Potentials Pauli

Differential Equation for Density Amplitude (p(r) and Concept of Pauli Potential... 

Though the Pauli potential Fpauii(r) = Fp(r) entering the Schrodinger equation at Eq. (45) is also only known approximately (for example, the von Weizsacker study of March and Murray [32] yielded Fp (r) = (5/3)c p(r) where the kinetic constant = (3h l0m) (3/Sn) ), the work of Lieb et al. [36] (see also [37]), who did not utilize the Pauli potential in their original paper, proves to be a splendid example of this approach, where important analytical progress proves possible. 

Regions (i), (ii) and (iii) correspond to the TF semiclassical form already discussed above. Region (iv), as Lieb et al. [36] point out, can be treated by a simple density matrix functional theory. It is, however, region (v) which can be discussed very naturally in terms of the Pauli potential introduced above. 

Nagy, A. AmoviUi, C. Electron-electron cusp condition and asymptotic behavior for the Pauli potential in pair density functional theory. J. Chem. Phys. 2008,128, 114115. 

Goddard et al. studied a case of type (d) in materials science. All particles including electrons and nuclei are treated classically. This may be efficient for a liquid like situation where the system is at a high electron temperature. The antisymmetry effect and the exchange repulsion, related to the fermion nature of electrons is included through an effective electron potential called the Pauli potential. They especially explored the Auger induced chemical process caused by drastic rearrangement of the electronic structure in a materials system after deep orbital ionization [396, 397]. 

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

For two and three dimensions, it provides a erude but useful pieture for eleetronie states on surfaees or in erystals, respeetively. Free motion within a spherieal volume gives rise to eigenfunetions that are used in nuelear physies to deseribe the motions of neutrons and protons in nuelei. In the so-ealled shell model of nuelei, the neutrons and protons fill separate s, p, d, ete orbitals with eaeh type of nueleon foreed to obey the Pauli prineiple. These orbitals are not the same in their radial shapes as the s, p, d, ete orbitals of atoms beeause, in atoms, there is an additional radial potential V(r) = -Ze /r present. However, their angular shapes are the same as in atomie strueture beeause, in both eases, the potential is independent of 0 and (j). This same spherieal box model has been used to deseribe the orbitals of valenee eleetrons in elusters of mono-valent metal atoms sueh as Csn, Cun, Nan and their positive and negative ions. Beeause of the metallie nature of these speeies, their valenee eleetrons are suffieiently deloealized to render this simple model rather effeetive (see T. P. Martin, T. Bergmann, H. Gohlieh, and T. Lange, J. Phys. Chem. 6421 (1991)). 

This term is essential to obtain the correct geometry, because there is no Pauli repulsion between quantum and classical atoms. The molecular mechanics energy tenn, E , is calculated with the standard potential energy term from CHARMM [48], AMBER [49], or GROMOS [50], for example. 

Figure 6.2. Potential energy diagram showing the attractive Van der Waals interaction and the repulsive interaction due to Pauli repulsion,...

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