The nuclear electrostatic potential

For every nuclear charge density distribution the corresponding electrostatic potential must be determined, since it is the latter quantity which subsequently enters an electronic structure calculation. With the standardization done as described above we obtain a well-defined sequence for the depth of the corresponding nuclear electrostatic potentials, V (0), namely 

Depth of electrostatic potentials, V(0) = t o = — Zja) f (in atomic units), with / = for four different nuclear charge density distribu- 

The actual numerical values for V (0) are obtained easily from the entries in Table 1, and provide lower boundaries for eigenvalues of single-particle functions, whenever one of these nuclear models is used. 

of course, the complete potential F(r) which is needed to determine these eigenvalues, and the corresponding eigenfunctions, in electronic structure calculations. The short-range behaviour of the electrostatic potential V r) for our four chosen nuclear models is shown in Fig. 7 for the case of mercury, Z = 80, A — 200 (the long-range behaviour of V(r) is the same for all nuclear models, and is known from Eq. (63)). 

For both the top slice model (T) and the homogeneous model (H), the potential is represented exactly by the Coulomb potential of a pointlike nucleus, —Z/r, beyond their respective nuclear size parameter. This function is approached only asymptotically by the potentials obtained from the Gaussian and Fermi-type models (G and F). In practice, however, the absolute deviation of these latter potentials from the Coulomb potential of the point-like nucleus is lower than machine precision (double precision) beyond a radius of about ten times the rms radius at most. 

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The external potential Vext(r) contains the nuclear or ionic contributions and possible external field contributions. The Hartree term Vuif) is the classic electrostatic potential of the electronic cloud... 

It is of considerable importance to note that the density-potential relationship (3) of the TF theory follows from a variational principle for the total energy. To see this, we note first that the classical electrostatic potential energy U consists of the sum of two terms in an atomic ion, the electron-nuclear potential energy Ken and the electron-electron potential energy Kee. We can write... 

Methods to Reproduce the Molecular Electrostatic Potential (MEP). The electrostatic potential surrounding the molecule that is created by the nuclear and electronic charge distribution of the molecule is a dominant feature in molecular recognition. Williams reviews (42) methods to calculate charge models to accurately represent the MEP as calculated by ab initio methods by use of large basis sets. The choice between models (monopole, dipole, quadrapole, bond dipole, etc.. Fig. 3.12) depends on the accuracy with which one desires to reproduce the MEP. This desire has to be balanced by the increased complexity of the model and its resulting computational costs when implemented in molecular mechanics. 

The first term in the definition (4.4) of the molecular electrostatic potential is the potential Vn(r) generated by the nuclei, also called the "bare nuclear potential" ... 

Figure 7. Short-range behaviour of nuclear electrostatic potentials V r) (in atomic units) for different finite nuclear charge density distributions in the case of mercury, Z = 80, A = 200 (dashed curve PNC, solid curves FNCs). The FNC curves may be identified from their labels at the origin, see also Eq. (109). The corresponding charge density distributions are standardized to a common value of the rms radius, a 5.4590 fin, determined from Eq. (54). The ground state energies for Hg and Hg /i" (only PNC) are indicated, together with the lower continuum threshold for the relativistic one-electron states (horizontal dashed lines). In the present scale, the full spectrmn of bound electronic states practically coincides with the horizontal axis. The conversion factor from atomic units of length to femtometer is la.u. = 52917.7249frn, and the myon-electron mass ratio is = 206.768262 [1].

The total electrostatic potential equals the sum of the nuclear and the electronic contributions ... 

In a somewhat different area of similarity research, chemists use not the electron density, but rather the electrostatic potential. It is somewhat different in the sense that the entire electrostatic potential also contains a contribution not related directly to the electron density and is therefore not positive definite. The total electrostatic potential contains two contributions. One originates from the nuclei, and one from the electron density. For a system with M nuclei with nuclear charges Z positioned at locations R , we obtain ... 

The point r is the position of a positive probe charge. is the nuclear charge on atom Alocated at position R. The function p(r ) is the electronic density. In the above equation, the first term represents the contribution of the nuclei to the electrostatic potential and the second term is the electronic contribution. Substituting the electron density expression ... 

If we consider a molecule as having a static but continuous distribution of electronic charge around a rigid nuclear framework, then its electrical or electrostatic potential will have a term similar to Eq. (3.2), with Q. being the positive charges of the nuclei, ZA, and a... 

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