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

This work proposes a new molecular orbital localization procedure. The approach is based on the decomposition of the overlap matrix in accordance with the partitioning of the three‐dimensional physical space into basins with clear chemical meaning arising from the topological analysis of the electron localization function. The procedure is computationally inexpensive, provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and basin occupancies, and preserves the σ/π‐separability in planar N‐electron systems. The localization algorithm is tested on selected molecular systems. © 2012 Wiley Periodicals, Inc.     

Electron localization functions and local measures of the covariance

The electron localization measure proposed by Becke and Edgecombe is shown to be related to the covariance of the electron pair distribution. Just as with the electron localization function, the local covariance does not seem to be, in and of itself, a useful quantity for elucidating shell structure. A function of the local covariance, however, is useful for this purpose. A different function, based on the hyperbolic tangent, is proposed to elucidate the shell structure encapsulated by the local covariance; this function also seems to work better for the electron localization measure of Becke and Edgecombe. In addition, we propose a different measure for the electron localization that incorporates both the electron localization measure of Becke and Edgecombe and the Laplacian of the electron density; preliminary indications are that this measure is especially good at elucidating the shell structure in valence regions. Methods for evaluating electron localization functions directly from the electron density, without recourse to the Kohn-Sham orbitals, are discussed.     

Enzyme function is regulated by its localization

To better understand how enzyme localization affects enzyme activity we studied the cellular localization of the glycosyltransferase MurG, an enzyme necessary for cell wall synthesis at the spore during sporulation in the bacterium Bacillus subtilis. During sporulation MurG was gradually enriched to the membrane at the forespore and point mutations in a MurG helical domain disrupting its localization to the membrane caused severe sporulation defects, but did not affect localization nor caused detectable defects during exponential growth. We found that this localization is dependent on the phospholipid cardiolipin, as in strains where the cardiolipin-synthesizing genes were deleted, MurG levels were diminished at the forespore. Furthermore, in this cardiolipin-less strain, MurG localization during sporulation was rescued by external addition of purified cardiolipin. These results support localization as a critical factor in the regulation of proper enzyme function and catalysis.     

Charge decomposition analysis of the electron localizability indicator: a bridge between the orbital and direct space representation of the chemical bond

The novel functional electron localizability indicator is a useful tool for investigating chemical bonding in molecules and solids. In contrast to the traditional electron localization function (ELF), the electron localizability indicator is shown to be exactly decomposable into partial orbital contributions even though it displays at the single-determinantal level of theory the same topology as the ELF. This approach is generally valid for molecules and crystals at either the single-determinantal or the explicitly correlated level of theory. The advantages of the new approach are illustrated for the argon atom, homonuclear dimers N2 and F2, unsaturated hydrocarbons C2H4 and C6H6, and the transition-metal-containing molecules Sc(2)2+ and TiF4.     

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.     

Electron localization and delocalization in open-shell molecules

Localization and delocalization indices derived in the framework of the quantum Atoms in Molecules theory have recently been used to analyze the electron-pair structure of closed-shell molecules. Here we report calculations of localization and delocalization indices for open-shell molecules at the Hartree-Fock (HF) level. Several simple doublet and triplet radical molecules are studied. In general, interatomic delocalization between bonded atoms is heavily dependent on the order and polarity of the bond. Unpaired electrons also have a significant effect on the interatomic delocalization indices. Indeed, for many radicals, the analysis of the spin components reveals that the interatomic delocalization is very different for alpha and beta spin electrons in many cases. In general, at the HF level, the results can be rationalized in terms of orbital contributions. However, the definition of localization and delocalization indices is completely general, and they could be calculated at any level of theory, provided that the one- and two-electron densities are available.     

Electron localization function from density components

This work addresses the decomposition of the Electron Localization Function (ELF) into partial density contributions using an appealing split of kinetic energy densities. Regarding the degree of the electron localization, the relationship between ELF and its usual spin‐polarized formula is discussed. A new polarized ELF formula, built from any subsystems of the density, and a localization function, quantifying the measure of electron localization for only a subpart of the total system are introduced. The methodology appears tailored to describe the electron localization in bonding patterns of subsystems, such as the local nucleophilic character. Beyond these striking examples, this work opens up opportunities to describe any electronic properties that depend only on subparts of the density in atoms, molecules, or solids. © 2016 Wiley Periodicals, Inc.     

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.     

A practical and efficient method to calculate AIM localization and delocalization indices at post-HF levels of theory

A practical and efficient method is proposed for calculating localization and delocalization indices at post-Hartree-Fock levels, and the method is tested at the CISD/6-311G++(2d, 2p) level for a large set of molecules. Our method, which utilizes wave functions written in the natural molecular orbital format and obtained with GAUSSIAN 94 or GAUSSIAN 98, convincingly extends the concepts established at the HF level.     

Efficient use of Jacobi rotations for orbital optimization and localization

Summary Quantum chemical orbital optimizations can be accomplished by Newton-type iterations, whereall orbitals are improved ateach step; or by a succession of Jacobi rotations, where onlytwo orbitals are improved at any one step. In both schemes, the iterative updating of the four-index two-electron integrals requires a large computational effort. We show that the four-index transformation due to a Jacobi rotation can be simplified to such a degree that the successive execution of the four-index transformations ofN(N–1)/2 different Jacobi rotations requires no greater computational effort than that required by theone full orthogonal transformation which is the product of allN(N–1)/2 Jacobi rotations. The four-index updating has therefore no bearing on the relative merit of the Newton approach versus the Jacobi approach. The Jacobi approach has, however, an advantage if the optimization of each Jacobi rotation angle is simple and if the effectiveness of the individual Jacobi rotations can be assessed without the execution of four-index transformations. For, in that case, all ineffectual rotations are easily deleted from the iterative sequence. Whether convergence can be guaranteed for one or the other approach is also relevant. Our conclusions are illustrated by application to the problem of intrinsic orbital localization where the succession of Jacobi rotations is the more effective strategy.Dedicated to Professor Per Olov LöwdinAmes Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract No. W-7405-Eng-82. This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences     

Fast noniterative orbital localization for large molecules

We use Cholesky decomposition of the density matrix in atomic orbital basis to define a new set of occupied molecular orbital coefficients. Analysis of the resulting orbitals ("Cholesky molecular orbitals") demonstrates their localized character inherited from the sparsity of the density matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a number of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addition, since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.     

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.     

On the importance of orbital localization in QC‐DMRG calculations

We investigate the importance of orbital localization in the application of the Density Matrix Renormalization Group (DMRG) technique to ab initio studies of molecular electronic structure. Our previous implementation of DMRG has been generalized to allow for the use of localized nonorthogonal orbitals. Simple cycles of equidistant hydrogen atoms, which are good examples of one dimensional metal‐like lattices with fully delocalized electronic structures, have been taken as test models. In this study, the efficiency of the DMRG method and the importance of orbital localization for the generation of the DMRG building blocks are confirmed. However, it is found that the convergence of the procedure based on nonorthogonal orbitals is slower and requires more DMRG components than the standard orthogonal formulation. Symmetrically orthonormalized atomic orbitals are shown to be a good compromise solution: they satisfy the requirement of orbital localization for the generation of the DMRG blocks and improve the convergence, reducing the number of components of the DMRG expansion. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Multiconfiguration self-consistent field,theory of localized orbitals: Choice of localization potential

We investigate the properties of two different choices for localization potentials for the direct construction of localized fixed orbitals by multiconfiguration self-consistent field theory. The first potential yields maximally screened orbitals by solution of a complicated orbital equation which depends explicitly on the complete set of orbitals for the system, and contains both one-and two-center matrix elements. The second localization potential yields somewhat less well screened orbitals by solution of a considerably simpler orbital equation which only contains simple one-center matrix elements.     

The Kohn-Sham kinetic energy density as indicator of the electron localization: atomic shell structure

In this report, it is shown that the Kohn-Sham (KS) kinetic energy density (KED) contains the average local electrostatic potential (ALEP) and the average local ionization energy (ALIE); the shell structure in atomic systems is presented as one application of the KS-KED. By writing the KS-KED from the KS equations, this quantity was divided in three contributions: orbital, Coulomb, and exchange correlation. By studying several closed and open shell atoms, the shell structure was established by the maxima presented by the Coulomb contribution and the minima in the orbital contribution of the KS-KED. The exchange-correlation contribution to the KS-KED does not show maxima or minima, but this quantity shows bumps where the division between shells is expected. The results obtained in this work were compared with other shell structure indicators such as the electron localization function, the ALEP, the ALIE, and the radial distribution function. The most important result in this work is related to the fact that even when the ALEP and the ALIE functions were built with different arguments to each other, they are contained in the KS-KED. In this way, the KS-KED shows its importance to reveal the electron localization in atomic systems.     

New insights on the bridge carbon-carbon bond in propellanes: a theoretical study based on the analysis of the electron localization function

The nature of the bonding between bridgehead carbon atoms (Ca, Ca') as well as the ring strain in a family of 10 propellanes formed by three-, four-, or five-member rings: [1.1.1] (I), [2.1.1] (II), [3.1.1] (III), [2.2.1] (IV), [3.2.1] (V), [2.2.2] (VI), [3.3.1] (VII), [3.2.2] (VIII), [3.3.2] (IX), and [3.3.3] (X) are studied by means of the electron localization function (ELF) at the DFT level (B3LYP/cc-pVTZ). The ELF analysis of smaller propellanes (I, II, and III) reveals the coexistence of two resonance forms: one with a nonbonding electron pair partially delocalized between Ca and Ca' atoms outside the cage (ionic) and the other with a bridge bond between the same atoms (covalent). The weights of each form are calculated according to the ELF-basin populations, yielding 94, 88, and 53% for the ionic structure of I, II, and III, respectively, while larger propellanes (IV-X) present only the covalent form. The question of the s-character of the bridge bond is addressed by dissecting the bridge-bond ELF basin into the molecular orbital contributions. Finally, sigma-aromaticity associated to surface electron delocalization has been analyzed by means of nucleus-independent chemical shift (NICS) calculations. The results point out that the stability of the fused ring structure of propellanes I, II, and III, can be assigned to the remarkable sigma-aromaticity of the involved three-member rings.     

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.     

Electron localization function in full-potential representation for crystalline materials

The electron localization function (ELF) is implemented in the first-principles, all-electron, full-potential local orbital method. This full-potential implementation increases the accuracy with which the ELF can be computed for crystalline materials. Some representative results obtained are presented and compared with the results of other methods. Although for crystal structures with directed bonding only minor differences are found, in simple elemental metals, there are differences in the valence region, which give rise to different ELF topologies.     

Determination of substitutional sites in heterocycles from the topological analysis of the electron localization function (ELF)

The topological analysis of the electron localization function ELF has been carried out on five‐membered (C₄H₄NH, C₄H₄PH, C₄H₄O, C₄H₄S) and six‐membered (C₅H₅N, C₅H₅P) heterocycles. The bonding in these molecules is discussed on the basis of the valence basin populations. It is shown that the values of the ELF function at the (3,−1) critical points between disynaptic basins related to a given center provide a criterion to determine substitutional sites. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 509–514, 2000     

Electron localization function for transition-metal compounds

 The analysis of the electron localization function of molecules and solids needs to involve the atomic core regions as well to reveal a more detailed insight into the bonding situation. Received: 20 February 2002 /Accepted: 14 June 2002 /Published online: 19 August 2002