Vibrational Eigenfunctions Research Articles

Revisiting Numerical Solutions of Weakly Bound Noble Gases' Vibrational Energy Levels Modeled by the Improved Lennard-Jones Potential.

We revisit the numerical solutions of vibrational eigenstates of weakly bound homonuclear and heteronuclear noble gas pairs by applying a Fortran program based on the Numerov method. The harmonic, Lennard-Jones (LJ), Morse, Tang-Toennies (TT), and Improved Lennard-Jones (ILJ) potential models have been implemented to represent the potential energy curves (PECs). The obtained vibrational energies spectrum was tested on the experimental data and accurate ab initio calculations at CCSD(T)/CBS level. The vibrational eigenvalues and eigenfunctions can be reproduced accurately within the ILJ potential model. Moreover, considering the calculated lifetime of van der Waals (vdW) complexes, the implementation of ILJ rather than standard LJ potential model has a significant impact on the systems dynamics by providing more representative atomic trajectories when the function is incorporated in force fields for molecular dynamics (MD) simulations. Overall, the ILJ function is the best suited potential model for the representation of vibrational motions and the determination of vibrational energy levels of weakly bound systems, both at equilibrium and non-equilibrium conditions.

Sensitivity of Intra- and Intermolecular Interactions of Benzo[h]quinoline from Car-Parrinello Molecular Dynamics and Electronic Structure Inspection.

The O-H...N and O-H...O hydrogen bonds were investigated in 10-hydroxybenzo[h]quinoline (HBQ) and benzo[h]quinoline-2-methylresorcinol complex in vacuo, solvent and crystalline phases. The chosen systems contain analogous donor and acceptor moieties but differently coupled (intra- versus intermolecularly). Car–Parrinello molecular dynamics (CPMD) was employed to shed light onto principle components of interactions responsible for the self-assembly. It was applied to study the dynamics of the hydrogen bonds and vibrational features as well as to provide initial geometries for incorporation of quantum effects and electronic structure studies. The vibrational features were revealed using Fourier transformation of the autocorrelation function of atomic velocity and by inclusion of nuclear quantum effects on the O-H stretching solving vibrational Schrödinger equation a posteriori. The potential of mean force (Pmf) was computed for the whole trajectory to derive the probability density distribution and for the O-H stretching mode from the proton vibrational eigenfunctions and eigenvalues incorporating statistical sampling and nuclear quantum effects. The electronic structure changes of the benzo[h]quinoline-2-methylresorcinol dimer and trimers were studied based on Constrained Density Functional Theory (CDFT) whereas the Electron Localization Function (ELF) method was applied for all systems. It was found that the bridged proton is localized on the donor side in both investigated systems in vacuo. The crystalline phase simulations indicated bridged proton-sharing and transfer events in HBQ. These effects are even more pronounced when nuclear quantization is taken into account, and the quantized Pmf allows the proton to sample the acceptor area more efficiently. The CDFT indicated the charge depletion at the bridged proton for the analyzed dimer and trimers in solvent. The ELF analysis showed the presence of the isolated proton (a signature of the strongest hydrogen bonds) only in some parts of the HBQ crystal simulation. The collected data underline the importance of the intramolecular coupling between the donor and acceptor moieties.

Anharmonic quantum nuclear densities from full dimensional vibrational eigenfunctions with application to protonated glycine

The interpretation of molecular vibrational spectroscopic signals in terms of atomic motion is essential to understand molecular mechanisms and for chemical characterization. The signals are usually assigned after harmonic normal mode analysis, even if molecular vibrations are known to be anharmonic. Here we obtain the quantum anharmonic vibrational eigenfunctions of the 11-atom protonated glycine molecule and we calculate the density distribution of its nuclei and its geometry parameters, for both the ground and the O-H stretch excited states, using our semiclassical method based on ab initio molecular dynamics trajectories. Our quantum mechanical results describe a molecule elongated and more flexible with respect to what previously thought. More importantly, our method is able to assign each spectral peak in vibrational spectroscopy by showing quantitatively how normal modes involving different functional groups cooperate to originate that spectroscopic signal. The method will possibly allow for a better rationalization of experimental spectroscopy.

Anharmonic vibrational eigenfunctions and infrared spectra from semiclassical molecular dynamics.

We describe a new approach based on semiclassical molecular dynamics that allows simulating infrared absorption or emission spectra of molecular systems with inclusion of anharmonic intensities. This is achieved from semiclassical power spectra by computing first the vibrational eigenfunctions as a linear combination of harmonic states, and then the oscillator strengths associated with the vibrational transitions. We test the approach against a 1D Morse potential and apply it to the water molecule with results in excellent agreement with discrete variable representation quantum benchmarks. The method does not require any grid calculations, and it is directly extendable to high dimensional systems. The usual exponential scaling of the basis set size with the dimensionality of the system can be avoided by means of an appropriate truncation scheme. Furthermore, the approach has the advantage to provide IR spectra beyond the harmonic approximation without losing the possibility of an intuitive assignment of absorption peaks in terms of normal modes of vibration.

Vibrational treatment of the formic acid double minimum case in valence coordinates.

One single full dimensional valence coordinate HCOOH ground state potential energy surface accurate for both cis and trans conformers for all levels up to 6000 cm-1 relative to trans zero point energy has been generated at CCSD(T)-F12a/aug-cc-pVTZ level. The fundamentals and a set of eigenfunctions complete up to about 3120 and 2660 cm-1 for trans- and cis-HCOOH, respectively, have been calculated and assigned using the improved relaxation method of the Heidelberg multi-configuration time-dependent Hartree package and an exact expression for the kinetic energy in valence coordinates generated by the TANA program. The calculated trans fundamental transition frequencies agree with experiment to within 5 cm-1. A few reassignments are suggested. Our results discard any cis trans delocalization effects for vibrational eigenfunctions up to 3640 cm-1 relative to trans zero point energy.

Closed-Form Exact Solutions for Viscously Damped Free and Forced Vibrations of Longitudinal and Torsional Bars

This paper studies the viscously damped free and forced vibrations of longitudinal and torsional bars. The method is exact and yields closed form solution for the vibration displacement in contrast with the well-known eigenfunction superposition (ES) method, which requires expression of the distributed forcing functions and displacement response functions as infinite series sums of free vibration eigenfunctions. The viscously damped natural frequency equation and the critical viscous damping equation are exactly derived for the bars. Then the viscously damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted, aside from the undamped free vibration and corresponding mode shapes typically computed and used in vibration problems. The longitudinal or torsional amplitude versus forcing frequency curves showing the forced response to distributed loadings are plotted for various viscous damping parameters. It is found that the viscous damping affects the natural frequencies and the corresponding mode shapes of longitudinal and torsional bars, especially for the fundamental frequency.

The “bound wavefunction” on the repulsive excited () state of the HD+ molecule

AbstractThe time‐independent Schrödinger equation for the HD+ molecule is solved beyond the Born–Oppenheimer (B‐O) approximation. In the adiabatic representation, the wavefunction of the ground vibrational eigenstate is found to contain two parts: One is on the ground ( ) state which is dominant, and the other is on the repulsive excited ( ) state in the range from R = 0.0 to R = 5.0 Bohr. This is because the nonadiabatic coupling between the ground ( ) and excited ( ) states is strong in that region. The influences of the nonadiabatic coupling on the vibrational eigenfunctions are discussed in detail.

Closed form exact solutions of viscously damped free and forced vibrations of rectangular membranes

In the present study, viscously damped free and forced vibrations of a rectangular membrane are investigated using a closed form exact method. The method is exact and yields closed form expressions for the vibratory displacements. This is in contrast with the well known eigenfunction superposition method which requires expressing the distributed forcing functions and the displacement response functions as infinite sums of free vibration eigenfunctions. Instead of undamped natural frequencies which are typically computed and applied in the free and forced vibrations, damped natural frequencies are done. The damped natural frequency equation and the critical viscous damping equation are exactly derived. In the damped free vibration, effects of viscous damping on natural frequencies and mode shapes are studied. Some contour plots of damped mode shapes are given. Accurate displacement amplitude vs. forcing frequency curves showing the forced response due to distributed loading are displayed.

Hysterically damped free and forced vibrations of axial and torsional bars by a closed form exact method

Hysterically damped free and forced vibrations of axial and torsional bars are investigated using a closed form exact method. The method is exact and yields closed form expressions for the vibratory displacements. This is in contrast with the well known eigenfunction superposition method which requires expressing the distributed forcing functions and the displacement response functions as infinite sums of free vibration eigenfunctions. The hysterically damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted instead of undamped free vibration and mode shapes which is typically computed and applied in vibration problems. The hysterically damped natural frequency equations are exactly derived. Accurate axial or torsional amplitude vs. forcing frequency curves showing the forced response due to distributed loading are displayed with various hysteretic damping parameters.

Shear buckling of rotationally-restrained composite laminated plates

Based on the Galerkin method, a semi-analytical solution for the shear buckling of composite laminated plates with all four edges elastically-restrained against rotation is presented. The considered laminated plates are loaded in pure shear or combined shear and compression. The deformation shape function is constructed through a unique weighting combination of vibration eigenfunctions of simply-supported and clamped conditions, and it is an effective method for solving the considered eigenvalue problem. A parametric study is conducted to evaluate the effect of the rotational restraint stiffness, plate aspect ratio, material orthotropy and anisotropy on the buckling behavior of the rotationally-restrained laminated plates under pure shear or combined shear and compression action. The semi-analytical solution presented can be used to analyze the restrained laminated plates under shear-dominated loading and applied to predict the web local buckling of thin-walled composite beams.

Post-buckling behavior of imperfect laminated composite plates with rotationally-restrained edges

The nonlinear governing equations of rotationally-restrained laminated composite plates with imperfection are presented by the Galerkin method, and they are solved by employing the Newton–Raphson method for the post-buckling analysis. The considered laminates are symmetric, and they are loaded in pure in-plane shear or combined in-plane shear and compression. The deformation shape function of the restrained plates is obtained through a linear combination of vibration eigenfunctions of simply supported and clamped beams along either the longitudinal or transverse direction of plates. The validity study shows that the presented method is effective for performing the nonlinear analysis of laminates with all four edges elastically-restrained against rotation. A parametric study is conducted to evaluate the effect of rotational spring stiffness, material properties, and fiber orientation under pure in-plane shear as well as the loading ratio under combined shear and compression on the nonlinear static and post-buckling behavior of rotationally-restrained laminates. The proposed solution for nonlinear static analysis of rotationally-restrained composite plates with imperfection is accurate and effective, as demonstrated by the comparisons with the predictions by the finite element analysis, and combined with the discrete plate analysis technique, it can be potentially applied to post-buckling analysis of FRP structural shapes.

Excited-state wavepacket and potential reconstruction by coherent anti-Stokes Raman scattering

Among the major challenges in the chemical sciences is controlling chemical reactions and deciphering their mechanisms. Since much of chemistry occurs in excited electronic states, in the last three decades scientists have employed a wide variety of experimental techniques and theoretical methods to recover excited-state potential energy surfaces and the wavepackets that evolve on them. These methods have been partially successful but generally do not provide a complete reconstruction of either the excited state wavepacket or potential. We have recently proposed a methodology for reconstructing excited-state molecular wavepackets and the corresponding potential energy surface [Avisar and Tannor, Phys. Rev. Lett., 2011, 106, 170405]. In our approach, the wavepacket is represented as a superposition of the set of vibrational eigenfunctions of the molecular ground-state Hamiltonian. We assume that the multidimensional ground-state potential surface is known, and therefore these vibrational eigenfunctions are known as well. The time-dependent coefficients of the basis functions are obtained by experimental measurement of the resonant coherent anti-Stokes Raman scattering (CARS) signal. Our reconstruction strategy has several significant advantages: (1) the methodology requires no a priori knowledge of any excited-state potential. (2) It applies to dissociative as well as to bound excited-state potentials. (3) It is general for polyatomics. (4) The excited-state potential surface is reconstructed simultaneously with the wavepacket. Apart from making a general contribution to the field of excited-state spectroscopy, our method provides the information on the excited-state wavepacket and potential necessary to design laser pulse sequences to control photochemical reactions.

Correlation between Franck-condon Factors and Average Internulcear Separations for Diatomics Using the Fourier Grid Hamiltonian Method

The Fourier Grid Hamiltonian (FGH) method is used to compute the vibrational eigenvalues and eigenfunctions of bound states of diatomic molecules. For these computations, the Hulburt and Hirschfelder (HH) potential model for diatomics is used. These potential energy functions are used for constructing and diagonalizing the molecular Hamiltonians. The vibrational wave functions for the ground and the excited states are used to calculate the Franck-Condon factors (FCFs), r-Centroids and average internuclear separations which play a significant role in determining the intensity of the bands in electronic transitions. The results of FCFs and r-Centroids for diatomic molecules such as H2, N2, CO, I2 and HF using the FGH method are compared with other methods. The Franck-Condon profiles indicate a strong correlation between the values of Franck-Condon factors and the mean internuclear separations for the corresponding transitions.

Free vibration analysis of a variable cross-section column partially immersed in a liquid

In the present paper, the free vibration analysis of an offshore structure, having the form of a column partially immersed in a liquid, is presented. For convenience, a beam without contacting with water is called the dry beam, while the one partially immersed in water is called the wet beam. The non-uniform dry beam, with circular cross-section, carries a concentrated mass with eccentricity e and rotary inertia Jm, at its free end, and it rests on a uniform beam – wet beam. The column is modelled as a Bernoulli–Euler beam supported by a translational and rotational springs, at the bottom.For investigating the dynamic behaviour of the structure under consideration, the effect of the translational and rotational elastic supports, the rotary inertia of the concentrated mass and its eccentricity are all taken into account. The analysis of free vibration frequencies and eigenfunctions of the model, presented in this paper, enables one to obtain very accurate results. The roots of the transcendental frequency equations are obtained by means of the improved conventional analytical (closed-form) solution. The non-dimensional frequency coefficients are given in tabular form. Some numerical examples are presented and the influence of different non-dimensional parameters on frequency values is discussed.

First principles intensity calculations of the methane rovibrational spectra in the infrared up to 9300 cm−1

We report global calculations of rovibrational spectra and dipole transition intensities of methane using our recent ab initio dipole moment and potential surfaces [Nikitin et al., Chem. Phys. Lett., 2011, 501, 179; 2013, 565, 5]. For the full symmetry account, a recently published variational tensor formalism in normal modes [Rey et al., J. Chem. Phys., 2012, 136, 244106] is applied, the convergence of high-J calculations being improved by the use of vibrational eigenfunctions to make a compressed basis set for solving the rovibrational problem. Comparisons of theoretical predictions up to J = 25 for various complex polyads of methane involving strongly coupled vibration-rotation bands support the validity of this new approach. For the first time, positions and line intensities at 80 K and 296 K are shown to be in excellent agreement with raw experimental data, even for high energy ranges. The theoretical predictions also correctly describe the isotopic effects in line positions and intensities due to the CH4 → CD4 substitution which is considered as the test for the method. This work is a first step toward the theoretical interpretation of numerous methane bands which remain still unassigned and detailed line-by-line absorption/emission spectra analyses for atmospheric and planetological applications.

Vibrational excitation resulting from electron capture in LUMO of F2 and HCl – A treatment using the time-dependent wave packet approach#

Vibrational excitation cross-sections \(\sigma_{\nu_f \leftarrow \nu_i } (E)\) in resonant e-F2 and HCl scattering are calculated from transition matrix elements \(T_{\nu_f \leftarrow \nu_i } (E)\) obtained using Fourier transform of the cross correlation function \(\left\langle {\phi_{\nu_f } \left( R \right)\vert \Psi_{\nu_i } \left( {R,t} \right)} \right\rangle \) where \(\Psi_{\nu_i } \left( {R,t} \right)\approx e^{-i\hbar H_{AB^-} \left( R \right)t}\phi_{\nu { }_i} \left( R \right)\). Time evolution under the influence of the resonance anionic Hamiltonian H AB - (AB=F2/HCl) is effected using Lanczos reduction technique followed by fast Fourier transform and the target (AB) vibrational eigenfunctions \(\phi _{\nu_i } \left( R \right)\) and \(\phi_{\nu_f } \left( R \right)\) are calculated using Fourier grid Hamiltonian method applied to potential energy (PE) curve of the neutral target. The resulting vibrational excitation cross-sections provide reasonable agreement with experimental and other theoretical results. The vibrational excitation cross-sections in e-F2 and e-HCl scattering have been calculated with local complex potential based time dependent wave packet approach and compared with those of other theoretical/experimental results.

First principles semiclassical calculations of vibrational eigenfunctions

Vibrational eigenfunctions are calculated on-the-fly using semiclassical methods in conjunction with ab initio density functional theory classical trajectories. Various semiclassical approximations based on the time-dependent representation of the eigenfunctions are tested on an analytical potential describing the chemisorption of CO on Cu(100). Then, first principles semiclassical vibrational eigenfunctions are calculated for the CO(2) molecule and its accuracy evaluated. The multiple coherent states initial value representations semiclassical method recently developed by us has shown with only six ab initio trajectories to evaluate eigenvalues and eigenfunctions at the accuracy level of thousands trajectory semiclassical initial value representation simulations.

Franck–Condon factors for molecule–ion reactions of [formula omitted] and its isotopomers

An existing, systematic, fundamental database of Franck–Condon factors and transition probabilities for all isotopomers of the hydrogen molecule (H 2, D 2, T 2, HD, HT, and DT) has been extended for reactions which connect the molecule with its ion H 2 + ( v ) (or the corresponding ion of the specific isotopomer). The most recent Born–Oppenheimer potential curves were used to calculate the vibrational eigenvalues and eigenfunctions, from which the Franck–Condon factors were obtained by convolution. The accuracy of the results has been assessed and a comparison with results available in the literature was performed.

Transverse Vibration Analysis of Euler-Bernoulli Beams Carrying Concentrated Masses with Rotatory Inertia at Both Ends

Transverse vibration analysis is presented for Euler-Bernoulli beams carrying concentrated masses and taking into account their rotatory inertia at both ends. The dimensionless eigenfunctions for the problems are first obtained using the differential equations of motion and considering translational and rotatory springs at both ends. A numerical technique, the Newton–Raphson algorithm, is then used to solve vibration eigenfunctions of the beams. Finally, the influences of different non-dimensional parameters on frequencies are discussed.

A New Java Program for Graphical Illustration of the Franck−Condon Principle: Application to the I2 Spectroscopy Experiment in the Undergraduate Physical Chemistry Laboratory

Requires FCIntensity.jar Themeasurement and subsequent analysis of I2 absorption and emission spectra is a standard experiment in the undergraduate physical chemistry laboratory (1-5). The observation of interesting intensity alternations in the dispersed fluorescence spectrum and the lack of clear alternations in the absorption spectrum are a manifestation of the Franck-Condon principle, which is an important concept in physical chemistry. Discussion of the Franck-Condon principle, although often neglected in the undergraduate curriculum, can illustrate the connection between vibrational eigenfunctions and transition probabilities. We have developed a portable, flexible, and user-friendly program to illustrate the relationship between Franck-Condon factors and the observed spectral intensities. Students also explore the effects of changing spectroscopic parameters on the predicted spectra. FCIntensity is a Java program that computes the eigenfunctions and eigenvalues of the Morse oscillator Hamiltonian using numerical methods. The program is easily uploaded and implemented on a personal computer and permits facile manipulation of input parameters (Figure 1). As part of the analysis of our I2 laboratory, the students are given the ground state Re (2.6664 A) value and asked to obtain the best-fit B-state Re by comparing the calculated intensity distribution to experiment. Although the calculated intensity patterns do not exactly match experiment (given the Morse approximation that can be discussed), only B-state Re values near 3 A provide an acceptable fit. The remarkable sensitivity is surprising to most students. The comparison between calculation and experiment also reveals the dependence of the spectrum to other parameters. For example, changes in Te, ve, and De for the B-state result in shifts of the spectral positions of all emission lines, while changes in De and ve for the ground state result in changes in the spacing between emission lines. Once the emission spectrum has been analyzed, we ask the students to use the program to simulate the measured absorption spectrum using the same parameters. Calculations using v00 =0, 1, 2 provide visual confirmation of the location of the hotbands and, through inspection of the vibrational wave functions, the origins of the nonoscillatory behavior relative to the emission spectrum.