Fuzzy space periodic symmetries for polyynes and their cyano-compounds

In our previous papers on the molecular fuzzy symmetry, we analyzed the basic characterization in connection with the fuzzy point group symmetry. In this paper, polyynes and their cyano-derivatives are chosen as a prototype of linear molecules to probe the one-dimensional fuzzy space group of parallel translation. It is notable that the space group is an infinite group whereas the point group is a finite group. For the fuzzy point group, we focus on considering the fuzzy characterization introduced due to the difference of atomic types in the monomer through point symmetry transformation in the beginning; and then we consider the difference between the infinity of space group and the finite size of real molecules. The difference between the point group and the space group lies in the translation symmetry transformation. This is the theme of this work. Starting with a simple case, we will only analyze the one-dimensional translation transformation and space fuzzy inversion symmetry transformation in this paper. The theory of the space group is often used in solid state physics; and some of its conclusions will be referred to. More complicated fuzzy space groups will be discussed in our future papers.     

Optimal non-linear dimension reduction scheme for classical molecular dynamics

The optimal creation of a reduced space that effectively captures the long timescale dynamics of a non-linear molecular system over a range of frequencies is described. The technique builds on a previously developed subspace method based on linear constant projective transformation of the original full space. The present work attempts to propose transformation that are spatially dependent thereby leading to an effective subspace for better representing the dynamics of interests. The algorithm seeks out an optimal transformation consistent with desired low frequency motion in a rather general way. The method is demonstrated for a six-dimensional nonlinear system reduced to two-dimensions. Superior performance is found in evaluating ensemble-averaged classical dynamical properties.     

直链共轭多烯的模糊ta/2对称性

近年来关于分子模糊对称性的工作多属于模糊点对称性的研究．关于模糊空间对称性探讨较少．只曾对线状一维模糊周期分子进行过一些分析．本文在此基础上进一步对于较复杂的平面一维模糊周期分子——直链共轭多烯（简称为共轭多烯）分子进行了较仔细的探讨．除模糊平移变换外,这里还将涉及模糊的螺旋旋转和滑移反映等空间变换．此外,还讨论了存在其中的其他模糊点对称变换．对于点对称元素的变动导致的模糊对称性特征,往往和某种空间对称变换的模糊对称性特征相关．对于分子轨道,除模糊对称变换的隶属函数外,分析了所属不可约表示成分．对这些分子的某些性质和其模糊对称性特征之间的相关性进行探讨．     

Fuzzy ta/2 symmetries of straight chain conjugate polyene molecules

On the basis of our recent studies on the molecular fuzzy point group symmetry, we further probe into the more complicated planar one-dimensional fuzzy periodic molecules—straight chain conjugate polyene. Except for the fuzzy translation transformation, the space transformation of the fuzzy screw rotation and the glide plane will be referred to. In addition, other fuzzy point symmetry transformation lain in the space transformation is discussed. Usually there is a correlation between the fuzzy symmetry characterization caused by the transition of the point symmetry elements and by certain space symmetry transformation. For the molecular orbital, the irreducible representation component is analyzed besides the membership function of the fuzzy symmetry transformation. Also, we inquire into the relativity between some molecular property and the fuzzy symmetry characterization.     

Fukutome classes in momentum space

Summary Fukutome's group theoretical classification scheme for determinants, based on the transformation properties of the Fock-Dirac density matrix under spin rotations and time reversal, has been extended to momentum space. Particular attention is paid to the transformation properties of orbitals and density matrices under inversion in momentum space.     

Mapping between local potentials and ground state densities

The mapping between the local one-body potential and the ground state density is discussed. The role of degenerate ground states, which form a set of measure zero in potential space, is shown to be amplified by the unitary transformation between degenerate densities, forming a space filling set in density space. A numerical experiment demonstrating these features is carried out. No indication of non-V representable densities is encountered.     

The Wigner function of the rotating Kratzer oscillator mapped onto the Morse oscillator

In this article, the rotating Kratzer oscillator in quantum phase space is studied. The Langer transformation is used to map the Kratzer oscillator with centrifugal term onto a one‐dimensional Morse oscillator. As a result, the Wigner distribution functions for the Morse oscillator are obtained. The quantum states of the system are visualized in the phase space for a few vibrational and rotational quantum numbers. The results obtained in the phase space correspond to those derived in the standard quantum theory. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

New space warping method for the simulation of large-scale macromolecular conformational changes

A space warping method, facilitating the modeling of large-scale conformational changes in mesoscopic systems, is presented. The method uses a set of "global (or collective) coordinates" that capture overall behavior, in conjunction with the set of atomic coordinates. Application of the space warping method to energy minimization is discussed. Several simulations where the method is used to determine the energy minimizing structures of simple central force systems are analyzed. Comparing the results and behavior of the space warping method to simulations involving atomic coordinates only, it is found that the space warping method scales better with system size and also finds lower minima when the potential energy surface has multiple minima. It is shown that the transformation of [Ala16]+ in vacuo from linear to globular is captured efficiently using the space warping method.     

The fuzzy symmetries for linear tri-atomic $${\rm B} \cdots {\rm A} \cdots {\rm C}$$ dynamic systems

Based on our previous study on the elementary characterization of the fuzzy symmetry, we inquired the static state fuzzy symmetry of some molecules and their molecular orbitals (MO). Now we will analyze the fuzzy symmetry of some simple linear tri-atomic dynamic systems in connection with the reaction. Three related transformations will mainly be studied in detail. These three transformations are (1) the space inversion transformation about the mid-atom as the center, (2) the reaction reversal transformation in relation to the reaction B + AC→BA + C and (3) the joint transformation of the above two. We examined the variation for the internal configuration of these systems owing to the operation of above three transformations, and then establish methods to calculate the fuzzy symmetry characterization, such as the membership functions for the MOs of such linear tri-atomic dynamic systems in relation to these transformations. We examined the variation regularity in relation to the fuzzy symmetry characterization for the MOs of these systems along the intrinsic reaction coordinate (IRC) and dividing line. The variation regularity and the distribution for the fuzzy symmetry characterization in related internal configuration coordinate space are also analyzed. An IRC-scale is suggested for internal configuration coordinate space in this paper.     

Decomposition of the first-order reduced density matrix: an isopycnic localization treatment

In this work, we propose a partitioning of the first-order reduced density matrix corresponding to an N-electron system into first-order reduced density matrices associated with regions defined in the real space (regional matrices). The treatment is based on an isopycnic orbital localization transformation that provides regional matrices that are diagonalized by identical localized orbitals, having many attributes associated with chemical concepts (appropriate localization in space, high transferability, etc.). Although the obtained numerical values are similar to those arising from previous studies, their interpretation is more rigorous and the computational cost is much lower.     

What Differentiates Free Amino Acids and Aminoacyl Residues? An Exploration of Conformational and Lipophilicity Spaces

The objective of this study was to unravel the changes in property space resulting from the amino-acid-to-residue transformation. Conformation-dependent lipophilicity was chosen as the metric to assess changes in property spaces. Phe, Ala-Phe-Ala, Gln, and Ala-Gln-Ala were first submitted to a conformational search strategy using quenched molecular dynamics in order to obtain an efficient sampling of a conformational space. This search was performed for the four electrical forms of the compounds (cationic, zwitterionic, uncharged, and anionic). The virtual lipophilicity (logP) of each conformer was then calculated by the Molecular Lipophilicity Potential (MLP). Similarly, the lipophilicity increment of the Phe and Gln residues in all electrical states and conformers of Ala-Phe-Ala and Ala-Gln-Ala, respectively, were calculated by the MLP. As expected, the results showed a marked expansion in the property space of a tripeptide compared to an amino acid. However, they also revealed a marked reduction in property space resulting from the amino-acid-to-residue transformation.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

Cost reduction of high-order coupled-cluster methods via active-space and orbital transformation techniques

We discuss several techniques which have the potential to decrease the computational expenses of high-order coupled-cluster (CC) methods with a reasonable loss in accuracy. In particular, the CC singles, doubles, and triples (CCSDT) as well as the CC singles, doubles, triples, and perturbative quadruples [CCSDT(Q)] methods are considered, which are frequently used in high-precision model chemistries for the calculation of iterative triples and quadruples corrections. First, we study the possibilities for using active spaces to decrease the computational costs. In this case, an active space is defined and some indices of cluster amplitudes are restricted to be in the space. Second, the application of transformed virtual orbitals is investigated. In this framework, to reduce the computation time the dimension of the properly transformed virtual one-particle space is truncated. We have found that the orbital transformation techniques outperform the active-space approaches. Using the transformation techniques, the computational time can be reduced in average by an order of magnitude without significant loss in accuracy. It is demonstrated that high-order CC calculations are possible for considerably larger systems than before using the implemented techniques.     

On the relationship between quantum control landscape structure and optimization complexity

It has been widely observed in optimal control simulations and experiments that state preparation is surprisingly easy to achieve, regardless of the dimension N of the system Hilbert space. In contrast, simulations for the generation of targeted unitary transformations indicate that the effort increases exponentially with N. In order to understand such behavior, the concept of quantum control landscapes was recently introduced, where the landscape is defined as the physical objective, as a function of the control variables. The present work explores how the local structure of the control landscape influences the effectiveness and efficiency of quantum optimal control search efforts. Optimizations of state and unitary transformation preparation using kinematic control variables (i.e., the elements of the action matrix) are performed with gradient, genetic, and simplex algorithms. The results indicate that the search effort scales weakly, or possibly independently, with N for state preparation, while the search effort for the unitary transformation objective increases exponentially with N. Analysis of the mean path length traversed during a search trajectory through the space of action matrices and the local structure along this trajectory provides a basis to explain the difference in the scaling of the search effort with N for these control objectives. Much more favorable scaling for unitary transformation preparation arises upon specifying an initial action matrix based on state preparation results. The consequences of choosing a reduced number of control variables for state preparation is also investigated, showing a significant reduction in performance for using fewer than 2N-2 variables, which is consistent with the topological analysis of the associated landscape.     

Solvent‐Dependent Delamination,Restacking, and Ferroelectric Behavior in a New Charge‐Separated Layered Compound: [NH4][Ag3(C9H5NO4S)2(C13H14N2)2]⋅8 H2O

A new anionic coordination polymer, [NH₄][Ag₃(C₉H₅NO₄S)₂(C₁₃H₁₄N₂)₂] ⋅ 8 H₂O, with a two‐dimensional structure, has been synthesized by a reaction between silver nitrate, 8‐hydroxyquinoline‐5‐sulfonic acid (HQS), and 4,4′‐trimethylene dipyridine (TMDP). The compound stabilizes in a noncentrosymmetric space group, and the lattice water molecules and the charge‐compensating [NH₄]⁺ group occupy the inter‐lamellar spaces. The lattice water molecules can be fully removed and reinserted, which is accompanied by a crystalline–amorphous–crystalline transformation. This transformation resembles the collapse/delamination and restacking of the layers. To the best of our knowledge, this is the first observation of delamination and restacking in an inorganic coordination polymer that contains silver. The presence of a natural dipole (the anionic framework and cationic ammonium ions) along with the noncentrosymmetric space group gives rise to the room‐temperature ferroelectric behavior of the compound. The ferroelectric behavior is also water‐dependent and exhibits a ferroelectric–paraelectric transformation. The temperature‐dependent dielectric measurements indicate that the ferroelectric/ paraelectric transformation occurs at 320 K. This transformation has also been investigated by using in‐situ IR spectroscopy and PXRD studies. The second‐harmonic generation (SHG) study indicated values that are comparable to some of the known SHG solids, such as potassium dihydrogen phosphate (KDP) and urea.     

Direct minimization of the energy functional in LCAO-MO density matrix formalism

The general multiconfiguration self-consistent-field method is presented along the density matrix formalism. The proposed optimization procedure for orbitals makes use of an orthogonal transformation in the space spanned by the fixed basis set. Acting on the unconstrained parameters of the transformation a direct minimization of the energy expression is performed using a gradient approach. A similar method may also be applied to the optimization of the expansion coefficients. The method works not only for the ground state of a given system, but also for any excited state, yielding an upper bound to the true energy of the considered state.     

Optimization with a direct search for orbital localization

Two direct-search approaches to obtain the SCF solution for a true maximum of the self-repulsion energy with energy-localized orbitals are proposed: an approach based on the level-shifted second-order method and another approach based on scaling of the orbital transformation vector to obtain an approximate solution for a true maximum. The latter is more advantageous to obtain convergence for large systems. Both methods involve calculation of the exact self-repulsion energy hypersurface in the controlling parameter space via a set of unitary transformations and selection of the unitary transformation which increases the self-repulsion energy. These approaches are found to converge efficiently even when started from a point far from convergence.     

Thermodynamic analysis and thermodynamic efficiency of chemical reactors

In this paper, conditions of minimal dissipation for processes occurring in chemical reactors with a given duration and a given degree of transformation are obtained. It is demonstrated that the derived entropy production can be used to construct the space of realizable (thermodynamically feasible) regimes of a chemical reactor.