Vibrational Phase Space Research Articles

Phase stability of Fe from first principles: Atomistic spin dynamics coupled with ab initio molecular dynamics simulations and thermodynamic integration

The calculation of free energies from first principles in materials is a formidable task which enables the prediction of phase stability with high accuracy; these calculations are complicated in magnetic materials by the interplay of electronic, magnetic, and vibrational degrees of freedom. In this work, we show the feasibility and accuracy of the calculation of phase stability in magnetic systems with ab initio methods and thermodynamic integration by sampling the magnetic and vibrational phase space with coupled atomistic spin dynamics-ab initio molecular dynamics (ASD-AIMD) simulations [Stockem et al., PRL 121, 125902 (2018)], where energies and interatomic forces are calculated with density functional theory (DFT). We employ the method to calculate the phase stability of Fe at ambient pressure from 800 K up to 1800 K. The Gibbs free energy difference between fcc and bcc Fe at zero pressure as a function of temperature is calculated carrying out thermodynamic integration over temperature on the energies at the DFT level from ASD-AIMD, using a reference free energy difference calculated in the paramagnetic state at temperatures much higher than the magnetic transition temperatures with thermodynamic integration over stress-strain variables with disordered local moment (DLM)-AIMD simulations. We show the importance of the magnetic ordering temperature of bcc Fe on the $\alpha$ to $\gamma$ structural transition temperature, whereas the $\gamma$ to $\delta$ transition is well reproduced independently of the exchange interactions. The Gibbs free energy difference between the two structures is within 5 meV/atom from the CALPHAD estimate, and both transition temperatures are reproduced within 150 K. The present work paves the way to free energy calculations in magnetic materials from first principles with accuracy in the order of 1 meV/atom.

Correlation effects on electron-phonon coupling in semiconductors: Many-body theory along thermal lines

A method is proposed for the inclusion of electron correlation in the calculation of the temperature dependence of band structures arising from electron-phonon coupling. It relies on an efficient exploration of the vibrational phase space along the recently introduced thermal lines. Using the $G_0W_0$ approximation, the temperature dependence of the direct gaps of diamond, silicon, lithium fluoride, magnesium oxide, and titanium dioxide is calculated. Within the proposed formalism, a single calculation at each temperature of interest is sufficient to obtain results of the same accuracy as in alternative, more expensive methods. It is shown that many-body contributions beyond semi-local density functional theory modify the electron-phonon coupling strength by almost $50$% in diamond, silicon, and titanium dioxide, but by less than $5$% in lithium flouride and magnesium oxide. The results reveal a complex picture regarding the validity of semi-local functionals for the description of electron-phonon coupling.

Vibrational averages along thermal lines

A method is proposed for the calculation of vibrational quantum and thermal expectation values of physical properties from first principles. Thermal lines are introduced: these are lines in configuration space parametrized by temperature, such that the value of any physical property along them is approximately equal to the vibrational average of that property. The number of sampling points needed to explore the vibrational phase space is reduced by up to an order of magnitude when the full vibrational density is replaced by thermal lines. Calculations of the vibrational averages of several properties and systems are reported, namely the internal energy and the electronic band gap of diamond and silicon, and the chemical shielding tensor of L-alanine. Thermal lines pave the way for complex calculations of vibrational averages, including large systems and methods beyond semi-local density functional theory.

Oscillatory reaction cross sections caused by normal mode sampling in quasiclassical trajectory calculations.

The quasiclassical trajectory (QCT) method is an efficient and important tool for studying the dynamics of bimolecular reactions. In this method, the motion of the atoms is simulated classically, and the only quantum effect considered is that the initial vibrational states of reactant molecules are semiclassically quantized. A sensible expectation is that the initial ensemble of classical molecular states generated this way should be stationary, similarly to the quantum state it is supposed to represent. The most widely used method for sampling the vibrational phase space of polyatomic molecules is based on the normal mode approximation. In the present work, it is demonstrated that normal mode sampling provides a nonstationary ensemble even for a simple molecule like methane, because real potential energy surfaces are anharmonic in the reactant domain. The consequences were investigated for reaction CH4 + H → CH3 + H2 and its various isotopologs and were found to be dramatic. Reaction probabilities and cross sections obtained from QCT calculations oscillate periodically as a function of the initial distance of the colliding partners and the excitation functions are erratic. The reason is that in the nonstationary ensemble of initial states, the mean bond length of the breaking C-H bond oscillates in time with the frequency of the symmetric stretch mode. We propose a simple method, one-period averaging, in which reactivity parameters are calculated by averaging over an entire period of the mean C-H bond length oscillation, which removes the observed artifacts and provides the physically most reasonable reaction probabilities and cross sections when the initial conditions for QCT calculations are generated by normal mode sampling.

Spectral Intensity Patterns and Vibrational Phase Space Structure

We examine patterns of absorption intensity in two-mode systems with a 2:1 Fermi resonance in an intensity model based on the effective fitting Hamiltonian. We relate these patterns to the Fermi resonance phase space structure and catastrophe map. Each of the four zones of the catastrophe map has a phase sphere structure with a certain number of distinct regions. For each zone, we find that for every region on the sphere, there is a distinct region in the intensity pattern.

Molecular rotation–vibration coupling by quantum wavepacket interference at potential energy curve crossings

Quantum wavepacket interference is shown to result in modulation of the ultrafast transient anisotropy of a diatomic molecule. The molecule is in a state comprising two bound electronic states at intermediate coupling strength and the anisotropy modulation enters via variations in the instantaneous contributions of the initial states to the total anisotropy. A necessary condition for the appearance of the modulation is that the involved rotational states possess different vibrational dynamics. The wavepacket interference mediates this coupling by imposing dynamical restrictions on the accessible vibrational phase space which depend on the rotational state.

Theoretical Photoabsorption Spectra of ClOOCl and Cl2O

We calculated transition energies and oscillator strengths of ClOOCl and Cl2O in the Franck−Condon region with ab initio techniques. We describe 8 singlet excited states for ClOOCl and 13 excited states for Cl2O. The photoabsorption spectra are simulated by a semiclassical approach, which takes into account the vibrational phase space distribution in the ground state and the geometry dependence of excitation energies and transition dipoles. We present an unambiguous assignation of the available UV spectra, which is preliminary to the interpretation of the photochemistry of these compounds.

Competing mechanisms for intramolecular vibrational redistribution in the ν14 asymmetric methyl stretch band of trans-ethanol

The extensively perturbed spectrum of the asymmetric methyl stretching vibration of trans-ethanol near 2990 cm−1 has been reinvestigated via direct absorption infrared spectroscopy at a resolution of 30 MHz. A ground state combination difference analysis of the vibrational state mixing is presented for the upper state levels Ka′ = 0–2 and J′=0–4. The analysis indicates that the rotationless 000 level is anharmonically coupled to the dark bath states. The effective number of perturbing states in each rovibrational transition increases with both J and Ka providing evidence for rotational involvement in intramolecular vibrational redistribution (IVR). The decrease of the average dilution factor from φd=0.41 at Ka′ = 0 to φd=0.09 at Ka′ = 2 and the increase of the average interaction width from Δε=0.04 cm−1 at Ka′ = 0 to Δε=0.19 cm−1 at Ka′ = 2 indicate an a-type Coriolis component to the bright-bath coupling. In the Ka′ = 0 series the dilution factor decreases rapidly from φd=0.92 at J′=0 to φd=0.14 at J′=3 indicating that b,c-type Coriolis coupling also plays a significant role in the IVR process. The effective level density ρeffc for all of the observed transitions lie above the total vibrational state density ρvib=9 levels per cm−1 and most are closer to the total rovibrational state density ρrovib=(2J+1)ρvib. This suggests that following a coherent preparation of the asymmetric methyl stretching vibration, the ensuing dynamics explores all of the energetically accessible vibrational phase space of both the gauche and trans forms and much of the accessible rovibrational phase space, i.e., that the Ka quantum number is at least partially destroyed. The C–H stretch is deduced to decay with a 59 ps IVR lifetime to the asymptotic probability of 0.24.

Rate and mechanism of intramolecular vibrational redistribution in the ν16 asymmetric methyl stretch band of 1-butyne

The spectrum of the ν16 asymmetric methyl stretch vibration of 1-butyne near 2991 cm−1 has been studied via direct absorption infrared spectroscopy at a resolution of 35 MHz. Analysis by ground state combination difference indicates that the ν16 band is extensively perturbed by dark vibrational bath states. All of the transitions appear as multiplets of about five eigenstates in a window of about 0.017 cm−1. A detailed analysis is presented for the upper state levels K′a = 0–2 and J′=0–6. A lack of J′ dependence implies anharmonic coupling is dominant and that b- and c-type Coriolis interactions are not important at these low J′ values. However, the average dilution factor goes from 0.72 at the K′a = 0 to 0.46 at the K′a = 2 suggesting weak a-type Coriolis interactions. For the K′a = 0 levels, the measured average level density of 17 states/cm−1/symmetry species is comparable to the value of 14 vibrational states/cm−1/symmetry species obtained from a symmetry specific direct count. This is an indication that the dynamics explore all of the energetically available vibrational phase space. The nearly Gaussian distribution of matrix elements suggests that there is significant coupling among the bath states. At the K′a = 0 level, the rms anharmonic coupling matrix element is <vsj2≳1/2=0.0125 cm−1. From the frequency-resolved data, a coherently prepared asymmetric methyl stretch in 1-butyne is deduced to decay with a 276 ps time constant to the asymptotic probability of 0.6.

Nonlinear dynamics of methyl rotation and intramolecular energy diffusion in p-fluorotoluene

This paper examines the effect of large amplitude internal rotation on the rate and extent of intramolecular vibrational energy redistribution (IVR). We study a classical Hamiltonian modeling the vibrations of p-fluorotoluene in its first excited singlet (S1) electronic state. We find that the full many-dimensional vibrational phase space of this system can be approximately decomposed into two subsystems. The first consists of the methyl rotor and the lowest-frequency ring modes, which interest strongly and chaotically with the methyl rotor. Within this subsystem, energy is rapidly exchanged. The second subsystem consists of the remaining high-frequency modes, which do not strongly couple to the methyl rotor directly. The chaotic low-frequency ring–rotor dynamics generate an effectively random force on the remaining degrees of freedom. This intrinsically stochastic perturbation induces slower intramolecular energy diffusion and relaxation of nonequilibrium initial distributions in the higher-frequency ring modes.

ChemInform Abstract: The Evolution of Vibrational Phase Space During the Collisional Relaxation of Highly Excited Collinear CS2

ChemInformVolume 21, Issue 40 Physical Organic Chemistry ChemInform Abstract: The Evolution of Vibrational Phase Space During the Collisional Relaxation of Highly Excited Collinear CS2 M. BRUEHL, M. BRUEHL Dep. Chem., Northwest. Univ., Evanston, IL 60208, USASearch for more papers by this authorG. C. SCHATZ, G. C. SCHATZ Dep. Chem., Northwest. Univ., Evanston, IL 60208, USASearch for more papers by this author M. BRUEHL, M. BRUEHL Dep. Chem., Northwest. Univ., Evanston, IL 60208, USASearch for more papers by this authorG. C. SCHATZ, G. C. SCHATZ Dep. Chem., Northwest. Univ., Evanston, IL 60208, USASearch for more papers by this author First published: October 2, 1990 https://doi.org/10.1002/chin.199040034Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume21, Issue40October 2, 1990 RelatedInformation

The evolution of vibrational phase space during the collisional relaxation of highly excited collinear CS2

This paper presents a detailed theoretical analysis of the vibrational relaxation of highly excited CS2 (initially 32 640 cm−1) in collinear collisions with a thermal bath of He atoms. The relaxation is simulated by a classical molecular dynamics method in which CS2 undergoes successive collisions with thousands of He atoms. In most of our studies the CS2 coordinates and momenta at the end of one collision are used as input to the next collision, so it is possible to examine the detailed evolution of the CS2 vibrational phase space during the relaxation process. By restricting motion to being collinear, it is possible to characterize this evolution using surfaces of section and other methods. Comparisons of our collinear results with corresponding three-dimensional simulations indicates that the collinear restriction does not alter the relaxation process significantly. Our phase space analysis indicates that individual relaxation sequences can evolve in a variety of different ways depending on the initial location in phase space and on the details of subsequent collisions. Much of the initial phase space is chaotic, and if a sequence starts in such a region then after usually less than 30 collisions, the CS2 has moved into a nonlinear resonance zone where the antisymmetric and symmetric stretch modes have frequency ratios of 5:2, 7:3, or 9:4. These nonlinear resonances do not greatly change the ensemble averaged energy transfer per collision 〈ΔE〉 compared to the chaotic regions, but they are collisionally stable relative to these regions. As a result, it takes an energetic collision to kick the molecule out of a nonlinear resonance. If kicked out, then usually within a few more collisions another nonlinear resonance (or perhaps the same) has been entered. As relaxation progresses molecules caught in nonlinear resonances eventually drop down to simple quasiperiodic regions where the frequency ratio is not constrained to be a ratio of integers. We do find a region of phase space that is quasiperiodic even at 32 640 cm−1, corresponding to a ‘‘hyperspherical mode’’ in which most of the vibrational energy is locked up in antisymmetric stretch motion. Molecules in this region of phase space relax much more slowly than in chaotic and resonant regions. In addition, molecules starting initially in a chaotic region can be kicked into this hyperspherical mode region, leading to an additional slowing of the relaxation as the molecule drops down the well. This additional slowing plays an important role in determining the dependence of 〈ΔE〉 on the molecular vibrational energy E. In particular, we find that 〈ΔE〉 varies linearly with E if phase space undergoes forced randomization after each collision, but it shows a stronger than linear dependence when redistribution is not forced. This implies that deviations from linearity in the dependence of 〈ΔE〉 on E provide a measure of the division of phase space into regions that have very different relaxation characteristics.

Picosecond photofragment spectroscopy. III. Vibrational predissociation of van der Waals’ clusters

This paper, last in this series, reports on the picosecond dynamics of vibrational predissociation in beam-cooled van der Waals’ clusters. Reaction rates have been measured for clusters (1:1) of phenol and cresol (p-methylphenol) with benzene by the picosecond pump–probe photoionization mass-spectrometry technique. Dissociation to form phenol (cresol) and benzene takes place from vibrational levels of the S1 state of phenol (cresol) prepared by the pump laser. The predissociation rates were measured for a number of different excess energies upto ∼2500 cm−1, and the reaction threshold was found to be 1400 cm−1 above the S1 origin for phenol–benzene and ∼1795 cm−1 for cresol–benzene, respectively. For phenol–benzene, the predissociation rates, following excitation of ring-type modes, vs excess energy vary more or less smoothly. Cresol–benzene exhibits biexponential decay, with the fast component becoming more dominant at higher energies. A non-RRKM model involving division of the vibrational phase space is discussed to explain this observation.

Picosecond photofragment spectroscopy. II. The overtone initiated unimolecular reaction H2O2(vOH=5)→2OH

This paper, second in the series, reports on the picosecond time-resolved photofragmentation of the overtone (vOH=5) initiated reaction: HOOH+hν→2OH. The hydrogen peroxide is initially excited by way of a picosecond laser pulse to the fourth overtone level of the OH-stretch local mode. The subsequent unimolecular reaction behavior is obtained by monitoring the laser-induced fluorescence, caused by the picosecond probe–pulse electronic excitation of the OH radical photoproduct (in a given rotational state). The two pulses are scanned relative to one another in time thereby mapping out the product yield for the given delay-time interval. The resultant product formation behavior is found to be nonexponential, and may be modeled as a biexponential rise. Furthermore, the quasibiexponential behavior is sensitive to the exact excitation wavelength—slight variations of which result in large changes in the two time constants and the relative amplitudes of the fast and slow components. These experiments give direct evidence for the inhomogeneous nature of the overtone transition on the picosecond time scale, and provide the dissociation rate contribution to the homogeneous width (0.05–0.15 cm−1).The apparent width for the main band feature is about 200 cm−1. The rate of product formation (magnitude and form) is interpreted in terms of statistical and nonstatistical theories. The limitations of the applicability of each model is discussed. The fluctuations of the fitting parameters as a function of excitation wavelength may be simulated by a statistical model which considers all possible discrete optical transitions within the simulated laser bandwidth and the details of product formation from each state. For a nonstatistical interpretation, the biexponential form reflects a division of the vibrational phase space, and this is discussed in the spirit of a kinetic model. Finally, experimental results are reported for direct UV initiated photofragmentation. The observed dynamics indicate that a very different type of potential surface (repulsive) is involved, in contrast to the overtone initiated dissociation, which takes place on the ground state surface.

Incomplete vibrational state mixing in the J′ = 0 triplet manifold of pyrazine at an excess energy of ≈ 4000 cm −1. Possible evidence for non-ergodicity

By combining data obtained from a deconvolution of the ultra-high-resolution J′ = 0 LIF spectrum of the mixed singlet-triplet states of pyrazine with some simple ideas regarding spin-orbit coupling in polyatomic molecules, we show that the “prediagonalized” triplet levels which are coupled to the zero-order singlet retain their vibrational identity (i.e. are incompletely mixed). This suggests that, when the rotationless molecule is excited to its lowest triplet state with an excess energy of = 4000 cm −1, it does not explore uniformly all regions of the available vibrational phase space. Some possible reasons for this behavior are discussed.

The harmonic mean field model for molecular solvents: A liquid state analog of the harmonic oscillator-rigid top approximation

A liquid state analog of the gas phase harmonic oscillator-rigid top model is developed. This analog, which we call the harmonic mean field model, permits one to compute the equilibrium and dynamic properties of real, i.e., vibrating, molecular solvents from the structure and dynamics of the corresponding rigid solvents. The harmonic mean field model is based on: (i) A mean field harmonic model for the solvent vibrational (V) force field. (ii) A rigid solvent model treatment of translational–rotational (TR) fluctuations. (iii) Complete neglect of explicit coupling between V and TR fluctuations. (Implicit coupling is included in the vibrational force field.) The model is developed for statics via a sequence of physically motivated approximations to the exact canonical ensemble phase space distribution function of the solvent, fCA[S]. This yields a model distribution function f(0)CA[S] =f(0)CA[pyy]fCA[pww; v0], where f(0)CA[pyy] is an effective harmonic vibrational phase space distribution function which describes mean field harmonic V fluctuations and where f(0)CA[pww; v0] is the rigid solvent canonical ensemble distribution function. The nonequilibrium version of the model is defined as the solvent dynamics generated by a model Liouville operator L(0). This is defined via the model equilibrium Liouville equation L(0)f(0)CA [S]=0. Explicit results for equilibrium averages and time correlation functions of molecular solvents are obtained. The frequency spectra of the time correlation functions contain a low frequency ‘‘acoustic’’ branch arising from solvent TR motions and high frequency ‘‘optical’’ branches arising from collective solvent V motions. A detailed analysis of the frequency spectra of autocorrelation functions of diatomic solvents is presented.

Generalized Langevin theory for many-body problems in chemical dynamics: Formulation for molecular solvents, equilibrium aspects

This paper deals with equilibrium statistical mechanical aspects of the extension of the MTGLE approach for chemical reaction dynamics in liquids [S. A. Adelman, Adv. Chem. Phys. 53, 611 (1983)] so that it is applicable to molecular as well as monatomic solvents. This extension is necessary in order to conveniently treat energy exchange between the solute molecules and the solvent vibrational degrees of freedom. The analysis yields a separation of equilibrium solvent density fluctuations into translation-rotational (TR) and vibrational (V) contributions. The TR fluctuations are treated within the rigid solvent molecule model. The V fluctuations are treated within the harmonic oscillator approximation. The analysis is carried out in terms of a set of generalized solvent phase space coordinates S=pvvpww where v and w are, respectively, the V and TR coordinates and where pv and pw are the corresponding conjugate momenta. In this coordinate system, canonical ensemble distribution function for the solvent given that the solute is fixed at configuration point r0, denoted by fCA[S;r0], may be factorized as fCA[S;r0] =fCA[ pvv] fCA[ pw w;v0r0] where fCA[ pww;v0r0] and fCA[pvv] are respectively: the rigid solvent model approximation to fCA[S;r0]; a vibrational phase space probability distribution function which governs equilibrium mean field V fluctuations in the pure solvent. This vibrational probability distribution function may be written as fCA[ pvv] =Z−1 exp[−β(Tv+W(v))] where Z=pure solvent partition function, where Tv =pure solvent kinetic energy, and where W[v] is a vibrational potential of mean force. The above factorization of fCA[S;r0] holds if: (i) Coriolis coupling between the v and w coordinates is ignored; (ii) The solvent vibrational frequencies are sufficiently high. Within the harmonic approximation to W[v], fCA[ pvv] =[2πkβT]−rNs det ω2 exp [−β(Tv+ 1/2 yTω2y)] where y are the mass-weighted displacements of the v coordinates from equilibrium, where rNs =number of solvent normal modes, and where ω2 is an (rNs×rNs) -dimensional dynamical matrix which determines the vibrational frequency spectrum of the pure solvent within the harmonic approximation. This dynamical matrix may be constructed from the canonical ensemble distribution function of the pure rigid molecule solvent. Thus given the harmonic approximation to W[v] the canonical ensemble distribution function of the nonrigid solvent fCA[S;r0] may be constructed from the equilibrium properties of the rigid molecule solvent.

Very long-lived doubly vibrationally excited states above dissociation threshold: non-RRKM behavior in model triatomic systems

For couplings and mass ratios appropriate to molecules such as H 2O, D 2O, and CH 2 large volumes of the classical vibrational phase space are found to be non-dissociating even well above threshold. Fully quantum calculations yield families of exceptionally long-lived quantum states corresponding to these trapped volumes.

THE CORRELATION OF THE SYMMETRY CLASSIFICATIONS OF STATES OF NONRIGID MOLECULES

The rovibronic and vibronic energy levels of some nonrigid molecules occur in nearly degenerate sets, whose splitting is due to tunneling between symmetrically equivalent regions of vibrational phase space. When the splitting is negligible, it is convenient to classify the different sets of energy levels according to a symmetry group that is determined by any one of the symmetrically equivalent regions, whereas when the splitting is appreciable, the individual split levels can be classified further according to a larger group that is determined in a similar way by a region containing all these equivalent regions. The former group is a subgroup of the latter.A simple correlation rule is proved, by means of which the symmetry species (in the larger group) of the individual split levels can be obtained from the symmetry species (in the smaller group) of the nearly degenerate set of levels. Some examples of the application of this rule to rovibronic and vibronic species are given.