Collectivity, shell openness indices, and complexity measures of multiconfigurational states: Computations within full CI scheme

Abstract

ABSTRACT:It is shown that the multiconfigurational character of the N-electron wave function can be treated by a new quantitative index, N eff , computed as a special matrix invariant of the full wave function. Along with N eff , a simpler index N eff av is given due to the Kutzelnigg and Smith open-shell theory. The indices are compared with an effective number of unpaired electrons introduced by Takatsuki, Fueno, and Yamaguchi. Full configuration interaction (FCI) computations on such quantities realized in the present work for atoms and dissociation reactions lead to the conclusion that all indices give a qualitatively similar picture. It is stressed that N eff as a more complicated quantity reflects a global complexity of the electronic state, while other indices represent the electron structure peculiarities only at the one-electron level.

Electron invariants and excited state structural analysis for electronic transitions within CIS, RPA, and TDDFT models

“…Apparently, the collectivity measure for configuration interaction wave functions was firstly given in 34 for CIS and then the similar approach was extended to full CI (FCI) wave functions in 53, 54. Here we elucidate how to define the analogous collectivity measure for RPA and TDDFT models considered as according to the preceding section, that is, as two‐component CIS‐like states governed by Eqs.…”

On the measure of electron correlation and entanglement in quantum chemistry based on the cumulant of the second-order reduced density matrix

“…͑5͔͒ should be regarded as the effective number of correlated electrons. 21 It is evident from ⑀ definition ͓cf. Eq.…”

“…2,3 There are also measures of electron correlation, such as the statistical correlation coefficients based on the structure of the reduced density matrices 24,25 and recently others supported by entropic concepts. 20,21,23 Entanglement definition shares the same difficulty and may be defined according to physical considerations. Therefore, to give a measure of entanglement is related with the idea of quantifying the correlation ͑its nonclassical component͒ of the molecular electronic states, which simplest example in quantum mechanics is the singlet state of two spins.…”

“…29 In such cases, it is necessary to specify the pure states of the system as being spin-adapted, i.e., the wave functions are eigenfunctions of the squaredand projection-spin operators, Ŝ 2 and Ŝ z , with eigenvalues S͑S +1͒ and S z , respectively. We will restrict ourselves to such kind of states because spin purity is of fundamental importance for a correct interpretation of highly correlated entangled states, despite in general a pure state may not be necessarily spin-adapted 20,21 for most of cases in which entanglement is defined. Regarding these previous considerations, we propose a new measure of electron correlation in molecular electronic states based on the many-body spin-free cumulant of the 2-RDM which collects spin and Coulomb correlation effects and allows one to quantity spin entanglement.…”

“…3,10,11 Nevertheless, electron densities are not enough for a complete description of the electron distribution of molecular systems, especially those undergoing chemical reactions and electron rearrangements ͑nonequilibrium geometrical conformations͒ such as formation and breaking of bonds. In these cases, higher-order density matrices are relevant [20][21][22][23] and particularly the second-order reduced density matrices ͑2-RDM͒ are essentially needed. 3,10 These latter types of system states permit to introduce a natural linkage between the quantum chemistry phenomena of correlation and entanglement 23 ͑the correlation that has not a classical counterpart͒.…”