Charge transfer associated with the physical process of hardness equalization and the chemical event of the molecule formation and the dipole moments

In this article, we have basically launched a search whether the dipole charge and dipole moment of heteronuclear diatomics can be justifiably evaluated in terms of charge transfer kernel using the hardness equalization principle as basis. We have derived a formula for computing dipole charge (q) on the basis of hardness equalization principle as q = aδ + b, where “a” and “b” are the constants and “δ” is the kernel of charge transfer from less hard atom to more hard atom during the rearrangement of charge on molecule formation. We have computed the dipole charges and dipole moments of as many as six different sets of compounds of widely diverse physicochemical behavior in terms of the algorithm derived in the present work. The computed dipole charge nicely reveals the known chemicophysical behavior of such compounds as are brought under the study. A comparative study of the nature of variation of theoretically evaluated and experimentally determined dipole moments reveals that there is an excellent agreement between the two sets of dipole data. Thus, the new algorithm derived for the calculation of the dipole charge using the hardness equalization principle as a basis is efficacious in computing the distribution and rearrangement of charge associated with the chemical event of molecule formation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Frontier orbital and density functional study of the variation of the hard–soft behavior of monoborane (BH3) and boron trifluoride (BF3) as a function of angles of reorganization from planar (D3h) to pyramidal (C3v) shape

In chemical response the BH₃ and BF₃ molecules undergo the physical process of planar (D_3h) to pyramidal (C_3v) reorganization in shape as the condition precedent to the event of chemical reaction under the requirement of symmetry. A frontier orbital and density functional study of the variation of the stability of electronic structures and chemical reactivity of associated with the physical process of D_3h to C_3v geometry reorganization has been performed. The theoretical parameters viz. eigenvalues of HOMO and LUMO, the HOMO and LUMO energy gap, the global hardness and global softness, the chemical potential, the condensed Fukui function, and local softness of B atom, the reaction site, have been computed over a wide range of ∠XBX angles. The nature of variation in the intrinsic chemical reactivity, global and local, of the molecules associated with their geometry reorganization during the chemical event of charge transfer interaction involving their frontier molecular orbitals has been quantitatively explored. The hardness profiles as a function of reaction coordinates are consistent with the principle of maximum hardness (PMH). Results demonstrate that the hardness and softness are not a static and invariable property of molecules but a dynamic and variable function of molecular structure. The hardness parameters and the HOMO–LUMO gap of the molecules are so modified with the distortion of molecular geometry that, after a certain stage of molecular deformation, the profiles of such parameters of the molecules intersect and cross each other, signifying that the relative order of the intrinsic hardness of their equilibrium geometry is reversed. The intrinsically hard molecule BF₃ becomes softer than the intrinsically soft molecule BH₃ as a consequence of structural distortion. The increase in chemical reactivity computed in terms of density functional parameters are transparent and justified in terms of the profiles of the eigenvalues of the frontier orbitals. The profiles of chemical potential reveal the inherent difference in the tendency of backdonation from two molecules. The computed values of Fukui functions and local softness parameters of the B atom site demonstrate that the concept of local softness can be exploited for a theoretical analysis and understanding of the characteristic chemical events of the molecules under consideration. The profiles of the Fukui functions and local softness parameters of the two molecules seem to reflect and reveal their intrinsic difference in the tendency of receiving donation in the LUMO (electrophilicity) and that of backdonation from the HOMO (nucleophilicity) and the inherent difference of overall reactivity of the two molecules by a simultaneous operation of two opposing processes of charge transfer. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

A DFT study on the correlation between topology and Bader charges: Part IV,on the change of atomic charges in polymorphic transitions – A case study on CaCl2

Referring to the experimental results of high pressure experiments of Léger et al. (1998) we have calculated the energies of all phases observed for CaCl₂ within the DFT formalism using the VASP package, and we have retrieved enthalpies and transition pressures. All phases can be considerably compressed or dilated without much change in energy. This energetic “softness” could even be quantified. We classify the high temperature TiO₂-type structure and the PbCl₂-type one at highest pressures as the energetically “softest” ones and the SrI₂-type one as the “hardest”. We furthermore discuss the energy density (E/V) of the different phases and redefine it as a fictive cohesive pressure within these structures.Pursuing our earlier approaches we have analysed the charges of the atoms in the different CaCl₂ phases and their change on compression or dilation. On comparing the gradients of the charge curves we define a sort of “charge hardness” which will generally depend on the type of cation–anion pair but also on their topological connection in the respective structures. We speculate that exhausting the “charge softness or hardness” of individual ions in such arrangements may initiate the structural reorganization at the transition pressures.     

Revealing the Unusual Rigid Boron Chain Substructure in Hard and Superconductive Tantalum Monoboride

Poor electrical conductivity severely limits the diverse applications of high hardness materials in situations where electrical conductivities are highly desired. A “covalent metal” TaB with metallic electrical conductivity and high hardness has been fabricated by a high pressure and high temperature method. The bulk modulus, 302.0(4.9) GPa, and Vickers hardness, 21.3 GPa, approaches and even exceeds that of traditional insulating hard materials. Meanwhile, temperature-dependent electrical resistivity measurements show that TaB possesses metallic conductivity that rivals some widely-used conductors, and it will transform into a superconductor at T_c=7.8 K. Contrary to common understanding, the hardness of TaB is higher than that of TaB₂, which indicates that low boron concentration borides could be mechanically better than the higher boron concentration counterparts. Compression behavior and first principles calculations denote that the high hardness is associated with the ultra-rigid covalent boron chain substructure. The hardness of TaB with different topologies of boron substructure shows that besides incorporating higher boron content, manipulating light element backbone configurations is also critical for higher hardness amongst transition metal borides with identical boron content.     

Products of class sums of the symmetric group: Rules of partial elimination

Progress in the formulation of a procedure for the combinatorial evaluation of the product of a single-cycle and an arbitrary class sum in the symmetric group algebra is presented. The procedure consists of a “global conjecture” concerning the representation of the product [(p)]_n·[*]_n in terms of a set of operators referred to as reduced class sums, and of an (incomplete) set of rules for the evaluation of the (n-independent!) coefficients of these operators. Two new types of index elimination rules are suggested, and some properties of the formalism are explored. These include useful sum rules as well as a certain “detailed balance” property that sheds some light on a combinatorial aspect of the global conjecture. The present results account for several new types of reduced class coefficients and suggest some feasible further developments. Some outstanding open problems are pointed out. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 961–979, 1997     

Determination of one-centre core integrals from the average energies of atomic configurations

One-center core integrals for valence orbitals are determined from the experimental average energies of neutral atomic configurations from Li through Zn. These values are compared with those estimated from CNDO /1, “INDO /1”, CNDO /2, “INDO /2” and with theoretical values calculated from a pseudo-potential method. The agreement is good between values obtained from neutral atoms and from the psuedo-potential calculation except for the 3d orbitals of the transition elements where the theoretically calculated integrals over single ξ functions are not realistic. These two methods reproduce both term and average configuration energies for the first two rows of atoms; the semiempirical method reliably reproduces them for the third row. The CNDO /1 and INDO /1 methods underestimate atomic energies, while the CNDO /2 and INDO /2 procedures fail rather poorly. The propriety of using core integrals estimated semiempirically in molecular orbital calculations is discussed.     

Intermetallic Compounds and the Use of Atomic Radii in Their Description

Metallic radii, which are obtained from atomic distances in the pure elements, are generally used for the calculation of distances in intermetallic compounds. However, the procedure for using such radii depends on the individual structural type: (a) For high coordination numbers and only slightly differing distances between atoms of the same kind and different atoms, all distances in a structure are proportional to the sum of radii, weighted according to the compositon. Such a “Vegard” relationship for ordered compounds is obeyed by intermetallic compounds with topological close packings, but strictly only if the various kinds of distances are correlated via symmetry relationships. For compounds with low coordination numbers the simple sum of radii holds for atoms participating in the shortest bond (e.g. in ionic crystals).-(b) The number of neighbors determines the size of each atom. It can be shown that the bond strength-bond length concept, developed for valence compounds, and often dealt with in the literature over the last ten years, is also applicable for alloys. On this basis a formalism is developed which uniformly describes the size of the atoms as a function of the coordination number for both the limiting cases of multiple bonds in molecules and for close packed atomic arrangements in alloys.     

New optimization method for conformational energy calculations on polypeptides: Conformational space annealing

A new optimization method is presented to search for the global minimum-energy conformations of polypeptides. The method combines essential aspects of the build-up procedure and the genetic algorithm, and it introduces the important concept of “conformational space annealing.” Instead of considering a single conformation, attention is focused on a population of conformations while new conformations are obtained by modifying a “seed conformation.” The annealing is carried out by introducing a distance cutoff, D_cut, which is defined in the conformational space; D_cut effectively divides the whole conformational space of local minima into subdivisions. The value of D_cut is set to a large number at the beginning of the algorithm to cover the whole conformational space, and annealing is achieved by slowly reducing it. Many distinct local minima designed to be distributed as far apart as possible in conformational space are investigated simultaneously. Therefore, the new method finds not only the global minimum-energy conformation but also many other distinct local minima as by-products. The method is tested on Met-enkephalin, a 24-dihedral angle problem. For all 100 independent runs, the accepted global minimum-energy conformation was obtained after about 2600 minimizations on average. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1222–1232     

New global constraints on electronic energy hypersurfaces

Improved global energy bounds — valid for the entire electronic energy hypersurfaces of a variety of polyatomic molecular systems — are proposed. The new constraints are applicable to a larger class of molecules and are tighter than the constraints proposed earlier. The new global bounds are easily applicable using readily available energy values of atoms and atom-ions. The actual evaluation of these constraints typically involves only “back-of-the-envelope” calculations, providing both upper and lower bounds for a complete energy hypersurface, even for very large molecules. Such global energy bounds are of some importance in theoretical studies of chemical reactions and conformational changes. The proposed bounds are likely to find some practical applications in computer-based quantum-chemical synthesis planning, using multidimensional potential surfaces.     

Influence of alloying elements on the mechanical and thermodynamic properties of ZrB12 ceramics from first-principles calculations

Although ZrB₁₂ is a promising advanced material because of the boron cuboctahedron cages, the hardness of ZrB₁₂ remains controversy. Here, we apply first-principles calculations to study the influence of transition metals (4d- and 5d-) on the hardness and thermodynamic properties of ZrB₁₂. The calculated hardness of ZrB₁₂ is 32.9 GPa, which is in good agreement with the previous theoretical result. Importantly, the calculated hardness of Re-doped ZrB₁₂ is up to 40.0 GPa, which is a potential superhard material. The essential reason is that the alloying element of Re enhances the localized hybridization of B B and Zr B atoms, and then forms the strong B B covalent bond and Zr B bond. The result is well demonstrated by the chemical bonding and lattice parameter. Here, our work shows that the alloying elements of Nb, Mo, and Re enhance the thermodynamic properties of ZrB₁₂. The Debye temperature of Re-doped ZrB₁₂ is 1225.2 K, which is larger than that of the parent ZrB₁₂ (1213.5 K).     

Locally dense basis sets for chemical shift calculations

Calculations of chemical shifts have been carried out using “locally dense” basis sets for the resonant atom of interest, and smaller, attenuated sets on other atoms in the molecule. For carbon, calculations involving a 6-311G(d) triply split valence set with polarization on the resonant atom and 3-21G atomic bases on other heavy atoms result in good agreement with experiment, and are virtually identical to those found employing the larger basis on all atoms. For species such as nitrogen, oxygen, and fluorine where standard balanced basis sets do not agree well with experiment, use of attenuated sets fail as well. The use of locally dense basis sets permits calculations previously impractical, and the successful application to carbon suggests that the chemical shift is most dependent on the local basis set, and less so on whether or not a balanced or unbalanced calculation is being carried out.     

On rotations as special cases of unitary transformations with some applications to the theory of spin

Some problems in elementary geometry are approached from the point of view of linear algebra and generalized to the theory of linear spaces of finite or infinite dimensions having a positive definite binary product. The angle ω between two elements of the linear space is defined from the concept of length by means of the cosine-theorem. A rotation is then defined as a special case of a unitary transformation moving all elements the same angle ω, except that under certain circumstances, some elements may stay invariant. In the former case, one speaks of a rotation around an “external axis,” and in the latter case, of a rotation around an “internal axis” defined by the invariant elements. It is shown that the finite rotations U of both types may be expressed in the simple exponential form U = exp(iωm), where the “generator” m in the former case is an operator satisfying the relation m² = 1, and in the latter case, m³ = m. The structure of the group of finite rotations in the former case is clarified in some detail. As an illustration of the theory, some applications to the three- and two-dimensional spaces as well as to the theory of spin are given. The coupling between the ordinary three-dimensional rotations and the spinor transformations is considered in somewhat greater detail.