Tensors In Cartesian Coordinates Research Articles

Advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase.

The purpose of this review is to demonstrate advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase. Molecular spectroscopy, particularly vibrational spectroscopy and electronic spectroscopy, has been used extensively for a wide range of areas of chemical sciences and materials science as well as nano- and biosciences because it provides valuable information about structure, functions, and reactions of molecules. In the meantime, quantum chemical approaches play crucial roles in the spectral analysis. They also yield important knowledge about molecular and electronic structures as well as electronic transitions. The combination of spectroscopic approaches and quantum chemical calculations is a powerful tool for science, in general. Thus, our article, which treats various spectroscopy and quantum chemical approaches, should have strong implications in the wider scientific community. This review covers a wide area of molecular spectroscopy from far-ultraviolet (FUV, 120-200 nm) to far-infrared (FIR, 400-10 cm-1)/terahertz and Raman spectroscopy. As quantum chemical approaches, we introduce several anharmonic approaches such as vibrational self-consistent field (VSCF) and the combination of periodic harmonic calculations with anharmonic corrections based on finite models, grid-based techniques like the Numerov approach, the Cartesian coordinate tensor transfer (CCT) method, Symmetry-Adapted Cluster Configuration-Interaction (SAC-CI), and the ZINDO (Semi-empirical calculations at Zerner's Intermediate Neglect of Differential Overlap). One can use anharmonic approaches and grid-based approaches for both infrared (IR) and near-infrared (NIR) spectroscopy, while CCT methods are employed for Raman, Raman optical activity (ROA), FIR/terahertz and low-frequency Raman spectroscopy. Therefore, this review overviews cross relations between molecular spectroscopy and quantum chemical approaches, and provides various kinds of close-reality advanced spectral simulation for condensed phases.

Stress-induced crystal transition of poly(butylene succinate) studied by terahertz and low-frequency Raman spectroscopy and quantum chemical calculation

We measured terahertz (THz) and low-frequency Raman spectra of Poly (butylene succinate) (PBS) which shows the crystal transition from α to β by stretching. For the assignment of the absorption peaks in the low-frequency region, we performed quantum chemical calculations with Cartesian-coordinate tensor transfer (CCT) method. Four major peaks appeared in the THz spectra of PBS at around 58, 76, 90, and 100cm−1, and in the low-frequency Raman spectra a peak was observed at 88cm−1. The THz peak at 100cm−1 and the Raman peak at 88cm−1 show a shift to a lower wavenumber region with increasing temperature. The quantum chemical calculation of β crystal form reveals the new peak appears above 100cm−1. It was found that two kinds of peaks overlapped at around 100cm−1 in the THz spectra of PBS. One of them can be assigned to a weak hydrogen bond between the C=O and CH2 groups in the intermolecular chains, which is perpendicular to the molecular chain of the α crystal form. Another one showed a parallel polarization which can be assigned to the intramolecular interaction between O (ether) and H-C groups in the β crystal form. The position of the peak at around 100cm−1 in the perpendicular polarization changed to a lower wavenumber region with stretching, because of the weakening of the intermolecular hydrogen bonding by increasing the interatomic distances. On the other hand, that of the parallel polarization shifts to a higher wavenumber region because of the shortening of the interatomic distance from α to β crystal form (the strength of the intramolecular hydrogen bonding became stronger) by stretching.

Three different kinds of weak C-H⋯O=C inter- and intramolecular interactions in poly(ε-caprolactone) studied by using terahertz spectroscopy, infrared spectroscopy and quantum chemical calculations

The long range structure of poly(ε-caprolactone) (PCL) has been studied by terahertz (THz) spectroscopy, infrared (IR) spectroscopy and quantum chemical calculations (QCCs). Natural bond orbital (NBO) analysis and the calculation of the interatomic distances between C-H and O=C groups in PCL crystalline indicate that the PCL chain has three kinds of weak intermolecular interactions between the CH2 and C=O groups. In the IR spectra, significant changes due to the influence of hydrogen bondings were observed in the CH2 and C=O stretching vibration regions. The results of QCCs performed by using the Cartesian Coordinate Tensor Transfer (CCT) method to assign the THz spectra of PCL suggest that the peaks at 47 and 67 cm−1 reflect the atomic motions of the C=O + CH2 moiety derived from the weak C-H⋯O=C hydrogen bondings. THz spectroscopy allows us to observe the intermolecular hydrogen bondings of the polymer separately, since hydrogen bondings with different strengths yield different peaks in a THz spectrum. The results of THz and IR spectral analysis, and QCCs all indicate that PCL forms three kinds of weak intermolecular C-H⋯O=C hydrogen bondings between the CH2 and C=O groups. This may be one of the causes for the low melting temperature but high crystallinity of PCL. There are about six interactions in PCL and about sixteen interactions in polyglycolic acid (PGA), within the lamellar thickness. Therefore, it is suggested that the difference in the number of intermolecular interactions within the lamellar thickness leads to the difference in melting points between the two polymers.

Low-Frequency Vibrational Modes of Poly(glycolic acid) and Thermal Expansion of Crystal Lattice Assigned On the Basis of DFT-Spectral Simulation Aided with a Fragment Method.

Low-frequency vibrational modes of lamellar crystalline poly(glycolic acid) (PGA) were measured on Raman and far-infrared (FIR) spectra. Among the observed bands, an FIR band at ∼70 cm-1 and a Raman band at 125 cm-1 showed a gradual lower-frequency shift with increasing temperature from 20 °C to the melting point at ∼230 °C. Their polarization direction was perpendicular to the chain axis of PGA. Both spectra were quantum-mechanically simulated with the aid of a fragment method, the Cartesian-coordinate tensor transfer, which enabled an explicit consideration of molecular interactions between two adjacent polymer chains. Good agreement was achieved between the experiment and theory in both spectra. The temperature-sensitive bands at ∼70 cm-1 in FIR and at 125 cm-1 in Raman comprise the out-of-plane C═O bending motion. The temperature-dependent shifts of the low-frequency bands were successfully simulated by the DFT-spectral calculation, exploring that the main origin of the shifts is the thermal expansion of the crystal lattice. This result indicates that the thermally shifted bands may be used as an indicator of the lattice expansion of PGA. Possible changes in intermolecular interactions of PGA under temperature rising were ascribed on the basis of natural bond orbital theory. The steric repulsion between the carbonyl O atom in one chain and the H-C bond in the adjacent chain will be a dominant interaction in the lattice-expanding process, which would cause the observed thermal shifts of the bending modes. Comparisons of the spectral assignment for PGA obtained in this study and that for poly-(R)-3-hydroxybutyrate (PHB) reported by us suggest that crystalline polyesters give vibrational modes composed of out-of-plane bending motion of C═O groups between ∼70 and ∼125 cm-1, the modes of which are sensitive to the thermal expansion of crystal lattice and its concomitant changes in their intermolecular interactions.

LayerOptics: Microscopic modeling of optical coefficients in layered materials

Theoretical spectroscopy is a powerful tool to describe and predict optical properties of materials. While nowadays routinely performed, first-principles calculations only provide bulk dielectric tensors in Cartesian coordinates. These outputs are hardly comparable with experimental data, which are typically given by macroscopic quantities, crucially depending on the laboratory setup. Even more serious discrepancies can arise for anisotropic materials, e.g., organic crystals, where off-diagonal elements of the dielectric tensor can significantly contribute to the spectral features. Here, we present LayerOptics, a versatile and user-friendly implementation, based on the solution of the Maxwell’s equations for anisotropic materials, to compute optical coefficients in anisotropic layered materials. We apply this tool for post-processing full dielectric tensors of molecular materials, including excitonic effects, as computed from many-body perturbation theory using the exciting code. For prototypical examples, ranging from optical to X-ray frequencies, we show the importance of combining accurate ab initio methods to obtain dielectric tensors, with the solution of the Maxwell’s equations to compute optical coefficients accounting for optical anisotropy of layered systems. Good agreement with experimental data supports the potential of our approach, in view of achieving microscopic understanding of spectroscopic properties in complex materials.

Terahertz Spectroscopy in Polymer Research: Assignment of Intermolecular Vibrational Modes and Structural Characterization of Poly(3-Hydroxybutyrate)

This paper reviews our recent spectroscopic study on poly(3-hydroxybutyrate) (PHB) for the terahertz (THz) frequency region. Clear differences are observed between the absorption spectra of crystalline and amorphous PHB, indicating that the higher order conformation of PHB determines the THz spectra. The THz polarization spectra are measured for a stretched PHB sample and the directions of the vibrational transition moments are determined. The spectra are found to have temperature dependences owing to which the peaks at lower frequencies shift with temperature, suggesting a large anharmonicity of the vibrational potential. To investigate low frequency vibrational modes more thoroughly, low-frequency Raman spectra are also measured for crystalline PHB. The observed vibrational modes in both THz and Raman spectra were successfully assigned by a quantum mechanical simulation of spectra with the Cartesian coordinate tensor transfer (CCT) method, which can provide realistic spectra of the PHB by considering the intermolecular interactions among polymer chains. The potential of CCT for the assignment of THz spectra is demonstrated. On the basis of these results, the structural change in PHB is studied. The isothermal crystallization of PHB is monitored with the help of THz absorption spectra. The observed spectra clearly show the crystal growth, and the crystallization rate is estimated by Avrami's model. Next, the isothermal crystallization spectra are analyzed using two-dimensional correlation spectroscopy (2DCOS). By using the synchronous and asynchronous data plots, the correlation between the absorption peaks and the temporal order of the changes in spectral intensity is determined.

Quantum Mechanical Interpretation of Intermolecular Vibrational Modes of Crystalline Poly-(R)-3-Hydroxybutyrate Observed in Low-Frequency Raman and Terahertz Spectra

Low-frequency vibrational bands observed in the Raman and terahertz (THz) spectra in the region of 50-150 cm(-1) of crystalline powder poly-(R)-3-hydroxybutyrate (PHB) were assigned based on comparisons of the Raman and THz spectra, polarization directions of THz absorption spectra, and their congruities to quantum mechanically (QM) calculated spectra. This combination, Raman and THz spectroscopies and the QM simulations, has been rarely adopted in spite of its potential of reliable assignments of the vibrational bands. The QM simulation of a spectrum has already been popular in vibrational spectroscopies, but for low-frequency bands of polymers it is still a difficult task due to its large scales of systems and a fact that interactions among polymer chains should be considered in the calculation. In this study, the spectral calculations with the aid of the Cartesian-coordinate tensor transfer (CCT) method were applied successfully to the crystalline PHB, which include the explicit consideration of an intermolecular interaction among helical polymer chains. The agreements between the calculations and the experiments are good in both the Raman and THz spectra in terms of spectral shapes, frequencies, and intensities. A Raman active band at 79 cm(-1) was assigned to the intermolecular vibrational mode of the out-of-plane C═O + CH(3) vibration. A polarization state of the corresponding far-infrared absorption band at ∼82 cm(-1), perpendicular to the helix-elongation direction of PHB, was reproduced only under the explicit correction, which indicates that this polarized band originates from the interaction among the polymer chains. The calculation explored that the polarization direction of this band was along the a axis, which is consistent with the direction in which weak intermolecular hydrogen bonds are suggested between the C═O and CH(3) groups of two parallel polymer chains. The results obtained here have confirmed sensitivity of the low-frequency vibrational bands to the weak hydrogen bonds among the polymer chains.

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.

Structure and Vibrational Motion of Insulin from Raman Optical Activity Spectra

The Raman optical activity (ROA) spectroscopic technique has been applied in the past to many biologically relevant systems including peptides, proteins, sugars, and even viruses. However, theoretical interpretation of the spectra relies on lengthy quantum-chemical computations, which are difficult to extend to larger molecules. In the present study, ROA and Raman spectra of insulin under a range of various conditions were measured and interpreted with the aid of the Cartesian-coordinate tensor transfer (CCT) method. The CCT methodology yielded spectra of insulin monomer and dimer of nearly ab initio quality, while at the same time reproducing the experimental data very well. The link between the spectra and the protein structure could thus be studied in detail. Spectral contributions from the peptide backbone and the amino acid side chains were calculated. Likewise, specific intensity features originating from the α-helical, coil, β-sheet, and 3(10)-helical parts of the protein could be deciphered. The assignment of the Raman and ROA bands to intrinsic molecular coordinates as based on the harmonic force field calculation revealed their origin and degree of locality. Alternatively, the relation of the structural flexibility of insulin to the inhomogeneous broadening of spectral bands was studied by a combination of CCT and molecular dynamics (MD). The present study confirms the sensitivity of the ROA technique to some subtle static and dynamic changes in molecular geometry, and many previous ad hoc or semiempirical spectral-structure assignments could be verified. On the other hand, a limitation in longer-range tertiary structure sensitivity was revealed. Unlike for smaller molecules with approximately equal contributions of the electric dipole (α), quadrupole (A), and magnetic dipole (G') polarizabilities, only the electric dipolar polarization (α) interactions seem to dominate in the protein ROA signal. The simulations concern the largest molecule for which such spectra were interpreted by a priori procedures and significantly enhance protein folding studies undertaken by this technique.

Applications of the Cartesian coordinate tensor transfer technique in the simulations of vibrational circular dichroism spectra of oligonucleotides

The application of the Cartesian coordinate tensor transfer (CCT) technique for simulations of the IR absorption and vibrational circular dichroism (VCD) spectra of relatively large nucleic acid fragments is demonstrated on several case studies. The approach is based on direct ab initio calculations of atomic tensors, determining molecular properties, for relatively small fragments, and subsequent transfer of these tensors to the larger systems in Cartesian coordinates. This procedure enables precise computations of vibrational spectra for large biomolecular systems, currently with up to several thousands of atoms. The versatile ability of the CCT methods is emphasized on the examples of VCD and IR absorption spectra calculations for B- and Z-forms of DNA, single-, double-, and triple-stranded RNA helices and DNA structures with different base content and sequences. The development and recent improvements of the methodology are followed, including utilization of the constrained normal mode optimization (NMO) strategy and combined quantum mechanics and molecular dynamics simulations. Advantages, drawbacks, and recommendations for future improvements of the CCT method as applied to nucleic acid spectra calculations are discussed.

A method to derive new Green’s tensors for polar domains

An efficient unified method to derive Green’s tensors in Cartesian coordinates, called the incompressible influence elements method (IIEM), had been elaborated and published earlier. When extending this method for the domains of polar system of coordinates the additional difficulties have appeared, because the incompressible Green’s tensor does not satisfy, as in Cartesian coordinates, the Poisson’s equation. This paper describes general integral representations for Green’s tensors via special introduced scalar Green’s functions in polar coordinates. Using them the IIEM has been extended to polar domains. To this end, using the IIEM, for deriving the Green’s tensors in polar coordinates, it is necessary either to compute some convolutions of product of scalar Green’s functions or to solve boundary integral equations about elastic dilatation. The paper deals with closed-form Green’s tensors in polar coordinates for an angle and a circle. As particular cases, the respective Green’s tensors for quadrant and half-plane with twice-mixed boundary conditions also are derived.

Interpretation of Raman and Raman Optical Activity Spectra of a Flexible Sugar Derivative, the Gluconic Acid Anion

Raman scattering and its polarized extension, Raman optical activity (ROA), are commonly used for monitoring of molecular conformational equilibria in solutions. This is complicated for saccharides due to extensive motions of the hydroxyl groups and other molecular parts. Standard interpretation procedures involving ab initio spectral simulations for a limited set of conformers are not adequate. In this study, a more general approach is proposed for the gluconic acid anion taken as a model compound, where quantum simulations of the spectra are directly coupled with molecular dynamics (MD) techniques. Such a multiscale approach reveals how the structural information is encoded in the broadened spectral lines. The spectra were simulated for solvent-solute clusters generated by MD. Conformational averaging was enabled by a limited library of conformers for which the spectral parameters could be calculated ab initio and moved on the MD geometries by Cartesian coordinate tensor transfer techniques. The B3LYP/CPCM/6-31+G** approximation was used as a default for computation of the source force fields and polarizability derivatives. The spectra thus obtained relatively faithfully reproduced most of the experimental features. The Amber and polarizable Amoeba MD force fields produced similar results; application of the latter, however, was limited by the long time necessary to achieve a converged conformational equilibrium. Both MD simulation and spectral averaging suggest that the hydroxyl groups as well as the backbone C-C bonds rotate relatively freely, with some restrictions in the vicinity of the carboxyl group. In spite of the averaging, spectral response of characteristic vibrational normal mode families, such as CH and OH bending, can clearly be identified in the spectra. The simulations thus confirm the experimental fact that flexible saccharides exhibit significant vibrational activity that reveals precious information about molecular structure and dynamics encoded in the Raman and ROA spectral shapes.

Transfer of molecular property tensors in cartesian coordinates: A new algorithm for simulation of vibrational spectra

A direct transfer of Cartesian molecular force fields FF and electric property tensors is tested on model systems and compared to transfer in internal coordinates with an aim to improve simulation of vibrational spectra for larger molecules. This Cartesian transformation can be implemented easily and offers greater flexibility in practical computations. It can be also applied for transfer of anharmonic derivatives. The results for model calculations of the force field and vibrational frequencies for N-methylacetamide show that our method removes errors associated with numerical artifacts caused by nonlinearity of the otherwise required Cartesian to internal coordinate transformation. For determination of IR absorption and vibrational circular dichroism intensities, atomic polar and axial tensors were also transferred in the Cartesian representation. For the latter, which are dependent upon the magnetic dipole operator, a distributed origin gauge is used to avoid an origin dependence. Comparison of the results of transferring ab initio FF and intensity parameters from an amide dimer fragment onto a tripeptide with those from a conventionally determined tripeptide FF document some limitations of the transfer method and its possible applications in the vibrational spectroscopy. Finally, application to determination of the FF and spectra for helical heptapeptide are presented and compared to experimental results. Q 1997 by John Wiley & Sons, Inc.