Coulomb electron-nucleus interaction

Clearly, a complete numerical solution of the coupled KS equations (67,68) for electrons and nuclei will be rather involved. Usually only the valence electrons need to be treated dynamically. The core electrons can be taken into account approximately by replacing the electron-nucleus interaction (65) by suitable pseudopotentials and by replacing the nuclear Coulomb potential in Eq. (64) by the appropriate ionic Coulomb potential [38]. This procedure reduces the number of electronic KS equations and hence the numerical effort considerably. 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

Of note, the solution to Eq. 12.23 is the exact total Coulombic potential of the electron-electron and electron-nucleus interactions. Therefore, we do not need to include Uee[n] and Uen [n] terms in the total free energy functional. 

Hence, outside the spherical nucleus, for r > R, the ordinary Coulomb attraction governs the electron-nucleus interaction. 

Within the Born-Oppenheimer approximation, we still need to know that the nuclear position parameters really correspond to the distances and angles of a classical molecular framework. Our choice of the Coulomb gauge ensures this—the nuclear positions only appear in the electron-nucleus interaction terms, and the derivation of this potential from relativistic field theory shows us that it is indeed the quantities of normal 3-space that appear here. Thus, any potential surface that we might calculate on the basis of the Born-Oppenheimer-separated electronic molecular Dirac equation is indeed spanned by the variations of molecular structural parameters in the usual meaning. 

If we consider a nucleus being not a point but a volumetric nucleus, additional effects in the electron-nucleus interaction appear. They play a very important role in the physical description of an atom. These additional effects are exceedingly small in comparison with the main Coulomb and even with fine interactions (refer to Section 7.5.4). So, they refer to the number of intraatomic superfine interactions. ... 

Wlien the potential consists of electron-electron and electron-nucleus Coulombic interactions,... 

The GEM force field follows exactly the SIBFA energy scheme. However, once computed, the auxiliary coefficients can be directly used to compute integrals. That way, the evaluation of the electrostatic interaction can virtually be exact for an perfect fit of the density as the three terms of the coulomb energy, namely the nucleus-nucleus repulsion, electron-nucleus attraction and electron-electron repulsion, through the use of p [2, 14-16, 58],... 

Electrostatic electron-nucleus attractive Coulomb interaction energy ... 

The terms in this Hamiltonian are successively, the kinetic energy associated with the nuclei, the kinetic energy associated with the electrons, the Coulomb interaction between the electrons, the electron-nucleus Coulomb interaction and finally, the nuclear-nuclear Coulomb interaction. Note that since we have assumed only a single mass M, our attention is momentarily restricted to the case of a one-component system. 

The indices and v indicate atomic orbitals, and the quantities/, and (ju, v) are, respectively, the sum of the kinetic and the potential energies in the field of the nucleus, the electrostatic interaction between electrons in orbital and the Coulomb and exchange interactions between electrons in orbitals /r and u, averaged over all possible pairs of quantum numbers Wj and of the two electrons (57). It is easily shown (57, 52) that the binding energy of a doubly occupied orbital, given by the... 

The electron-electron interaction is usually supposed to be well described by the instantaneous Coulomb interaction operator l/rn. Also, all interactions with the nuclei whose internal structure is not resolved, like electron-nucleus attraction and nucleus-nucleus repulsion, are supposed to be of this type. Of course, corrections to these approximations become important in certain cases where a high accuracy is sought, especially in computing the term values and transition probabilities of atomic spectroscopy. For example, the Breit correction to the electron-electron Coulomb interaction should not be neglected in fine-structure calculations and in the case of highly charged ions. However, in general, and particularly for standard chemical purposes, these corrections become less important. 

In the nonrelativistic case V v has to account for all interactions of the valence electron i with the nucleus and the (removed) core electron system, i.e., at the independent particle level for Coulomb and exchange interaction as well as... 

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