Application of the electron localization function to radical systems

An application of the topological electron localization function (ELF) analysis to free radical systems is presented. A separation of the ELF function into its α‐spin and β‐spin contributions has been performed. Methyl and phenyl radicals, ortho‐, meta‐, and para‐benzyne biradicals, and their corresponding radical anions have been chosen with the aim to validate the new ELF_α and ELF_β proposed functions. The results show that the ELF separation yields complementary information about the localization of the unpaired electron. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Electron pair localization function: a practical tool to visualize electron localization in molecules from quantum Monte Carlo data

In this work we introduce an electron localization function describing the pairing of electrons in a molecular system. This function, called "electron pair localization function," is constructed to be particularly simple to evaluate within a quantum Monte Carlo framework. Two major advantages of this function are the following: (i) the simplicity and generality of its definition; and (ii) the possibility of calculating it with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, multiconfigurational wave functions, valence bond, density functional theory, variational Monte Carlo with explicitly correlated trial wave functions, fixed-node diffusion Monte Carlo, etc). A number of applications of the electron pair localization function to simple atomic and molecular systems are presented and systematic comparisons with the more standard electron localization function of Becke and Edgecombe are done. Results illustrate that the electron pair localization function is a simple and practical tool for visualizing electronic localization in molecular systems.     

Partitioning of the orbital overlap matrix and the localization criteria

A new molecular orbital localization procedure is proposed. The approach is based on partitioning of the overlap matrix into atomic contributions in accordance with Bader's topological theory of atoms in molecules. The new procedure has several advantages over other schemes. It preserves the 6/7c-separability in planar systems and allows for a straightforward interpretation of the localized orbitals in terms of their localization indices and atomic occupancies. The new procedure is tested on the H2O, LiF, N2, CO, BH₃ · CO and Li₂ molecules.Research partially supported by US DOE.     

Asymptotic analysis of the localization spread and polarizability of 1‐D noninteracting electrons

According to the modern Theory of the Insulating State [Resta, J Chem Phys 2006, 124, 104104], the metallic behavior of a N‐electron system with open boundary conditions is characterized by a localization spread λ_βγ diverging in the thermodynamic limit. This quantity, which is the second‐moment cumulant of the position operator (per electron), cannot in general be evaluated in closed form but for simple model systems. In this article, we perform an asymptotic analysis of λ_βγ for a gas of N non‐interacting electrons in a 1‐Dimensional box and a Hückel chain of N equivalent sites. The asymptotic behavior of the closely related polarizability tensor is also investigated for these exactly solvable models. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Electron localization and delocalization indices for solids

The electron localization and delocalization indices obtained by the integration of exchange‐correlation part of pair density over chemically meaningful regions of space, e.g., QTAIM atoms are valuable tools for the bonding analysis in molecular systems. However, among periodic systems only few simplest models were analyzed with this approach until now. This contribution reports implementation and evaluation of the localization and delocalization indices on the basis of solid state DFT calculations. A comparison with the results of simple analytical model of Ponec was made. In addition, a small set of compounds with ionic (NaCl), covalent (diamond, graphite), and metallic (Na, Cu) bonding interactions was characterized using this method. Typical features of different types of bonding were discussed using the delocalization indices. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

An orbital localization criterion based on the topological analysis of the electron localization function at correlated level

This work describes a procedure for localizing orbitals based on the topological analysis of the electron localization function at correlated level. The decomposition of the overlap matrix according to the partitioning of the three dimensional physical space into basins provided by that function allows us to define a localization index to be maximized using isopycnic orbital transformations. The localization algorithm has been computationally implemented and its efficiency tested on selected molecular systems at equilibrium, stretched, and twisted geometries. We report results which allow to analyze the influence of the correlated and uncorrelated treatments on the orbital localization.     

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many‐electron dynamics of molecular systems, where an explicitly time‐dependent electronic wave packet is represented as a linear combination of N‐electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one‐electron operator by reducing the multideterminantal, many‐electron wave packet using the Slater‐Condon rules. Here, we introduce a general tool for post‐processing multideterminant configuration‐interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which shares and builds on the modular design of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N‐electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 Wiley Periodicals, Inc.     

Approximate charge density localization of molecular orbitals

A NDO approximate procedure based on the indirect intrinsic ab initio localization method of von Niessen is developed. It is shown that only when the NDO approximations are introduced at the two electron level, expressions are obtained which are the charge density counterpart of those found in the approximate energy localization methods. The results of these two methods are quite similar both in the CNDO and INDO approximations. The indeterminacies observed in the CNDO localization for unsaturated systems and for molecules with two or three lone pairs on the same atom, are removed by localizing up to an INDO level. The approximate charge density localization is however computationally much easier than the approximate energy localization method and should be more appropriate in LMO studies of large organic molecules.     

An orbital localization criterion based on the theory of "fuzzy" atoms

This work proposes a new procedure for localizing molecular and natural orbitals. The localization criterion presented here is based on the partitioning of the overlap matrix into atomic contributions within the theory of "fuzzy" atoms. Our approach has several advantages over other schemes: it is computationally inexpensive, preserves the sigma/pi-separability in planar systems and provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and atomic occupancies. The corresponding algorithm has been implemented and its efficiency tested on selected molecular systems.     

Orbital localization in transition metal molecules

In multiply bonded, weakly interacting systems the excessive electron repulsion associated with the non-dynamical correlation error can be reduced within the Hartree Fock approximation by localizing the bonding orbitals. The mechanism behind this (unphysical) orbital localization is studied through calculations on a model system, and SCF and CI calculations on the MnO⁺ ion. It is shown, from a pair-population analysis of the two-particle density matrix (which is analogous to a Mulliken population analysis of the one-density) that the orbital localization is a two-electron effect. Transition metal molecules often exhibit this kind of orbital localization which may (or may not) require symmetry breaking. The special characteristics of transition metal molecules that makes them suitable candidates for orbital localization will be discussed.     

Ultrafast Electron Diffraction (UED)

With properly timed sequences of ultrafast electron pulses, it is now possible to image complex molecular structures in the four dimensions of space and time with resolutions of 0.01 Å and 1 ps, respectively. The new limits of ultrafast electron diffraction (UED) provide the means for the determination of transient molecular structures, including reactive intermediates and non‐equilibrium structures of complex energy landscapes. By freezing structures on the ultrafast timescale, we are able to develop concepts that correlate structure with dynamics. Examples include structure‐driven radiationless processes, dynamics‐driven reaction stereochemistry, pseudorotary transition‐state structures, and non‐equilibrium structures exhibiting negative temperature, bifurcation, or selective energy localization in bonds. These successes in the studies of complex molecular systems, even without heavy atoms, and the recent development of a new machine devoted to structures in the condensed phase, establish UED as a powerful method for mapping out temporally changing molecular structures in chemistry, and potentially, in biology. This review highlights the advances made at Caltech, with emphasis on the principles of UED, its evolution through four generations of instrumentation (UED‐1 to UED‐4) and its diverse applications.     

Pipek–Mezey localization of occupied and virtual orbitals

Recent advances in orbital localization algorithms are used to minimize the Pipek–Mezey localization function for both occupied and virtual Hartree–Fock orbitals. Virtual Pipek–Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek–Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state‐of‐the‐art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state‐of‐the‐art localized orbitals. © 2013 Wiley Periodicals, Inc.     

Charge localization by molecular orbital calculations. II. Formamide,thioformamide and N-methylated analogues

Ab initio molecular orbital calculations have been used to determine the degree of charge localization associated with the formation of radical cations on electron impact of formamide, thioformamide and their N-methyl and N, N-dimethyl analogues. The charge favours the nitrogen of formamides and the sulphur of thioformamides, but the degree of localization is calculated to be lower than in the radical cations of urea and thiourea.     

Electron localization function as information measure

The conditional two-electron probability function, which defines the electron localization function (ELF) of Becke and Edgecombe in the Kohn-Sham theory, is interpreted as the nonadditive (interorbital) Fisher information contained in the electron distribution. The probability normalization considerations suggest a use of the related information measure defined in terms of the unity-normalized probability distributions (shape factors of the electron densities), as the key ingredient of the modified information-theoretic ELF. This modified Fisher information density is validated by a comparison with the original two-electron probability function. Illustrative applications to typical molecular systems demonstrate the adequacy of the modified information-theoretic ELF in extracting the key features of the electron distributions in molecules. The overall Fisher information itself and the associated information-distance quantities are also proposed as complementary localization functions.     

Electron localization function and electron localizability indicator applied to study the bonding in the peroxynitrous acid HOONO

The ground‐state electronic structure of peroxynitrous acid (HOONO) and its singlet biradicaloid form (HO ··· ONO) have been studied using topological analysis of the electron localization function (ELF), together with the electron localizability indicator (ELI‐D), at the DFT (B3LYP, M05, M052X, and M06), CCSD, and CASSCF levels. Three isomers of HOONO (cis‐cis, cis‐perp, and trans‐perp) have been considered. The results show that from all functionals applied, only B3LYP yields the correct geometrical structure. The ELF and ELI‐D‐topology of the O? O and central N? O bonds strongly depends on the wave function used for analysis. Calculations carried out at CAS (14,12)/aug‐cc‐pVTZ//CCSD(T)/aug‐cc‐pVTZ level reveal two bonds of the charge‐shift type: a protocovalent N? O bond with a basin population of 0.82–1.08e, and a more electron depleted O? O bond with a population of 0.66–0.71e. The most favorable dissociation channel (HOONO → HO + ONO) corresponds to breaking of the most electron‐deficient bond (O? O). In the case of cis‐ and trans‐HO ··· ONO, the ELF, ELI‐D, and electron density fields results demonstrate a closed‐shell O ··· O interaction. The α‐spin electrons are found mainly (0.64e) in the lone pairs of oxygen V_{i = 1,2} (O) from the OH group. The β‐spin electrons are delocalized over the ONO group, with the largest concentration (0.34e) on the lone pair of nitrogen V(N). © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

A study of the molecular orbital localization into an extended Hartree—Fock approach: Application to the BeH2 ground state

In this article, we present a study of the localization and properties of the molecular orbitals (MOs) of polyatomic systems by using a comprehensive version of the G1 model. In this version, the wave function is written as a DODS product of univocally determined spin orbitals (MOs), “projected” on the singlet ground state. A procedure for determining the MOs is given and applied to the BeH₂ ground state. Equivalent split shell and localized MOs are found. The Be orbitals are seen to exhibit sp hybridization and the localized valence MOs are found to produce − 13.7 kcal/mol localization energy. Multistructural calculations are carried out and show that the present approach is able to describe localized and well-oriented bonds whenever the molecule under study presents only a single well-defined nonresonant chemical structure. © 1996 John Wiley & Sons, Inc.     

Electron-pairing analysis from localization and delocalization indices in the framework of the atoms-in-molecules theory

This article presents an overview of recent advances in the study of electron pairing through the use of localization and delocalization indices obtained from double integration over atomic basins of the exchange–correlation density in the framework of the atoms-in-molecules theory. These localization and delocalization indices describe the intra- and interatomic distribution of the electron pairs in a molecule. The main results of the application of these second-order indices to the analysis of molecular structure and chemical reactivity are briefly reviewed. It is shown that localization and delocalization indices represent a powerful tool to describe the electron-pair structure of molecules, which, in turn, provides deeper insight into relevant chemical phenomena such as electron correlation effects and the formation of localized α, β electron pairs. Received: 8 April 2002 / Accepted: 26 June 2002 / Published online: 6 September 2002 Acknowledgements. Financial help was furnished by the Spanish DGES projects no. PB98-0457-C02-01 and BQU2002-04112-C02-02. J.P. thanks the Departament d'Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya for benefiting from a doctoral fellowship, no. 2000FI-00582. M.S. is indebted to the Departament d'Universitats, Recerca i Societat de la Informació of the Generalitat de Catalunya for financial support through the Distinguished University Research Promotion, 2001. We also thank the Centre de Supercomputació de Catalunya for providing us with computing facilities. Correspondence to: M. Solà e-mail: miquel.sola@udg.es     

Electron localization function at the correlated level

The electron localization function (ELF) has been proven so far a valuable tool to determine the location of electron pairs. Because of that, the ELF has been widely used to understand the nature of the chemical bonding and to discuss the mechanism of chemical reactions. Up to now, most applications of the ELF have been performed with monodeterminantal methods and only few attempts to calculate this function for correlated wave functions have been carried out. Here, a formulation of ELF valid for mono- and multiconfigurational wave functions is given and compared with previous recently reported approaches. The method described does not require the use of the homogeneous electron gas to define the ELF, at variance with the ELF definition given by Becke. The effect of the electron correlation in the ELF, introduced by means of configuration interaction with singles and doubles calculations, is discussed in the light of the results derived from a set of atomic and molecular systems.