Spatial density distribution

It is noteworthy that even in the isotropic phase (in the coil as well as in the globule) there exists a weak orientational ordering (not of the liquid-crystalline nature), due to the inhomogeneity of the spatial density distribution in the macromolecule. This ordering will be considered in the Appendix. 

Further contributions to SEC theory were made by Glandt (32) for the description of the spatial density distribution for crowded pores. This work contrasts with earlier studies based solely on dilute solutions of solutes, in which solute-wall effects were primarily considered. 

The relative intensity of the light beam passed through the dusty gas measured by photodiode is consistent with the spatial density distribution. It means that the high density "p-layer does really exist in the flow. Thus, the considered physical efifect discovered earlier numerically was confirmed by direct experiments. This effect is inherent also in flows of spherical and cylindrical symmetry. 

If we multiply the probability density P(x, y, z) by the number of electrons N, then we obtain the electron density distribution or electron distribution, which is denoted by p(x, y, z), which is the probability of finding an electron in an element of volume dr. When integrated over all space, p(x, y, z) gives the total number of electrons in the system, as expected. The real importance of the concept of an electron density is clear when we consider that the wave function tp has no physical meaning and cannot be measured experimentally. This is particularly true for a system with /V electrons. The wave function of such a system is a function of 3N spatial coordinates. In other words, it is a multidimensional function and as such does not exist in real three-dimensional space. On the other hand, the electron density of any atom or molecule is a measurable function that has a clear interpretation and exists in real space. 

Two objects are similar and have similar properties to the extent that they have similar distributions of charge in real space. Thus chemical similarity should be defined and determined using the atoms of QTAIM whose properties are directly determined by their spatial charge distributions [32]. Current measures of molecular similarity are couched in terms of Carbo s molecular quantum similarity measure (MQSM) [33-35], a procedure that requires maximization of the spatial integration of the overlap of the density distributions of two molecules the similarity of which is to be determined, and where the product of the density distributions can be weighted by some operator [36]. The MQSM method has several difficulties associated with its implementation [31] ... 

The preceding section may be concluded by the statement that the experimental studies pubhshed hitherto did not come to a clear conclusion regarding the radial density distribution of dendrimers. It is therefore interesting to delineate the main problems of scattering studies as applied to small dissolved objects and enumerate possible sources of scattering intensity not related to the spatial structure of the particles [5,23,24] ... 

Figure 6.6 Two-state quantum system driven on resonance by an intense ultrashort (broadband) laser pulse. The power spectral density (PSD) is plotted on the left-hand side. The ground state 11) is assumed to have s-symmetry as indicated by the spherically symmetric spatial electron distribution on the right-hand side. The excited state 12) is ap-state allowing for electric dipole transitions. Both states are coupled by the dipole matrix element. The dipole coupling between the shaped laser field and the system is described by the Rabi frequency Qji (6 = f 2i mod(6Iti-...

Local chemical composition from areas less than 1 nm in diameter can be measured by energy dispersive X-ray spectroscopy (EDS) or electron energy loss spectroscopy (EELS). Such spectroscopic information may be presented in 2D maps showing the spatial element distribution in the specimen (13). Furthermore, information about the local density of unoccupied electron states of a specific element can be extracted from EELS data and used to estimate the oxidation state and the local coordination geometry of the excited atoms (14). In some favorable cases, electronic structure information with a resolution of about 1 eV from individual atomic columns has been attained (15,16). Recent developments of monochromators and spectrometers have brought the resolution down to 0.1 eV (17,18), and this capability may offer new opportunities to determine relationships between electronic structure information, the atomic arrangements and the catalytic activities of solids. 

In metals, the concentration of mobile electrons is enormously high so that the excess charge is confined to a region very close to the surface, within atomic distances [14, 15]. In semiconductors with substantially less charge carrier density, on the other hand, a region of spatial charge distribution can be found [16, 17]. 

The description, in MO theory, of the electronic structure of a system is given in terms of molecular orbitals. A molecular orbital, tp, is a function of the spatial coordinates of the electron. The product tp rp, where cp denotes the complex conjugate of electronic density distribution. 

On the basis of their spatial symmetry, the molecular orbitals are denoted as a- and re-type orbitals, the main difference being that ff-type orbitals have cylindrical symmetry around the bond axes, while re-type orbitals have nodal planes (i.e., they vanish) at the internuclear axes. The er-type orbitals can be assumed to be localized, i.e., they define electronic density distributions only over specific regions of the molecule, while the re-type orbitals are delocalized, extending over the whole molecule. 

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