Hardness Kernel Research Articles

Variational Principle for Eigenmodes of Reactivity in Conceptual Density Functional Theory.

In conceptual density functional theory, reactivity indexes as the Fukui function, the global hardness/softness, and hardness/softness kernels are fundamental linear responses extensively studied to predict the nucleophilic and electrophilic propensities of atoms in molecules. We demonstrate that the hardness/softness kernels of an isolated system can be expanded in eigenmodes, solutions of a variational principle. These modes are divided into two groups: the polarization modes and the charging modes. The eigenvectors of the polarization modes are orthogonal to the Fukui function and can be interpreted as densities induced at a constant chemical potential. The charging modes of an isolated system are associated with virtual charge transfers weighted by the Fukui function and obey an exact nontrivial sum rule. The exact relation between these charging eigenmodes and those of the polarizability kernel is established. The physical interpretation of the modes is discussed. Applications of the present findings to the Thomas–Fermi and von Weizacker kinetic energy functionals are presented. For a confined free quantum gas, described by the von Weizacker kinetic energy functional, we succeed to derive an approximate analytical solution for the Fukui function and for hardness/softness and polarizability kernels. Finally, we indicate how numerical calculations of the hardness kernel of a molecule could be performed from the Kohn–Sham orbitals.

Differences in seed properties and germination between native and introduced populations of Triadica sebifera

Abstract Aims Seeds of many invasive plants germinate more quickly than those of native species, likely facilitating invasion success. Assessing the germination traits and seed properties simultaneously for introduced and native populations of an invasive plant is essential to understanding biological invasions. Here, we used Triadica sebifera as a study organism to examine intraspecific differences in seed germination together with seed characteristics. Methods We measured physical (volume, mass, coat hardness and coat thickness of seeds) and chemical (crude fat, soluble protein, sugar, gibberellins [GA] and abscisic acid [ABA] of kernels) properties of T. sebifera seeds collected in 2017 from 12 introduced (United States) populations and 12 native (China) populations and tested their germination rates and timing in a greenhouse experiment in China. Furthermore, we conducted an extra experiment in the United States using seeds collected in 2016 and 2017 to compare the effects of study sites (China vs. United States) and seed collection time (2016 vs. 2017) on seed germination. Important Findings Seeds from the introduced range germinated faster than those from the native range. Physical and chemical measurements showed that seeds from the introduced range were larger, had higher GA concentrations and GA:ABA ratio, but lower crude fat concentrations compared to those from the native range. There were no significant differences in seed mass, coat hardness and coat thickness or kernel ABA, soluble protein or sugar concentrations between seeds from introduced vs. native ranges. Germination rates were correlated between United States and China greenhouses but germination rates for populations varied between collection years. Our results suggest that larger seeds and higher GA likely contribute to faster germination, potentially facilitating T. sebifera invasion in the introduced range.

Reply to the 'Comment on "Revisiting the definition of local hardness and hardness kernel"' by C. Morell, F. Guégan, W. Lamine, and H. Chermette, Phys. Chem. Chem. Phys., 2018, 20, DOI.

This reply complements the comment of Guégan et al. about our recent work on the revision of the local hardness and the hardness kernel concepts. Guegan et al. analyze our work using a Taylor series expansion of the energy as a functional of the electron density, to show that our procedure opens a new way to define local descriptors. In this contribution we show that the strategy we followed for the local hardness and the hardness kernel is even more general, and that it can be used to derive from a global response function its corresponding local and non-local counterparts by: (1) requiring that the integral over one of the two variables that characterizes the non-local function leads to the local function, and that the integral over the local function leads to the global response index, and (2) assuming that the global and local functions are related through the electronic density, by making use of the chain rule for functional derivatives.

Maize Kernel Hardness, Endosperm Zein Profiles, and Ethanol Production

Maize (Zea mays L.) grain is an important feedstock for the ethanol-producing industry. However, little is known about the optimum grain quality for optimizing ethanol yielding efficiencies. We specifically investigated the response of ethanol yields (L Mg−1) to kernel hardness, and its physiological determinant endosperm zein protein profiles, as affected by genotype selection, field nitrogen (N) fertilization, and crop growth environment. We measured ethanol yield and related this to different kernel hardness indicators, kernel composition, and zein profiles. We also described changes in field ethanol yield (L ha−1), by taking into account the crop yield (Mg ha−1). Hard endosperm genotypes always yielded less ethanol than softer endosperm ones per grain mass (L Mg−1). Higher N fertilization rates increased kernel hardness and decreased ethanol yield (L Mg−1) on soft endosperm dented genotypes but had no effect on hard endosperm ones. Ethanol yield was negatively correlated with kernel density, kernel protein concentration, and Z1 and Z2 zein fractions. Within Z2, 15 kDa β-zein explained the largest ethanol yield variation generated by genotypes, N fertilizations, and growth environments. However, and although these differences were as large as 10%, ethanol field yield (L ha−1) was mainly driven by crop yields (r 2 0.98) due to the large crop yield (Mg ha−1) differences observed across treatments. Together, our results helped describe the magnitude that changes in maize kernel hardness can have over ethanol yield, both through genotype selection or crop management. A particular Z2 zein protein rises as relevant for future genetic manipulations of maize ethanol yield determination.

Condensed-to-atoms hardness kernel from the response of molecular fragment approach

Condensed reactivity descriptors obtained from the response of molecular fragment (RMF) approach are analyzed within the variational formulation of conceptual density functional theory. It is shown that this approach can serve as the basis of a coherent formulation of the hardness kernel.

Modeling the electron density kernels

Existing approximation to the softness kernel, successfully explored in earlier work, has been extended; the normal Gauss distribution function has been used instead of the Dirac delta. The softness kernel becomes continuous functions in space and may be used to calculate the linear response function of the electron density. Three-dimensional visualization of the softness kernel and the linear response function are presented for a nitrogen atom as a working example. By using a single parameter of the spatial Gauss distribution, the novel softness kernel has been adjusted to be consistent with the standard form of the hardness kernel, representing the leading fraction of the electronic interactions in the system.

The unconstrained local hardness: an intriguing quantity, beset by problems

Developing a mathematical approach to the local hard/soft acid/base principle requires an unambiguous definition for the local hardness. One such quantity, which has aroused significant interest in recent years, is the unconstrained local hardness. Key identities are derived for the unconstrained local hardness, δμ/δρ(r). Several identities are presented which allow one to determine the unconstrained local hardness either explicitly using the hardness kernel and the inverse-linear response function, or implicitly by solving a system of linear equations. One result of this analysis is that the problem of determining the unconstrained local hardness is infinitely ill-conditioned because arbitrarily small changes in electron density can cause enormous changes in the chemical potential. This is manifest in the exponential divergence of the unconstrained local hardness as one moves away from the system. This suggests that one should be very careful when using the unconstrained local hardness for chemical interpretation.

Fukui functions for atoms and ions: Polarizability justified approach

AbstractRecently proposed Fukui function has been tested for a collection of atoms and atomic ions. The key point of the analysis is an approximation for the softness kernel, as an extended and parameterized version of the local approximation s(r,r′) = s(r)δ(r − r′). The new Fukui functions lead to exact reproduction of global electronic dipole polarizability and atomic softness. The global hardness calculated with the classic definition using new Fukui functions and the hardness kernel |r − r′|−1 matches the η = I − A value with linear correlation superior to any other published hitherto. Applied approximations for the softness and hardness kernels were proved to be in perfect coherence. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Potentialphilicity and potentialphobicity: Reactivity indicators for external potential changes from density functional reactivity theory

In analogy to the electrophilicity, we define potentialphilicity indicators that represent energetically favorable ways to change the external potential of a molecule at fixed electron number. Similarly, we define a potentialphobicity to represent the least favorable way to change the external potential of a molecule. The resulting indicators should be useful for describing how molecular geometries change and predicting favorable and unfavorable ways for a reagent to approach a molecule. The linear response function enters plays a very important role in this approach, analogous to the role of the hardness for the electrophilicity or the hardness kernel for the Fukui function. The mathematical properties of the response function and its implications for these reactivity indicators are discussed in depth.

Automated Single‐Kernel Sorting to Select for Quality Traits in Wheat Breeding Lines

ABSTRACTAn automated single kernel near‐infrared system was used to select kernels to enhance the end‐use quality of hard red wheat breeder samples. Twenty breeding populations and advanced lines were sorted for hardness index, protein content, and kernel color. To determine whether the phenotypic sorting was based upon genetic or environmental differences, the progeny of the unsorted control and sorted samples were planted at two locations two years later to determine whether differences in the sorted samples were transmitted to the progeny (e.g., based on genetic differences). The average hardness index of the harvested wheat samples for segregating populations improved significantly by seven hardness units. For the advanced lines, hardness index was not affected by sorting, indicating little genetic variation within these lines. When sorting by protein content, a significant increase from 12.1 to 12.6% was observed at one location. Purity of the red samples was improved from ≈78% (unsorted control) to ≈92% (sorted samples), while the purity of the white samples improved from 22% (control) to ≈62% (sorted samples). Similar positive results were found for sorting red and blue kernel samples. Sorting for kernel hardness, color, and protein content is effective and based upon genetic variation.

Chargephilicity and chargephobicity: Two new reactivity indicators for external potential changes from density functional reactivity theory

In analogy to the electrophilicity and the potentialphilicity, we define the chargephilicity and chargephobicity to describe the energetic response of a molecule to a change in its charge distribution. The chargephilicity captures the ‘best’ way to change the underlying charge distribution; the chargephobicity captures the ‘worst’ way to change the underlying charge distribution. The Coulomb potential induced by the linear-response function plays a very important role in this approach, analogous to the role of the hardness for the electrophilicity, the hardness kernel for the Fukui function, and the linear-response function for the potentialphilicity and potentialphobicity.

The hardness kernel as the basis for global and local reactivity indices

In a very recent article (Torrent-Sucarrat et al., J Comput Chem 2007, 28, 574), we have shown how to evaluate the global hardness for polyatomic molecules using a hardness kernel approximation. We present here an extension of this previous work by improving the model used to evaluate the hardness kernel and the Fukui function. In addition, the concept of the local hardness is analyzed in detail, and for the first time, profiles of local hardness with kinetic and exchange-correlation contributions for polyatomic molecules are reported. Finally, the concept of condensed atomic hardness is introduced and its usefulness as chemical reactivity descriptor is examined.

Local hardness: a critical account

This paper gives a critical account on the present status of the local hardness. We analyze the behavior of hardness related global, local, and nonlocal properties, paying particular attention to local hardness. Although this reactivity index has become very useful in predicting the regioselectivity of chemical reactions, the lack of a rigorous definition in the literature has shown that different approximations can yield different and sometimes contradictory conclusions. The present paper tries to provide insights into this controversial issue, starting from an approximate model of the hardness kernel and analyzing the features of the different models to evaluate the local and global hardnesses.

A Device for the Preparation of Cereal Endosperm Bricks

sp. L.) endosperm hardness (kernel texture) has been a topic of intense interest for the past ≈100 years (Pomeranz and Williams 1990; Morris 2002). One of the first recorded devices for measuring wheat kernel texture was that of Roberts (1910), which determined the force required to crush individual kernels. There are two reasons why the assessment of kernel texture in wheat, in particular, is of great interest: 1) wheat exhibits distinct classes of kernel texture, namely soft, hard, and durum; and 2) the differences in kernel texture among these classes has a profound effect on flour milling, starch damage, particle size distribution, water absorption, and end-use quality (Morris and Rose 1996; Morris 2002). As we will describe, our objective was to produce a geometrically defined subsample of the endosperm as opposed to studying the whole kernel. The measurement of wheat kernel texture has been largely empirical because kernels and endosperm are difficult to work with due largely to their diminutive size. Current methods (AACC International 2000) such as near-infrared reflectance (Approved Method 39-70A), particle size index (Approved Method 55-30), and single kernel characterization System (Approved Method 55-31) provide measurements in arbitrary unitless proportions or scales. Generally, to obtain objective measures of material proper-ties in universal units of force, work, etc., complex geometries must be simplified, and in the case of wheat, the bran, germ, and pigment strand should also be eliminated. A few researchers have been successful in achieving this goal. Glenn of the USDA devel-oped a method of “turning” on a lathe endosperm cylinders of defined geometry, and then subjecting them to testing in both com-pression and tension (Glenn et al 1991; Jolly 1991; Jolly et al 1996; Delwiche 2000; Osborne et al 2001; Dobraszczyk et al 2002). This same technique was studied in the senior author’s lab and found to be exceedingly tedious. The centering of the kernel “cheek” on the lathe stub was particularly problematic. Haddad, Abecassis and co-workers (Haddad et al 1998, 1999, 2001; Sam-son et al 2005) eliminated the problems associated with “turning” cylinders, by preparing through sanding rectangular parallelepi-pedal test samples. Here we term such specimens “bricks”. We have devised and constructed a similar device that eliminates the need for adhesive paper and glue to hold specimens during their preparation (see Haddad et al 1998). Furthermore, one of the pro-cessing steps in the technique described by Haddad et al (1998) as “Two machined half-grains are placed between two sheets, and a wedge is set at height l of the extremities of the sheets” has been eliminated. We have used the device described here to prepare hundreds of bricks of various wheat (

Analogies and differences between two ways to evaluate the global hardness

The weight of the energetic components (electronic kinetic, electron-nucleus and electron-electron Coulombic, and correlation energies) of the ionization potential, electron affinity, chemical potential, and global hardness is evaluated and contrasted with the energetic components of the hardness kernel and the experimental values of these properties for 40 systems. The contrast of the hardness terms obtained from finite difference and hardness kernel gives some insight on the possible implications to differentiate the electronic energy with respect to the number electrons or the electron density.

On the quality of the hardness kernel and the Fukui function to evaluate the global hardness

An approximated hardness kernel, which includes the second derivative with respect to the density of the kinetic energy, the electron-electron coulomb repulsion, and the exchange density functionals, has been tested for the calculation of the global hardness. The results obtained for a series of 40 cations and neutral systems and 16 anions represent in most cases an improvement of the results obtained using the HOMO-LUMO gap approach and indicate the viability of this approach to evaluate global hardness. In addition, the relevance of the Fukui function approximation and the role of the three components of the hardness kernel in the evaluation of the global hardness have been analyzed.

An assessment of a simple hardness kernel approximation for the calculation of the global hardness in a series of Lewis acids and bases

A simplified model of the hardness kernel, η ( r → , r → ′ ) ≅ δ ( r → − r → ′ ) , has been evaluated for the calculation of the global hardness in a series of Lewis acids and bases using the MP2 method. It is found that this simple model yields the appropriate ordering of the molecules according to their hardness values when compared to experimental available results.

Puroindoline Genotype of the U.S. National Institute of Standards & Technology Reference Material 8441, Wheat Hardness

ABSTRACTGrain hardness (kernel texture) is of central importance in the quality and utilization of wheat (Triticum aestivum L.) grain. Two major classes, soft and hard, are delineated in commerce and in the Official U.S. Standards for Grain. However, measures of grain hardness are empirical and require reference materials for instrument standardization. For AACC Approved Methods employing near‐infrared reflectance (NIR) and the Single Kernel Characterization System (39‐70A and 55‐31, respectively), such reference materials were prepared by the U.S. Dept. of Agriculture Federal Grain Inspection Service. The material was comprised of genetically pure commercial grain lots of five soft and five hard wheat cultivars and was made available through the National Institute of Standards and Technology (SRM 8441, Wheat Hardness). However, since their establishment, the molecular‐genetic basis of wheat grain hardness has been shown to result from puroindoline a and b. Consequently, we sought to define the puroindoline genotype of these 10 wheat cultivars and more fully characterize their kernel texture through Particle Size Index (PSI, Method 55‐30) and Quadrumat flour milling. NIR, SKCS, and Quadrumat break flour yield grouped the hard and soft cultivars into discrete texture classes; PSI did not separate completely the two classes. Although all four of these methods of texture measurement were highly intercorrelated, each was variably influenced by some minor, secondary factors. Among the hard wheats, the two hard red spring wheat cultivars that possess the Pina‐D1b (a‐null) hardness allele were harder than the hard red winter wheat cultivars that possess the Pinb‐D1b allele based on NIR, PSI, and break flour yield. Among the soft wheat samples, SKCS grouped the Eastern soft red winter cultivars separate from the Western soft white. A more complete understanding of texture‐related properties of these and future wheat samples is vital to the use and calibration of kernel texture‐measuring instruments.

Negative and Infinite Fukui Functions: The Role of Diagonal Dominance in the Hardness Matrix

The possible genesis of negative atom condensed Fukui functions is discussed based on hardness kernel matrix relationships. The recent hypothesis that diagonal dominance of the hardness matrix is a requirement for positive Fukui functions is proven, and general considerations also predict the possibility of regions with numerically unstable Fukui functions, including discontinuities.

Variational principles for describing chemical reactions: Condensed reactivity indices

Two recent papers [P. W. Ayers and R. G. Parr, J. Am. Chem. Soc. 122, 2010 (2000); 123, 2007 (2001)] have shown how variational principles for the energy may be used to derive and elucidate the significance of the chemical reactivity indices of density-functional theory. Here, similar ideas are applied, yielding a systematic, mathematically rigorous, and physically sound approach to condensed reactivity indices. First, we use the variational principle for the energy to derive an expression for the condensed Fukui function index in terms of the condensed hardness kernel. Next, we address an important open problem pertaining to condensed reactivity indices: when (if ever) is the condensed Fukui function for an atom in a molecule negative? In particular, our analysis confirms the observation, hitherto based only on computational evidence, that the Hirshfeld partitioning is optimal for obtaining non-negative Fukui functions. We also hypothesize that the strong diagonal dominance of the condensed hardness kernel is sufficient for the non-negativity of the Fukui function. Errors in the partitioning of molecules into atoms and inadequate treatment of correlation are pinpointed as the most likely causes of negative condensed Fukui functions. We conclude by noting that the condensed Fukui functions are, in some respects, more appropriate indicators of a molecular site’s reactivity than the Fukui function itself.