Configuration mixing

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 C F2, therefore the moments Hk(lN) are mainly conditioned by the non-spherical part of the 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. 

Asymmetry of the spectrum, defined by skewness jq in (32.7), is positive, if the majority of levels are in the lower part of the spectrum, and negative, if the opposite is true. The excess K2 of the spectra of the majority of atoms with one open shell for 1 N 41 + 1 is also positive hence, the density of the levels, while approaching the average energy, grows more rapidly in comparison to the case of normal distribution of the levels. The results of the Hartree-Fock calculations of a2, q and K2 for isoelectronic sequences with an open ndN shell indicate that a grows almost linearly with Z, whereas jq and K2 decrease with increase of Z, more rapidly for larger n. 

32 Global methods in the theory of many-electron atoms 

The total moment of a complex of configurations may be presented as a sum of two terms nl(K) representing the moment of the total spectrum in single-configuration approximation, and the second, An (K), describing the correction due to superposition of configurations, i.e. 

For estimation of the degree of configuration mixing as well as for evaluation of the configurations, which must be accounted for first, the special mean characteristic - the strength of configuration mixing - can be adopted. It is defined as follows  

The order of perturbation theory in which each block contributes to the correlation energy is indicated. The structure of the Hamiltonian matrix when the single determinantal reference function is constructed from bare nucleus orbitals is displayed in Table 13, where again the order of perturbation theory in 

---

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

The effect of configurational mixing of higher-lying s orbitals into the ligand field d-orbital basis set is also likely to favour elongation rather than contraction. ... 

Some rather heavily-truncated versions of the more complex configurations have been used at ANL to assist in level identifications. A summary of the fitted parametric results for the lowest configurations of Pu I and Pu II is given in Table II. In some cases configuration mixing has been included. The numbers enclosed in parentheses following the... 

Configuration mixing model a general approach to organic reactivity, 21,99 Conformations of polypeptides, calculations of, 6,103 Conjugated molecules, reactivity indices, in, 4,73... 

We can now consider explicitly how configurations interact to produce electronic states. Our first task is to define the Hamiltonian operator. In order to simplify our analysis, we adopt a Hamiltonian which consists of only one electron terms and we set out to develop electronic states which arise from one electron configuration mixing. 

We show another aspect of these numbers in Table 15.3, where we detail the effects of resonance between the two Kekule structures and among all of the covalent structures for the two sorts o 2pz orbitals. The results suggest that within one structure the SCVB orbitals duplicate, to some extent, the effect of multiple structures, and the configurational mixing produces less energy lowering with them. 

---