Configuration general open-shell

Electronic resonances are generally open shell states. The simplest case is that of a single electron outside of a closed shell for which a single configuration Is also a single determinant,... 

Abstract This contribution reviews a selection of findings on atomic density functions and discusses ways for reading chemical information from them. First an expression for the density function for atoms in the multi-configuration Hartree-Fock scheme is established. The spherical harmonic content of the density function and ways to restore the spherical symmetry in a general open-shell case are treated. The evaluation of the density function is illustrated in a few examples. In the second part of the paper, atomic density functions are analyzed using quantum similarity measures. The comparison of atomic density functions is shown to be useful to obtain physical and chemical information. Finally, concepts from information theory are introduced and adopted for the comparison of density functions. In particular, based on the Kullback-Leibler form, a functional is constructed that reveals the periodicity in Mendeleev s table. Finally a quantum similarity measure is constructed, based on the integrand of the Kullback-Leibler expression and the periodicity is regained in a different way. 

S. Yamamoto, H. Tatewaki, and T. Saue, Dipole allowed transitions in GdF A four-component relativistic general open-shell configuration interaction study, J. Chem. Phys. 129, 44505 ( 8 pages) (2008). 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

Similar expressions for submatrix elements of the operators in question for transitions of the type (25.26) may be easily obtained from general formula (25.28) as special cases. Analogous equalities for the transition (25.27) already follow from the cases of configurations consisting of three open shells. We shall consider them only in LS coupling. Thus, the submatrix element of the operator under consideration for the transition... 

Such general expressions for matrix elements of electrostatic interactions, covering the cases of three and four open shells, may be found in Chapter 25 of [14]. However, they are rather cumbersome and, therefore of little use for practical applications. Quite often sets of simpler formulas, adopted for particular cases of configurations, are employed. Below we shall present such expressions only for the simplest interconfigurational matrix elements occurring while improving the description of a shell of equivalent electrons (the appropriate formulas for the more complex cases may be found in [14]) ... 

MBPT significantly improves the electron transition wavelengths, line and oscillator strengths, transition probabilities as well as the lifetimes of excited levels. Therefore, it seems promising to generalize such an approach to cover the cases of more complex electronic configurations having several open shells, even with n > 2. 

The majority of the above-mentioned problems will be discussed in more or less detail in this book. The book is based on the variational approach, which is the most universal and efficient method of theoretical study of the spectra of any atom or ion of the Periodical Table. However, generalization of the perturbation theory to cover the case of configurations with several open shells is also presented [26, 51, 52],... 

The UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130],... 

Although the closed shell CC theory has been sometimes used to describe some special open-shell situations/39,40/ (like single configuration triplets/39/) or dissociating species/40-42/, from the point of view of generality, and on physical grounds it is natural to look for open-shell generalizations of the cluster expansion formalisms as developed for the closed shells. 

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