Extracting Covalent and Ionic Structures from Usual Delocalized Wave Functions: The Electron-Expansion Methodology

Abstract

We present easily programmable expansions, allowing the calculation of the weights of local covalent and ionic structures of a chemical bond from usual delocalized wave functions; they are obtained in the framework of the electron-expansion methodology, in which the hole conditions (involved by definition in a covalent or ionic structure) are expanded in terms involving only electrons. From the derived relations, true for both HF and correlated levels, one can also express the covalency/ionicity and the localization of a usual two-electron two-center (2e/2c) bond in terms of electronic populations. The three-electron populations are crucial for bond localization. On the contrary, in 2e/2c bonding, and particularly in Charge-Shift bonds (which show enhanced covalent-ionic interactions) although the three-electron populations can be non-negligible, they are not important for the covalency/ionicity of these bonds. Numerical applications and discussion are given for correlated MO wave functions of butadiene, hexatriene, and pyrrole molecules on the basis of both natural atomic orbitals (NAOs) (orthogonal orbitals) and pre-NAOs (nonorthogonal orbitals).

Unpaired electrons, spin polarization, and bond orders in radicals from the 2‐RDM in orbital spaces: Basic notions and testing calculations

“…The basic calculations and results are obtained in the NAO basis sets, which are orthogonal orbitals as mathematically required for eigenfunctions of any physical Hermitian operator. In the calculated quantities, P E;H (where E is the number of electrons and H is the number of holes), one can give a probabilistic as well as an occupation number interpretation; alternatively, P E;H can be interpreted as weights of VB‐type local structures . Additionally, we used the nonorthogonal PNAO basis sets, because VB concepts and perspective are based on interatomic nonorthogonality, and VB resonance structures are often recalled to understand and rationalize the behavior of conjugated π‐systems.…”

“…In the calculated quantities, P E;H (where E is the number of electrons and H is the number of holes), one can give a probabilistic as well as an occupation number interpretation; alternatively, P E;H can be interpreted as weights of VB-type local structures. [35,51,54] Additionally, we used the nonorthogonal PNAO basis sets, because VB concepts and perspective are based on interatomic nonorthogonality, and VB resonance structures are often recalled to understand and rationalize the behavior of conjugated p-systems. Using the PNAO basis sets the probabilistic interpretation does not hold, but only the occupation number one; in this case an emphasis to the interpretation as VB-type weights (vide infra) must be given.…”

“…Such information can be obtained from molecular orbital (MO) wave functions by means of the population analysis of 1‐RDMs in natural orbitals, the decomposition of MO Slater determinants, a higher‐order population analysis method which uses localized bonds, and other well elaborated methods providing local descriptions . In this work, the local information is obtained by means of the PEPA , . The initial MO wave function, Ψ(ΜΟ), which can be either uncorrelated or correlated, is rewritten to a totally local wave function, Ψ(TL), which is a linear combination of Slater determinants, , involving local orbitals, as, for example, AOs.…”

“…[45][46][47][48] In this work, the local information is obtained by means of the PEPA. [35], [49][50][51] The initial MO wave function, W(MO), which can be either uncorrelated or correlated, is rewritten to a totally local wave function, W(TL), which is a linear combination of Slater determinants, jU K j, involving local orbitals, as, for example, AOs. The transformation of W(MO) to W(TL) is achieved by means of Moffitt's theorem, [42,43,52] which guarantees that the approximation level of the initial W(MO) is not altered:…”

Domain‐averaged Fermi hole and domain‐restricted reduced density matrices: A critical comparison

“…In the second example, a quasi‐analytic VB model for a two‐electron system is implemented to show in a graphic way the physical localization of the electron cloud. For that goal, we use a simple model52, which has recently been subject of interest in studies of population analysis53. This model allows us to complement this critical discussion by analyzing the behavior of both densities in terms of the spatial localization of the electron cloud.…”