Number seniority

Out of the set of additional quantum numbers a, the seniority number v is separated in an explicit form. 

Expressions (16.5) and (16.6) include only wave functions where the number of electrons equals the seniority number. 

The last relation, together with (16.20), corresponds to the set of equations (9.9) and (9.10), which is generally used to find (by the recurrence method on the number of electrons N) numerical values of CFP. Using formulas (16.24)-( 16.26), we can construct a system of equations that enable us, using the recurrence (in seniority number v) method, to find numerical values of SCFP. However, if the CFP are known, then the numerical values of the SCFP are determinable in the simplest way from (16.17). Tables 16.1 and 16.2 summarize the numerical values of the latter for the p- and d-electrons, taken from [91]. The coefficients that are related to the transposition condition (16.18) are omitted. 

This expression for /-electrons can be derived using the phase relations established for isoscalar parts of factorized CFP with different parities of the seniority number [24]. It turned out [91] that phase (16.55) provides sign relations between the CFP in the tables for d- and /-electrons, but it is unsuitable for the p-electrons. In this connection, in what follows all the relationships derived using the symmetry properties under transposition of the quantum numbers of spin and quasispin are provided up to the sign. 

It is well known that even for three /-electrons the classification of terms by the seniority number turns out to be insufficient. In this case, the exclusive group G2 has been applied with success [24], although it is still... 

The objective of this section is to give algebraic formulae for evaluating the matrix elements of the interaction operators in the basis set of ATs. First, we need to introduce the seniority number and the coefficients of the fractional parentage (CFPs). Then we will use to full advantage the ITO... 

The seniority number (seniority for short), v, is a quantum number related to eigenvalues of the Racah seniority operator . The seniority matches the electron configuration ln in which the particular term first appears. We will see later that some matrix elements of the operators included in the Hamiltonian are diagonal in v so that their off-diagonal counterparts vanish. For some operators, however, there are crossing terms in v. 

The seniority number (shortly seniority ), v, is a number associated with each term, viz. L, and it behaves like a quantum number there exists an operator (Racah seniority operator) whose eigenvalues are functions of v. The kets of many-electron atoms may be indexed as vLSJM) and for dn configurations they are evolved according to the scheme... 

The label g can distinguish between the different eigenstates of the hamiltonian with a common particle number and a common seniority number. One can also define states... 

According to Racah, a unique characterization of the terms of these configurations is achieved by introducing only one quantum number in addition to the set (nd) SL. For this purpose the irreducible representations of the rotation group in five dimensions R can be used, but it is more common to use the seniority number v. In the latter case the states are classified according to the eigenvalues of the seniority operator... 

A study of the compactness of wave functions based on Shannon entropy indices a seniority number approach... 

Abstract This work reports the formulation of Shannon entropy indices in terms of seniority numbers of the Slater determinants expanding an A-electron wave fimc-tion. Numerical determinations of those indices prove that they provide a suitable quantitative procedure to evaluate compactness of wave functions and to describe their configurational structures. An analysis of the results, calculated for full configuration interaction wave functions in selected atomic and molecular systems, allows one to compare and to discuss the behavior of several types of molecular orbital basis sets in order to achieve more compact wave... 

Keywords Compactness of wave functions Seniority number Shannon enttopy indices Optimization of molecular orbital basis sets... 

This article has been organized as follows. Section 2 summarizes the notation and formulation of the main concepts used in this work it also reports the formulation of the Shannon entropy indices in terms of the seniority numbers of the Slater determinants. In Sect. 3, we present numerical values of those indices for wave functions of selected atomic and molecular systems these values allow one to characterize the compactness of the wave function expansions. The calculation level and the computational details are also indicated in this section. An analysis and discussion of these results are reported in Sect. 4. Finally, in the last section, we highlight the main conclusions and perspectives of this work. 

The K orbitals of an orthonormal basis set will be denoted by i,j,and their corresponding spin-orbitals by f,f, ... (a and a mean the spin coordinates a or P). The spin-free version of the N-electron seniority number operator has been formulated as [6, 9, 11 ]... 

These weights provide the definition of the cumulative index I, which in the seniority number approach is... 

We have determined expansions of wave functions of several atomic and molecular systems in their ground states, at FCI level. These wave functions have been expressed in the three mentioned molecular basis sets CMO, NO, and in order to study their compacmess in different molecular orbital basis sets. Our aim is to analyze the structure and compactness of those expansions by means of the entropic indices proposed in Eqs. (5), (7), and (8) according to the seniority numbers. We have mainly chosen the systems of... 

Table 1 Calculated values of the and I, quantities (Eqs. (5) and (CMO), in the orbitals which minimize the seniority number (7)) for the ground states of atomic and molecular systems described and in the natural orbitals (NO) by FCI expansions expressed in the canonical molecular orbitals ...

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