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Theoretical calculations on the Raman and infrared (IR) vibrational modes in the tetragonal structure of the Ruddlesden-Popper (RP) phase Sr2BO4 (B = Ti, V, and Mn) have been done in this paper. The Raman and IR vibrational modes have been investigated using Wilson’s GF matrix method for the Sr2BO4 (B = Ti, V, and Mn) compounds having space group I4/mmm and symmetry D4h17 by analyzing the normal coordinates. The measurements of the zone centre vibrational modes have been calculated including three stretching and six bending force constants. In this work, the infrared frequencies for the Sr2MnO4 compound in phase I4/mmm have been assigned first and foremost. There is an extremely strong concurrence between the observed Raman and IR vibrational modes and our calculated results. To determine the role of interatomic forces in the calculated Raman and IR vibrational modes, the potential energy distributions for the Sr2BO4 (B = Ti, V, and Mn) compounds have also been investigated.

In this article, we present a refined mathematical model of the stress–strain state of multilayer composite cylindrical shells taking into account the piezoelectric effect. The displacements and the electric potential of the shell are represented as polynomials in the normal coordinates two degrees higher in comparison with the classical Kirchhoff–Love type of theory. A mathematical model of the electromechanical state of composite shells is obtained using the variational Lagrange principle. The formulated electroelasticity boundary value problem is solved by reducing three-dimensional equations to two-dimensional ones. An example of calculating the stress state of the boundary layer type of composite cylindrical shells with symmetric and asymmetric distributions of layers under the action of arbitrary mechanical and electrical loads is treated taking into account the piezoelectric effect.

Spatially localized vibrational normal mode coordinates are shown to reduce the importance of calculating the full set of two-electron terms in the molecular electronic Schrödinger equation. Electron correlation and dispersion interactions become less significant in (E,E)-1,3,5,7-octatetraene vibrational self-consistent field calculations when displacing remote atoms along multiple coordinates. Electron correlation interactions between spatially remote modes are also found to be less important compared to their corresponding uncorrelated interaction terms. Attenuation of the Coulomb operator indicates that the two-electron terms between remote electrons become less important for accurately describing the strongly contributing mode-coupling terms between sets of localized vibrational modes.

We derive how to incorporate topological features of Riemann surfaces in string amplitudes by insertions of bi-local operators called handle operators. The resulting formalism is exact and globally well-defined in moduli space. After a detailed and pedagogical discussion of Riemann surfaces, complex structure deformations, global vs local aspects, boundary terms, an explicit choice of gluing-compatible and global (modulo U(1)) coordinates (termed `holomorphic normal coordinates'), finite changes in normal ordering, and factorisation of the path integral measure, we construct these handle operators explicitly. Adopting an offshell local coherent vertex operator basis for the latter, and gauge fixing invariance under Weyl transformations using holomorphic normal coordinates (developed by Polchinski), is particularly efficient. All loop amplitudes are gauge-invariant (BRST-exact terms decouple up to boundary terms in moduli space), and reparametrisation invariance is manifest, for arbitrary worldsheet curvature and topology (subject to the Euler number constraint). We provide a number of complementary viewpoints and consistency checks (including one-loop modular invariance, we compute all one- and two-point sphere amplitudes, glue two three-point sphere amplitudes to reproduce the exact four-point sphere amplitude, etc.)., Comment: 328 pages, 34 figures. Various typos and an inconsequential error in Sec. 6.5 have been corrected. This version is an update to the published version

Vibrational configuration interaction calculations (VCI) have been performed to study the impact of the nature of the underlying coordinate systems, i.e., canonical vs localized normal coordinates, on accurate vibrational structure calculations for non-Abelian molecules. Once the correlation space is represented by real-valued primitive Hartree products, the assignment of vibrational states beyond the fundamentals is usually a tedious task and is further complicated by the use of non-symmetry-adapted coordinates. Our recently presented approach based on sparse grid integration of overlap integrals of the VCI wave function with the corresponding harmonic oscillator wave function has been used to determine and assign all fundamentals and vibrational overtones of allene and its deuterated isotopologue. Excellent agreement with available experimental results is observed in all cases.

A planar restricted elliptic three-body problem is considered. The motions close to the triangular libration points are studied. The problem parameters (the eccentricity of the orbit of the main attracting bodies and the ratio of their masses) are assumed to lie inside the linear stability region of the libration points. The magnitude of eccentricity is considered small. A linear canonical, periodic in true anomaly transformation is obtained analytically up to the second degree of eccentricity inclusive that reduces the Hamiltonian function of the linearized equations of perturbed motion to real normal form in the vicinity of the libration points. This form corresponds to two harmonic oscillators not connected to one another, with frequencies depending on the problem parameters. In constructing the normalizing canonical transformation, the Depri-Hori method of the perturbation theory of Hamiltonian systems is used. Its implementation in the problem under study relies heavily on computer systems of analytical calculations.

The singular structure of lower vibrational states of the difluorosilylene molecule (up to four total excitation quanta) is studied by expansion of energies of each state into series of the high-order Rayleigh–Schrodinger perturbation theory and analysis of their implicit properties of multiple-valuedness using Pade–Hermite fourth-order approximants. The quartic surface of potential energy in dimensionless normal coordinates is calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of values of multiple-valued approximants reproduces the variational solution with a high accuracy, while other values (beginning with the fourth polyad) in many cases coincide with energies of other states of the same polyad. The Fermi and Darling–Dennison resonances are analyzed based on facts of coincidence of singular complex branch points of the approximants for mutually interacting states inside a unit circle or near it on the complex plane. It is found that a pair of states can have several coinciding branch points of the solutions (in particular, inside the unit circle). A conclusion is made that this approach is an effective method for the determination of the polyad structure of vibrational states. Calculation parameters necessary for the reproducibility of key results are selected. The calculations are carried out by means of a Fortran-based software package using an arithmetic calculation package with a long mantissa of real numbers (200 digits).

International audience; We present the first use of curvilinear vibrational coordinates, specifically poly-spherical coordinates, in combination with vibrational coupled cluster theory. The polyspherical coordinates are used in the context of both the adaptive density-guided approach to potential energy surface construction and in the subsequent vibrational coupled cluster calculations of anharmonic vibrational states. Results obtained based on the polyspherical coordinate parametrization are compared to results obtained with the use of rectilinear vibrational coordinates, namely normal coordinates and hybrid optimized and localized coordinates for the formaldehyde molecule. This comparison is carried out with the full vibrational configuration interaction model, using the respective fully coupled potential energy surfaces and untruncated kinetic energy operators. The polyspherical coordinates are shown to facilitate an acceleration of convergence for truncated methods, when compared to the use of normal coordinates. We furthermore report on calculations on the hydrogen peroxide molecule in polyspherical coordinate 1 parametrization. The polyspherical vibrational coordinates are shown to perform very well, even for truncated methods, especially when considering the difficulty that rec-tilinear vibrational coordinates can exhibit in treating complicated internal molecular motion.

Density Functional Theory (DFT) method of the type (B3LYP) and a Gaussian basis set (6-311G) were applied for calculating the vibration frequencies and absorption intensities for normal coordinates (3N-6) at the equilibrium geometry of the Di and Tetra-rings layer (6, 0) zigzag single wall carbon nanotubes (SWCNTs) by using Gaussian-09 program. Both were found to have the same symmetry of D6d point group with C--C bond alternation in all tube rings (for axial bonds, which are the vertical C--Ca bonds in rings layer and for circumferential bonds C—Cc in the outer and mid rings bonds). Assignments of the modes of vibration IR active and inactive vibration frequencies (symmetric and asymmetric modes) based on the image modes applied by the Gaussian 09 display. The whole relations for the vibration modes were also done including \(\nu\)CH stretching, \(\nu\)C--C stretching, \(\delta\)CH, \(\delta\)ring (\(\delta\)C--C--C) deformation in plane of the molecule) and \(\gamma\)CH, \(\gamma\)ring (\(\gamma\)C--C--C) deformation out of plane of the molecule. The assignment also included modes of puckering, breathing and clock-anticlockwise bending vibrations.

DFT (3-21G, 6-31G and 6-311G/ B3LYP) and Semi-empirical PM3 methods were applied for calculating the vibration frequencies and absorption intensities for normal coordinates (3N-6) of the Tri-rings layer (6,0) Zigzag single wall carbon nanotube (SWCNT) at their equilibrium geometries which was found to have D6h symmetry point group with C-C bond alternation in all tube rings. As well as mono ring layer. Assignments of the modes of vibration were done depending on the pictures of their modes applying by Gaussian 03 program. The whole relations for the vibration modes were also done including (CH stretching, \(\nu\)CC stretching, deformation in plane of the molecule (\(\delta\)CH, \(\delta\)ring and \(\delta\)CCC), deformation out of plane of the molecule (\(\gamma\)CH and gring (\(\gamma\)CCC). Also include the assignment of puckering, breathing and clock-anticlockwise bending vibrations. Comparison for the geometry (the relations for axial bonds, which are the vertical C-C bonds (linear bonds) in the rings layer and for circumferential bonds which are the outer ring bonds), electronic properties and IR active vibration frequencies (asymmetric modes) of (Mono and Tri) rings layer were done. Clear relationships were found in the results of an odd layer number (Mono and Tri-rings layer). The theoretical results allow a comparative view of the charge density at the carbon atoms too.

Atomic Force Microscopy coupled with Tip Enhanced Raman Spectroscopy (AFM-TERS) was applied to obtain information about the structure and surface composition of single nano co-crystals. For this purpose, a co-crystalline system consisting of 2,4,6,8,10,12-hexanitro-2,4,6,8,10,12-hexaazatetracyclo-[5.5.0.03,11.05,9]-dodecane (CL-20) and 1,3,5,7-tetranitro-1,3,5,7-tetrazocane (HMX) in a molar ratio of 2 : 1 (CL-20/HMX) was chosen. CL-20/HMX nano-plates were prepared by spray flash evaporation. To ensure co-crystallinity and nanostructures, powder X-ray diffraction and AFM investigations were performed. The results demonstrate that coherence lengths and particle dimensions are on a similar level though coherence lengths appear shorter than measured particle dimensions. According to this fact, defects inside the nano co-crystals are minimized. The co-crystallinity was additionally proven by confocal Raman spectroscopy. Here, marker bands for pristine CL-20 and HMX were chosen which appear in the CL-20/HMX spectrum in an intensity ratio of ∼2.5 : 1 (CL-20 : HMX). Afterwards surface investigations of single CL-20/HMX nano-plates were performed by AFM-TERS. Due to the surface sensitivity of TERS, these experiments reveal that the ratio of the Raman intensities between CL-20 and HMX inverts at CL-20/HMX nano-plate surfaces. Therefore, it is concluded that nano co-crystal surfaces consist of molecular layers of HMX. A theoretical approximation of the normal coordinates of the investigated marker vibrations supports this conclusion since it can exclude the occurrence of the intensity ratio inversion because of the given orientation between CL-20/HMX nano-plates and the Raman scattering system. Based on this finding, an impact ignition mechanism is proposed, enabling explanation of the close impact sensitivity values of β-HMX and CL-20/HMX.

The impact of localized CH-stretching normal coordinates in comparison to canonical normal coordinates on the performance of accurate vibrational structure calculations has been studied for simple molecules of up to eight atoms. Two aspects have been considered in detail, (a) the (pre)screening of coupling terms within an n-mode expansion of the multidimensional potential energy surface and (b) the demands in vibrational configuration interaction calculations (VCI). All calculations have been performed in a realistic setup, and the effect of any approximation has been measured in deviations of the final VCI frequencies, which allows for a direct comparison with experimental data.

Let $c$ be a piecewise smooth wave speed on $\mathbb R^n$, unknown inside a domain $\Omega$. We are given the solution operator for the scalar wave equation $(\partial_t^2-c^2\Delta)u=0$, but only outside $\Omega$ and only for initial data supported outside $\Omega$. Using our recently developed scattering control method, we prove that piecewise smooth wave speeds are uniquely determined by this map, and provide a reconstruction formula. In other words, the wave imaging problem is solvable in the piecewise smooth setting under mild conditions. We also illustrate a separate method, likewise constructive, for recovering the locations of interfaces in broken geodesic normal coordinates using scattering control., Comment: 18 pages, 3 figures

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times\partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suitable neighborhood of $p$ in $M$, up to the natural diffeomorphism invariance of the problem. We also consider the closely related lens rigidity problem which is a more natural formulation if the boundary distance is not realized by unique minimizing geodesics. The lens relation measures the point and the direction of exit from $M$ of geodesics issued from the boundary and the length of the geodesic. The lens rigidity problem is whether we can determine the metric up to isometry from the lens relation. We solve the lens rigidity problem under the assumption that there is a function on $M$ with suitable convexity properties relative to $g$. This can be considered as a complete solution of a problem formulated first by Herglotz in 1905. We also prove a semi-global results given semi-global data. This shows, for instance, that simply connected manifolds with strictly convex boundaries are lens rigid if the sectional curvature is non-positive or non-negative or if there are no focal points. The key tool is the analysis of the geodesic X-ray transform on 2-tensors, corresponding to a metric $g$, in the normal gauge, such as normal coordinates relative to a hypersurface, where one also needs to allow weights. This is handled by refining and extending our earlier results in the solenoidal gauge.

Domain motions involved in the function of proteins can often be well described as a combination of motions along a handfull of low-frequency modes, that is, with the values of a few normal coordinates. This means that, when the functional motion of a protein is unknown, it should prove possible to predict it, since it amounts to guess a few values. However, without the help of additional experimental data, using normal coordinates for generating accurate conformers far away from the initial one is not so straightforward. To do so, a new approach is proposed: instead of building conformers directly with the values of a subset of normal coordinates, they are built in two steps, the conformer built with normal coordinates being just used for defining a set of distance constraints, the final conformer being built so as to match them. Note that this approach amounts to transform the problem of generating accurate protein conformers using normal coordinates into a better known one: the distance-geometry problem, which is herein solved with the help of the ROSETTA software. In the present study, this approach allowed to rebuild accurately six large amplitude conformational changes, using at most six low-frequency normal coordinates. As a consequence of the low-dimensionality of the corresponding subspace, random exploration also proved enough for generating low-energy conformers close to the known end-point of the conformational change of the LAO binding protein, lysozyme T4 and adenylate kinase., 11 pages, 5 figures

The assignment of vibrational states is an integral part of quantum chemical calculations, which supports the analysis of experimental infrared spectra. In variational calculations, usually, it is the leading coefficient of the configuration interaction vector, which provides the state identity. However, this concept will possibly fail in case of special coordinate systems, such as, for example, localized normal coordinates, or within calculations for overtones of non-Abelian molecules, when a real valued configuration basis has been employed. A combination of both renders a proper assignment fairly tedious. We present a route to overcome this problem by using a highly efficient calculation of multidimensional overlap integrals based on the Smolyak quadrature. Beside this, a general protocol for the symmetry assignment of vibrational states will be discussed, which completes a general assignment. Extensive benchmark calculations are provided for the fundamental modes and overtones of chloromethane, CH3Cl, in canonical and localized normal coordinates based on accurate potential energy surfaces obtained from explicitly correlated coupled-cluster theory. In addition, the linear CNNC molecule has been studied, for which hardly any reference data do exist.

The vibronic coupling constant (VCC), as well as the vibronic coupling density (VCD), will be exactly defined in this chapter. Before the precise definition of VCC and VCD, we will introduce Dirac’s notation and the basic concept of group and symmetry. The definition of the normal coordinates will be given, and the characters and the direct products will be tabulated for the treatments of group and symmetry.

In the final chapter of this manuscript we turn our attention towards traces on submanifolds N in fractional Sobolev, Besov- and Triebel-Lizorkin spaces defined on noncompact Riemannian manifolds M. Here M is of dimension d and N of dimension k with k < d, respectively. Usually, these spaces are defined via geodesic normal coordinates. However, for our investigations we require more flexibility in the choice of coordinates and therefore rely on the results obtained in Chap. 2, where we studied which conditions are sufficient such that different local coordinates give equivalent norms and therefore yield the same spaces.

Small oscillations of mechanical systems are considered. The corresponding linear differential equations with constant coefficients are derived using the two methods: the linearization of the initial nonlinear system of equations or preliminary reducing of the expressions for kinetic and potential energies to quadratic forms with constant coefficients. General solutions to equations of small oscillation are found, the notions of natural frequencies and normal modes of oscillation are introduced, their properties are studied. The case of zero frequency and the case when several natural frequencies coincide are examined with the help of normal coordinates. The Rayleigh theorem and the Courant theorem are proved. Free small oscillations in the presence of resistance are considered. The Thomson and Tait theorems on the influence of dissipative and gyroscopic forces on the stability of state of equilibrium are presented. Forced oscillations under the action of arbitrary and periodic forces are considered. The relationship between the impulse transient function and the transfer function is established.

The investigation of the Raman and Infrared wavenumbers has been made in tetragonal structure of the complex Sr2TiO4. This is a member of the Ruddlesden-Popper series Srn+1TinO3n+1 with n=1. Five stretching and four bending force constants were used for the calculation of zone center phonons in this paper by applying a short range force constant model, using normal coordinates having space group I4/mmm and symmetry D4h17. The Calculated frequencies have been assigned 4h and are in a very good agreement with the experimental frequencies in the literature. The contribution from each force constant towards the Raman and infrared wave numbers has also been determined with the investigation of the potential energy distribution.

International audience; Vibrational energy levels of H 2 CO are reported using global variational nuclear motion calculations from new ab initio and empirically optimized full 6-dimensional ab initio potential energy surfaces in ground electronic state of the formaldehyde molecule. Ab initio calculations were carried out using extended electronic structure coupled-cluster calculations accounting for dynamic electron correlations including triple and quadruple excitations as well as relativistic and diagonal Born-Oppenheimer corrections. Variational nuclear motion calculations are compared in different set of coordinates with exact kinetic energy operator and in normal coordinates with Watson-Eckart kinetic energy operator. Our best ab initio potential energy surface including the above mentioned contributions provides the RMS (obs.-calc.) errors of 0.25 cm-1 for fifteen vibrational band centers without empirically adjusted parameters. The average error drops down to 0.08 cm-1 for an empirically optimized potential energy function with six adjusted parameters corresponding to quadratic force field terms. The estimation of the accuracy for the calculated vibrational levels in an extended range up to 4500 cm-1 shows that the set of ab initio vibrational levels could be more consistent than experimental levels obtained from earlier studies of low resolution spectra. The comparison of the calculated and experimental vibrational energy levels of the D 2 CO isotopologue is also reported.

Spectral and dynamical properties of molecular donor-acceptor systems strongly depend on the steric arrangement of the constituents with exciton coupling J as a key control parameter. In the present work we study two peri-arylene based dyads with orthogonal and parallel transition dipoles for donor and acceptor moieties, respectively. We show that the anharmonic multi-well character of the orthogonal dyad's intramolecular potential explains findings from both stationary and time-resolved absorption experiments. While for a parallel dyad, standard quantum chemical estimates of J at 0 K are in good agreement with experimental observations, J becomes vanishingly small for the orthogonal dyad, in contrast to its ultrafast experimental transfer times. This discrepancy is not resolved even by accounting for harmonic fluctuations along normal coordinates. We resolve this problem by supplementing quantum chemical approaches with dynamical sampling of fluctuating geometries. In contrast to the moderate Gaussian fluctuations of J for the parallel dyad, fluctuations for the orthogonal dyad are found to follow non-Gaussian statistics leading to significantly higher effective J in good agreement with experimental observations. In effort to apply a unified framework for treating the dynamics of optical coherence and excitonic populations of both dyads, we employ a vibronic approach treating electronic and selected vibrational degrees on an equal footing. This vibronic model is used to model absorption and fluorescence spectra as well as donor-acceptor transport dynamics and covers the more traditional categories of Förster and Redfield transport as limiting cases.

We introduce a procedure to determine the size and shape of normal neighborhoods in any spacetimes and their dependence on the precision of the measurements performed by arbitrary observers. As an example, we consider the Schwarzschild geometry in Riemann and Fermi normal coordinates and determine the size and shape of normal neighborhoods in the vicinity of the event horizon. Depending on the observers, normal neighborhoods extend to the event horizon and even beyond into the black hole interior. It is shown that the causal structure supported by normal neighborhoods across an event horizon is consistent with general relativity. In particular, normal neighborhoods reaching over an event horizon are void of the Schwarzschild coordinate singularity. In addition, we introduce a new variant of normal coordinates which we call Fermi normal coordinates around a point, unifying features of Riemann and Fermi normal coordinates, and analyze their neighborhoods., Comment: 21 pages, 4 figures, minor spelling change, agrees with publication in Physical Review D

The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and simulation free, in the sense that it allows to pass from physical coordinates (FE nodes) to normal coordinates, describing the dynamics in an invariant-based span of the phase space. The number of master modes for the ROM is not a priori limited since a complete change of coordinate is proposed. The underlying theory ensures the quality of the predictions thanks to the invariance property of the reduced subspace, together with their curvatures in phase space that accounts for the nonresonant nonlinear couplings. The method is applied to a beam discretised with 3D elements and shows its ability in recovering internal resonance at high energy. Then a fan blade model is investigated and the correct prediction given by the ROMs are assessed and discussed. A method is proposed to approximate an aggregate value for the damping, that takes into account the damping coefficients of all the slave modes, and also using the Rayleigh damping model as input. Frequency-response curves for the beam and the blades are then exhibited, showing the accuracy of the proposed method., 34 pages, 10 figures, 2 tables, submitted to CMAME

Using quantum chemistry computations TDDFT//B3LYP/6-31G(d), we study resonance Raman spectra of the Q band for meso-tetrakis(3,5-di-tertiarybutylphenyl)-porphyrin (H2TBPP) molecule due to the Franck-Condon and non-Condon mechanisms including the Herzberg-Teller and second-order Herzberg-Teller terms. Generally, the Herzberg-Teller terms are large. However, for some vibrational modes, the second-order Herzberg-Teller terms are the strongest and dominate resonance Raman spectra, which may also impact on fluorescence and absorption spectra. Hence, the Taylor expansion of the electric dipole transition moment with respect to the normal coordinates at the equilibrium structure of the ground electronic state may not converge for H2TBPP. A method to solve this problem is suggested.

Experimental analysis of the motion in a system of two coupled oscillators with arbitrary initial conditions was performed and the normal coordinates were obtained directly. The system consisted of two gliders moving on an air track, joined together by a spring and joined by two other springs to the fixed ends. From the positions of the center of mass and the relative distance, acquired by analysis of the digital video of the experiment, normal coordinates were obtained, and by a non linear fit the normal frequencies were also obtained. It is shown that although the masses of the springs are relatively small compared to that of the gliders, it is necessary to take them into consideration to improve the agreement with the experimental results. This experimental-theoretical proposal is targeted to an undergraduate laboratory., 13 pages, 4 figs, minor changes

Mid-Infrared spectra of pyrimidine (PM) and pyrazine (PZ) were recorded in the gas phase using a multi-pass long path gas cell. The IR band structure of these compounds above and below 3000 cm−1 is very broad and contains many humps and shoulders. These humps and shoulders are due to various higher quantum excitation of low-frequency vibrational modes, which participate in Fermi resonance with the nearby C H stretch fundamentals and appears in this region. We constructed an Effective Spectroscopic Hamiltonian (ESH) in a mixed local mode (LM) normal mode (NM) basis to assign the various overtone and combination bands in the C H stretching region of these compounds. The C H stretching vibrations of both PZ and PM were treated as symmetrized anharmonic Morse oscillators in local coordinates and the in-plane deformations down to 1000 cm−1 were treated as normal coordinates. The ESHs were diagonalized and the resulting eigenvalues were subsequently fitted in a given parameter space with the experimentally observed bands. The eigenvalues of the converged Hamiltonian are the anharmonic frequencies and the transition intensities were obtained by summing the squared eigenvector components. The overtone and combination transitions near 3000 cm−1 of both PM and PZ were identified and assigned from the eigenvector coefficients of the ESH matrix. The wavefunctions of a pure C H stretch, overtone of the H C C in-plane bend and due to Type 1 Fermi coupling (resonance between a fundamental with an overtones of a low frequency mode, in this case resonance between the C H strech and the overtone of H C C in-plane bending modes) has been demonstrated pictorially.

Manifolds endowed with an affine geometry of general type with nontrivial metric, torsion, and nonmetricity tensor are considered. Such manifolds have recently attracted much attention due to the construction of generalized gravity models. Under the assumption that all geometric objects are real analytic functions, normal coordinates in a neighborhood of an arbitrary point are constructed by expanding the connection and the metric in Taylor series. It is shown that the normal coordinates are a generalization of a Cartesian coordinate system in Euclidean space to the case of manifolds with any affine geometry. Moreover, the components of any real analytic tensor field in a neighborhood of any given point are represented in the form of a power series whose coefficients are constructed from the covariant derivatives and the curvature and torsion tensors evaluated at this point. For constant curvature spaces, these series are explicitly summed, and an expression for the metric in normal coordinates is found. It is shown that normal coordinates determine a smooth surjective mapping of Euclidean space to a constant curvature manifold. The equations for extremals are explicitly integrated in normal coordinates for constant curvature spaces. A relationship between normal coordinates and the exponential mapping is analyzed.

An atom-atom interaction potential for the H2O–A system is proposed in a form that depends on the normal coordinates q of an H2O molecule. Vibrational and rotational corrections to the effective potential for H2O–He and H2O–Ar systems are calculated and their impact on the calculated broadening coefficients γ is analysed for the absorption lines of different vibrational H2O bands in the case of broadening by helium and argon. It is shown that excitation of the stretching modes of vibrations in the H2O molecule leads to an increase in the calculated broadening coefficients γ. Accounting for rotational corrections, γ increases by 15% for the lines with rotational quantum number K a = 9 for the lower transition state in the case of broadening by He and by 4% when broadening by Ar.

The sampling of saddle points (SPs) on a potential energy surface (PES) is critical for describing the dynamics and transport properties of materials. Here, we propose a saddle point search (SPS) method that enables efficient sampling of practically meaningful SPs around a local minimum by converting the PES to the scaled normal coordinates (SNCs), together with the dimer method. We find that the pathway leading to a SP using SNCs is highly efficient and is independent of the system size. This results in the significantly increased SPS efficiency and the enhanced possibility of a complete catalog of the SPs. In addition, we perform SPSs using the SNCs for the diffusion of a vacancy and a dumbbell in body-centered cubic Fe and demonstrate (1) the required number of system force evaluations for a SPS decreases by at least an order of magnitude, (2) the effects of the system size on the number of force evaluations are greatly decreased, and (3) the effects of the number of atoms involved in the SPS on the probability of finding practically meaningful SPs is mostly eliminated. We also discuss the computational cost of introducing the SNCs. It should be highlighted that the SNCs could also be applied to other minimum-mode following methods, demonstrating the general versatility of the proposed method.