Orbitals electrostatic interaction integrals

Because of the greater extension of the 5f radial wavefunctions with respect to those of the shielding 7s and 7p shells, they are more sensitive to changes in the valence-electron situation than for the corresponding lanthanide cores. Nevertheless, their rigidity is remarkable as compared to that for the valence electrons themselves. This can be seen quantitatively in the plots for the Slater electrostatic interaction integrals (Sf, Sf) and spin-orbit radial integral C, which dominate the atomic Hamiltonian for all cases of interest to us here. 

In the original, elementary treatment governed by Eq. 4 above, one might initially expect contributions to the barrier from several sources. There is first the Coulomb integral Q, which will contain angle dependent terms from the electrostatic interaction of the electrons and protons ar the two ends of the molecule. In this treatment the only orbitals used are Is on each H atom and tetra-... 

Model Hartree-Fock calculations which include only the electrostatic interaction in terms of the Slater integrals F0, F2, F and F6, and the spin-orbit interaction , result in differences between calculated and experimentally observed levels596 which can be more than 500 cm-1 even for the f2 ion Pr3. However, inclusion of configuration interaction terms, either two-particle or three-particle, considerably improves the correlations.597,598 In this way, an ion such as Nd3+ can be described in terms of 18 parameters (including crystal field... 

The Slater integrals of the energy of electrostatic interaction and the spin-orbit interaction constant are the main parameters searched. It is obvious that the number of known levels must be equal to or exceed the number of unknown parameters. 

In a single-configurational non-relativistic approach, the integrals of electrostatic interactions and the constant of spin-orbit interactions compose the minimal set of semi-empirical parameters. Then for pN and dN shells we have two and three parameters, respectively. However, calculations show that such numbers of parameters are insufficient to achieve high accuracy of the theoretical energy levels. Therefore, we have to look for extra parameters, which would be in charge of the relativistic and correlation effects not yet described. 

Here the sum is over all Z values for which there is a known R(Z) value. As expected, the values of the parameter q, obtained in this way, as a rule are very close to the corresponding quantities, following from the hydrogenic approximation q = 1 for the integrals of the electrostatic interaction, 2 for average energies, 4 for spin-orbit parameter and q = —k for mean values of rk. 

It is interesting to notice that the numerical coefficients at F2 rapidly decrease with growth of the power of this integral. For atoms with small Z values t]p electrostatic interaction. In the case of neutral atoms the spin-orbit contribution together with the mixed term exceeds the electrostatic contribution to the variance and excess of the atomic spectrum of the configuration njF only for n> 6. 

The variance may be adopted to evaluate the accuracy of the coupling scheme used. Indeed, its formula (32.21) consists of two terms, corresponding to spin-orbit and electrostatic interactions, therefore the quantity aso/ae may also serve as a measure to estimate the coupling scheme. It is preferred over the ratio rj i/F2(nl, nl), because the coupling scheme depends not only on radial integrals, but also on spin-angular parts of them. Let us also mention that a characterizes the width of the spectrum AE, i.e. the energy separation between upper and lower levels. 

The symbol Jy is often used to represent this Coulomb interaction between electrons in spin orbitals i andand is unfavourable (i.e. positive). The total electrostatic interaction between the electron in orbital Xi and the other N — 1 electrons is a sum of all such integrals, where the summation index j runs from 1 to N, excluding i ... 

The neglect of diatomic differential overlap (NDDO) method [236] is an improvement over the INDO approximation, since the ZDO approximation is applied only for orbital pairs centered at different atoms. Thus, all integrals pv Xa) are retained provided p and v are on the same atomic center and A and a are on the same atomic center, but not necessarily the center hosting p and v. In principle, the NDDO approximation should describe long-range electrostatic interactions more accurately than INDO. Most modern semiempirical models (MNDO, AMI, PM3) are NDDO models. 

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