A fundamental connection between symmetry and spatial localization properties of basis sets

The problem of the compatibility between symmetry and localization properties of basis sets is addressed here. It is shown that both concepts are closely related from a fundamental point of view through the notion of invariance extent. This quantity is a functional that depends on the symmetry group and the basis set choices, and it is shown that all basis sets adapted in a general way to symmetry, i.e. induced from irreducible bases of the subgroups, are stationary points of it. In particular, the usual irreducible bases of the full group display a maximal invariance extent, while those symmetry-adapted basis sets that display a minimal value of this quantity feature in most cases the same symmetry properties as localized functions obtained by means of the Boys scheme. The most relevant conclusions are illustrated by means of simple molecular and periodic examples.     

Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

Harmonic oscillator tensors. V. The doubly degenerate harmonic oscillator

The step operators of the two-dimensional isotropic harmonic oscillator are shown to be separable into the basis elements of two disjoint Heisenberg Lie algebras. This separability leads to two sets of irreducible tensors, each of which is based upon its associated underlying Heisenberg Lie algebra. The matrix elements of these tensors are evaluated, along with those of some vibrational operators of physical interest. The possibility of other irreducible tensors are discussed and their usefulness is compared with that of those found here. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 343–357, 1998     

Finite group theory for large systems. 1. Symmetry-adaptation

The challenge of symmetry-adapting large basis sets to finite groups, apart from extensive calculations with large matrices, is obtaining linearly independent bases for frequently repeated irreducible representations, a process that is not determined by the group theory. The usual projection method is modified here to solve this problem efficiently and systematically. The resulting basis is suitably conditioned so that repeated irreducible representations are identical as required by the symmetry-generation theorem.     

Splicing I: Using mixed basis sets in ab initio calculations

The effect of mixing (or “splicing”) extended and minimal basis sets on molecular properties such as geometries, Mulliken charges, dipoles, and internal rotation barriers was studied for several test molecules. The effect is gauged by comparison with full extended basis set calculations. It is found that splicing improves most properties relative to full minimal basis set calculations, and little accuracy is lost if the splicing is done in a judicious manner.     

A simple proof for the formula to get symmetrized powers of group representations

A general formula to decompose the p-power of irreducible representations of an arbitrary space group into sum of sets of irreducible representations of such a group, having identical permutational symmetry, is presented. Its proof is based upon a straightforward application of the properties of the generalized projection (shift) operators. © 1993 John Wiley & Sons, Inc.     

Robotic hierarchical mixing for the production of combinatorial libraries of proteins and small molecules

We present a method to automatically plan a robotic process to mix individual combinations of reactants in individual reaction vessels (vials or wells in a multiwell plate), mixing any number of reactants in any desired stoichiometry, and ordering the mixing steps according to an arbitrarily complex treelike assembly protocol. This process enables the combinatorial generation of complete or partial product libraries in individual reaction vessels from intermediates formed in the presence of different sets of reactants. It can produce either libraries of chimeric genes constructed by ligation of fragments from different parent genes or libraries of chemical compounds constructed by convergent synthesis. Given concentrations of the input reactants and desired amounts or volumes of the products, our algorithm, RoboMix, computes the required reactant volumes and the resulting product concentrations, along with volumes and concentrations for all intermediate combinations. It outputs a sequence of robotic liquid transfer steps that ensures that each combination is correctly mixed even when individualized stoichiometries are employed and with any fractional yield for a product. It can also account for waste in robotic liquid handling and residual volume needed to ensure accurate aspiration. We demonstrate the effectiveness of the method in a test mixing dyes with different UV-vis absorption spectra, verifying the desired combinations spectroscopically.     

Response function theory of electron correlation

The full perturbation expansion for the response (or density—density correlation) function is examined in order to provide a useful general theory of excitation energies, oscillator strengths, dynamic polarizabilities, etc., that is more accurate than the random phase approximation. It is first shown how the formal partition of the diagrammatic version of the perturbation expansion into reducible and irreducible diagrams is generally useless as the latter category contains all the difficult terms which have heretofore resisted analysis in all but a haphazard form. It is then shown how the diagram for the response function can be partitioned into “correlated” and “uncorrelated” subsets. Restricting attention to the particle—hole blocks of the full response function, the “uncorrelated” diagrams desecribe the propagation of a particle—hole pair in an N-electron system where the particle and hole are each interacting with the remaining electrons but they are not interacting with each other. The “correlated” diagrams are those containing the hole—particle interactions, and, by defining a new class of reducible and irreducible diagrams, these are all summed to provide a perturbation expansion of the effective two-body hole—particle interaction that appears in the inverse of the response function. The “uncorrelated” diagrams are further partitioned into two sets, one of which is summed to all orders, while the other set is inverted in an order by order fashion. The final result presents a perturbation expansion for the inverse of the response function that is analogous to the Dyson equation for one-electron Green functions. Maintaining the perturbation expansion through first order for the inverse of the response function yields the eigenvalue equation of the familiar random phase approximation, while truncation at second order provides the most advanced theories that have been generated by the equations-of-motion method.     

A theory for the construction of the irreducible representations of finite groups

It is shown how the irreducible representations of a finite group can be calculated from the irreducible characters (the latter can be calculated exactly by using Dixon's method). All elements of the matrix, representing a group element, lie in the rational field of polynomials of ξ = exp (2πi/e), where e is the exponent of the group.     

Singlet-triplet splittings and ground- and excited-state electron affinities of selected cyanosilylenes, XSiCN (X = H, F, Cl, CH3, SiH3, CN)

Several cyanosilylenes, XSiCN, (X = H, F, Cl, CH3, SiH3, CN) have been investigated using the RHF-ACPF and CAS(2,2)-ACPF methods in conjunction with the aug-cc-pVTZ basis sets. All silylenes are found to have singlet ground states. The ground-state electron affinities are found to be rather high, i.e., 1.832, 1.497, 1.896, 1.492, 2.235, and 2.631 eV for HSiCN, FSiCN, ClSiCN, H3CSiCN, H3SiSiCN, and Si(CN)2, respectively. The existence of bound excited negative ion states has been discovered for the first time within these silylenes. All these bound excited anion states belong to the totally symmetric irreducible representations and can be characterized as dipole-bound negative ion states. All triplet excited states have even larger dipole moments than the singlet states and are, therefore, "dressed" by dipole-bound negative ion states, which correspond to Feshbach resonances.     

Analysis of the transverse and the longitudinal pseudodiffusion of CO2 in sub- and supercritical states: a molecular-dynamics analysis

We have performed molecular-dynamics simulations of CO(2) system along the gas-liquid coexistence curve and on the isochore 94.22 cm(3) mol(-1) (which corresponds to the critical isochore). The calculation has been carried out in order to analyze the diffusion of CO(2) and particularly to figure out how the diffusion coefficient may be decomposed along the molecular axes. This makes it possible to analyze the anisotropy of the diffusion along these axes and to shed light on the microscopic changes which accompany such behavior. This anisotropy is traced back to the effect of the translation-rotation coupling (TRC) along the molecular axes. Along the liquid-gas coexistence curve, the pseudolongitudinal diffusion is found to be more rapid than the transverse one. The opposite trend is found along the isochore 94.22 cm(3) mol(-1). The role of the local structure was explored by calculating intermediate scattering function and the autocorrelation functions for the forces acting along the molecular axes. It is shown that the strength of the TRC effect is correlated to the difference between the relaxation times of the local structure, that of the reorientation along the molecular axes, and that of the translational motion. The analysis of the correlation time and the average mean square force along the longitudinal and transverse directions confirms the anisotropy of the local environment that determines the translational dynamics of a molecule.     

Preparation of Biporous Adsorbents with Controlled Structure

A new procedure was developed for the preparation of the adsorbents with a biporous structure by mixing metal hydroxides with fine- and coarse-porous structures. It was shown that structural parameters of the biporous samples can be changed significantly by varying the aging time of initial hydroxides or preparing them from salt solutions of different concentrations. This is confirmed by the positions of maxima on the curves of pore volume distribution over effective radii. It was found that the regularities of structural changes directly depend on the conditions of preparation of mixing hydroxides. A suggested mechanism of structural variations allowed us to explain the results obtained.     

Two-component natural spinors from two-step spin-orbit coupled wave functions

We developed an algorithm to obtain the natural orbitals (natural spinors) from the two-step spin-orbit coupled wave functions. These natural spinors are generally complex-valued, mixing two spin components, and they can have similar symmetry properties as the j-j spinors from the one-step spin-orbit coupling calculations, if the reduced density equally averages all the components of a multi-dimensional irreducible representation. Therefore, the natural spinors can serve as an approximation to the j-j spinors and any wave function analysis based on the j-j spinors can also be performed based on them. The comparison between the natural spinors and the j-j spinors of three representative atoms, Tl, At, and Lu, shows their close similarity and demonstrates the ability of the natural spinors to approximate the j-j spinors.     

Sequence-adapted molecular tensors: Algebraic methods and application to crystal field theory

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, T_d, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d¹ ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.     

Effects of mixing rules on phase equilibrium calculations

The density dependence in the mixing rule of the cohesion parameter of the cubic equation of state is found not to be necessary in vapour—liquid equilibrium (VLE) calculations. The effects of the functional form and the number of the adjustable parameters of the mixing rules on VLE calculations are examined. It is found that the functional form is less influential in VLE calculations than the number of adjustable parameters and that two adjustable parameters in the mixing rules are sufficient for practical applications.     

Protein dynamics from NMR: the slowly relaxing local structure analysis compared with model-free analysis

(15)N-(1)H spin relaxation is a powerful method for deriving information on protein dynamics. The traditional method of data analysis is model-free (MF), where the global and local N-H motions are independent and the local geometry is simplified. The common MF analysis consists of fitting single-field data. The results are typically field-dependent, and multifield data cannot be fit with standard fitting schemes. Cases where known functional dynamics has not been detected by MF were identified by us and others. Recently we applied to spin relaxation in proteins the slowly relaxing local structure (SRLS) approach, which accounts rigorously for mode mixing and general features of local geometry. SRLS was shown to yield MF in appropriate asymptotic limits. We found that the experimental spectral density corresponds quite well to the SRLS spectral density. The MF formulas are often used outside of their validity ranges, allowing small data sets to be force-fitted with good statistics but inaccurate best-fit parameters. This paper focuses on the mechanism of force-fitting and its implications. It is shown that MF analysis force-fits the experimental data because mode mixing, the rhombic symmetry of the local ordering and general features of local geometry are not accounted for. Combined multifield multitemperature data analyzed with the MF approach may lead to the detection of incorrect phenomena, and conformational entropy derived from MF order parameters may be highly inaccurate. On the other hand, fitting to more appropriate models can yield consistent physically insightful information. This requires that the complexity of the theoretical spectral densities matches the integrity of the experimental data. As shown herein, the SRLS spectral densities comply with this requirement.     

炭黑在异戊橡胶中的分散及其对动静态性能的影响北大核心CSCD

橡胶在使用时往往需要在生胶中加入填料颗粒以提高其力学强度,填料在橡胶中的分散状态对其使用性能有显著影响。为了明晰填料分散状态对橡胶动静态性能的影响规律,设计了不同的胶料混炼工艺,进而获得具有不同分散程度的橡胶材料,并对其分散等级、颗粒聚集体尺寸分布、胶料硫化性质、橡胶拉伸以及疲劳等性质进行了系统研究。结果表明,在所研究的密炼时间4~16 min范围内,随着密炼时间的延长,填料在基体中的分散度等级有较大程度的提高,密炼时间为16 min时填料分散等级达到了8.1级,胶料的门尼粘度下降显著,但胶料的拉伸性能变化不明显,而动态疲劳性能得以显著改善。随着开炼包辊时间的延长,填料在基体中的分散度等级呈先升高后降低的趋势,橡胶的动态疲劳性能也先升高后降低,当开炼包辊时间为20 min时,硫化胶的伸张疲劳寿命达到4×10~4次,继续延长包辊时间硫化胶的伸张疲劳寿命下降至2.3×10~4次;但是胶料整体的拉伸性能变化不明显。由此可见,通过调整混炼工艺可以改善填料在橡胶基体中的分散状态,进而提高胶料的疲劳性能。     

Ab initio investigation of vibrational spectra of water-(CO2)n complexes (n = 1, 2)

In this paper, we have calculated using the ab initio method the IR vibrational spectra of complexes of CO2 formed with water (sp3 O-donating atom). Binding energies and structures of the CO2-H2O and water-(CO2)2 complexes have been determined at the second-order level of the Moller-Plesset perturbation theory (MP2) using Dunning's basis sets. The results are presented and critically discussed in terms of the nature of the water-CO2 interactions, electron donor acceptor (EDA) and weak O...H-O interactions. For water-(CO2)2 trimer, it is also shown that the contribution to the interaction energy of the irreducible three-bodies remains relatively negligible. We have analyzed the frequency shifts and the IR and Raman intensity variations under the complex formation. We have particularly emphasized the splitting of the 2 bending mode of CO2 and stretching modes of water, which have been revealed as the most pertinent probes to assess the nature of the forces involved in the different complexes. Finally, because water can play the role of Lewis base and acid as well, we found that weak O...H-O interactions can cooperate with EDA interactions in trimer, leading to very specific spectral signatures that are further discussed.     

Effect of mixing on the formation of complexes of hyperbranched cationic polyelectrolytes and anionic surfactants

The effect of different mixing protocols on the charged nature and size distribution of the aqueous complexes of hyperbranched poly(ethylene imine) (PEI) and sodium dodecyl sulfate (SDS) was investigated by electrophoretic mobility and dynamic light scattering measurements at different pH values, polyelectrolyte concentrations, and ionic strengths. It was found that at large excess of the surfactant a colloidal dispersion of individual PEI/SDS nanoparticles forms via an extremely rapid mixing of the components by means of a stop-flow apparatus. However, the application of a less efficient mixing method under the same experimental conditions might result in large clusters of the individual PEI/SDS particles as well as in a more extended precipitation regime compared with the results of stop-flow mixing protocol. The study revealed that the larger the charge density and concentration of the PEI, the more pronounced the effect of mixing becomes. It can be concluded that an efficient way to avoid precipitation in the solutions of oppositely charged polyelectrolytes and surfactants might be provided by extending the range of kinetically stable colloidal dispersion of polyelectrolyte/surfactant nanoparticles via the application of appropriate mixing protocols.     

Real irreducible tensorial sets

A simple formalism of real irreducible tensorial sets of real bases is proposed. The definition of the real bases, the coupling of the real bases, and the transformation of the real bases in a group chain including the three-dimensional rotation group and the molecular point groups are studied. The double coset technique is used to derive the close formulas for generating the coupling coefficients and the transformation coefficients. © 1996 John Wiley & Sons, Inc.