New insights in quantum chemical topology studies using numerical grid-based analyses

New insights in Quantum Chemical Topology of one-electron density functions have been proposed here by using a recent grid-based algorithm (Tang et al., J Phys Condens Matter 2009, 21, 084204), initially designed for the decomposition of the electron density. Beyond the charge analysis, we show that this algorithm is suitable for different scalar functions showing a more complex topology, that is, the Laplacian of the electron density, the electron localization function (ELF), and the molecular electrostatic potential (MEP). This algorithm makes use of a robust methodology enabling to numerically assign the data points of three-dimensional grids to basin volumes, and it has the advantage of requiring only the values of the scalar function without details on the wave function used to build the grid. Our implementation is briefly outlined (program named TopChem), its capabilities are examined, and technical aspects in terms of CPU requirement and accuracy of the results are discussed. Illustrative examples for individual molecules and crystalline solids obtained with gaussian and plane-wave-based density functional theory calculations are presented. Special attention was given to the MEP because its topological analysis is complex and scarce.     

How do electron localization functions describe π-electron delocalization?

Scalar fields provide an intuitive picture of chemical bonding. In particular, the electron localization function (ELF) has proven to be highly valuable in interpreting a broad range of bonding patterns. The discrimination between enhanced or reduced electron (de)localization within cyclic π-conjugated systems remains, however, challenging for ELF. In order to clearly distinguish between the local properties of ten highly and weakly π-(de)localized prototype systems, we compare the ELFs of both the canonical wave functions and electron-localized states (diabatic) with those of two closely related scalar fields: the electron localizability indicator (ELI-D) and the localized orbital locator (LOL). The simplest LOL function distinguishes enhanced from weak π-(de)localization in an insightful and reliable manner. LOL offers the finest contrast between annulenes with 4n/4n + 2 π electrons and their inorganic analogues as well as between hyperconjugated cyclopentadiene derivatives. LOL(π) also gives an appealing and intuitive picture of the π-bond. In contrast, the most popular ELF fails to capture subtle contrasting local electronic properties and suffers from the arbitrariness of the σ/π dissection. The orbital separation of the most recent ELI-D is clear-cut but the interpretations sometime less straightforward in the present context.     

通过价层电子密度分析展现分子电子结构

迄今已有众多实空间函数被提出用来揭示化学上感兴趣的分子电子结构特征,例如化学键、孤对电子和多中心电子共轭。在这些分析方法中,电子定域化函数(ELF)、电子密度的拉普拉斯(∇²ρ)和变形密度(ρ_def)被广泛用于实际研究。众所周知,分析分子的总电子密度无法像以上提及的方法那样展现出与分子电子结构有关的丰富的信息。但是,在本文中,通过数个实例以及通过与ELF、∇²ρ和ρ_def的对比,我们指出若只关注价层电子密度分布,分子电子结构特征也是可能被探究的。我们发现对大多数情况,对非常简单的价层电子密度的分析也可以给出与ELF、∇²ρ和ρ_def分析类似的信息,并且这种分析具有计算复杂度更低的额外优点。我们希望本文的工作可以使得化学家们关注长期被忽视的价层电子密度所具有的重要价值。也值得注意的是,价层电子密度分析并非完全没有缺点,当这种方法无法提供丰富信息的时候,研究者仍需借助于其它类型的分析手段。     

Electron delocalization and bond formation under the ELF framework

A first approach to the relationship between the electron localization function (ELF) and electronic delocalization upon bond formation is provided. We show from first principles the ability of ELF at the bond critical points to act as an index of the electron reorganization involved in chemical bonding. Simultaneously, this index, that we shall call ELF delocalization index (EDI), constitutes a good measure of electron delocalization. We will show how the core of ELF is proportional to the Wiberg index under the valence bond approach. This relationship will be exploited for some representative examples where EDI is able to identify the stages of bond formation. Furthermore, a maximum in EDI along this process has been found to correlate with the molecular equilibrium configuration, allowing for a formulation of a ??maximal localization principle?? for the stable structure of covalent compounds in terms of ELF.     

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     

An analysis of two local measures of the electronic localization: a comparison with the ELF and the exchange-correlation density results

In this work we have explored the performance of two functions, recently proposed by Ayers [J. Chem. Sci., 2005, 117, 441], with the purpose of quantifying local electron localization. The first function, ζ(h), measures the total fluctuation per electron in the number of electrons at a given position r(1), while the second one, ζ(R), is a local representation of the minimum fluctuation criterion for electron localization. The study is carried out through a set of diatomic molecules that covers a wide range of covalent/polar character. Additionally, we have also calculated the electron localization function and the exchange-correlation hole along the internuclear axis. We have found that, for all the studied molecules, the numerical integration involved in computing ζ(h) did not converge. We think that this is so because the hole correlation calculations are not able to yield its correct asymptotic decaying behavior for large absolute values of the internuclear distances. On the other hand, the calculation of ζ(R) has proved to be feasible, and the information obtained from it has been concluded to be compatible to that rendered by the electron localization function (ELF) and the exchange-correlation density. Moreover, it has been also found that the results for ζ(R) allow to quantify the relative degree of electron localization within different molecular regions.     

A partially projected wave function for radicals

A partially projected wave function for odd electron systems with quantum number M=1/2, containing μu spin functions α and μ spin functions α, with fractional spin component αS_z=1/2 and 3/2 are derived from the totally projected wave function. To obtain these wave functions new symmetry relations between Sanibel coefficients for the odd electron case have been found, as well as the relations between primitive spin functions and their spin permutations. The wave function for the doublet state is shown not to contain contamination of the quadruplet state, and the wave function for the quadruplet does not have contamination of the duplet. Both wave functions exhibit equal forms except in the signs of their summation terms. The number of primitive spin functions depends on the number of electrons (n_s), it grows linearly as n_s=(N+3)/2. It can be considered as a generalization of the half projected Hartree–Fock wave function to the odd electron case. The HPHF wave function is defined for even electron systems and consists of only two Slater determinants, it has been shown to introduce some correlation effects and it has been successfully applied to calculate the low-lying excited states of molecules. Therefore, this investigation is the first step to propose a method to calculate the excited states of radicals when other methods are impracticable. This revised version was published online in July 2006 with corrections to the Cover Date.     

Maximum probability domains from Quantum Monte Carlo calculations

Although it would be tempting to associate the Lewis structures to the maxima of the squared wave function |Psi|2, we prefer in this paper the use of domains of the three-dimensional space, which maximize the probability of containing opposite-spin electron pairs. We find for simple systems (CH4, H2O, Ne, N2, C2H2) domains comparable to those obtained with the electron localization function (ELF) or by localizing molecular orbitals. The different domains we define can overlap, and this gives an interesting physical picture of the floppiness of CH5+ and of the symmetric hydrogen bond in FHF-. The presence of multiple solutions has an analogy with resonant structures, as shown in the trans-bent structure of Si2H2. Correlated wave functions were used (MCSCF or Slater-Jastrow) in the Variational Quantum Monte Carlo framework.     

Understanding reaction mechanisms in organic chemistry from catastrophe theory applied to the electron localization function topology

Thom's catastrophe theory applied to the evolution of the topology of the electron localization function (ELF) gradient field constitutes a way to rationalize the reorganization of electron pairing and a powerful tool for the unambiguous determination of the molecular mechanisms of a given chemical reaction. The identification of the turning points connecting the ELF structural stability domains along the reaction pathway allows a rigorous characterization of the sequence of electron pair rearrangements taking place during a chemical transformation, such as multiple bond forming/breaking processes, ring closure processes, creation/annihilation of lone pairs, transformations of C-C multiple bonds into single ones. The reaction mechanism of some relevant organic reactions: Diels-Alder, 1,3-dipolar cycloaddition and Cope rearrangement are reviewed to illustrate the potential of the present approach.     

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.     

A survey of wave function effects on theoretical calculation of gas phase F NMR chemical shifts using factorial design

The wave functions for calculating gas phase ¹⁹F chemical shifts were optimally selected using the factorial design as a multivariate technique. The effects of electron correlation, triple-ξ valance shell, diffuse function, and polarization function on calculated ¹⁹F chemical shifts were discussed. It is shown that of the four factors, electron correlation and the polarization functions affect the results significantly. B3LYP/6-31 + G(df,p) wave functions have been proposed as the best and the most efficient level of theory for calculating ¹⁹F chemical shifts. An additional series of fluoro compounds were used as a test set and their predicted ¹⁹F chemical shifts values confirmed the validity of the approaches.     

Sigma-pi separation of the electron localization function and aromaticity

The electron localization function (ELF) has been separated in its sigma and pi components. The topological analysis of the new ELFsigma and ELFpi functions has been used to quantify the concept of resonance. The highest bifurcation values of these functions describe in a correct way the aromaticity of classical ring molecules and some new aromatic compounds as B6CO6, Al4(2-), and N5-. In the case of Al4(2-), an important sigma delocalization contribution has been found, which is in agreement with previous interpretation.     

Maximal probability domains in linear molecules

Regions of space are defined to maximize the probability to find a given number of electrons within. Their chemical significance and their relationship to the electron localization function (ELF) are explored by analyzing the results for a few linear molecules: LiH, BH, N2, CO, CS, C2H2, and C4H2.     

Comparison of the electron localization function and deformation electron density maps for selected earth materials

The electron localization function (ELF) and experimental and theoretical deformation electron density maps are compared for several earth materials and one representative molecule. The number and arrangement of the localized one-electron probability density domains generated in a mapping of the ELF correspond to the number and arrangement of the localized electron density domains generated in a mapping of the deformation electron density distribution, a correspondence that suggests that the two fields are homeomorphically related. As a homeomorphic relationship has been established previously between the Laplacian of the electron density distribution and the ELF, the relationship suggests that the deformation electron density distribution is also homeomorphically related to the Laplacian of the distribution.     

Electron pairing and chemical bonds: pair localization in ELF domains from the analysis of domain averaged Fermi holes

This article reports the application of a recently proposed formalism of domain averaged Fermi holes to the problem of the localization of electron pairs in electron localization function (ELF) domains and its possible implications for the electron pair model of chemical bond. The main focus was on the systems, such as H2O or N2, in which the "unphysical" population of ELF domains makes the parallel between these domains and chemical bond questionable. On the basis of the results of the Fermi-hole analysis, we propose that the above problems could be due to the fact that in some cases the boundaries of the ELF domains need not be determined precisely enough.     

Theory of electron transfer in systems with intermediate link

A theory is proposed for electron transfer through an intermediate link, the theory being based on solution of the time-dependent wave equation of the system with the exact Hamiltonian by assigning a wave function in form of a linear combination of wave functions of the initial state (electron on the donor), intermediate state (electron on the intermediate link), and final state (electron on the acceptor). The squares of the moduli of the time-dependent coefficients in these wave functions represent the probabilities of finding electrons in the indicated states. The coefficients have been determined by means of Laplace transforms, and an expression has been obtained for the rate of electron transfer through the intermediate link.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 3, pp. 288–293, May–June, 1985.     

Electronic structure of germanium monohydrides Ge(n)H, n = 1-3

Quantum chemical calculations were applied to investigate the electronic structure of germanium hydrides, Ge(n)H (n = 1, 2, 3), their cations, and anions. Computations using a multiconfigurational quasi-degenerate perturbation approach (MCQDPT2) based on complete active space wave functions (CASSCF), multireference perturbation theory (MRMP2), and density functional theory reveal that Ge(2)H has a (2)B(1) ground state with a doublet-quartet gap of approximately 39 kcal/mol. A quasidegenerate (2)A(1) state has been derived to be 2 kcal/mol above the ground state (MCQDPT2/aug-cc-pVTZ). In the case of the cation Ge(3)H(+) and anion Ge(3)H(-), singlet low-lying electronic states are derived, that is, (1)A' and (1)A(1), respectively. The singlet-triplet energy gap is estimated to 6 kcal/mol for the cation. An "Atoms in Molecules" (AIM) analysis shows a certain positive charge on the Ge(n) (n = 1, 2, 3) unit in its hydrides, in accordance with the NBO analysis. The topologies of the electron density of the germanium hydrides are different from that of the lithium-doped counterparts. On the basis of our electron localization function (ELF) analysis, the Ge-H bond in Ge(2)H is characterized as a three-center-two-electron bond. Some key thermochemical parameters of Ge(n)H have also been derived.     

Nature of bonding in the cyclization reactions of (2-Ethynylphenyl)triazene and 2-Ethynylstyrene

The topological analysis of the electron localization function (ELF) has been applied to explore the nature of bonding in thermal cyclizations of (2-ethynylphenyl)triazene and 2-ethynylstyrene. These processes have been proposed to occur through both five- (i.e., coarctate) and six-membered (i.e., pericyclic) transition states. The analysis of electron delocalization, as measured from an irreducible ELF f-localization domain reduction diagram, allows us to characterize these cyclizations of 2-ethynylstyrene in terms of a more pronounced pericyclic or coarctate character than those associated with (2-ethynylphenyl)triazene. The latter evolve through pseudopericyclic and pseudocoarctate pathways. It is found that ELF results are also in good agreement with recent magnetic evidence data obtained from the anisotropy of induced current density (ACID) calculations.     

Electron group functions for the analysis of the electronic structures of molecules

The electronic structure of a vast majority of molecular systems can be understood in terms of electron groups and their wave functions. They serve as a natural basis for bringing intuitive chemical and physical concepts into quantum chemical calculations. This article considers the general electron group functions formalism as well as its simple geminal version. We try to characterize the wave function with the group structure and its capabilities in actual calculations. For this purpose we implement a variational method based on the wave function in the form of an antisymmetrized product of strongly orthogonal group functions and perform a series of electronic structure calculations for small molecules and model systems. The most important point studied is the relation between the choice of electron groups and the results obtained. We consider energetic characteristics as well as optimal geometry parameters. In view of practical importance, the structure of variationally optimized local one-electron states is considered in detail as well as intuitive characteristics of chemical bonds.