Degeneracy in one‐dimensional quantum mechanics: A case study

In this work we study the isotonic oscillator, V(x) = Ax² + Bx^?2, on the whole line ?∞ < x < + ∞ as an example of a one‐dimensional quantum system with energy level degeneracy. A symmetric double‐well potential with a finite barrier is introduced to study the behavior of energy pattern between both limit: the harmonic oscillator (i.e., a system without degeneracy) and the isotonic oscillator (i.e., a system with degeneracy). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Entropic measures of Rydberg‐like harmonic states

The Shannon entropy, the desequilibrium and their generalizations (Rényi and Tsallis entropies) of the three‐dimensional single‐particle systems in a spherically symmetric potential V(r) can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this article, we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillator‐like systems. The angular and radial contributions to these entropic measures are analytically expressed in terms of the quantum numbers which characterize the corresponding quantum states and, for the radial part, the oscillator strength. In the latter case, we use some recent powerful results of the information theory of the Laguerre polynomials and spherical harmonics which control the oscillator‐like wavefunctions.     

Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

A study of the two‐dimensional hydrogen atom confined within a circle of impenetrable walls is presented. The potential inside the box is Coulomb type, whereas outside it is infinite. The energy eigenvalues and some radial wave function properties are computed with high accuracy for different box sizes. We derive the polarizability in the Kirkwood approximation, calculate the Fermi contact term as a function of the confinement radius, and investigate the filling order of the one‐electron states. When the electronic configuration of many electrons is constructed by means of the Aufbau principle, the model predicts the inversion 2s–3d levels in the N atom. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Electric properties of molecules confined by the spherical harmonic potential

The harmonic oscillator potential is very often used in quantum chemical studies of electric properties to model the effect of spatial confinement. In the vast majority of works, the harmonic potential of cylindrical symmetry was applied. Thus far, its spherical counterpart was used mainly to describe properties of spatially restricted atomic systems. Therefore, our main goal was to study the molecular electric properties in the presence of the spherically symmetric harmonic oscillator potential and to characterize the impact of the relative position of the considered molecules and spherical confinement on these properties. Moreover, we analyzed how the topology of confining environment affects the dipole moment and (hyper)polarizability, by comparing the results obtained in the spherical and cylindrical harmonic potential. Based on the conducted research, it was found that the position of the molecules relative to the spherical confinement strongly influences their electric properties. The observed trends of changes in the electric properties, caused by increasing the confinement strength, vary significantly. Moreover, it was shown that in the vast majority of cases, significant differences in the values of electric properties, obtained in the cylindrical and spherical confinement of a given strength, occur.     

Massively parallel implementation of 3D‐RISM calculation with volumetric 3D‐FFT

A new three‐dimensional reference interaction site model (3D‐RISM) program for massively parallel machines combined with the volumetric 3D fast Fourier transform (3D‐FFT) was developed, and tested on the RIKEN K supercomputer. The ordinary parallel 3D‐RISM program has a limitation on the number of parallelizations because of the limitations of the slab‐type 3D‐FFT. The volumetric 3D‐FFT relieves this limitation drastically. We tested the 3D‐RISM calculation on the large and fine calculation cell (2048³ grid points) on 16,384 nodes, each having eight CPU cores. The new 3D‐RISM program achieved excellent scalability to the parallelization, running on the RIKEN K supercomputer. As a benchmark application, we employed the program, combined with molecular dynamics simulation, to analyze the oligomerization process of chymotrypsin Inhibitor 2 mutant. The results demonstrate that the massive parallel 3D‐RISM program is effective to analyze the hydration properties of the large biomolecular systems. © 2014 Wiley Periodicals, Inc.     

Information theory of D‐dimensional hydrogenic systems: Application to circular and Rydberg states

The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power‐like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D‐dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s‐states, circular states (i.e., single‐electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

3D‐Printed Microfluidics

The advent of soft lithography allowed for an unprecedented expansion in the field of microfluidics. However, the vast majority of PDMS microfluidic devices are still made with extensive manual labor, are tethered to bulky control systems, and have cumbersome user interfaces, which all render commercialization difficult. On the other hand, 3D printing has begun to embrace the range of sizes and materials that appeal to the developers of microfluidic devices. Prior to fabrication, a design is digitally built as a detailed 3D CAD file. The design can be assembled in modules by remotely collaborating teams, and its mechanical and fluidic behavior can be simulated using finite‐element modeling. As structures are created by adding materials without the need for etching or dissolution, processing is environmentally friendly and economically efficient. We predict that in the next few years, 3D printing will replace most PDMS and plastic molding techniques in academia.     

N‐dimensional switch function for energy conservation in multiprocess reaction dynamics

The MReaDy program was designed for studying Multiprocess Reactive Dynamic systems, that is, complex chemical systems involving different and concurrent reactions. It builds a global potential energy surface integrating a variety of potential energy surfaces, each one of them representing an elementary reaction expected to play a role in the chemical process. For each elementary reaction, energy continuity problems may happen in the transition between potential energy surfaces due to differences in the functional form for each of the fragments, especially if built by different authors. A N‐dimensional switch function is introduced in MReaDy in order to overcome such a problem. As an example, results of a collision trajectory calculation for H₂ + OH → H₃O are presented, showing smooth transition in the potential energy, leading to conservation in the total energy. Calculations for a hydrogen combustion system from 1000 K up to 4000 K shows a variation of 0.012% when compared to the total energy of the system. © 2016 Wiley Periodicals, Inc.     

An analytical approach for computing Franck‐Condon integrals of harmonic oscillators with arbitrary dimensions

We have developed an analytical approach for computing Franck‐Condon integrals (FCIs) of harmonic oscillators (HOs) with arbitrary dimensions in which the mode‐mixing Duschinsky effect is taken into account. A general formula of FCIs of HOs was obtained and was applied to study the photoelectron spectroscopy of vinyl alcohol and ovalene (C₃₂H₁₄). The equilibrium geometries, harmonic vibrational frequencies and normal modes of vinyl alcohol, ovalene, and their cations were computed at the B3LYP/aug‐cc‐pVTZ or the B3LYP/6‐31G(d) level, from which Franck‐Condon factors were calculated and photoelectron spectra were simulated. The adiabatic ionization energies of vinyl alcohol were also computed by extrapolating the CCSD(T) energies to the complete basis set limit with aug‐cc‐pVXZ (X = D, T, Q, 5). The simulated photoelectron spectra of both vinyl alcohol and ovalene are in agreement with the experiments. The computed adiabatic ionization energies of syn‐ and anti‐vinyl alcohol are in consistent with the experiment within 0.008 eV and 0.014 eV, respectively. We show, for the first time, that the analytical approach of computing FCIs is also efficient and promising for the studies of vibronic spectra of macrosystems. © 2012 Wiley Periodicals, Inc.     

Neue Zirconiumphosphatfluoride mit 3D‐Gerüststruktur

New Zirconium Phosphate Fluorides with 3D‐Framework From aqueous solutions of ZrOCl₂, H₃PO₄, HF, and various amines, two new compounds of the general formula [amH₂]_1/2[Zr₂(HPO₄)(PO₄)₂F] · nH₂O ( I : am = N,N‐dimethylethylenediamine, n = 0,5; II : am = N,N‐dimethyl‐1,3‐diaminopropane, n = 0) adopting the ZrPOF‐1 structure type have been synthesized under hydrothermal conditions. In contrast to the monoclinic ZrPOF‐1, both compounds crystallize in the space group P 1 with a = 6.611(3), b = 9.109(4), c = 11.560(5) Å, α = 85.62(4), β = 89.60(4), γ = 70.57(4)° in I , and a = 6.616(2), b = 9.045(3), c = 11.565(4) Å, α = 85.26(4), β = 88.86(4), γ = 71.46(4)° in II . Compound III (am = ethylenediamine, n = 0) has been obtained by dehydration of ZrPOF‐1 and occurs in the space group P1 with a = 6.605(2), b = 8.787(3), c = 11.499(5) Å, α = 93.07(4), β = 90.42(4) and γ = 104.66(4)°. The structural motifs of the frameworks of the three compounds have much in common. The template and the PO₃OH tetrahedra in I and II are disordered. Differences in the water content in both compounds are due to differences in the chain lengths of the amines. The absence of crystal water in compound III breaks the template disordering which is present in ZrPOF‐1. The rotation of the PO₃OH tetrahedra in II and III compared with I and ZrPOF‐1 is discussed in regard with the absence of stabilizing H‐bridges in the former compounds.     

Vibrational spectrum of a 1D oscillator: The quantum,the Wigner,and the classical ways

Infrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included “absolute” intensities in the evaluation. The infrared spectrum of a one-dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity.     

Separation of D‐psicose and D‐fructose using simulated moving bed chromatography

Simulated moving bed (SMB) processes have been widely used in the sugar industries with ion‐exchange resin as a stationary phase. D ‐Psicose, a rare monosaccharide known as a valuable pharmaceutical substrate, was synthesized by the enzymatic conversion from D ‐fructose. The SMB process was adopted to separate D ‐psicose from D ‐fructose. Before the SMB experiment, the reaction mixture including D ‐psicose and D ‐fructose was treated by a deashing process to remove contaminants, such as buffers, proteins, and other organic materials. Four columns packed with Dowex 50WX4‐Ca²⁺ (200–400 mesh) ion‐exchange resins were used in the four‐zone SMB. Single‐step frontal analysis was performed to estimate the isotherm parameters of each monosaccharide. The operating conditions of the SMB process were determined based on the Equilibrium Theory. According to the simulation of the SMB process, the purity and yield of extract product (D ‐psicose) achieved were 99.04 and 97.46%, respectively and those of raffinate product (D ‐fructose) were 99.06 and 99.53%, respectively. Under the optimized operating condition, complete separation (extract purity = 99.36%, raffinate purity = 99.67%) was achieved experimentally.     

An Improved Preparation of D‐Glyceraldehyde 3‐Phosphate and Its Use in the Synthesis of 1‐Deoxy‐D‐xylulose 5‐Phosphate

D ‐Glyceraldehyde 3‐phosphate (=D ‐GAP; 2 ) was prepared by an improved chemical method (Scheme 2), and it was then employed to synthesize 1‐deoxy‐D ‐xylulose 5‐phosphate (=DXP; 3 ) which is enzymatically one of the key intermediates in the MEP ( 4 ) terpenoid biosynthetic pathway (Scheme 1). The recombinant DXP synthase of Rhodobacter capsulatus was used to catalyze the condensation of D ‐glyceraldehyde 3‐phosphate ( 2 ) and pyruvate (=2‐oxopropanoate; 1 ) to produce the sugar phosphate 3 (Scheme 2). The simple two‐step chemoenzymatic route described affords DXP ( 3 ) with more than 70% overall yield and higher than 95% purity. The procedure may also be used for the synthesis of isotope‐labeled DXP ( 3 ) by using isotope‐labeled pyruvate.     

Derivation of shell effects and magic numbers in metal clusters by the application of Strutinsky's method to the Clemenger–Nilsson and q‐deformed 3‐D harmonic oscillator models

Strutinsky's standard averaging method is applied to metal clusters described by two different potentials—Clemenger–Nilsson (CN) and q‐deformed 3‐D (Q3D) harmonic oscillator (HO). In addition, a new approximate fitting formula—of the liquid drop model type—is derived for the smooth part of the energy. The results obtained for the oscillating part of the energy (shell correction term) with the two cluster potentials through the two averaging methods are compared and discussed. It is found that with both CN and Q3D HO potentials the standard and approximate methods give similar results for clusters with a large number of particles, whereas for smaller clusters significant differences appear. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

DFT‐EPR study of radiation‐induced radicals in α‐D‐glucose

The structures of two radiation‐induced radicals in solid‐state α‐D ‐glucose have been identified by means of single‐molecule density function theory (DFT) calculations. Using the original crystalline structure as input, several radical models were created and their geometries optimized. Subsequently, electron paramagnetic resonance (EPR) parameters were calculated. During these calculations, the global orientation of the radical structure was kept fixed with respect to the crystal axes reference frame. This was essential to allow for an easy analysis of the hyperfine tensor principal directions, besides the isotropic and anisotropic coupling constants. By comparing these calculated EPR parameters with their experimentally determined counterparts, a plausible identification of two carbon‐centered glucose radicals was possible. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Unified analytical treatment for calculation of the two‐dimensional Franck–Condon factors using the duschinsky transformation

Including binomial expansion theorems, we present an analytical formula for calculating Franck–Condon (FC) factors of two‐dimensional (2D) harmonic oscillators including the Duschinsky effect. The FC principle has various practical applications in quantum modeling of electronic spectra of polyatomic molecules. The 2D FC factors are expressed through the binomial coefficients. Use of the memory of the computer for the calculation of binomial coefficients may extend the limits to large arguments for users and result in speeder calculation, should such limits be required in practice. Accurate numerical results are provided to validate the proposed algorithm. © 2012 Wiley Periodicals, Inc.