Vibrational Basis Research Articles

Using nested tensor train contracted basis functions with group theoretical techniques to compute (ro)-vibrational spectra of molecules with non-Abelian groups.

In this paper, we use nested tensor-train contractions to compute vibrational and ro-vibrational energy levels of molecules with five and six atoms. At each step, we fully exploit symmetry by using symmetry adapted basis functions obtained from an irreducible tensor method. Contracted basis functions are determined by diagonalizing reduced dimensional Hamiltonian matrices. The size of matrices of eigenvectors, used to account for coupling between groups of coordinates, is reduced by discarding rows and columns. The size of the matrices that must be diagonalized is thus substantially reduced, making it possible to use direct eigensolvers, even for molecules with five and six atoms. The symmetry-adapted contracted vibrational basis functions have been used to compute J = 0 energy levels of the CH3CN (C3v) and J > 0 levels of CH4.

Analytical and numerical investigation of beam-spring systems with varying stiffness: a comparison of consistent and lumped mass matrices considerations

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system.

Teknik Pengobatan Tradisional Pada Masyarakat Melayu di Semendo Darat Ulu Kabupaten Muara Enim

<p><strong>Malay people have a tendency to choose traditional medicine as an alternative in treating diseases. Treatment is a process of healing by using tools. The types of traditional medicine in Indonesia broadly consist of traditional medicine with medicinal herbs derived from nature, while traditional kebatinan medicine consists of traditional medicine on the basis of belief, religion, and on the basis of magnetic vibrations. Traditional medicine used by people in several regions in Indonesia is very diverse, including traditional medicine practiced by one of the Malay tribes in South Sumatra Province, Muara Enim Regency, namely the Semendo tribe. This research aims to complement existing references regarding traditional medicine, this research focuses on traditional medicine techniques in the Semendo tribe community, Muara Enim Regency. This descriptive qualitative research aims to explore the sources of knowledge, types and practices of traditional medicine in the Semendo tribe. The results showed that traditional medicine techniques in the Semendo tribe are carried out in 3 ways, namely, drinking, smearing or sticking using plants from nature, and internally. Traditional medicine in the Semendo tribe, Muara Enim Regency is still preserved to this day. This is due to the very high level of public trust in the efficacy of traditional medicine.</strong></p><p><strong><em>Keywords</em></strong><strong> - </strong><em>Malay, Nature, Prayer</em><em>, </em><em>Semendo, Traditional medicine</em><em>.</em></p>

Studies towards investigation of Naphthoquinone-based scaffold with crystal structure as lead for SARS-CoV-19 management

In this work, 1-(4-bromophenyl)-2a,8a-dihydrocyclobuta[b]naphthalene-3,8‑dione (1-(4-BP)DHCBN-3,8-D) has been characterized by single crystal X-ray to get it's crystal structure with R(all data) - R1 = 0.0569, wR2 = 0.0824, 13C and 1HNMR, as well as UV–Vis and IR spectroscopy. Quantum chemical calculations via DFT were used to predict the compound structural, electronic, and vibrational properties. The molecular geometry of 1-(4-BP)DHCBN-3,8-Dwas optimized utilizing the B3LYP functional at the 6–311++G(d,p) level of theory. The Infrared spectrum has been recorded in the range of 4000–550 cm−1. The Potential Energy Distribution (PED) assignments of the vibrational modes were used to determine the geometrical dimensions, energies, and wavenumbers, and to assign basic vibrations. The UV–Vis spectra of the titled compound were recorded in the range of 200-800 nm in ACN and DMSO solvents. Additionally, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy gap and electronic transitions were determined using TD-DFT calculations, which also simulate the UV–Vis absorption spectrum. Natural Bond Orbital (NBO) analysis can be used to investigate electronic interactions and transfer reactions between donor and acceptor molecules. Temperature-dependent thermodynamic properties were also calculated. To identify the interactions in the crystal structure, Hirshfeld Surface Analysis was also assessed. The Molecular Electrostatic Potential (MEP) and Fukui functions were used to determine the nucleophilic and electrophilic sites. Additionally, the biological activities of 1-(4-BP)DHCBN-3,8-D were done using molecular docking. These results demonstrate a significant therapeutic potential for 1-(4-BP)DHCBN-3,8-D in the management of Covid-19 disorders. Molecular Dynamics Simulation was used to look at the stability of biomolecules.

Diagnosis of the Pneumatic Wheel Condition Based on Vibration Analysis of the Sprung Mass in the Vehicle Self-Diagnostics System.

This paper presents a method for the multi-criteria classification of data in terms of identifying pneumatic wheel imbalance on the basis of vehicle body vibrations in normal operation conditions. The paper uses an expert system based on search graphs that apply source features of objects and distances from points in the space of classified objects (the metric used). Rules generated for data obtained from tests performed under stationary and road conditions using a chassis dynamometer were used to develop the expert system. The recorded linear acceleration signals of the vehicle body were analyzed in the frequency domain for which the power spectral density was determined. The power field values for selected harmonics of the spectrum consistent with the angular velocity of the wheel were adopted for further analysis. In the developed expert system, the Kamada-Kawai model was used to arrange the nodes of the decision tree graph. Based on the developed database containing learning and testing data for each vehicle speed and wheel balance condition, the probability of the wheel imbalance condition was determined. As a result of the analysis, it was determined that the highest probability of identifying wheel imbalance equal to almost 100% was obtained in the vehicle speed range of 50 km/h to 70 km/h. This is known as the pre-resonance range in relation to the eigenfrequency of the wheel vibrations. As the vehicle speed increases, the accuracy of the data classification for identifying wheel imbalance in relation to the learning data decreases to 50% for the speed of 90 km/h.

Dinuclear Phenoxo-Bridged Nickel(II) and Copper(II) Complexes of Phenolate-Based Tripodal Ligand: Theoretical and Experimental Insights

Three dinuclear complexes of composition [NiII 2(L)2][ClO4]2 and [CuII 2(L)2(OClO3)2] . 3H2O have been synthesized using a new tripodal ligand [(2-pyridyl)methyl](2-benzyl)-aminomethyl}-phenol (HL)], in its deprotonated form, providing a N2O donor set. Crystallographic analyses reveal that [CuII 2(L)2(OClO3)2] . 3H2O has a diphenoxo-bridged structure. In [CuII 2(L)2(OClO3)2] . 3H2O, each metal center is MIIN2O3-coordinated with a square-pyramidal environment for each copper(II) center. In this work, the molecular structure, harmonic vibrational frequencies and UV-Vis of [NiII 2(L)2]2+ and [CuII 2(L)2]2+ has been explored. With the help of density functional theory (DFT)/B3LYP techniques and LANL2DZ as a basis set, the ground-state molecule shape and vibrational frequencies were computed. The basic vibrations were allocated using the VEDA program to compute the potential energy distribution (PED) of the vibrational modes. The band gap energies of the title complexes ([NiII 2(L)2]2+ and [CuII 2(L)2]2+) are 3.21 eV and 1.59 eV, respectively, according to HOMO-LUMO energies. The maximal absorption wavelength and band gap energy of the title complexes were calculated theoretically using the UV absorption spectra. MEP analysis identifies electrophilic and nucleophilic sites. Hirshfeld surface analysis was used to characterize the 3D intermolecular interactions in ([NiII 2(L)2]2+ and [CuII 2(L)2]2+) of the crystal surface, whereas fingerprint plots were used to explain the 2D interactions. The biological activity of the complexes was investigated using molecular docking.

Intermolecular Interactions of Edaravone in Aqueous Solutions of Ethaline and Glyceline Inferred from Experiments and Quantum Chemistry Computations.

Edaravone, acting as a cerebral protective agent, is administered to treat acute brain infarction. Its poor solubility is addressed here by means of optimizing the composition of the aqueous choline chloride (ChCl)-based eutectic solvents prepared with ethylene glycol (EG) or glycerol (GL) in the three different designed solvents compositions. The slurry method was used for spectroscopic solubility determination in temperatures between 298.15 K and 313.15 K. Measurements confirmed that ethaline (ETA = ChCl:EG = 1:2) and glyceline (GLE = ChCl:GL = 1:2) are very effective solvents for edaravone. The solubility at 298.15 K in the optimal compositions was found to be equal xE = 0.158 (cE = 302.96 mg/mL) and xE = 0.105 (cE = 191.06 mg/mL) for glyceline and ethaline, respectively. In addition, it was documented that wetting of neat eutectic mixtures increases edaravone solubility which is a fortunate circumstance not only from the perspective of a solubility advantage but also addresses high hygroscopicity of eutectic mixtures. The aqueous mixture with 0.6 mole fraction of the optimal composition yielded solubility values at 298.15 K equal to xE = 0.193 (cE = 459.69 mg/mL) and xE = 0.145 (cE = 344.22 mg/mL) for glyceline and ethaline, respectively. Since GLE is a pharmaceutically acceptable solvent, it is possible to consider this as a potential new liquid form of this drug with a tunable dosage. In fact, the recommended amount of edaravone administered to patients can be easily achieved using the studied systems. The observed high solubility is interpreted in terms of intermolecular interactions computed using the Conductor-like Screening Model for Real Solvents (COSMO-RS) approach and corrected for accounting of electron correlation, zero-point vibrational energy and basis set superposition errors. Extensive conformational search allowed for identifying the most probable contacts, the thermodynamic and geometric features of which were collected and discussed. It was documented that edaravone can form stable dimers stabilized via stacking interactions between five-membered heterocyclic rings. In addition, edaravone can act as a hydrogen bond acceptor with all components of the studied systems with the highest affinities to ion pairs of ETA and GLE. Finally, the linear regression model was formulated, which can accurately estimate edaravone solubility utilizing molecular descriptors obtained from COSMO-RS computations. This enables the screening of new eutectic solvents for finding greener replacers of designed solvents. The theoretical analysis of tautomeric equilibria confirmed that keto-isomer edaravone is predominant in the bulk liquid phase of all considered deep eutectic solvents (DES).

Ro-vibrational levels and their (e-f) splitting of acetylene molecule calculated from new potential energy surfaces

Ro-vibrational energy levels of acetylene are reported using variational nuclear motion calculations from new ab initio and empirically optimized full six-dimensional potential energy surfaces in the ground electronic state of the acetylene molecule. The calculations account for the triple, quadruple and quintuple excitations as well as relativistic and diagonal Born-Oppenheimer corrections. Variational nuclear motion calculations were performed using the exact kinetic energy operator in orthogonal coordinates. The convergence of energy levels calculations versus the size of the vibrational basis set functions was verified. Our best ab initio potential energy surface that includes the above-mentioned contributions provides the RMS (obs.-calc.) errors of 0.95 cm−1 for five fundamental energy levels. The largest contribution to the RMS error is caused primarily by a significant deviation of the ν4 fundamental frequency. Experimental values of 120 vibrational band origins were used to empirically adjust few lower-order parameters of the potential energy surface. The average error drops down to 0.45 cm−1 or 0.25 cm−1 for empirically optimized potential energy function with two or seven adjusted parameters corresponding to quadratic force field terms and one third order term. The splitting of (e – f) rovibrational doublets and their J dependence were calculated with high accuracy due to the full account of Coriolis interactions. Computed (e – f) splittings allow one to check the correctness of the assignment of empirical energy levels. The estimation of the accuracy for the calculated vibrational levels in an extended range up to 9500 cm−1 shows that the set of ab initio vibrational levels can be used for future assignments of empirically not observed ro-vibrational energy levels. The comparison of the calculated and experimental ro-vibrational energy levels of the C2D2 and 13C2H2 isotopologues is also reported.

Dynamics characteristics of axial–torsional–lateral drill string system under wellbore constraints

The combination of air and liquid drilling technologies is usually used for increasing rate of penetration (ROP) and protecting reservoir in drilling engineering. Meanwhile, dynamics characteristics of drill string system are closely relevant to drilling safety and well trajectory quality in the practical drilling engineering. Hence, in this paper, dynamics characteristics of axial–torsional–lateral coupled drill string system under wellbore constraints are emphatically concerned in the air and liquid drilling process. Firstly, a rotor dynamics model for revealing the dynamics characteristics of bottom hole assembly (BHA) with single stabilizer is established by the lumped parameter method. Moreover, two novel contact models for describing real-time interactions of BHA-sidewall and bit-formation are established through Hertzian contact theory and discontinuous support model, respectively. Then normal, stick–slip and bit-bounce modes of BHA are discussed on basis of coupling vibrations of drilling string in lateral, axial, torsional and whirling motion. Meanwhile, the dynamics responses of the BHA in the processes of air and liquid drilling are conducted by contrast analysis. Subsequently, dynamics behaviors of the BHA at different hardness formations are further revealed through numerical computations. The results indicate that the lateral, axial and torsional vibrations of BHA in the air drilling process are obviously stronger than the liquid drilling process in the same operation conditions. When drill bit suddenly encounters hard formation, especially in air drilling process, the dynamics characteristics of drill string system are easily disturbed, and the coexistence motion of stick–slip and bit-bounce is appeared in BHA, thereby leading to severe whirling and lateral impact of drilling string. Besides, when the BHA is out of contact with the sidewall, only the forward whirling of BHA is implemented; when the sidewall is frequently impacted by the BHA, forward and backward whirling of drilling string are easily happened; when the BHA is contact with the sidewall, the backward whirling of drilling string is easily appeared. Therefore, the proposed model in this paper may offer a comprehensive understanding for dynamics of BHA, and is expected to further develop dynamics studies of the BHA with double or multiple stabilizers.

An innovative piezoelectric energy harvester inspired by a line tooth: design, dynamic model and broadband harvesting conditions

Piezoelectric energy harvesting is commonly considered to be a promising field of development for microelectronic devices due to its potential to address a variety of key supply problems. However, due to their geometric designs, traditional piezoelectric energy harvesters (PEHs) tend to only be able to cultivate energy from vibrations flowing in one direction. The results of PEHs only capable of harvesting mono-directional vibrations are that they suffer from narrow resonance frequency bands and low energy conversion efficiency. To overcome these difficulties, this paper proposes a PEH inspired by a line tooth (PEH-ILT) with the ability to collect three-dimensional stochastic vibrations. To do so, the PEH-ILT possesses a nonlinear geometric shape which can, in theory, be designed arbitrarily. An example PEH-ILT is illustrated in this paper as well corresponding nonlinear piezoelectric constitutive equations. The cylindrical spiral curve is inspired by the line tooth design and is intended to replicate a nonlinear electro-mechanical model and its electrical output. Furthermore, the PEH-ILT is evaluated in this study by interacting with the four basic vibrations such devices are expected to encounter. In addition, the broadband piezoelectric energy harvesting conditions of the PEH-ILT are parsed and determined through the Melnikov theory, providing a theoretical explanation to the broadband conditions of the harvester. And this study can lay the theoretical basis for practical applications.

Theoretical and Experimental Study on the Influence of Ultrasonic Vibration on Contact Friction

The effects of punch radius, deep-drawing speed and amplitude on the friction coefficient were studied on an improved drawing-bulging friction coefficient testing device on basis of ultrasonic vibration. A contact friction model based on the tribology theory of adhesion and plowing was constructed and used to explain the friction reduction phenomenon of applied out-of-plane normal vibration. The results show that the friction coefficient decreases with the increase of ultrasonic vibration amplitude. At the same deep-drawing height, the friction coefficient decreasing rates at amplitude of 7.8um and 10.1 um on the deep-drawing speed of 0.1 mm/s and 10 mm/s, were 6.7% and 18.8%, respectively. the friction coefficients at the punch radii R0.3 and R1.5 declined from 0.18 to 0.13 and from 0.12 to 0.11. The friction coefficients of thin specimens were larger than thick specimens whether ultrasonic vibration was applied or not. The average friction coefficient from theoretical modeling (μ_v) was smaller than the friction coefficient without ultrasonic vibration (μ_0), and the relative friction coefficient ratio declined with the rise of amplitude and was inversely proportional to time.

A local mode study of ring puckering effects in the infrared spectra of cyclopentane.

We report and interpret recently recorded high-resolution infrared spectra for the fundamentals of the CH2 scissors and CH stretches of gas phase cyclopentane at -26.1 and -50 °C, respectively. We extend previous theoretical studies of this molecule, which is known to undergo barrierless pseudorotation due to ring puckering, by constructing local mode Hamiltonians of the stretching and scissor vibrations for which the frequencies, couplings, and linear dipoles are calculated as functions of the pseudorotation angle using B3LYP/6-311++(d,p) and MP2/cc-pVTZ levels of theory. Symmetrization (D5h) of the vibrational basis sets leads to simple vibration/pseudorotation Hamiltonians whose solutions lead to good agreement with the experiment at medium resolution, but which miss interesting line fractionation when compared to the high-resolution spectra. In contrast to the scissor motion, pseudorotation leads to significant state mixing of the CH stretches, which themselves are Fermi coupled to the scissor overtones.

Reduced-dimensional vibrational models of the water dimer.

A model based on the finite-basis representation of a vibrational Hamiltonian expressed in internal coordinates is developed. The model relies on a many-mode, low-order expansion of both the kinetic energy operator and the potential energy surface (PES). Polyad truncations and energy ceilings are used to control the size of the vibrational basis to facilitate accurate computations of the OH stretch and HOH bend intramolecular transitions of the water dimer (H2 16O)2. Advantages and potential pitfalls of the applied approximations are highlighted. The importance of choices related to the treatment of the kinetic energy operator in reduced-dimensional calculations and the accuracy of different water dimer PESs are discussed. A range of different reduced-dimensional computations are performed to investigate the wavenumber shifts in the intramolecular transitions caused by the coupling between the intra- and intermolecular modes. With the use of symmetry, full 12-dimensional vibrational energy levels of the water dimer are calculated, predicting accurately the experimentally observed intramolecular fundamentals. It is found that one can also predict accurate intramolecular transition wavenumbers for the water dimer by combining a set of computationally inexpensive reduced-dimensional calculations, thereby guiding future effective-Hamiltonian treatments.

The Neural Basis of Skull Vibration Induced Nystagmus (SVIN).

I list a summary of the major clinical observations of SVIN in patients with total unilateral vestibular loss (TUVL) and show how basic results from neurophysiology can explain these clinical observations. The account integrates results from single neuron recordings of identified semicircular canal and otolith afferent neurons in guinea pigs in response to low frequency skull vibration with evidence of the eye movement response in cats to selective semicircular canal stimulation (both individual and combined) and a simple model of nystagmus generation to show how these results explain most of the major characteristics of SVIN.

Study on noise characteristics of 500kV oil-immersed transformer under no-load voltage condition

With the expansion of urban construction scale, substation noise has attracted more and more attention. Noise characteristics and propagation of the oil-immersed transformers are made important theoretical basis of vibration and noise reduction measures. Based on the 500kV oil-immersed transformer, the noise test of transformer structures had been completed. An advanced optical fiber testing system is adopted in the test, which solves the problem of accuracy and reliability in high electromagnetic field and solid-liquid coupling environment. Time domain analysis and frequency analysis for the test data were carried out. By test and analysis, noise characteristics are obtained by the large power transformers. The frequency spectrum of noise and winding vibration in transformer insulation oil is mainly 100Hz and frequency doubling. With the increase of load, the noise presents obvious nonlinear change process.

The Possibility to Recreate the Shapes of Objects on the Basis of Printer Vibration in the Additive Printing Process

Information protection is an important safety issue in many human activity fields. Technological advances and related ubiquitous computerization bring new challenges in this area. In particular, the problem concerns the protection of devices against non-invasive acquisition processed information in ICT systems. It is known that, e.g., VGA, DVI/HDMI interfaces or laser printer formatter systems that process visual signals are the effective sources of compromising electromagnetic emanations. The emission safety of the more and more commonly used 3D printers is less known. In many cases, the disclosure of information about printed objects might constitute an infringement of the state/industrial/commercial secret, copyright, patent protection, etc. In order to analyze the existing threat, a selected printer using FDM technology was tested. During the tests, simple objects were printed to identify the operation of the stepper motor and the movements of the printer head and the printer platform, which are sources of emissions in the secondary channels. The analyses performed focused on finding the correlation between the recorded vibration signals and the printer head movements when an object was printed. It was shown that the analysis of the registered sensitive signal runtimes and their spectrograms allowed to recreate the printed object shape. Three simple objects (a trihedron, a tube and a tetrahedron) were selected for testing because they include elements that allow the four major movements of the printer head to be easily recognized: along the X axis, along the Y axis, along an arc and diagonally (between the X and Y axes). The paper presents the test results and their analyses.

An alternative formulation of vibrational heat-bath configuration interaction

Recently, a selected vibrational configuration interaction method was introduced where the selection of the vibrational basis was performed using the heat-bath algorithm (VHCI). This method was adapted from the heat-bath configuration interaction (HCI) method used to solve the electronic Schrödinger equation. The selection algorithm in electronic HCI exploits the fact that most nonzero Hamiltonian matrix elements correspond to a double electronic excitation from one determinant to another and have values equal to the two-electron molecular orbital integrals such that they can be computed once, sorted, stored, and easily accessed later using a dictionary data structure. However, the Hamiltonian matrix elements in VHCI are more complicated than their electronic HCI counterparts, which lead to the possibility of different VHCI implementations. We explore one such implementation. The most significant differences compared to the original VHCI implementation is that we (i) explicitly compute all one- and two-quanta excitations, (ii) include all occupation-number contributions to the Hamiltonian matrix elements, and (iii) add individual Hartree product basis functions to the variational space rather than multiple basis functions as determined by operator products. We apply the new VHCI algorithm to ethylene oxide and naphthalene and achieve good accuracy relative to previous results at a modest computational cost.

Nonlinear free and forced vibration of porous piezoelectric doubly-curved shells based on NUEF model

In this paper, the nonlinear free and forced vibration of porous piezoelectric doubly-curved shells resting on visco-Pasternak foundation is performed within the framework of the nonuniform electric field (NUEF) model and von Kármán geometric nonlinearity assumption. Three types of porosity distribution are adopted to study the effect of porosity on the nonlinear dynamic responses of the current system. According to Hamilton’s principle, the nonlinear governing differential equations are obtained. For the nonlinear free vibration of porous piezoelectric doubly-curved shells, an analytical solution is achieved by utilizing the harmonic balance method. For the nonlinear forced vibration of the present system, the numerical solution is obtained by means of Runge–Kutta method. Through numerical examples, the relationship between nonlinear free vibration responses and nonlinear forced vibration responses is explored under different amplitudes of external excitation, width to thickness ratios, types of porosity distribution, electric field models and elastic foundation parameters. The main novelty of the present study is that the nonlinear frequency ratio that is calculated based on the nonlinear free vibration, is in excellent agreement with the resonant frequency ratio which is determined on the basis of the nonlinear forced vibration.

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

The main structural element of a reinforced concrete supports: mathematical modeling of mechanical vibrations

The problem of mathematical modeling of a reinforced concrete supports mechanical vibrations is considered. The basic vibrations of the main structural element (a road) are taken into account. The eigenfrequencies of the first three vibration modes of the rod are determined without and with prestressing of the rod. Participation factor of the basic, second and third modes is investigated.