Electron density, positive nuclei

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

The electronic densities on the thiazole nucleus or on the thiazolium ion suggest that electrophilic substitution should occur preferentially at the 5-position. The order of tt electronic densities is 2<4<5 (163,164). 

For XH bonds, where X is any heavy atom, the hydrogen electron density is not thought to be centered at the position of the hydrogen nucleus but displaced along the bond somewhat, towards X. The MMh- force field reduces the XH bond length by a factor of 0.915 strictly for the purposes of calculating van der Waals interactions with hydrogen atoms. 

Several calculations of the electronic structure of isoindoles have Ijeen published, and the distribution of charge density around the isoindole nucleus calculated by these methods is summarized in X able I. A common prediction of the calculations, which are based on tlie LCAO-MO method or the frontier electron concept, is the relatively high electron density to bo found at position 1, and the expectation, thei efore, i.s that electrophilic substitution on carbon... 

Figure 1.2 A schematic view of an atom. The dense, positively charged nucleus contains most of the atom s mass and is surrounded by negatively charged electrons. The three-dimensional view on the right shows calculated electron-density surfaces. Electron density increases steadily toward the nucleus and is 40 times greater at the blue solid surface than at the gray mesh surface.

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

The electron density in d-orbitals is low near the nucleus, and so they are not very effective at shielding other electrons from the positive nuclear charge. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

The isomer shift, d, arises from the Coulomb interaction between the positively charged nucleus and the negatively charged s-electrons, and is thus a measure for the s-electron density at the nucleus, yielding useful information on the oxidation state of the iron in the absorber. An example of a single line spectrum is fee iron, as in stainless steel or in many alloys with noble metals. 

The relative change of the mean-square nuclear radius in going from the excited to the ground state, A r )/ r ), is positive for u. An increase in observed isomer shifts S therefore reflects an increase of the s-electron density at the Ru nucleus caused by either an increase in the number of s-valence electrons or a decrease in the number of shielding electrons, preferentially of d-character. 

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