Volume 117, Issue 5
September 2005, pages 367-615
pp 367-367 September 2005
pp 369-377 September 2005
Chemical hardness and density functional theory
The concept of chemical hardness is reviewed from a personal point of view.
pp 379-386 September 2005
DFT reactivity indices in confined many-electron atoms
Jorge Garza Rubicelia Vargas Norberto Aquino K D Sen
The density functional descriptors of chemical reactivity given by electronegativity, global hardness and softness are reported for a representative set of spherically confined atoms of IA, IIA, VA and VIIIA series in the periodic table. The atomic electrons are confined within the impenetrable spherical cavity defined by a given radius of confinement satisfying the Dirichlet boundary condition such that the electron density vanishes at the radius of confinement. With this boundary condition the non-relativistic spin-polarized Kohn-Sham equations were solved. The electronegativity in a confined atom is found to decrease as the radius of confinement is reduced suggesting that relative to the free state the atom loses its capacity to attract electrons under confined conditions. While the global hardness of a confined atom increases as the radius of confinement decreases, due to the accompanying orbital energy level crossing, it does not increase infinitely. At a certain confinement radius, the atomic global hardness is even reduced due to such crossover. General trends of the atomic softness parameter under spherically confined conditions are reported and discussed.
pp 387-400 September 2005
The electron-propagator approach to conceptual density-functional theory
Junia Melin Paul W Ayers J V Ortiz
Both electron propagator theory and density-functional theory provide conceptually useful information about chemical reactivity and, most especially, charge transfer. This paper elucidates the qualitative and quantitative links between the two theories, with emphasis on how the reactivity indicators of conceptual density-functional theory can be derived from electron propagator theory. Electron propagator theory could be used to compute reactivity indices with high accuracy at reasonable computational cost.
pp 401-409 September 2005
The electron density changes in molecular systems in the presence of external electric fields are modeled for simplicity in terms of the induced charges and dipole moments at the individual atomic sites. A chemical potential equalisation scheme is proposed for the calculation of these quantities and hence the dipole polarizability within the framework of density functional theory based linear response theory. The resulting polarizability is expressed in terms of the contributions from individual atoms in the molecule. A few illustrative numerical calculations are shown to predict the molecular polarizabilities in good agreement with available results. The usefulness of the approach to the calculation of intermolecular interaction needed for computer simulation is highlighted.
pp 411-418 September 2005
Relation between the Fukui function and the Coulomb hole
By using a coarse-grain representation of the molecular electronic density, we demonstrate that the value of the condensed Fukui function at an atomic site is directly related to the polarization charge (Coulomb hole) induced by a test electron removed (or added) from (at) the atom. The link between the formation of an electron-hole pair and the condensed Fukui function provides insights on the possible negativity of the Fukui function which is interpreted in terms of two phenomena: overscreening and over-strengthening.
pp 419-424 September 2005
E Chamorro M Duque C Cárdenas J Santos W Tiznado P Fuentealba
Use of regions of space defined by topological analysis of electron localization function (ELF) as reactivity descriptors is explored. By starting from the fact that the ELF presents high values in the regions where it is most probable to find an electron pair and that the square of the HOMO is a good measure of the reactivity of a molecule, it is proposed that the integration of the square of the HOMO over the volumes defined by the ELF should be a local index for predicting the most nucleophilic site of a molecule. We present here some computational results on simple systems in order to get some insights about this possibility.
pp 425-435 September 2005
Molecular quantum similarity using conceptual DFT descriptors
Patrick Bultinck Ramon Carbó-Dorca
This paper reports a Molecular Quantum Similarity study for a set of congeneric steroid molecules, using as basic similarity descriptors electron density ρ(r), shape function σ(r), the Fukui functions f^{+}(r) and f^{−}(r) and local softness s^{+}(r) and s^{−}(r). Correlations are investigated between similarity indices for each couple of descriptors used and compared to assess whether these different descriptors sample different information and to investigate what information is revealed by each descriptor.
pp 437-440 September 2005
Hardness and excitation energy
The concept of the ensemble Kohn-Sham hardness is introduced. It is shown that the first excitation energy can be given by the Kohn-Sham hardness (i.e. the energy difference of the ground-state lowest unoccupied and highest occupied levels) plus an extra term coming from the partial derivative of the ensemble exchange-correlation energy with respect to the weighting factorw in the limitw → 0. It is proposed that the first excitation energy can be used as a reactivity index instead of the hardness.
pp 441-454 September 2005
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.
pp 455-466 September 2005
Geometric interpretation of density displacements and charge sensitivities
The “geometric” interpretation of the electronic density displacements in the Hilbert space is given and the associated projection-operator partitioning of the hardness and softness operators (kernels) is developed. The eigenvectors |Æ〉= |α〉 of the hardness operator define the complete (identity) projector P =Σ_{α}| α〉〈α | = 1 for general density displacements, including thecharge-transfer (CT) component, while the eigenvectors |i〉= |i〉 of the linear response operator determine thepolarizational P-projector, P_{p} = Σ_{i} |i〉〈i| . Their difference thus defines the complementary CT-projector: P_{CT} = 1-P_{P}. The complete vector space for density displacements can be also spanned by supplementing the P-modes with the homogeneous CT-mode. These subspaces separate the integral (normalization) and local aspects of density shifts in molecular systems.
pp 467-472 September 2005
The reaction force: Three key points along an intrinsic reaction coordinate
Peter Politzer Alejandro Toro-Labbé Soledad Gutiérrez-Oliva Bárbara Herrera Pablo Jaque Monica C Concha Jane S Murray
The concept of the reaction force is presented and discussed in detail. For typical processes with energy barriers, it has a universal form which defines three key points along an intrinsic reaction coordinate: the force minimum, zero and maximum. We suggest that the resulting four zones be interpreted as involving preparation of reactants in the first, transition to products in the second and third, and relaxation in the fourth. This general picture is supported by the distinctive patterns of the variations in relevant electronic properties. Two important points that are brought out by the reaction force are that (a) the traditional activation energy comprises two separate contributions, and (b) the transition state corresponds to a balance between the driving and the retarding forces.
pp 473-475 September 2005
On the significance of ELF basins
For perfectly localized orbitals, the basins of ELF are the domains in which the probability of finding a pair of electrons is maximal.
pp 477-483 September 2005
Dynamic behaviors of chemical concepts in density functional theory such as frontier orbitals (HOMO/LUMO), chemical potential, hardness, and electrophilicity index have been investigated in this work in the context of Bohn-Oppenheimer quantum molecular dynamics in association with molecular conformation changes. Exemplary molecular systems like CH_{5}^{+}, Cl^{−} (H_{2}O)_{30} and Ca^{2+} (H_{2}O)1_{5} are studied at 300 K in the gas phase, demonstrating that HOMO is more dynamic than LUMO, chemical potential and hardness often fluctuate concurrently. It is argued that DFT concepts and indices may serve as a good framework to understand molecular conformation changes as well as other dynamic phenomena.
pp 485-490 September 2005
Third-order energy derivative corrections to the Kohn-Sham orbital hardness tensor
The third term in the Taylor expansion of the total energy functional around the number of electronsN is evaluated as the second-order derivative of orbital Kohn-Sham energies with respect to orbital occupancy. Present approach is an extension of an efficient algorithm to compute densityfunctional based orbital reactivity indices. Various energy derivatives used to approximate orbital reactivity indices are defined within the space spanned by the orbital occupation numbers and the Kohn-Sham one-electron energies. The third-order energy functional derivative has to be considered for singular hardness tensor ([η]). On the contrary, this term has negligible influence on the reactivity index values for atomic or molecular systems with positively defined hardness tensors. In this context, stability of a system in equilibrium state estimated through the eigenvalues of [η] is discussed. Numerical illustration of the Kohn-Sham energy functional derivatives in orbital resolution up to the third order is shown for benchmark molecules such as H_{2}O, H_{2}S, and OH^{−}.
pp 491-495 September 2005
Equalization equations in reactant resolution
The chemical system can be analyzed in different resolutions. The assumed resolution imposes a given partitioning of the system in physical or functional space. The most frequently explored are global, reactant, atoms-in-molecule, orbital, and local resolutions. In this paper we have considered reactant resolution, i.e., the mutually polarized reactants before the charge-transfer among them. We have demonstrated that a certain type of generalized sensitivity, the system responses to the population variables, is equalized throughout the space up to the infinite order in the perturbation expansion.
pp 497-505 September 2005
Separability of local reactivity descriptors
The size-dependence of different local reactivity descriptors of dimer A_{2} and AB type of systems is discussed. We derive analytic results of these descriptors calculated using finite difference approximation. In particular, we studied Fukui functions, relative electrophilicity and relative nucleophilicity, local softness and local philicity. The results are explained using the example of the dimer of BH_{3}NH_{3}.
pp 507-514 September 2005
Using the constrained search and Legendre-transform formalisms, one can derive “generalized” density-functional theories, in which the fundamental variable is either the electron pair density or the second-order reduced density matrix. In both approaches, theN-representability problem is solved by the functional, and the variational principle is with respect to all pair densities (density matrices) that are nonnegative and appropriately normalized. The Legendre-transform formulation provides a lower bound on the constrained-search functional. Noting that experience in density-functional and density-matrix theories suggests that it is easier to approximate functionals than it is to approximate the set ofN-representable densities sheds some light on the significance of this work.
pp 515-524 September 2005
G I Cárdenas-Jirón F Espinoza-Leyton T L Sordo
The structure and coordination chemistry of boron porphyrin complexes B_{2}OX_{2} (T_{p}YPP) (X = OH, F; Y = Cl, CH_{3}) in connection with its chemical reactivity are analyzed atab initio density functional theory B3LYP/6-31G* and restricted Hartree-Fock RHF/6-31G* levels of theory. Global reactivity and local selectivity descriptors are used as adequate tools to analyze the isomerism effect (Z orE isomer) and the substitution effect (X: in axial ligand; or Y: in porphyrin ligand). In all the cases, we find that theE conformation is the most stable one, in agreement with X-ray results, and that a principle of maximum hardness in the isomerism analysis is fullfilled. In the substitution analysis, we find that the three global reactivity indexes (η, Ω, ϰ) and the two local reactivity indexes (f_{k}, electrostatic potential) used in this paper predict the same trend when an electron-withdrawing substituent is replaced by an electron donor. Finally, we show that substitution in the porphyrin ligand is slightly more significant than that in the axial ligand.
pp 525-531 September 2005
Francisco Méndez María De L Romero José L Gazquez
The silicon atom may increase its coordination number to values greater than four, to form pentacoordinated compounds. It has been observed experimentally that, in general, pentacoordinated compounds show greater reactivity than tetracoordinated compounds. In this work, density functional theory is used to calculate the global softness and the condensed softness of the silicon atom for SiH_{n}F_{4−n} and SiH_{n}F_{5−n}^{1−}. The values obtained show that the global and condensed softness are greater in the pentacoordinated compounds than in the tetracoordinated compounds, a result that explains the enhanced reactivity. If the results are analysed through a local version of the hard and soft acids and bases principle, it is possible to suggest that in nucleophilic substitution reactions, soft nucleophiles preferably react with SiH_{n}F_{5−n}^{1−}, and hard nucleophiles with SiH_{n}F_{4−n}.
pp 533-539 September 2005
Clays are layered alumino-silicates. Clays swell and expand in aqueous solution. This property governs the usage of these materials in synthesis of nano-composites and is a source of many of its catalytic applications. We used both localized and periodic calculations within the realm of density functional theory (DFT) on a series of monovalent (Li^{+}, Na^{+}, K^{+}, Rb^{+}, Cs^{+}), and divalent (Mg^{2+}, Ca^{2+}, Sr^{2+}, Ba^{2+}) cations, to monitor their effect on the swelling of clays. The activity order obtained for the exchangeable cations among all the monovalent and divalent series studied is: Mg^{2+} > Ca^{2+} > Sr^{2+} > Ba^{2+} > Cs^{+} > Rb^{+} > Na^{+} > Li^{+} > K^{+}. We have studied two types of clays, montmorillonite and beidellite, with different surface structures and with/without water using periodic calculations. We have calculated the layer spacing at the first, second and third hydration shells of exchangeable cation, to compare with the experimentald-spacing values to correlate with humidity. A novel quantitative scale is proposed in terms of the intermolecular relative nucleophilicity of the active cation sites in their hydrated state through Fukui functions using hard-soft acid base (HSAB) principle. Finally, a swelling mechanism is proposed. This is a unique study where a combination of periodic and localized calculations has been performed to validate the capability of reactivity index calculations in material designing.
pp 541-548 September 2005
A philicity based analysis of adsorption of small molecules in zeolites
Angeles Cuán Marcelo Galván Pratim Kumar Chattaraj
Adsorption of small molecules like CH_{4}, CO and NH_{3} into the acid sites of zeolites is analysed as an interaction between an electrophile and a nucleophile. Global reactivity descriptors like softness and electrophilicity, and local reactivity descriptors like the Fukui function, local softness and local philicity are calculated within density functional as well as Hartree-Fock frameworks using both Mulliken and Hirshfeld population analysis schemes. The HSAB principle and the best electrophile-nucleophile combination suggest that the reaction between the NH_{3} and Brönsted acid site of the zeolite is the strongest. Interaction between the zeolite and a small probe molecule takes place through the most electrophilic atom of one with the most nucleophilic atom of the other. This result is in conformity with those provided by the frontier orbital theory and the local HSAB principle.
pp 549-554 September 2005
Basis set effects on the energy and hardness profiles of the hydrogen fluoride dimer
Miquel Torrent-Sucarrat Miquel Duran Josep M Luis Miquel Solà
In earlier work, the present authors have shown that hardness profiles are less dependent on the level of calculation than energy profiles for potential energy surfaces (PESs) having pathological behaviors. At variance with energy profiles, hardness profiles always show the correct number of stationary points. This characteristic has been used to indicate the existence of spurious stationary points on the PESs. In the present work, we apply this methodology to the hydrogen fluoride dimer, a classical difficult case for the density functional theory methods.
pp 555-560 September 2005
Performance of density functional theory methods to describe intramolecular hydrogen shifts
Nelly González-Rivas Andrés Cedillo
The performance of three exchange and correlation density functionals, LDA, BLYP and B3LYP, with four basis sets is tested in three intramolecular hydrogen shift reactions. The best reaction and activation energies come from the hybrid functional B3LYP with triple-ζ basis sets, when they are compared with high-level post-Hartree-Fock results from the literature. For a fixed molecular geometry, the electrophilic Fukui function is computed from a finite difference approximation. Fukui function shows a small dependence with both the exchange and correlation functional and the basis set. Evolution of the Fukui function along the reaction path describes important changes in the basic sites of the corresponding molecules. These results are in agreement with the chemical behavior of those species.
pp 561-571 September 2005
B Pintér F De Proft T Veszprémi P Geerlings
Regioselectivity of the photochemical [2 + 2] cyclo-addition of triplet carbonyl compounds with a series of ground state electron-rich and electron-poor alkenes, the Paterno-Büchi reaction, is studied. Activation barriers for the first step of the triplet reaction are computed in the case of the O-attack. Next, the observed regioselectivity is explained using a series of DFT-based reactivity indices. In the first step, we use the local softness and the local HSAB principle within a softness matching approach, and explain the relative activation barriers of the addition step. In the final step, the regioselectivity is assessed within the framework of spin-polarized conceptual density functional theory, considering response functions of the system’s external potential v, number of electronsN and spin numberN_{s}, being the difference between the number ofα andβ electrons in the spin-polarized system. Although the concept of local spin philicity, introduced recently within this theory, appears less suited to predict the regioselectivity in this reaction, the correct regioselectivity emerges from considering an interaction between the largest values of the generalized Fukui functionsf_{ss} on both interacting molecules.
pp 573-582 September 2005
B3LYP/6-31G(d) calculations were performed to obtain all the transition states and products for the 128 distinct reaction channels of Diels-Alder reactions by taking all possible combinations from a series of dienes (1N-a, 1N-b, 2N, 1P-a, 1P-b, 2P, 1O, 1S) and dienophiles (NE, PE, OE, SE, AE, OHE, MeE, CNE). The predictive ability of the values to gauge the regioselectivity of the putative [4 + 2] cycloaddition reactions is analysed. No correlation is obtained between the reaction energies and activation energies. The extent of asynchronicity is measured based on the bond order analysis. DFT-based descriptors such as the local softness (s_{k}^{+} ands_{k}^{−}), Fukui function indices (f_{k}^{+} andf_{k}^{−}), global electrophilicity index (Ω) and local electrophilicity index (Ω_{k}) were found to be better than the conventional FMO predictions.
pp 583-589 September 2005
Investigation of the role of the C-PCM solvent effect in reactivity indices
It has been shown recently, how the coupling between electronic degrees of freedom and vibrational modes is reflected in the properties of molecules. The necessary derivatives have been analyzed and their thermodynamic relations were demonstrated. This present work is focused on the analysis of a molecular system, under the influence of C-PCM induced solvent effect. The analysis is based on reactivity indices derived from DFT. The shift of frequency for diatomic molecules has been obtained. It has been identified as chemical force effect. The role of nuclear reactivity indices has been emphasized. This concept has been extended to obtain regional chemical potential values within C-PCM model for polyatomic molecules.
pp 591-598 September 2005
Study of atomic and condensed atomic indices for reactive sites of molecules
P Kolandaivel G Praveena P Selvarengan
In this paper, we have introduced the atomic descriptors s(f)_{k} to determine the local reactive sites of the molecular systems during electrophilic, nucleophilic and radical attacks. The condensed Fukui function and the newly introduced condensed atomic descriptor have been calculated for six different systems, namely glycine, alanine, aniline, BH_{2}Cl,trans-FC(O)OF andm-anisidine. The individual atomic charges (gross charge) calculated by the MPA scheme have been used to calculate the condensed Fukui functions (f_{k}) and the newly derived condensed atomic descriptors (sf)_{k}^{α} at B1-DZP level of theory. We carried out the calculation using the “stockholders” charge partitioning technique (i.e., Hirshfeld population scheme). The newly derived quantity gives the same reactive sites as the condensed Fukui functions, and the complexities associated with the negative Fukui functions are removed.
pp 599-612 September 2005
A conceptual DFT approach towards analysing toxicity
U Sarkar D R Roy P K Chattaraj R Parthasarathi J Padmanabhan V Subramanian
The applicability of DFT-based descriptors for the development of toxicological structure-activity relationships is assessed. Emphasis in the present study is on the quality of DFT-based descriptors for the development of toxicological QSARs and, more specifically, on the potential of the electrophilicity concept in predicting toxicity of benzidine derivatives and the series of polyaromatic hydrocarbons (PAH) expressed in terms of their biological activity data (pIC_{50}). First, two benzidine derivatives, which act as electron-donating agents in their interactions with biomolecules are considered. Overall toxicity in general and the most probable site of reactivity in particular are effectively described by the global and local electrophilicity parameters respectively. Interaction of two benzidine derivatives with nucleic acid (NA) bases/selected base pairs is determined using Parr’s charge transfer formula. The experimental biological activity data (pIC_{50}) for the family of PAH, namely polychlorinated dibenzofurans (PCDF), poly-halogenated dibenzo-p-dioxins (PHDD) and polychlorinated biphenyls (PCB) are taken as dependent variables and the HF energy (E), along with DFT-based global and local descriptors, viz., electrophilicity index (Ω) and local electrophilic power (Ω^{+}) respectively are taken as independent variables. Fairly good correlation is obtained showing the significance of the selected descriptors in the QSAR on toxins that act as electron acceptors in the presence of biomolecules. Effects of population analysis schemes in the calculation of Fukui functions as well as that of solvation are probed. Similarly, some electron-donor aliphatic amines are studied in the present work. We see that global and local electrophilicities along with the HF energy are adequate in explaining the toxicity of several substances, both electron donors or acceptors when they interact with biosystems, in gas as well as solution phases.
pp 613-615 September 2005
Some late-term thoughts of a density-functional theorist
Fourteen problems are stated within the density-functional theory of molecular electronic structure. Their alleviation will bring the subject closer to maturity.
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