Coulomb interaction first-order derivatives

Oppenheimer approximation, 517-542 Coulomb interaction, 527-542 first-order derivatives, 529-535 second-order derivatives, 535-542 normalization factor, 517 nuclei interaction terms, 519-527 overlap integrals, 518-519 permutational symmetry, group theoretical properties, 670—674... 

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

This formula was first derived in [25]. Note, that the reference state correction is absent for the Coulomb-Coulomb second order interaction. The contribution of the "cross Coulomb-Coulomb interaction (Fig.7b) is irreducible and can be obtained with the use of the formulas (153) and (154). 

For further details the reader is referred to, e.g., a review article by Kutzel-nigg [67]. The Gaunt- and Breit-interaction is often not treated variationally but rather by first-order perturbation theory after a variational treatment of the Dirac-Coulomb-Hamiltonian. The contribution of higher-order corrections such as the vaccuum polarization or self-energy of the electron can be derived from quantum electrodynamics (QED), but are usually neglected due to their negligible impact on chemical properties. 

The Dirac-Coulomb-Breit Hamiltonian is derived pertmbationally and thus it is frequently suggested that the Breit correction to the Coulomb interaction should be considered in the perturbation framework and evaluated as the first-order contribution to the energy which follows from the Dirac-Coulomb calculations, and this is the way how it is done, as example, in the atomic GRASP2K package. 

Ewald summation presented above calls for the calculation of AP terms for each of the periodic boxes, a computationally demanding requirement for large biomolecular systems. Recently, Darden et al. proposed an N log N method, called particle mesh Ewald (PME), which incorporates a spherical cutoff R. This method uses lookup tables to calculate the direa space sum and its derivatives. The reciprocal sum is implemented by means of multidimensional piecewise interpolation methods, which permit the calculation of this sum and its first derivative at predefined grids with fast Fourier transform methods. The overhead for this calculation in comparison to Coulomb interactions ranges from 16 to 84% of computer time, depending on the reciprocal sum grid size and the order of polynomial used in calculating this sum. 

The physical interpretation of the electron-interaction component Wge(r) was originally proposed by Harbola and Sahni [9], and derived by them via Coulomb s law. It is based on the observation that the pair-correlation density g(r,r ) is not a static but rather a dynamic charge distribution whose structure changes as a function of electron position. The dynamic nature of this charge then must be accounted for in the description of the potential. Thus, in order to obtain the local potential in which the electron moves, the force field due to this charge distribution must first be determined. According to Coulomb s law this field is... 

The / and K integrals derive from the electron-electron repulsion term in Figure 14.3. / is the called the Coulomb integral and K is the exchange integral. It is because of the deter-minantal nature of the wave function that these two types of electron-electron interaction terms arise. In order to understand this, let s consider a molecule with only two electrons. We first introduce some new symbolism to simplify the notation (Eq. 14.25). 

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