Normal Mode Basis Research Articles

Mesoscale Molecular Simulations of Fabry-Pérot Vibrational Strong Coupling.

Developing theoretical frameworks for vibrational strong coupling (VSC) beyond the single-mode approximation is crucial for a comprehensive understanding of experiments with planar Fabry-Pérot cavities. Herein, a generalized cavity molecular dynamics (CavMD) scheme is developed to simulate VSC of a large ensemble of realistic molecules coupled to an arbitrary 1D or 2D photonic environment. This approach is built upon the Power-Zienau-Woolley Hamiltonian in the normal mode basis and uses a grid representation of the molecular ensembles to reduce the computational cost. When simulating the polariton dispersion relation for a homogeneous distribution of molecules in planar Fabry-Pérot cavities, our data highlight the importance of preserving the in-plane translational symmetry of the molecular distribution. In this homogeneous limit, CavMD yields the consistent polariton dispersion relation as an analytic theory, i.e., incorporating many cavity modes with varying in-plane wave vectors (k∥) produces the same spectrum as the system with a single cavity mode. Furthermore, CavMD reveals that the validity of the single-mode approximation is challenged when nonequilibrium polariton dynamics are considered, as polariton-polariton scattering occurs between modes with the nearest neighbor k∥. The procedure for numerically approaching the macroscopic limit is also demonstrated with CavMD by increasing the system size. Looking forward, our generalized CavMD approach may facilitate understanding vibrational polariton transport and condensation.

First-Principles Anharmonic Infrared and Raman Vibrational Spectra of Materials: Fermi Resonance in Dry Ice.

We introduce a computational tool for the quantum-mechanical simulation of anharmonic infrared and Raman vibrational spectra of materials. The approach, implemented in the CRYSTAL software, stems from Taylor's expansion of the potential energy surface (PES) on the basis of normal modes up to cubic and quartic terms. The PES can be sampled with four different numerical schemes at the level of density functional theory (DFT), with local, generalized-gradient, and hybrid density functional approximations. Anharmonic states are obtained by solving Shrödinger's nuclear equation with either the vibrational self-consistent field (VSCF) or vibrational configuration interaction (VCI) methods. Nuclear quantum effects (NQEs) are thus fully accounted for. Infrared intensities are computed numerically through a Berry phase approach or analytically through a coupled-perturbed (CP) approach. Raman intensities are computed analytically via the CP approach. A variety of anharmonic features of vibrational spectra of materials can be simulated, including band shifts, combination bands, overtones, resonances (first-order Fermi, second-order Darling-Dennison), and hot bands. We showcase the effectiveness of the approach on the description of a first-order Fermi resonance (FR) in CO2 dry ice: a challenging test-case given that the FR occurs in the Raman spectrum, requires NQEs, and involves two- and three-mode couplings. Fundamental mechanistic differences with respect to the well-known FR in molecular CO2 are addressed. This application represents the first quantum-mechanical, periodic description of FR in dry ice.

Random Matrix Theory for Sound Propagation in a Shallow-Water Acoustic Waveguide with Sea Bottom Roughness

The problem of sound propagation in a shallow sea with a rough sea bottom is considered. A random matrix approach for studying sound scattering by the water–bottom interface inhomogeneities is developed. This approach is based on the construction of a statistical ensemble of the propagator matrices that describe the evolution of the wavefield in the basis of normal modes. A formula for the coupling term corresponding to inter-mode transitions due to scattering by the sea bottom is derived. The Weisskopf–Wigner approximation is utilized for the coupling between waterborne and sediment modes. A model of a waveguide with the bottom roughness described by the stochastic Ornstein–Uhlenbeck process is considered as an example. Range dependencies of mode energies, modal cross coherences and scintillation indices are computed using Monte Carlo simulations. It is shown that decreasing the roughness correlation length enhances mode coupling and facilitates sound scattering.

Observed Oceanic Surface Modes in the Northern South China Sea

Abstract Using observations and theoretical models, a substantial topographic modulation on the quasigeostrophic (QG) dynamics, which results in a primary surface mode distinct from the classic first baroclinic (BC1) mode with a flat bottom, is revealed in the northern South China Sea (NSCS). In contrast to open oceans, the surface-intensified modes decay downward more rapidly over the continental slope of the NSCS, with a mean e-folding scale of approximately 1/5 of water depth. The subinertial flow variability appears to be vertically incoherent, with planetary and topographic Rossby waves dominating in the upper and deep layers, respectively. With a larger deformation radius (Rd), the surface-mode Rossby waves propagate at a phase speed ∼1.5 times of that of the BC1 mode. Moreover, the modal structures can be substantially modified by seasonal NSCS circulation, which is significantly enhanced over continental slopes. Analysis of the triad interactions further implies that the short waves tend to transfer energy to larger scales via the inverse cascade and only those with wavelengths larger than Rd ≈ 70 km in the NSCS can persist because of a slower unstable growth rate but a higher fraction of upscale energy transfer. The present theory excludes the bottom-trapped mode, which is closely associated with topographic Rossby waves and is observed to be significant in the abyssal NSCS. Hence, a complete normal-mode basis for any QG state is required for a study that focuses on flow variability throughout the water column. Our findings provide an insight into the vertical partition of horizontal kinetic energy for QG motions, as well as the relevant oceanic variation in marginal seas.

Anharmonic Terms of the Potential Energy Surface: A Group Theoretical Approach

In the framework of density functional theory (DFT) simulations of molecules and materials, anharmonic terms of the potential energy surface are commonly computed numerically, with an associated cost that rapidly increases with the size of the system. Recently, an efficient approach to calculate cubic and quartic interatomic force constants in the basis of normal modes [Theor. Chem. Acc., 120, 23 (2008)] was implemented in the Crystal program [J. Chem. Theory Comput., 15, 3755-3765 (2019)]. By applying group theory, we are able to further reduce the associated computational cost, as the exploitation of point symmetry can significantly reduce the number of distinct atomically displaced nuclear configurations to be explicitly explored for energy and forces calculations. Our strategy stems from Wigner's theorem and the fact that normal modes are bases of the irreducible representations (irreps) of the point group. The proposed group theoretical approach is implemented in the Crystal program and its efficiency assessed on six test case systems: four molecules (methane, CH4; tetrahedrane, C4H4; cyclo-exasulfur, S6; cubane, C8H8), and two three-dimensional crystals (Magnesium oxide, MgO; and a prototypical Zinc-imidazolate framework, ZIF-8). The speedup imparted by this approach is consistently very large in all high-symmetry molecular and periodic systems, peaking at 76% for MgO.

Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra

The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is common to apply the quantum harmonic-oscillator model, whose eigenfunctions intimately involve combinatorics. Each electronic state has distinct force constants; therefore, each normal-mode basis is distinct. Duschinsky proposed a linearized approximation to the transformation between the normal-mode bases of two electronic states using a rotation matrix. The rotation angles are typically obtained by using quantum-chemical computations or via gas-phase spectroscopy measurements. Quantifying the rotation angles in the condensed phase remains a challenge. Here, we apply a two-dimensional harmonic model that includes a Duschinsky rotation to condensed-phase femtosecond coherence spectra (FCS), which are created in transient–absorption spectroscopy measurements through impulsive excitation of coherent vibrational wavepackets. Using the 2D model, we simulate spectra to identify the signatures of Duschinsky rotation. The results suggest that peak multiplicities and asymmetries may be used to quantify the rotation angle, which is a key advance in condensed-phase molecular spectroscopy.

On the Discrete Normal Modes of Quasigeostrophic Theory

AbstractThe discrete baroclinic modes of quasigeostrophic theory are incomplete, and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal-mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and the fact that their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin-space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert-space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly nonlinear wave-interaction theory of geostrophic turbulence that takes topography into account.

First-principles calculations of vibrational spectra of CdSe/CdS superlattices

The vibrational spectra of CdSe/CdS superlattices (SLs) with different layer thicknesses are calculated from first principles within the density functional theory. It is shown that, along with folded acoustic and confined optical modes, a number of confined acoustic modes appear in SLs. In structures with a minimum thickness of one of the layers, microscopic interface modes similar to local and gap modes in crystals appear. An analysis of projections of the eigenvectors of vibrational modes in SLs onto the orthonormal basis of normal modes in binary compounds enables to establish the details of formation of these vibrational modes and, in particular, to determine the degree of intermixing of acoustic and optical modes. A comparison of the frequencies of vibrational modes in CdSe/CdS SLs and CdSe/CdS nanoplatelets enables to separate the influence of size quantization and surface relaxation on the vibrational frequencies in the nanoplatelets. Keywords: phonon spectra, semiconductor superlattices, cadmium selenide, cadmium sulfide, nanostructures.

Potential allosteric sites captured in glycolytic enzymes via residue-based network models: Phosphofructokinase, glyceraldehyde-3-phosphate dehydrogenase and pyruvate kinase

Likelihood of new allosteric sites for glycolytic enzymes, phosphofructokinase (PFK), glyceraldehyde-3-phosphate dehydrogenase (GADPH) and pyruvate kinase (PK) was evaluated for bacterial, parasitic and human species. Allosteric effect of a ligand binding at a site was revealed on the basis of low-frequency normal modes via Cα-harmonic residue network model. In bacterial PFK, perturbation of the proposed allosteric site outperformed the known allosteric one, producing a high amount of stabilization or reduced dynamics, on all catalytic regions. Another proposed allosteric spot at the dimer interface in parasitic PFK exhibited major stabilization effect on catalytic regions. In parasitic GADPH, the most desired allosteric response was observed upon perturbation of its tunnel region which incorporated key residues for functional regulation. Proposed allosteric site in bacterial PK produced a satisfactory allosteric response on all catalytic regions, whereas in human and parasitic PKs, a partial inhibition was observed. Residue network model based solely on contact topology identified the ‘hub residues’ with high betweenness tracing plausible allosteric communication pathways between distant functional sites. For both bacterial PFK and PK, proposed sites accommodated hub residues twice as much as the known allosteric site. Tunnel region in parasitic GADPH with the strongest allosteric effect among species, incorporated the highest number of hub residues. These results clearly suggest a one-to-one correspondence between the degree of allosteric effect and the number of hub residues in that perturbation site, which increases the likelihood of its allosteric nature.

PorphyStruct: A Digital Tool for the Quantitative Assignment of Non-Planar Distortion Modes in Four-Membered Porphyrinoids.

PorphyStruct, a new digital tool for the analysis of non‐planar distortion modes of different porphyrinoids, and its application to corrole structures is reported. The program makes use of the normal‐coordinate structure decomposition technique (NSD) and employs sets of normal modes equivalent to those established for porphyrins in order to describe the out‐of‐plane dislocation pattern of perimeter atoms from corroles, norcorroles, porphycenes and other porphyrinoids quantitatively and in analogy to the established terminology. A comparative study of 17 porphyrin structures shows very similar results to the original NSD analysis and no systematic error. Application to corroles is successful and reveals the necessity to implement an extended basis of normal modes for a large share of experimental structures. The results frequently show the concomitant occurence of several modes but remain interpretable. For group XI metal corroles the phenomenon of supersaddling was unravelled, allowing for more in‐depths discussions of structure‐function correlations.

Stratospheric ozone and quasi-biennial oscillation (QBO) interaction with the tropical troposphere on intraseasonal and interannual timescales: a normal-mode perspective

Abstract. The Madden–Julian oscillation (MJO) is the main controller of the weather in the tropics on intraseasonal timescales, and recent research provides evidence that the quasi-biennial oscillation (QBO) influences the MJO interannual variability. However, the physical mechanisms behind this interaction are not completely understood. Recent studies on the normal-mode structure of the MJO indicate the contribution of global-scale Kelvin and Rossby waves. In this study we test whether these MJO-related normal modes are affected by the QBO and stratospheric ozone. The partial directed coherence method was used and enabled us to probe the direction and frequency of the interactions. It was found that equatorial stratospheric ozone and stratospheric zonal winds are connected with the MJO at periods of 1–2 months and 1.5–2.5 years. We explore the role of normal-mode interactions behind the stratosphere–troposphere coupling by performing a linear regression between the MJO–QBO indices and the amplitudes of the normal modes of the atmosphere obtained by projections on a normal-mode basis using ERA-Interim reanalysis data. The MJO is dominated by symmetric Rossby modes but is also influenced by Kelvin and asymmetric Rossby modes. The QBO is mostly explained by westward-propagating inertio-gravity waves and asymmetric Rossby waves. We explore the previous results by identifying interactions between those modes and between the modes and the ozone concentration. In particular, westward inertio-gravity waves, associated with the QBO, influence the MJO on interannual timescales. MJO-related modes, such as Kelvin waves and Rossby waves with a symmetric wind structure with respect to the Equator, are shown to have significantly different dynamics during MJO events depending on the phase of the QBO.

Расчеты из первых принципов колебательных спектров сверхрешеток CdSe/CdS

The vibrational spectra of CdSe/CdS superlattices (SLs) with different layer thicknesses are calculated from first principles within the density functional theory. It is shown that, along with folded acoustic and confined optical modes, a number of confined acoustic modes appear in SLs. In structures with a minimum thickness of one of the layers, microscopic interface modes similar to local and gap modes in crystals appear. An analysis of projections of the eigenvectors of vibrational modes in SLs onto the orthonormal basis of normal modes in binary compounds enables to establish the details of formation of these vibrational modes and, in particular, to determine the degree of intermixing of acoustic and optical modes. A comparison of the frequencies of vibrational modes in CdSe/CdS SLs and CdSe/CdS nanoplatelets enables to separate the influence of size quantization and surface relaxation on the vibration frequencies in the nanoplatelets.

Modelling of sound propagation in the ocean using the matrix propagator

The problem of sound propagation in an oceanic waveguide is considered. Attention is concentrated on properties of a wavefield propagator that governs transformation of an arbitrary wavefield in course of propagation. Using the basis of normal modes, we can represent the propagator in the matrix form. If horizontal inhomogeneity of a waveguide is weak, the propagator can be calculated by means of the first-order perturbation theory. In the case of random inhomogeneity, the apparatus of random matrix theory can be exploited. We demonstrate methods of propagator construction for realistic models of waveguides, including waveguides with adiabatic inhomogeneities and shallow-sea waveguides. A scheme of experimental measurement of a propagator in a shallow-sea waveguide is proposed.

A New Fast Track to Nonlinear Modal Analysis of Power System Using Normal Form

The inclusion of higher-order terms in small-signal (modal) analysis augments the information provided by linear analysis and enables better dynamic characteristic studies on the power system. This can be done by applying Normal Form theory to simplify the higher order terms. However, it requires the preliminary expansion of the nonlinear system on the normal mode basis, which is impracticable with standard methods when considering large scale systems. In this paper, we present an efficient numerical method for accelerating those computations, by avoiding the usual Taylor expansion. Our computations are based on prescribing the linear eigenvectors as unknown field in the initial nonlinear system, which leads to solving linear-only equations to obtain the coefficients of the nonlinear modal model. In this way, actual Taylor expansion and associated higher order Hessian matrices are avoided, making the computation of the nonlinear model up to third order and nonlinear modal analysis fast and achievable in a convenient computational time. The proposed method is demonstrated on a single-machine-infinite-bus (SMIB) system and applied to IEEE 3-Machine, IEEE 16-Machine and IEEE 50-Machine systems.

Normal mode analysis of disordered random-matrix ensembles

The statistics of random-matrix spectra can be very sensitive to the unfolding procedure that separates global from local properties. In order to avoid the introduction of possible artifacts, recently it has been applied to ergodic ensembles of Random Matrix Theory (RMT) the singular value decomposition (SVD) method, based on normal mode analysis, which characterizes the long-range correlations of the spectral fluctuations in a direct way without performing any unfolding. However, in the case of more general ensembles, the ergodicity property is often broken leading to ambiguities between spectrum-unfolded and ensemble-unfolded fluctuation statistics. Here, we apply SVD to a disordered random-matrix ensemble with tunable nonergodicity, as a mathematical framework to characterize the nonergodicity. We show that ensemble-averaged and individual-spectrum averaged statistics are calculated consistently using the same normal mode basis, and the nonergodicity is explained as a breakdown of this common basis.

Superconducting qubits beyond the dispersive regime

We present a theoretical description for circuits consisting of weak anharmonic qubits coupled to cavity multimodes. We obtain a unitary transformation that diagonalizes harmonic sector of the circuit. Weak anharmonicity does not alter the normal mode basis, however it can modify energy levels. We study two examples of a transmon and two transmons coupled to bus resonator, and we determine dressed frequencies and Kerr nonlinearities in closed form formulas. Our results are valid for arbitrary frequency detuning and coupling within and beyond dispersive regime.

Frequency-adaptive bilinear reduced-order model for structures with intermittent contacts

Computing the nonlinear forced response of structures with localized nonlinearity, such as intermittent contacts, is a time-intensive task. Temporal and spatial reduction methods are beneficial in making nonlinear analyses affordable especially during design iterations. In this paper, reduced-order models for structures with intermittent contacts are presented. Models for such systems are reduced by projection onto a basis of normal modes computed by enforcing special boundary conditions at the contact surfaces. The resulting low-order models are used to obtain the forced response by the harmonic balance method. A frequency-adaptive method is used to efficiently estimate the contact area during vibrations and establish the special boundary conditions to be used for reduction. Furthermore, the novel approach is developed to be adaptive in that it creates reduced-order models of different sizes at different frequencies. The proposed method is applied to three test cases to demonstrate its effectiveness and its numerical efficiency.

Infrared Spectral Assignment of Pyrimidine and Pyrazine in the C[sbnd]H Stretching Region by an Effective Spectroscopic Hamiltonian

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 CH 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 CH stretching region of these compounds. The CH 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 CH stretch, overtone of the HCC 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 CH strech and the overtone of HCC in-plane bending modes) has been demonstrated pictorially.

Crossover in nonstandard random-matrix spectral fluctuations without unfolding.

Recently, singular value decomposition (SVD) was applied to standard Gaussian ensembles of random-matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the β-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a crossover between soft and rigid behavior. In this way, possible artifacts introduced applying unfolding techniques are avoided. By using the trend modes, we perform data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes.

Normal-mode-based analysis of electron plasma waves with second-order Hermitian formalism

The classic problem of the dynamic evolution and Landau damping of linear Langmuir electron waves in a collisionless plasma with Maxwellian background is cast as a second-order, self-adjoint problem with a continuum spectrum of real and positive squared frequencies. The corresponding complete basis of singular normal modes is obtained, along with their orthogonality relation. This yields easily the general expression of the time-reversal-invariant solution for any initial-value problem. Examples are given for specific initial conditions that illustrate different behaviors of the Landau-damped macroscopic moments of the perturbations.