surface, MIDCO G(K,a)y of nuclear configuration K and density threshold a is defined as the collection of all points r of the 3D space where the electronic density is equal to the threshold value a. In the notation for electron density it is useful to specify the nuclear configuration K of the molecule and in the forthcoming discussion the notation p(K, r) will be used for the electron density of nuclear configuration K. Accordingly, the MIDCO G(K,a) is defined as... 

In the notation of Fig. 7, the straight bond model requires an electronic configuration of type (I)2(II)2(IV)2 while the bent bond model configuration is (1)2(11)2(111)2. While there is no difference in symmetry type orbitals III and IV, the electron density distributions on these orbitals are quite different. It follows that the distinction between the two extremes is meaningful, although intermediate situations are possible in the low-symmetry cases under dispute. 

The characterization of the interrelations between chemical bonding and molecular shape requires a detailed analysis of the electronic density of molecules. Chemical bonding is a quantum mechanical phenomenon, and the shorthand notations of formal single, double, triple, and aromatic bonds used by chemists are a useful but rather severe oversimplification of reality. Similarly, the classical concepts of body and surface , the usual tools for the shape characterization of macroscopic objects, can be applied to molecules only indirectly. The quantum mechanical uncertainty of both electronic and nuclear positions within a molecule implies that valid descriptions of both chemical bonding and molecular shape must be based on the fuzzy, delocalize properties of electronic density distributions. These electron distributions are dominated by the nuclear arrangements and hence quantum mechanical uncertainly affects electrons on two levels by the lesser positional uncertainty of the more massive nuclei, and by the more prominent positional uncertainty of the electrons themselves. These two factors play important roles in chemistry and affect both chemical bonding and molecular shape. 

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