Electron-Nucleus Distances

It has been found possible to evaluate s0 theoretically by means of the following treatment (1) Each electron shell within the atom is idealised as a uniform surface charge of electricity of amount — zte on a sphere whose radius is equal to the average value of the electron-nucleus distance of the electrons in the shell. (2) The motion of the electron under consideration is then determined by the use of the old quantum theory, the azimuthal quantum number being chosen so as to produce the closest approximation to the quantum... 

For H2 to be a stable molecule, the sum of the attractive energies must exceed the sum of the repulsive energies. Figure 9A shows a static arrangement of electrons and nuclei In which the electron-nucleus distances are shorter than the electron-electron and nucleus-nucleus distances. In this arrangement, attractive interactions exceed repulsive interactions, leading to a stable molecule. Notice that the two electrons occupy the region between the two nuclei, where they can interact with both nuclei at once. In other words, the atoms share the electrons in a covalent bond. 

Several approaches have been made to calculate 13C chemical shifts of coumarins by MO methods. Good correlations were found between the 13C chemical shift values of coumarin (also protonated) and the n charge densities calculated by the CNDO/2 method [962], and of coumarins with it charge densities calculated by the Hiickel MO method (which, however, fails for methoxylated coumarins) [965]. Chemical shifts of mono- and dimethoxycoumarins have been correlated with parameters determined by refined INDO MO calculations, in which n bond orders, atom-atom polarizabilities, excitation energies and electron-nucleus distances were taken into consideration [966], In 3-substituted 4-hydroxy and 4-hydroxy-7-methoxycoumarins chemical shifts were found to be related to Swain and Lupton s parameters iF and M [388], according to equation 5.4 (SE = Substitution Effect) ... 

At this point we recall that the unpaired electron may have some probability to sit just on the nucleus. Type s orbitals have maximal electron density on the nucleus, as their wavefunction is of the type exp(—r), where r is the electron-nucleus distance. Therefore, if the unpaired electron occupies an s orbital, or an MO containing an s orbital, there will be a finite probability that the electron resides on that nucleus. The amount of unpaired electrons residing on the nucleus is the spin density p at the nucleus. The spin density at the nucleus or in... 

Relaxation measurements provide a wealth of information both on the extent of the interaction between the resonating nuclei and the unpaired electrons, and on the time dependence of the parameters associated with the interaction. Whereas the dipolar coupling depends on the electron-nucleus distance, and therefore contains structural information, the contact contribution is related to the unpaired spin density on the various resonating nuclei and therefore to the topology (through chemical bonds) and the overall electronic structure of the molecule. The time-dependent phenomena associated with electron-nucleus interactions are related to the molecular system, and to the lifetimes of different chemical situations, for the resonating nucleus. Obtaining either structural or dynamic information, however, is only possible if an in-depth analysis of a series of experimental results provides sufficient data to characterize the system within the theoretical framework discussed in this chapter. 

PROBLEM 3.1.2. Bohr s 1913 derivation of the energy of the hydrogen atom (nuclear charge = e, electron charge = e, reduced mass of the electron-nucleus couple = m, electron-nucleus distance = r, linear momentum = p) is based on the classical energy... 

It is noted that all s orbitals are functions only of r (and not of 6 and 0), i.e. each s function has the same value for all points of space at a given distance r from the nucleus. They have spherical symmetry. We see that - 0 as r—>oo, through a factor e ", where p = Zrlao. For Is, this is the only r-dependent factor, and the function decreases exponentially on increasing the electron-nucleus distance (Fig. 3.3). 

The radial wave functions i j(r) for = 1, 2, and 3 and 1 = 0 and 1 are shown plotted in Figure 21-2. The abscissas represent values of p hence the horizontal scale should be increased by the factor n in order to show R r) as functions of the electron-nucleus distance r. It will be noticed that only for s states (with 1 = 0) is the wave function different from zero at r = 0. The wave function crosses the p axis n — l — 1 times in the region between p = 0 and p = oo. 

The corresponding values of p are represented by vertical lines in Figure 21-3. From this expression it is seen that the size of the atom increases about as the square of the principal quantum number , fnjm being in fact proportional to n2 for the states with 1=0 and showing only small deviations from this proportionality for other states. This variation of size of orbit with quantum number is similar to that of the old quantum theory, the time-average electron-nucleus distance for a Bohr orbit being... 

In a more rigorous way, Grigor and Webb (65) reproduced the l3C shieldings of coumarin and some mono- and dimethoxycoumarins by refined INDO MO calculations and found that, in addition to atom-atom polarizabilities (14) and ti bond orders (20), other factors such as excitation energies and electron-nucleus distances, (r 3)2P, play an important role in the determination of the 13C chemical shifts of these compounds. 

Let us consider again the atom-atom interaction, but permit now the atoms to have a structure. Namely, allow now for inelasticities, and indicate with R the internuclear distance and with r the collection of vectors representing the electron-nucleus distances. We can write down the following Schrodinger equation ... 

The one-electron Hamiltonian only involves differences of geometrical parameters the nuclear attraction terms involve the electron-nucleus distance not the absolute position of either particle. Likewise the molecular integrals dependence on molecular geometry is only via inter-centre distance the fact that the basis functions are atom-centred does not induce any dependence of the integrals on absolute position of the integrals. 

Distance measurements by ESR have recently attracted attention particularly in biological systems. Detailed accounts of pulsed- and CW-ESR methods to determine electron-electron distances are given in [52], Measurements of electron-nucleus distances have been an essential part in the characterization of trapping sites of paramagnetic species first by ENDOR, later also by ESEEM. Table 2.4 provides a limited overview of methods and applications. 

Table 2.4 Measurement methods for electron-electron and electron-nucleus distances by ESR, see [52] for details...

We can now write down the Hartree-Fock one-electron hamiltonian in the D— oo limit. As usual, we will write the hamiltonian as one for the probability amplitude, and will remove the dominant dimension-dependence of the solutions through use of appropriately scaled units. As discussed in the previous section, this means that energies will be in units of 4/(D —1) haxtrees, and distances in units of D(D—l)/6 Bohr radii. The symmetry assumption allows us to equate all electron-nucleus distances, and to constrain the electrons to positions directly above the nuclei, prior to minimization of the hamiltonian. Also, the Hartree-Fock approximation allows dihedral angles to be fixed at 90°. With these scalings and simplifications, the D— oo limit hamiltonian can be written... 

The potential V in Eq. (3.64) will be taken as—Zfr, where r is the electron-nucleus distance. 3.4.1.2 The Large Component Spinor... 

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