114 research outputs found
Path integral evaluation of equilibrium isotope effects
A general and rigorous methodology to compute the quantum equilibrium isotope
effect is described. Unlike standard approaches, ours does not assume
separability of rotational and vibrational motions and does not make the
harmonic approximation for vibrations or rigid rotor approximation for the
rotations. In particular, zero point energy and anharmonicity effects are
described correctly quantum mechanically. The approach is based on the
thermodynamic integration with respect to the mass of isotopes and on the
Feynman path integral representation of the partition function. An efficient
estimator for the derivative of free energy is used whose statistical error is
independent of the number of imaginary time slices in the path integral,
speeding up calculations by a factor of 60 at 500 K. We describe the
implementation of the methodology in the molecular dynamics package Amber 10.
The method is tested on three [1,5] sigmatropic hydrogen shift reactions.
Because of the computational expense, we use ab initio potentials to evaluate
the equilibrium isotope effects within the harmonic approximation, and then the
path integral method together with semiempirical potentials to evaluate the
anharmonicity corrections. Our calculations show that the anharmonicity effects
amount up to 30% of the symmetry reduced reaction free energy. The numerical
results are compared with recent experiments of Doering and coworkers,
confirming the accuracy of the most recent measurement on
2,4,6,7,9-pentamethyl-5-(5,5-H)methylene-11,11a-dihydro-12H-naphthacene
as well as concerns about compromised accuracy, due to side reactions, of
another measurement on
2-methyl-10-(10,10-H)methylenebicyclo[4.4.0]dec-1-ene.Comment: 14 pages, 8 figures, 6 table
Relation of exact Gaussian basis methods to the dephasing representation: Theory and application to time-resolved electronic spectra
We recently showed that the Dephasing Representation (DR) provides an
efficient tool for computing ultrafast electronic spectra and that further
acceleration is possible with cellularization [M. \v{S}ulc and J.
Van\'i\v{c}ek, Mol. Phys. 110, 945 (2012)]. Here we focus on increasing the
accuracy of this approximation by first implementing an exact Gaussian basis
method, which benefits from the accuracy of quantum dynamics and efficiency of
classical dynamics. Starting from this exact method, the DR is derived together
with ten other methods for computing time-resolved spectra with intermediate
accuracy and efficiency. These methods include the Gaussian DR, an exact
generalization of the DR, in which trajectories are replaced by communicating
frozen Gaussian basis functions evolving classically with an average
Hamiltonian. The newly obtained methods are tested numerically on time
correlation functions and time-resolved stimulated emission spectra in the
harmonic potential, pyrazine S0/S1 model, and quartic oscillator. Numerical
results confirm that both the Gaussian basis method and the Gaussian DR
increase the accuracy of the DR. Surprisingly, in chaotic systems the Gaussian
DR can outperform the presumably more accurate Gaussian basis method, in which
the two bases are evolved separately.Comment: 15 pages, 7 figure
Finite-temperature vibronic spectra from the split-operator coherence thermofield dynamics
We present a numerically exact approach for evaluating vibrationally resolved
electronic spectra at finite temperatures using the coherence thermofield
dynamics. In this method, which avoids implementing an algorithm for solving
the von Neumann equation for coherence, the thermal vibrational ensemble is
first mapped to a pure-state wavepacket in an augmented space, and this
wavepacket is then propagated by solving the standard, zero-temperature
Schr\"odinger equation with the split-operator Fourier method. We show that the
finite-temperature spectra obtained with the coherence thermofield dynamics in
a Morse potential agree exactly with those computed by Boltzmann-averaging the
spectra of individual vibrational levels. Because the split-operator
thermofield dynamics on a full tensor-product grid is restricted to
low-dimensional systems, we briefly discuss how the accessible dimensionality
can be increased by various techniques developed for the zero-temperature
split-operator Fourier method.Comment: 5 pages, 4 figure
High-order geometric integrators for the variational Gaussian approximation
Among the single-trajectory Gaussian-based methods for solving the
time-dependent Schr\"{o}dinger equation, the variational Gaussian approximation
is the most accurate one. In contrast to Heller's original thawed Gaussian
approximation, it is symplectic, conserves energy exactly, and may partially
account for tunneling. However, the variational method is also much more
expensive. To improve its efficiency, we symmetrically compose the second-order
symplectic integrator of Faou and Lubich and obtain geometric integrators that
can achieve an arbitrary even order of convergence in the time step. We
demonstrate that the high-order integrators can speed up convergence
drastically compared to the second-order algorithm and, in contrast to the
popular fourth-order Runge-Kutta method, are time-reversible and conserve the
norm and the symplectic structure exactly, regardless of the time step. To show
that the method is not restricted to low-dimensional systems, we perform most
of the analysis on a non-separable twenty-dimensional model of coupled Morse
oscillators. We also show that the variational method may capture tunneling
and, in general, improves accuracy over the non-variational thawed Gaussian
approximation.Comment: 17 pages, 11 figure
Three applications of path integrals: equilibrium and kinetic isotope effects, and the temperature dependence of the rate constant of the [1,5] sigmatropic hydrogen shift in (Z)-1,3-pentadiene
Recent experiments have confirmed the importance of nuclear quantum effects
even in large biomolecules at physiological temperature. Here we describe how
the path integral formalism can be used to describe rigorously the nuclear
quantum effects on equilibrium and kinetic properties of molecules.
Specifically, we explain how path integrals can be employed to evaluate the
equilibrium (EIE) and kinetic (KIE) isotope effects, and the temperature
dependence of the rate constant. The methodology is applied to the [1,5]
sigmatropic hydrogen shift in pentadiene. Both the KIE and the temperature
dependence of the rate constant confirm the importance of tunneling and other
nuclear quantum effects as well as of the anharmonicity of the potential energy
surface. Moreover, previous results on the KIE were improved by using a
combination of a high level electronic structure calculation within the
harmonic approximation with a path integral anharmonicity correction using a
lower level method.Comment: 9 pages, 4 figure
Uniform semiclassical wave function for coherent 2D electron flow
We find a uniform semiclassical (SC) wave function describing coherent
branched flow through a two-dimensional electron gas (2DEG), a phenomenon
recently discovered by direct imaging of the current using scanned probed
microscopy. The formation of branches has been explained by classical
arguments, but the SC simulations necessary to account for the coherence are
made difficult by the proliferation of catastrophes in the phase space. In this
paper, expansion in terms of "replacement manifolds" is used to find a uniform
SC wave function for a cusp singularity. The method is then generalized and
applied to calculate uniform wave functions for a quantum-map model of coherent
flow through a 2DEG. Finally, the quantum-map approximation is dropped and the
method is shown to work for a continuous-time model as well.Comment: 9 pages, 7 figure
Efficient Estimators for Quantum Instanton Evaluation of theKinetic Isotope Effects: Application to the Intramolecular HydrogenTransfer in Pentadiene
The quantum instanton approximation is used to compute kinetic isotope effects for intramolecular hydrogen transfer in cis-1,3-pentadiene. Due to the importance of skeleton motions, this system with 13 atoms is a simple prototype for hydrogen transfer in enzymatic reactions. The calculation is carried out using thermodynamic integration with respect to the mass of the isotopes and a path integral Monte Carlo evaluation of relevant thermodynamic quantities. Efficient 'virial' estimators are derived for the logarithmic derivatives of the partition function and the delta-delta correlation functions. These estimators require significantly fewer Monte Carlo samples since their statistical error does not increase with the number of discrete time slices in the path integral. The calculation treats all 39 degrees of freedom quantum-mechanically and uses an empirical valence bond potential based on a modified general AMBER force field
Stability of Quantum Motion: Beyond Fermi-golden-rule and Lyapunov decay
We study, analytically and numerically, the stability of quantum motion for a
classically chaotic system. We show the existence of different regimes of
fidelity decay which deviate from Fermi Golden rule and Lyapunov decay.Comment: 5 pages, 5 figure
A Synthetic Earth Gravity Model Designed Specifically for Testing Regional Gravimetric Geoid Determination Algorithms
A synthetic [simulated] Earth gravity model (SEGM) of the geoid, gravity and topography has been constructed over Australia specifically for validating regional gravimetric geoid determination theories, techniques and computer software. This regional high-resolution (1-arc-min by 1-arc-min) Australian SEGM (AusSEGM) is a combined source and effect model. The long-wavelength effect part (up to and including spherical harmonic degree and order 360) is taken from an assumed errorless EGM96 global geopotential model. Using forward modelling via numerical Newtonian integration, the short-wavelength source part is computed from a high-resolution (3-arc-sec by 3-arc-sec) synthetic digital elevation model (SDEM), which is a fractal surface based on the GLOBE v1 DEM. All topographic masses are modelled with a constant mass-density of 2,670 kg/m3. Based on these input data, gravity values on the synthetic topography (on a grid and at arbitrarily distributed discrete points) and consistent geoidal heights at regular 1-arc-min geographical grid nodes have been computed. The precision of the synthetic gravity and geoid data (after a first iteration) is estimated to be better than 30 ÎŒ Gal and 3 mm, respectively, which reduces to 1 ÎŒ Gal and 1 mm after a second iteration.The second iteration accounts for the changes in the geoid due to the superposed synthetic topographic mass distribution. The first iteration of AusSEGM is compared with Australian gravity and GPS-levelling data to verify that it gives a realistic representation of the Earthâs gravity field. As a by-product of this comparison, AusSEGM gives further evidence of the northâsouth-trending error in the Australian Height Datum. The freely available AusSEGM-derived gravity and SDEM data, included as Electronic Supplementary Material (ESM) with this paper, can be used to compute a geoid model that, if correct, will agree to in 3 mm with the AusSEGM geoidal heights, thus offering independent verification of theories and numerical techniques used for regional geoid modelling
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