A canonical averaging in the second-order quantized Hamilton dynamics

Quantized Hamilton dynamics (QHD) is a simple and elegant extension of classical Hamilton dynamics that accurately includes zero-point energy, tunneling, dephasing, and other quantum effects. Formulated as a hierarchy of approximations to exact quantum dynamics in the Heisenberg formulation, QHD has been used to study evolution of observables subject to a single initial condition. In present, we develop a practical solution for generating canonical ensembles in the second-order QHD for position and momentum operators, which can be mapped onto classical phase space in doubled dimensionality and which in certain limits is equivalent to thawed Gaussian. We define a thermal distribution in the space of the QHD-2 variables and show that the standard beta=1/kT relationship becomes beta'=2/kT in the high temperature limit due to an overcounting of states in the extended phase space, and a more complicated function at low temperatures. The QHD thermal distribution is used to compute total energy, kinetic energy, heat capacity, and other canonical averages for a series of quartic potentials, showing good agreement with the quantum results.     

Classical and quantum mechanical infrared echoes from resonantly coupled molecular vibrations

The nonlinear response function associated with the infrared vibrational echo is calculated for a quantum mechanical model of resonantly coupled, anharmonic oscillators at zero temperature. The classical mechanical response function is determined from the quantum response function by setting variant Planck's over 2pi-->0, permitting the comparison of the effects of resonant vibrational coupling among an arbitrary number of anharmonic oscillators on quantum and classical vibrational echoes. The quantum response function displays a time dependence that reflects both anharmonicity and resonant coupling, while the classical response function depends on anharmonicity only through a time-independent amplitude, and shows a time dependence controlled only by the resonant coupling. In addition, the classical response function grows without bound in time, a phenomenon associated with the nonlinearity of classical mechanics, and absent in quantum mechanics. This unbounded growth was previously identified in the response function for a system without resonant vibrational energy transfer, and is observed to persist in the presence of resonant coupling among vibrations. Quantitative agreement between classical and quantum response functions is limited to a time scale of duration inversely proportional to the anharmonicity.     

Second-order quantized Hamilton dynamics coupled to classical heat bath

Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.     

Molecular dynamics with quantum transitions study of the vibrational relaxation of the HOD bend fundamental in liquid D2O

The molecular dynamics with quantum transitions method is used to study the vibrational relaxation of the HOD bend fundamental in liquid D(2)O. All of the vibrational bending degrees of freedom of the HOD and D(2)O molecules are described by quantum mechanics, while the remaining translational and rotational degrees of freedom are described classically. The effect of the coupling between the rotational and vibrational degrees of freedom of the deuterated water molecules is analyzed. A kinetic mechanism based on three steps is proposed in order to interpret the dynamics of the system. It is shown that intermolecular vibrational energy transfer plays an important role in the relaxation process and also that the transfer of energy into the rotational degrees of freedom is favored over the transfer of energy into the translational motions. The thermalization of the system after the relaxation is reached in a shorter time scale than that of the recovery of the hydrogen bond network. The relaxation and equilibration times obtained compare well with experimental and previous theoretical results.     

Influences of Quantum Mechanically Mixed Electronic and Vibrational Pigment States in 2D Electronic Spectra of Photosynthetic Systems: Strong Electronic Coupling Cases

In 2D electronic spectroscopy studies, long‐lived quantum beats have recently been observed in photosynthetic systems, and several theoretical studies have suggested that the beats are produced by quantum mechanically mixed electronic and vibrational states. Concerning the electronic‐vibrational quantum mixtures, the impact of protein‐induced fluctuations was examined by calculating the 2D electronic spectra of a weakly coupled dimer with the Franck‐Condon active vibrational modes in the resonant condition [Fujihashi et al., J. Chem. Phys.­ 2015 , 142, 212403.]. This analysis demonstrated that quantum mixtures of the vibronic resonance are rather robust under the influence of the fluctuations at cryogenic temperatures, whereas the mixtures are eradicated by the fluctuations at physiological temperatures. However, this conclusion cannot be generalized because the magnitude of the coupling inducing the quantum mixtures is proportional to the inter‐pigment electronic coupling. In this study, we explore the impact of the fluctuations on electronic‐vibrational quantum mixtures in a strongly coupled dimer with an off‐resonant vibrational mode. Toward this end, we calculate energy transfer dynamics and 2D electronic spectra of a model dimer that corresponds to the most strongly coupled bacteriochlorophyll molecules in the Fenna‐Matthews‐Olson complex in a numerically accurate manner. The quantum mixtures are found to be robust under the exposure of protein‐induced fluctuations at cryogenic temperatures, irrespective of the resonance. At 300 K, however, the quantum mixing is disturbed more strongly by the fluctuations, and therefore, the beats in the 2D spectra become obscure even in a strongly coupled dimer with a resonant vibrational mode. Further, the overall behaviors of the energy transfer dynamics are demonstrated to be dominated by the environment and coupling between the 0 0 vibronic transitions as long as the Huang‐Rhys factor of the vibrational mode is small. The electronic‐vibrational quantum mixtures do not necessarily play a significant role in electronic energy transfer dynamics despite contributing to the enhancement of long‐lived quantum beating in the 2D spectra.     

Nuclear quantum effects on an enzyme-catalyzed reaction with reaction path potential: proton transfer in triosephosphate isomerase

Nuclear quantum mechanical effects have been examined for the proton transfer reaction catalyzed by triosephosphate isomerase, with the normal mode centroid path integral molecular dynamics based on the potential energy surface from the recently developed reaction path potential method. In the simulation, the primary and secondary hydrogens and the C and O atoms involving bond forming and bond breaking were treated quantum mechanically, while all other atoms were dealt classical mechanically. The quantum mechanical activation free energy and the primary kinetic isotope effects were examined. Because of the quantum mechanical effects in the proton transfer, the activation free energy was reduced by 2.3 kcal/mol in comparison with the classical one, which accelerates the rate of proton transfer by a factor of 47.5. The primary kinetic isotope effects of kH/kD and kH/kT were estimated to be 4.65 and 9.97, respectively, which are in agreement with the experimental value of 4+/-0.3 and 9. The corresponding Swain-Schadd exponent was predicted to be 3.01, less than the semiclassical limit value of 3.34, indicating that the quantum mechanical effects mainly arise from quantum vibrational motion rather than tunneling. The reaction path potential, in conjunction with the normal mode centroid molecular dynamics, is shown to be an efficient computational tool for investigating the quantum effects on enzymatic reactions involving proton transfer.     

Nonadiabatic effects on peptide vibrational dynamics induced by conformational changes

Quantum dynamical simulations of vibrational spectroscopy have been carried out for glycine dipeptide (CH(3)-CO-NH-CH(2)-CO-NH-CH(3)). Conformational structure and dynamics are modeled in terms of the two Ramachandran dihedral angles of the molecular backbone. Potential energy surfaces and harmonic frequencies are obtained from electronic structure calculations at the density functional theory (DFT) [B3LYP/6-31+G(d)] level. The ordering of the energetically most stable isomers (C(7) and C(5)) is reversed upon inclusion of the quantum mechanical zero point vibrational energy. Vibrational spectra of various isomers show distinct differences, mainly in the region of the amide modes, thereby relating conformational structures and vibrational spectra. Conformational dynamics is modeled by propagation of quantum mechanical wave packets. Assuming a directed energy transfer to the torsional degrees of freedom, transitions between the C(7) and C(5) minimum energy structures occur on a sub-picosecond time scale (700...800 fs). Vibrationally nonadiabatic effects are investigated for the case of the coupled, fundamentally excited amide I states. Using a two state-two mode model, the resulting wave packet dynamics is found to be strongly nonadiabatic due to the presence of a seam of the two potential energy surfaces. Initially prepared adiabatic vibrational states decay upon conformational change on a time scale of 200...500 fs with population transfer of more than 50% between the coupled amide I states. Also the vibrational energy transport between localized (excitonic) amide I vibrational states is strongly influenced by torsional dynamics of the molecular backbone where both enhanced and reduced decay rates are found. All these observations should allow the detection of conformational changes by means of time-dependent vibrational spectroscopy.     

Mixed quantum-classical molecular dynamics simulation of vibrational relaxation of ions in an electrostatic field

The vibrational relaxation of ions in low-density gases under the action of an electrostatic field is reproduced through a molecular dynamics simulation method. The vibration is treated though quantum mechanics and the remaining degrees of freedom are considered classical. The procedure is tested through comparison against analytic results for a two-dimensional quantum model and by studying energy exchange during binary ion-atom collisions. Finally, the method has been applied successfully to the calculation of the mobility and the vibrational relaxation rate of O2+ in Kr as a function of the mean collision energy using a model interaction potential that reproduces the potential minimum of a previously known ab initio potential surface. The calculation of the steady mean vibrational motion of the ions in (flow) drift tubes seems straightforward, though at the expense of large amounts of computer time.     

Hybrid quantum/classical studies of photodissociation and recombination of I2 (A) in rare gas matrices: A linear chain model

A hybrid quantum/classical model is developed for the photodissociation and recombination dynamics of an I₂ molecule in low-temperature rare-gas (Rg) matrices. The simplified model consists of an I₂ molecule embedded in a linear chain of Rg atoms. The aggregate is partitioned into a quantum system and a classical bath, which are self-consistently coupled. Two partitioning schemes are used. The first treats the I-I coordinate quantum mechanically and the Rg coordinates classically. The second and more reliable scheme includes in the quantum system both the I-I mode and the symmetric motion of the two nearest Rg atoms. Both models show substantial energy transfer from the dissociating I₂ to the solvent, followed by coherent vibrational motion of the recombined I₂. It is found that the one-dimensional quantum/classical scheme is consistent with its higher dimensional counterpart, although the latter shows much faster dephasing. © 1996 John Wiley & Sons, Inc.     

Interpreting nonlinear vibrational spectroscopy with the classical mechanical analogs of double-sided Feynman diagrams

Observables in coherent, multiple-pulse infrared spectroscopy may be computed from a vibrational nonlinear response function. This response function is conventionally calculated quantum-mechanically, but the challenges in applying quantum mechanics to large, anharmonic systems motivate the examination of classical mechanical vibrational nonlinear response functions. We present an approximate formulation of the classical mechanical third-order vibrational response function for an anharmonic solute oscillator interacting with a harmonic solvent, which establishes a clear connection between classical and quantum mechanical treatments. This formalism permits the identification of the classical mechanical analog of the pure dephasing of a quantum mechanical degree of freedom, and suggests the construction of classical mechanical analogs of the double-sided Feynman diagrams of quantum mechanics, which are widely applied to nonlinear spectroscopy. Application of a rotating wave approximation permits the analytic extraction of signals obeying particular spatial phase matching conditions from a classical-mechanical response function. Calculations of the third-order response function for an anharmonic oscillator coupled to a harmonic solvent are compared to numerically correct classical mechanical results.     

Quantum tunneling effect in entanglement dynamics

Quantum tunneling effect in entanglement dynamics between two coupled particles with separable Gaussian initial state is investigated using entangled trajectory molecular dynamics method in terms of the reduced‐density linear entropy. It has been presented through showing distinguish contribution of single trajectory to linear entropy between classical trajectory and entangled trajectory with same initial state. We find that quantum tunneling effect makes single trajectory's contribution remarkably decrease under quantum dynamics compared to classical dynamics. The nonlocality of quantum entanglement is presented, and the energy transfer between two coupled particles through quantum correlations and classical ones is also discussed in the end. © 2015 Wiley Periodicals, Inc.     

Derivation of class II force fields: V. Quantum force field for amides,peptides, and related compounds

As the field of biomolecular structure advances, there is an ever-growing need for accurate modeling of molecular energy surfaces to simulate and predict the properties of these important systems. To address this need, a second generation amide force field for use in simulations of small organics as well as proteins and peptides has been derived. The critical question of what accuracy can be expected from calculations in general, and with this class II force field in particular, is addressed for structural, dynamic, and energetic properties. The force field is derived from a recent methodology we have developed that involves the systematic use of quantum mechanical observables. Systematic ab initio calculations were carried out for numerous configurations of 17 amide and related compounds. Relative energies and first and second derivatives of the energy of 638 structures of these compounds resulted in 140,970 ab initio quantum mechanical observables. The class II peptide quantum mechanical force field (QMFF), containing 732 force constants and reference values, was parameterized against these observables. A major objective of this work is to help establish the role of anharmonicity and coupling in improving the accuracy of molecular force fields, as these terms have not yet become an agreed upon standard in the ever more extensive simulations being used to probe biomolecular properties. This has been addressed by deriving a class I harmonic diagonal force field (HDFF), which was fit to the same energy surface as the QMFF, thus providing an opportunity to quantify the effects of these coupling and anharmonic contributions. Both force field representations are assessed in terms of their ability to fit the observables. They have also been tested by calculating the properties of 11 stationary states of these amide molecules. Optimized structures, vibrational frequencies, and conformational energies obtained from the quantum calculations and from both the QMFF and the HDFF are compared. Several strained and derivatized compounds including urea, formylformamide, and butyrolactam are included in these tests to assess the range of applicability (transferability) of the force fields. It was found that the class II coupled anharmonic force field reproduced the structures, energies, and vibrational frequencies significantly more faithfully than the class I harmonic diagonal force field. An important measure, rms energy deviation, was found to be 1.06 kcal/mol with the class II force field, and 2.30 kcal/mol with the harmonic diagonal force field. These deviations represent the error in relative configurational energy differences for strained and distorted structures calculated with the force fields compared with quantum mechanics. This provides a measure of the accuracy that might be expected in applications where strain may be important such as calculating the energy of a system as it approaches a (rotational) barrier, in ligand binding to a protein, or effects of introducing substituents into a molecule that may induce strain. Similar results were found for structural properties. Protein dynamics is becoming of ever-increasing interest, and, to simulate dynamic properties accurately, the dynamic behavior of model compounds needs to be well accounted for. To this end, the ability of the class I and class II force fields to reproduce the vibrational frequencies obtained from the quantum energy surface was assessed. An rms deviation of 43 cm⁻¹ was achieved with the coupled anharmonic force field, as compared to 105 cm⁻¹ with the harmonic diagonal force field. Thus, the analysis presented here of the class II force field for the amide functional group demonstrates that the incorporation of anharmonicity and coupling terms in the force field significantly improves the accuracy and transferability with regard to the simulation of structural, energetic, and dynamic properties of amides. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 430–458, 1998     

Collisional line shapes for low frequency vibrations of adsorbates on a metal surface

The dynamics of atoms or molecules adsorbed on a metal surface, and excited by collisions with an atomic beam, are treated within a theory that includes energy dissipation into lattice vibrations by means of a frequency and temperature dependent friction function. The theory provides dynamic structure factors for energy transfer derived from collisional time correlation functions. It describes the relaxation of a vibrationally excited atom or molecule within a model of a damped quantum harmonic oscillator bilinearly coupled to a bath of lattice oscillators. The collisional time correlation function is generalized to include friction effects and is applied to the vibrational relaxation of the frustrated translation mode of Na adsorbed on a Cu(001) surface, CO on Cu(001), and CO on Pt(111), following excitation by collisions with He atoms. Results for the frequency shift and width of line shapes versus surface temperature are in very good agreement with experimental measurements of inelastic He atom scattering. Our interpretation of the experimental results provides insight on the relative role of phonon versus electron-hole relaxation.     

Quantum and classical studies of the O(3P) + H2(v = 0-3,j = 0) --> OH + H reaction using benchmark potential surfaces

We present results of time-dependent quantum mechanics (TDQM) and quasiclassical trajectory (QCT) studies of the excitation function for O(3P) + H2(v = 0-3,j = 0) --> OH + H from threshold to 30 kcal/mol collision energy using benchmark potential energy surfaces [Rogers et al., J. Phys. Chem. A 104, 2308 (2000)]. For H2(v = 0) there is excellent agreement between quantum and classical results. The TDQM results show that the reactive threshold drops from 10 kcal/mol for v = 0 to 6 for v = 1, 5 for v = 2 and 4 for v = 3, suggesting a much slower increase in rate constant with vibrational excitation above v = 1 than below. For H2(v > 0), the classical results are larger than the quantum results by a factor approximately 2 near threshold, but the agreement monotonically improves until they are within approximately 10% near 30 kcal/mol collision energy. We believe these differences arise from stronger vibrational adiabaticity in the quantum dynamics, an effect examined before for this system at lower energies. We have also computed QCT OH(v',j') state-resolved cross sections and angular distributions. The QCT state-resolved OH(v') cross sections peak at the same vibrational quantum number as the H2 reagent. The OH rotational distributions are also quite hot and tend to cluster around high rotational quantum numbers. However, the dynamics seem to dictate a cutoff in the energy going into OH rotation indicating an angular momentum constraint. The state-resolved OH distributions were fit to probability functions based on conventional information theory extended to include an energy gap law for product vibrations.     

Quantum dynamics of inelastic scattering with a moving grid

The time‐dependent discrete variable representation (TDDVR) of a wave function with grid points defined by the Hermite part of the Gauss–Hermite (G‐H) basis set introduces quantum corrections to classical mechanics. The grid points in this method follow classical trajectory and the approach converges to the exact quantum formulation with sufficient trajectories (TDDVR points) but just with a single grid point; only classical mechanics performs the dynamics. This newly formulated approach (developed for handling time‐dependent molecular quantum dynamics) has been explored to calculate vibrational transitions in the inelastic scattering processes. Traditional quantum mechanical results exhibit an excellent agreement with TDDVR profiles during the entire propagation when enough grid points are included in the quantum‐classical dynamics. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Exploring the importance of quantum effects in nucleation: the archetypical Ne(n) case

The effect of quantum mechanics (QM) on the details of the nucleation process is explored employing Ne clusters as test cases due to their semi-quantal nature. In particular, we investigate the impact of quantum mechanics on both condensation and dissociation rates in the framework of the microcanonical ensemble. Using both classical trajectories and two semi-quantal approaches (zero point averaged dynamics, ZPAD, and Gaussian-based time dependent Hartree, G-TDH) to model cluster and collision dynamics, we simulate the dissociation and monomer capture for Ne(8) as a function of the cluster internal energy, impact parameter and collision speed. The results for the capture probability P(s)(b) as a function of the impact parameter suggest that classical trajectories always underestimate capture probabilities with respect to ZPAD, albeit at most by 15%-20% in the cases we studied. They also do so in some important situations when using G-TDH. More interestingly, dissociation rates k(diss) are grossly overestimated by classical mechanics, at least by one order of magnitude. We interpret both behaviours as mainly due to the reduced amount of kinetic energy available to a quantum cluster for a chosen total internal energy. We also find that the decrease in monomer dissociation energy due to zero point energy effects plays a key role in defining dissociation rates. In fact, semi-quantal and classical results for k(diss) seem to follow a common "corresponding states" behaviour when the proper definition of internal and dissociation energies are used in a transition state model estimation of the evaporation rate constants.     

Exploring the effects of intramolecular vibrational energy redistribution on the operation of the proton wire in green fluorescent protein

The operation of the proton wire in Green Fluorescent Protein has been simulated by quantum dynamics and considering the coupling to the protein environment by means of a bath of harmonic oscillators. The simulation consists of 36 explicit and fully quantum degrees of freedom: 6 degrees of freedom represent the configuration of the proton wire, which are coupled to 30 bath coordinates. Regimes of weak and strong coupling have been studied. It is found that presence of the bath induces a fast energy transfer from the proton wire to the bath, with characteristic times under 400 fs. This internal vibrational redistribution happens at the expense of the potential energy content of the proton wire, deformed through the interaction to the bath from its uncoupled state. Strong coupling induces a slowing-down of the operation of the wire because it hinders to some extent the approaching of donor and acceptor atoms to distances in which proton transfer can occur. Internal vibrational energy redistribution affects the dynamics, but from our simulations we conclude that it cannot be the only cause responsible for the experimentally reported fluorescence rise times.