Simple Coulombic correction

L. Visscher. Approximate molecular relativistic Dirac-Coulomb calculations using a simple Coulombic correction. Theor. Chem. Acc., 98(2/3) (1997) 68-70. 

T he core-core interaction between pairs of nuclei was also changed in MINDO/3 from the fiiriu used in CNDO/2. One way to correct the fundamental problems with CNDO/2 such as Ihe repulsion between two hydrogen atoms (or indeed any neutral molecules) at all di -l.inces is to change the core-core repulsion term from a simple Coulombic expression (/ ., ii = ZaZb/Rab) to ... 

Forsman, J. (2007). Simple correlation-corrected theory of systems described by screened coulomb interactions. Langmuir 23, 5515—5521. 

Retardation effects. If the phonon contribution dominates the bare vertex (6), the retardation effects associated with heavy ions can play an important role in the many-body theory. In order to develop this point in greater detail let us, for reasons of clarity, ignore the Coulomb contribution to the bare vertex (6). Some simple vertex corrections are shown in Fig. 1. These particular diagrams are chosen because in the one-dimensional case (ij = r/ — 0) they all yield the same > g6log2T contribution to the vertex, provided that the retardation effects are neglected. Such a degenerate situation is usually named parquet. 

Within the first-order estimations made here, it is apparent that no change in d-d repulsion energy accompanies the hydration process. Second-order adjustments would, of course, take account of the change in mean i/-orbital radius on complex formation. Let us agree to stop at the simple level of correction here. Overall, therefore, the significant Coulombic change on hydration concerns the loss of exchange stabilization. 

Note again the formal simplicity of equation (7-17) as compared to equation (7-18) in spite of the fact that the former is exact provided the correct Vxc is inserted, while the latter is inherently an approximation. The calculation of the formally L2/2 one-electron integrals contained in hllv, equation (7-13) is a fairly simple task compared to the determination of the classical Coulomb and the exchange-correlation contributions. However, before we turn to the question, how to deal with the Coulomb and Vxc integrals, we want to discuss what kind of basis functions are nowadays used in equation (7-4) to express the Kohn-Sham orbitals. 

Compared to all other intermolecular interactions, the Coulomb interaction is described by a simple law, i.e.. Equation 15.2. A theory for Coulombic interaction, therefore, uses the concepts and laws that have been developed in classical electrostatics. However, it is worth pointing out that the dielectric constant is a macroscopic property and it is therefore, in principle, not correct to describe the solvent as a dielectric continuum on the molecular level. Nevertheless, experience has shown that it is in fact a useful approximation. 

The expression for quantity g(Rs) in (19.52) follows directly from (5.40). Thus, for p- and d-shells we have simple algebraic formulas for the coefficients of radial integrals (actually, for matrix elements) of all relativistic corrections of the order a2 to the Coulomb energy. For the /-shell such a formula exists only in the case of the orbit-orbit interaction. For the almost filled shell we find... 

It is also noteworthy that, even for methods having an intrinsic high formal complexity, the PCM corrections are always quite simple , thanks to the electrostatic nature of the solute-solvent interactions, which allows us to write the PCM-related operators in analogy to Coulomb operator (and allows us, for instance, to have the same PCM contribution in HF and DFT). 

Let as apply this formalism to the case of a single level a in a one-electron atom. To find the Coulomb nuclear recoil correction we have to calculate the contribution of the diagram shown in Fig. 1. A simple calculation of this diagram... 

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