Harmonic Oscillator Approximation Research Articles

A general analytical expression for evaluation of an arbitrary n-dimensional Franck-Condon overlap integral including the Duschinsky effect

Evaluations of the multidimensional Frank-Condon (FC) overlapping integrals are needed theoretically to investigate the mechanism of radiative and non-radiative processes among the different electronic states of polyatomic molecules. We extended our recently reported approach to calculate an arbitrary n-dimensional Franck–Condon overlapping integrals based on harmonic oscillator approximation and taking into account the Duschinsky effects. An analytical formula was derived straightforwardly based on the Hermite polynomials expanding properties and directly evaluating Gaussian integrals. A method to determine the coefficient parameter matrices of four-dimensional Franck-Condon integrals by matrix operations was presented. In addition, as a practical application, we simulated and elucidated the vibrationally resolved peaks of the photodetachment spectra of SeO2‾. Moreover, in the spectral simulations, employing our iterative Franck-Condon analysis procedure, fairly reliable geometries of the ground state X~2B1 of the negative ion SeO2‾ are obtained.

Heavy quark radiation in the quark\u2013gluon plasma in the Moliere theory: angular distribution of the radiation

We study the effects of adding the Coulomb interactions to the harmonic oscillator (HO) approximation of the heavy parton propagating through the quark–gluon plasma (the extension to QCD of the Molliere theory). We explicitly find the expression for the transverse momentum distribution of the gluon radiation of the heavy quark propagating in the quark gluon plasma in the framework of the Moliere theory, taking into account the BDMPSZ radiation in the HO approximation, and the Coulomb logarithms described by the additional logarithmic terms in the effective potential. We show that these Coulomb logarithms significantly influence the HO distribution, derived in the BDMPSZ works, especially for the small transverse momenta, filling the dead cone, and reducing the dead cone suppression of the heavy quark radiation (dead cone effect). In addition we study the effect of the phase space constraints on the heavy quark energy loss, and argue that taking into account of both the phase space constraints and of the Coulomb gluons reduces the dependence of the heavy quark energy loss on its mass in the HO approximation.

Photoluminescence spectra of point defects in semiconductors: Validation of first-principles calculations

Optically and magnetically active point defects in semiconductors are interesting platforms for the development of solid-state quantum technologies. Their optical properties are usually probed by measuring photoluminescence spectra, which provide information on excitation energies and on the interaction of electrons with lattice vibrations. We present a combined computational and experimental study of photoluminescence spectra of defects in diamond and SiC, aimed at assessing the validity of theoretical and numerical approximations used in first principles calculations, including the use of the Franck-Condon principle and the displaced harmonic oscillator approximation. We focus on prototypical examples of solid-state qubits, the divacancy centers in SiC and the nitrogen-vacancy in diamond, and we report computed photoluminescence spectra as a function of temperature that are in very good agreement with the measured ones. As expected we find that the use of hybrid functionals leads to more accurate results than semilocal functionals. Interestingly our calculations show that constrained density functional theory (CDFT) and time-dependent hybrid DFT perform equally well in describing the excited state potential energy surface of triplet states; our findings indicate that CDFT, a relatively cheap computational approach, is sufficiently accurate for the calculations of photoluminescence spectra of the defects studied here. Finally, we find that only by correcting for finite-size effects and extrapolating to the dilute limit, one can obtain a good agreement between theory and experiment. Our results provide a detailed validation protocol of first principles calculations of photoluminescence spectra, necessary both for the interpretation of experiments and for robust predictions of the electronic properties of point defects in semiconductors.

Electronic spectroscopy of the A1̃2A′′/A2̃2A′−X̃2A′ transitions of jet-cooled calcium ethoxide radicals: Vibronic structure of alkaline earth monoalkoxide radicals of C s symmetry

Laser-induced fluorescence/dispersed fluorescence (LIF/DF) and cavity ring-down spectra of the A1̃2A''/A2̃2A'-X̃2A' electronic transition of the calcium ethoxide (CaOC2H5) radical have been obtained under jet-cooled conditions. An essentially constant Ã2-Ã1 energy separation for different vibronic levels is observed in the LIF spectrum, which is attributed to both the spin-orbit (SO) interaction and non-relativistic effects. Electronic transition energies, vibrational frequencies, and spin-vibrational eigenfunctions calculated using the coupled-cluster method, along with results from previous complete active space self-consistent field calculations, have been used to predict the vibronic energy level structure and simulate the recorded LIF/DF spectra. Although the vibrational frequencies and Franck-Condon (FC) factors calculated under the Born-Oppenheimer approximation and the harmonic oscillator approximation reproduce the dominant spectral features well, the inclusion of the pseudo-Jahn-Teller (pJT) and SO interactions, especially those between the A1̃2A″/A2̃2A' and the B̃2A' states, induces additional vibronic transitions and significantly improves the accuracy of the spectral simulations. Notably, the spin-vibronic interactions couple vibronic levels and alter transition intensities. The calculated FC matrix for the A1̃2A''/A2̃2A'-X̃2A' transition contains a number of off-diagonal matrix elements that connect the vibrational ground levels to the levels of the ν8 (CO stretch), ν11 (OCC bending), ν12 (CaO stretch), ν13 (in-plane CaOC bending), and ν21 (out-of-plane CaOC bending) modes, which are used for vibrational assignments. Transitions to the ν21(a″) levels are allowed due to the pJT effect. Furthermore, when LIF transitions to the Ã-state levels of the CaOC-bending modes, ν13 and ν21, are pumped, A1̃2A''/A2̃2A'→X̃2A' transitions to the combination levels of these two modes with the ν8, ν11, and ν12 modes are also observed in the DF spectra due to the Duschinsky mixing. Implications of the present spectroscopic investigation to laser cooling of asymmetric-top molecules are discussed.

Non-equilibrium dissociation and relaxation behind the shock wave within two-temperature approach

Within the recently proposed asymptotic method for solving the Boltzmann equation for chemically reacting gas mixture, the equations for a dissociating diatomic gas have been derived assuming two-temperature (translational-rotational + vibrational) approximation. Corresponding expressions for the reaction and relaxation rates, determined by the quasi-stationary vibrational distributions, have been obtained under assumption of dissociation from the highest vibrational level. Cut-off harmonic oscillator approximation for the diatomic molecules is assumed. It is shown that all reaction rates are the complex functions of the species densities. Analysis of a flow behind a shock wave is performed in a wide range of the flow parameters. It is shown that under strong non-equilibrium condition the dissociation rate is determined not by the dissociation probability, but by the vibration probability, since the excitation of highest vibrational levels is the bottleneck of dissociation process. This means that the procedure of obtaining the data on dissociation rates needs the accurate revision.

Development of a MATLAB Algorithm for Calculating Reorganization Energy Utilizing Rectilinear Normal Mode Displacements: Investigation of the Effect of Substituents on Electron and Hole Reorganization Energies of Styryl‐Capped Silicon Quantum Dots

We developed a MATLAB algorithm to calculate reorganization energy utilizing rectilinear normal mode displacements. Normal mode‐projected rectilinear displacements and the corresponding angular frequencies, required for evaluating charge transfer reorganization energy within the harmonic oscillator approximation, were obtained from Cartesian coordinates and Cartesian force constant matrices determined with respect to the principal axes. To verify the algorithm developed with MATLAB, we compared the computed charge transfer reorganization energies to those evaluated by the DUSHIN program, and there was no substantial difference, indicating that our algorithm guarantees the numerical accuracy of the calculations. This algorithm was applied to design silicon quantum dots (Si QDs) with low reorganization energies for charge transfer.

Fitting quantum machine learning potentials to experimental free energy data: predicting tautomer ratios in solution.

The computation of tautomer ratios of druglike molecules is enormously important in computer-aided drug discovery, as over a quarter of all approved drugs can populate multiple tautomeric species in solution. Unfortunately, accurate calculations of aqueous tautomer ratios—the degree to which these species must be penalized in order to correctly account for tautomers in modeling binding for computer-aided drug discovery—is surprisingly difficult. While quantum chemical approaches to computing aqueous tautomer ratios using continuum solvent models and rigid-rotor harmonic-oscillator thermochemistry are currently state of the art, these methods are still surprisingly inaccurate despite their enormous computational expense. Here, we show that a major source of this inaccuracy lies in the breakdown of the standard approach to accounting for quantum chemical thermochemistry using rigid rotor harmonic oscillator (RRHO) approximations, which are frustrated by the complex conformational landscape introduced by the migration of double bonds, creation of stereocenters, and introduction of multiple conformations separated by low energetic barriers induced by migration of a single proton. Using quantum machine learning (QML) methods that allow us to compute potential energies with quantum chemical accuracy at a fraction of the cost, we show how rigorous relative alchemical free energy calculations can be used to compute tautomer ratios in vacuum free from the limitations introduced by RRHO approximations. Furthermore, since the parameters of QML methods are tunable, we show how we can train these models to correct limitations in the underlying learned quantum chemical potential energy surface using free energies, enabling these methods to learn to generalize tautomer free energies across a broader range of predictions.

Heat Capacity and Bond Dissociation Energy Calculations of Some Fluorinated Ethanol’s and its Radicals: CH<sub>3-x</sub>CH<sub>2</sub>F<sub>x</sub>OH, CH<sub>3</sub>CH<sub>2-x</sub>F<sub>x</sub>OH

Structures and thermochemical properties of these species were determined by the gaussian M-062x/6-31 + g (d, p) calculation enthalpies of formation for 19 fluorinated ethanol and some radicals were calculated with a popular Ab initio and density functional theory methods: The gaussian M-062x/6-31 + g (d, p) via several series of isodesmic reactions. Entropies (S298°K in Cal·Mol-1 K-1) were estimated using the M-062x/6-31 + g (d, p) computed frequencies and geometries. Contributions of entropy, S298°K, and heat capacities, Cp(T) due to vibration, translation, and external rotation of the molecules were calculated based on the vibration frequencies and structures obtained from the M-062x/6-31 + g (d, p) Density Functional Method. Potential barriers are calculated using M-062x/6-31 + g (d, p) density functional method and are used to calculate rotor contributions to entropy and heat capacity using integration over energy levels of rotational potential. Rotational barriers were determined and hindered internal rotational contributions for S298° - 1500°K, and Cp (T) were calculated using the rigid rotor harmonic oscillator approximation, with direct integration over energy levels of the intramolecular rotation potential energy curves. Thermochemical properties of fluorinated alcohols are needed for understanding their stability and reactions in the environment and in thermal process

Магнитные и магнитокалорические эффекты в системах с реверсивными переходами первого рода

Within the framework of the model of interacting parameters of magnetic and structural orders, a theoretical analysis of magnetostructural reversible first-order phase transitions is carried out. Reversible phase transitions are characterized by a jump-like appearance of magnetic order with decreasing temperature (as in a first-order phase transition), and with a reverse increase in temperature, the magnetic order gradually disappears (as in a second-order phase transition). Such transitions are observed in some alloys of the Mn_{1-x}Cr_{x}NiGe magnetocaloric system under pressure (x = 0.11) and without (x = 0.18) and are accompanied by specific magnetic and magnetocaloric features. A phenomenological description of these features is carried out within the concept of a soft mode for the structural subsystem undergoing first-order structural phase transition (P6_{3}/mmc-P_{nma}) and the Heisenberg model for the spin subsystem. For systems with magnetostructural instability within the molecular field approximation for the spin subsystem and the shifted harmonic oscillator approximation for the lattice subsystem, it is shown that the reversible phase transitions arise when the temperature of magnetic disordering is in the temperature hysteresis region of the 1st order structural phase transition P6_{3}/mmc-P_{nma}. It is also shown that the two-peak form of the isothermal entropy, which is characteristic of reversible transitions, is due to the separation of the structural and magnetic entropy contributions.

Airy functions and transition between semiclassical and harmonic oscillator approximations for one-dimensional bound states

We consider the one-dimensional Schrodinger operator with a semiclassical small parameter $$h$$ . We show that the “global” asymptotic form of its bound states in terms of the Airy function “works” not only for excited states $$n\sim1/h$$ but also for semi-excited states $$n\sim1/h^\alpha$$ , $$\alpha>0$$ , and, moreover, $$n$$ starts at $$n=2$$ or even $$n=1$$ in examples. We also prove that the closeness of such an asymptotic form to the eigenfunction of the harmonic oscillator approximation.

Heavy quark energy loss in the quark-gluon plasma in the Moller theory

We study the energy loss of a heavy quark propagating in the quark-gluon plasma (QGP) in the framework of the Moller theory, including possible large Coulomb logarithms as a perturbation to BDMPSZ bremsstrahlung, described in the harmonic oscillator (HO) approximation. We derive the analytical expression that describes the energy loss in the entire emitted gluon frequency region. In the small frequencies region, for angles larger than the dead cone angle, the energy loss is controlled by the BDMPSZ mechanism, while for larger frequencies it is described by N = 1 term in the GLV opacity expansion. We estimate corresponding quenching rates for different values of the heavy quark path length and different m/E ratios.

On the Complementarity of the Harmonic Oscillator Model and the Classical Wigner-Kirkwood Corrected Partition Functions of Diatomic Molecules.

The vibrational and rovibrational partition functions of diatomic molecules are considered in the regime of intermediate temperatures. The low temperatures are those at which the harmonic oscillator approximation is appropriate, and the high temperatures are those at which classical partition function (with Wigner–Kirkwood correction) is applicable. The complementarity of the harmonic oscillator and classical integration over the phase space approaches is investigated for the CO and H molecules showing that those two approaches are complementary in the sense that they smoothly overlap.

Optical Observation of Quasiperiodic Heisenberg Antiferromagnets in Two Dimensions

A spin‐wave analysis of magnetic Raman scattering in an antiferromagnetic Heisenberg model on the 2D Penrose lattice of point symmetry is made. Following the Shastry–Shraiman perturbation scheme for a strongly correlated Hubbard model, fourth‐order effective Raman operators are obtained. Within the second‐order mechanism, there is one and only Raman active mode of symmetry, yielding a spectral weight independent of light polarization. Considering the fourth‐order scatterings as well, activates and as well as modes and therefore results in polarization‐dependent Raman spectra. With the use of linearly and circularly polarized lights, all the symmetry species can separately be extracted from observations. Though the linear spin‐wave theory, i.e., the harmonic oscillator approximation, is far from quantitative, the and Raman intensities are well describable with two magnon interactions. The Raman‐active mode owes much to higher‐order magnon–magnon interactions as well.

Nuclear Quantum Effects from the Analysis of Smoothed Trajectories: Pilot Study for Water.

Nuclear quantum effects have significant contributions to thermodynamic quantities and structural properties; furthermore, very expensive methods are necessary for their accurate computation. In most calculations, these effects, for instance, zero-point energies, are simply neglected or only taken into account within the quantum harmonic oscillator approximation. Herein, we present a new method, Generalized Smoothed Trajectory Analysis, to determine nuclear quantum effects from molecular dynamics simulations. The broad applicability is demonstrated with the examples of a harmonic oscillator and different states of water. Ab initio molecular dynamics simulations have been performed for ideal gas up to the temperature of 5000 K. Classical molecular dynamics have been carried out for hexagonal ice, liquid water, and vapor at atmospheric pressure. With respect to the experimental heat capacity, our method outperforms previous calculations in the literature in a wide temperature range at lower computational cost than other alternatives. Dynamic and structural nuclear quantum effects of water are also discussed.

Скачкообразные процессы магнитного разупорядочения, стимулированные магнитным полем в системах со структурной неустойчивостью

A theoretical analysis of the features of structural and magnetostructural first-order phase transitions in magnetocaloric helimagnetic alloys of the Mn_{1-x}Cr_{x}NiGe system has been carried out. To describe the observed displacive structural transitions hex(P6_{3}/mmc)<->orth(P_{nma}), we used the local soft mode model in the approximation of a biased harmonic oscillator. In the absence of a magnetic field, the emergence of a helimagnetic order as a structurally induced second-order transition was described in the framework of the Heisenberg model, taking into account the dependence of the exchange integrals on the structural order parameters and elastic strains. In the presence of a magnetic field, it was found that the approximation of the characteristic temperatures for the helimagnetic HM(P_{nma}) and lability temperatures of the hexagonal paramagnetic PM(P6_{3}/mmc) states, due to the influence of the magnetic field, leads to the appearance of previously unexplored peripheral magnetostructural first-order phase transitions with insignificant magnetization jumps that increase with increasing magnetic induction.In this case, as the pressure increases to 4 kbar with constant induction of the magnetic field, the peripheral transitions transform into reversible first-order magnetostructural transitions, and at even higher pressures (10-14 kbar) into full-fledged first-order magnetostructural transitions with magnetization jumps comparable with maximum value of magnetization. Experimental pressure studies of the temperature dependences of magnetization in static magnetic fields with an induction of up to 1 T and a pressure of up to 14 kbar confirm the theoretical results.

Molecular Modeling, Dimer Calculations, Vibrational Spectra, and Molecular Docking Studies of 5-Chlorouracil

The structure and vibrational calculations of 5-chlorouracil (5-ClU) and its most stable dimer have been analyzed using the DFT method with B3LYP/6-31++G(d,p) and wb97xd/6-31++G(d,p), respectively. Vibrational calculations of the monomeric and dimeric forms were performed using both harmonic and anharmonic oscillator approximations with the same basis sets. A complete vibrational analysis of the molecule has been performed by combining experimental Raman, FT-IR spectral data and quantum chemical calculations. In addition, the DNA docking analysis of 5-ClU molecule was performed. A 5-ClU molecule binds to the active site of DNA by hydrogen bonding interactions. The results show that the docked ligand formed a stable complex with DNA with binding affi nity of –5.3 kcal/mol.

Thermodynamic Functions of 1, 2, 3- tri-meth-oxy-benzene and m-methyl-meth-oxy benzene

The statistical computation of an ideal gas state thermodynamic functions namely enthalpy, entropy, free energy, heat capacity of 1,2,3 – Trimethoxy benzene, m-methyl methoxybenzene have been performed. These calculations have been made at a pressure of 1 atmosphere in the temperature range 100°K to 1500°K under rigid rotor harmonic oscillator approximation for 1 mole of perfect gas and on the basis of vibrational frequencies obtained from FTIR and Raman spectra and also with the help of moment of inertia.

Gaussian M-062x/6-31+g (d,p) Calculation of Standard Enthalpy, Entropy and Heat Capacity of Some Fluorinated Alcohol’s and Its Radicals at Different Temperatures

Thermochemical properties of fluorinated alcohols are needed for understanding their stability and reactions in the environment and in thermal process. Structures and thermochemical properties of these species were determined by the Gaussian M-062x/6-31+g (d,p) calculation. Contributions of entropy, S°298, and heat capacities, Cp(T) due to vibration, translation, and external rotation of the molecules were calculated based on the vibration frequencies and structures obtained from the M-062x/6-31+g (d,p) density functional method. Potential barriers are calculated using M-062x/6-31+g (d,p) density functional method and are used to calculate rotor contributions to entropy and heat capacity using integration over energy levels of rotational potential. Enthalpies of formation for 19 fluorinated ethanol and some radicals were calculated with a popular ab initio and density functional theory methods: the Gaussian M-062x/6-31+g (d,p) via several series of isodesmic reactions. The recommended ideal gas phase ΔHf298 ° (kcal mol−1) values calculated in this study are the following: -101.74 ± 0.72 for CH2FCH2OH; -113.51 ±1.39 for CH3CHFOH; -50.66 ± 0.75 for C•HFCH2OH; -56.05±0.62 for CH2FCH•OH; -45.00±1.31 for CH2FCH2O•; -59.61±1.20 for CH2•CHFOH; -67.99± 1.29 for CH3CF•OH; -58.76±1.20 for CH3CHFO•; -154.12±1.72 for CH2FCHFOH; -155.26±1.67 for CF2HCH2OH; -174.53±1.54 for CH3CF2OH; -104.07 ± 1.45 for CH2FC•FOH; -105.63±1.74 for C•HFCFHOH; -99.08±1.57 for CH2FCHFO•; -102.34±1.74 for CHF2C•HOH; -102.23±1.57 for C•F2CH2OH; -98.86±1.57 for CHF2CH2O•; -119.41±1.74 for CH2•CF2OH; -110.56±1.62 for CH3CF2O•. Entropies (S298° in cal mol−1 K−1) were estimated using the M-062x/6-31+g (d,p) computed frequencies and geometries. Rotational barriers were determined and hindered internal rotational contributions for S298°- 1500°, and Cp(T) were calculated using the rigid rotor harmonic oscillator approximation, with direct integration over energy levels of the intramolecular rotation potential energy curves.

Laser-induced fluorescence and dispersed-fluorescence spectroscopy of the Ã2E-X̃2A1 transition of jet-cooled calcium methoxide (CaOCH3) radicals.

Laser-induced fluorescence (LIF) and dispersed fluorescence (DF) spectra of the Ã2E-X̃2A1 electronic transition of the calcium methoxide (CaOCH3) radical have been obtained under jet-cooled conditions. Complete active space self-consistent field and coupled-cluster calculations on the free radical were performed to aid the assignment of vibronic transitions observed in the LIF/DF spectra. In addition to dominant spectral features that are well reproduced by vibrational frequencies and Franck-Condon (FC) factors calculated ab initio, the FC matrix for the Ã2E-X̃2A1 electronic transition contains considerable off-diagonal elements that connect (i) the CaO-stretch (ν4) mode and non-CaO stretch modes and (ii) the asymmetric CaOC stretch (ν3) and the CaOC bending (ν8) modes. The Jahn-Teller and pseudo-Jahn-Teller interactions involving the Ã2E state as well as the spin-orbit interaction induce additional vibronic transitions that are not allowed under the Born-Oppenheimer approximation. Additionally, anharmonic vibrational terms in the ground state induce transitions that are forbidden in the harmonic-oscillator approximation. Spin-orbit splitting has been observed for several vibrational levels of the Ã2E state, and an essentially constant value was measured at all levels accessed in the LIF experiment. Implications of the present spectroscopic investigation to the proposed schemes of laser-cooling MOCH3 (M = alkaline earth metals) molecules and detection of time-reversal-symmetry-violating interactions are discussed.

Vibrational spectroscopic characterization, electronic absorption, optical nonlinearity computation and terahertz investigation of (2E) 3-(4-ethoxyphenyl)-1-(3-bromophenyl) prop-2-en-1-one for NLO device fabrication

The present study spotlights the key structure-property features of a chalcone derivative (2E) 3-(4-ethoxyphenyl)-1-(3-bromophenyl) prop-2-en-1-one (E3BC) for future applications in nonlinear optical devices through dual approach involving experimental and theoretical techniques. The sample has been characterized by 1H NMR, FT-IR, FT-Raman and terahertz spectroscopies. The experimental results are supported by computational studies using density functional theory. To analyse the 1H NMR spectra, theoretical calculations on chemical shifts were done by GIAO method. Complete vibrational assignments of the vibrational modes have been made on the basis of the potential energy distribution in terms of internal coordinates. The low frequency vibrational modes between 0.2 and 1.8 THz were examined via terahertz time domain spectroscopy. The UV–vis spectrum was recorded in methanol solution. The energies, absorption wavelengths and oscillator strengths of electronic singlet-singlet transitions were determined by time dependent density functional theory. The calculated HOMO-LUMO energy gap is small which reveals that the charge transfer takes place within the molecule and it can be exploited for the nonlinear optical applications. Information about the charge density distribution and the sites of chemical reactivity of the molecule has been obtained by mapping electron density with molecular electrostatic potential. The MEP surface around the bromine atom exhibited the dual nature. Along the C-Br axis the positive electrostatic potential (the σ-hole) is present while the remainder of the surface is negative. Hence bromine atom can interact electrostatically with both nucleophiles and electrophiles. For other atoms no atom-specific effects are found. The stability of the molecule arising from hyper conjugative interactions and charge delocalization has been interpreted using natural bond orbital analysis. The study indicates that the intramolecular charge transfer (n → σ∗, n → π∗ and π → π∗) can induce nonlinearity in the molecule and validates E3BC's candidature to be used as a nonlinear optical active material. In order to understand the nonlinear optical activity of the molecule the calculations of electronic dipole moment, polarizability, anisotropy of polarizability as well as first and second hyperpolarizabilities are performed using finite field approach implemented in the DFT method. The findings suggested that the first order hyperpolarizability of titled molecule is 139 times greater than the value of urea. The high value of first order hyperpolarizability shows that E3BC can be a potential candidate in the nonlinear optical applications. The vibrational contribution to the above properties has also been computed using the self-consistent field wave functions within the double harmonic oscillator approximation.