Charge, nuclear distribution shape

A molecule contains a nuclear distribution and an electronic distribution there is nothing else in a molecule. The nuclear arrangement is fully reflected in the electronic density distribution, consequently, the electronic density and its changes are sufficient to derive all information on all molecular properties. Molecular bodies are the fuzzy bodies of electronic charge density distributions consequently, the shape and shape changes of these fuzzy bodies potentially describe all molecular properties. Modern computational methods of quantum chemistry provide practical means to describe molecular electron distributions, and sufficiently accurate quantum chemical representations of the fuzzy molecular bodies are of importance for many reasons. A detailed analysis and understanding of "static" molecular properties such as "equilibrium" structure, and the more important dynamic properties such as vibrations, conformational changes and chemical reactions are hardly possible without a description of the molecule itself that implies a description of molecular bodies. 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

Similar and additional other geometrical quantities for nuclear charge density distributions were defined by Myers [20], see also [21] and the comprehensive discussion in [22] which extends this subject to the case of non-spherical shapes. The most complete use of expectation values () (including the extension to arbitrary real powers of r) is made by the moment function M(p),... 

The shape of the nucleus is best described by a power series, the relevant term of which yields the nuclear quadrupole moment. In Cartesian coordinates, this is represented by a set of intricate integrals of the type J p (r)(3x,x, — 6-jr )Ax, where x, = x, y, z, and pfifi) is the nuclear charge distribution (4.12). The evaluation of Pn(r) for any real nucleus would be very challenging. 

The strength of the quadrupolar interaction is proportional to the quadrupole moment Q of a nucleus and the electric field gradient (EFG) [21-23]. The size of Q depends on the effective shape of the ellipsoid of nuclear charge distribution, and a non-zero value indicates that it is not spherically symmetric (Fig. 1). 

Based on Eq. 3, the factors that govern the oscillation frequency (and the observed color) are electron density (size/shape of the nanostructures. Figure 6.11), the effective nuclear charge of the nuclei, and the size/shape of the charge distribution (polarization effects, strongly affected by the dielectric constant of the metal). As you might expect, further effects toward the resonance frequency/intensity are... 

Quadrupole effects are due to the interaction of the nuclear quadrupole moment (caused by a non-spherical distribution of charge on the nucleus) with the electric field gradient at the nucleus. Quadrupole effects cause peak broadening, displacement of the peak from the isotropic (true) chemical shift, and distortion of the peak shape. These effects decrease in magnitude with the square of the Bq field strength, and spectra of quadrupolar nuclides are usually recorded at the highest field available. 

Basis sets taken from nonrelativistic calculations are a good first choice for most problems since they provide a good representation of the valence electron distribution. However, we know that the shape of the wavefunction is sensitive to the nuclear charge close to the nucleus, and it is therefore important to choose basis functions for the most penetrating s and p orbitals to improve the fit at short range. The simplest way to do this is to add some larger exponents. 

H. Behrens, W. Buhring, On the Sensitivity of /9-TVansitions to the Shape of the Nuclear Charge Distribution, Nucl. Phys. A 150 (1970) 481-496. 

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