Cayley modification for path-integral simulations
Abstract
Furthermore, it is shown that these numerical artifacts are removed by replacing the exact evolution of the free ring polymer with a second-order approximation based on the Cayley transform. Furthermore, it is shown that the improved numerical stability of the Cayley modification allows the use of larger MD time steps.
Introduction
An almost universal feature of MD numerical integration schemes for equations of motion based on imaginary-time path integrals is the use of harmonic normal modes for the accurate evolution of the free ring polymer positions and momenta. For thermostated RPMD and PIMD integration schemes involving a free ring-polymer step, the exact evolution of this step leads to the provable and numerically provable non-ergodicity.
Theory
To see why, note that the eigenvalues of the exponential matrix exp(Δ𝑡𝑨) are 𝑒±𝑖𝜔Δ𝑡and (P2) requires that For complete clarity, we now present a side-by-side comparison of the full RPMD time step (Eq. 1.2 ) with the ring-polymer free motion exp(Δ𝑡 𝐿0) implemented using a standard exponential map (i.e., the exact normal mode evolution ) or via the Cayley modification.
Results for RPMD
1.3(b) and (c) reveals clear numerical advantages of Cayley's modified RPMD integration scheme over the standard RPMD integration scheme. The improved stability of the modified Cayley integration scheme is seen to consistently allow the use of larger RPMD time steps.
Results for T-RPMD
Normalized histograms of the ring polymer normal mode displacement coordinates for a single trajectory (6 beads, 𝛽 =1) developed at a harmonic potential with a time step of Δ𝑡 =0.3. The same non-ergodicity problems arise for anharmonic potentials using the standard T-RPMD integrator and can be easily avoided using the Cayley modification.
Summary
We give the concentration of rare (in natural abundance), doubly substituted species relative to that of unsubstituted common isotopologues using 𝑋:. 3.4) The index 𝑖 refers to the type in the cluster. 6.2-6.7 and B.2-B.11 are based on harmonic frequencies, which are very sensitive to the correct description of the molecular potential.
Dimension-free path-integral molecular dynamics
Abstract
Convergence with respect to discretization in imaginary time (ie, the number of ring polymer beads) is an essential part of any path-integral-based molecular dynamics (MD) calculation. We show that these and other problems can be avoided through the introduction of "dimensionless" numerical integration schemes for which the sampled ring-polymer position distribution has non-zero overlap with the exact distribution in the limit of infinite bead in the case of a harmonic potential.
Introduction
It is further shown that the OBCBO scheme can be made dimensionless via the force softening technique. It is shown that the newly introduced BCOCB integrator provides better accuracy than all other non-preconditioned PIMD integrators considered and allows for significantly larger time steps in the calculation of both statistical and dynamical properties.
Non-preconditioned PIMD
Specifically, for the Cayley-modified OBABO scheme (called OBCBO), we replace the exact free-ring polymer update of the time step 𝜏 = Δ𝑡 with the Cayley transform given in Eq. For the Cayley-modified BAOAB scheme (called BCOCB), we replace the two exact free updates of the half-step polymer ring 𝜏 = Δ𝑡/2 by cay(Δ𝑡𝑨)1/2.
BCOCB avoids pathologies in the infinite bead limit
Due to the lack of strong stability in the exact evolution of the free ring polymer, Eq. 1.5 No such problem exists for the Cayley modification. Furthermore, by comparing the exact covariance matrix in Eq. 2.9) with the finite time step approximations in Eq.
Consequences for the primitive kinetic energy expectation value
For different MD time steps: the expected value of the primitive kinetic energy as a function of the number of ring polymer beads, where the exact kinetic energy is indicated as a gray dashed line. 2.1(a-d) is that the BCOCB shows no such deviation or error in the expectation value of the primitive kinetic energy at a high number of beads, regardless of the time step used.
Dimensionality freedom for OBCBO via force mollification
A simple choice for this matrix is 𝛀˜ =𝛀, such that softening is applied to all the internal states that are not zero-ring polymer. This observation points to a simple and general refinement of the OMCMO scheme, which we discuss in the following subsection.
Results for anharmonic oscillators
Attractively, the BCOCB scheme is consistently more accurate for the expected value of the virial kinetic energy, as it was for the expected value of the primitive kinetic energy. Also, the position autocorrelation function (e) for the quartic oscillator at room temperature calculated using T-RPMD with the BCOCB integrator.
Results for liquid water
Multi-nanosecond staging PIMD25,31 simulations at a time step of 0.1 fs were performed to obtain a bead-converged reference value for the H-atom kinetic energy, plotted as a dashed line in Figs. For the OBABO integrator, the calculated correlation function is qualitatively incorrect for time steps as large as 0.8 fs.
Summary
In the next section, we discuss the current knowledge of the equilibrium isotopic composition of methane vs. The values reported by [86] express enrichment of the center position relative to random distribution of isotopes – the same nomenclature typically used for the clumped isotope effect.
Application of path integrals to heavy isotope equilibria
Introduction
Since the development of the BCOCB integrator (Chapter 2), it has been included in the Python-based i-piwrapper package74, the Julia-based NQCDynamics75, the powerful Tinker-HP76, and used by others to investigate nuclear quantum effects on thermal conductivity,77 study reaction rates. ,78,79and electron transfer.80. PIMC is especially suitable for the study of heavy isotope equilibrium among isotopologists, since (i) the effect of interest is purely statistical in nature; (ii) the deviation from randomness is due to the nuclear quantum effects;
Nomenclature of stable isotope equilibria
As in the definition for delta notation (equation 3.2), in equation (3.3), M and N can represent two different phases (e.g. liquid water vs water vapor), species in a given phase (e.g. carbon dioxide and methane in gas phase for carbon isotopes), two sites within the same molecules (e.g. terminal methyl vs. center methylene carbon of propane), or the site of a molecule (e.g. the terminal methyl of propane) vs. Although we write (14C ) explicitly for clumping in single-site molecules (9A), an analogous expression is valid for ethane (reaction 10A-C) and propane (reaction 11A and others in Appendix A3). where𝑄 are the partition functions of isotopologues 𝑁 and𝑀, the product of the ratio of masses raised to the power 3/2 runs over all isotopes different between 𝑀 and 𝑁, and 𝜎 are the corresponding rotational symmetry numbers. 3.24) inherently normalizes both the mass terms and the symmetry numbers in𝑄, so that at infinite temperature (i.e. in the classical limit) RPFR is unity.
Methods
Second, including these effects reduces the statistical accuracy of the MC sampling due to the so-called. Again, there is no definitive convergence to the reference value when increasing the quality of the method beyond MP2.
Clumped isotope effects in methane
Abstract
The stable isotopic composition of methane has been used extensively for decades to limit the source of methane in the environment. Given that many natural occurrences of methane form below these temperatures, previous calibrations require extrapolation when calculating clumped-isotope-based temperatures outside this calibration range.
Introduction
Previous experimental and theoretical determinations of temperature dependence of Δ values for isotopically equilibrated systems. Previous calculations of equilibrium Δ13CH3D and ΔCH2D2 values are based on one of two theoretical approaches: (i) the Bigeleisen and Mayer/Urey model92,96 (BMU, which in practice involves calculations of so-called reduced partition function ratios (RPFRs) using a harmonic approximation for the treatment of the vibrational partition function and classical expressions for rotational and translational partition functions and (ii) Path Integral Monte Carlo (PIMC) simulations that avoid the large approximations of the BMU model that provide a fully anharmonic and quantum mechanical description of the partition function conditions.86.
Methods
Results
The number of beads for other temperatures is determined by interpolation between the last two points in each panel. The number of beads for calculations at intermediate temperatures is determined by interpolating between the last two points in each panel.
Discussion
The calculated ±1𝜎 of the residuals (Table 4.4) are ±0.22‰ and ±1.17‰ for the Δ13CH3D and ΔCH2D2 values, respectively, and are comparable to the external precision estimated solely from the experimental replications at a given temperature (±0.28‰for Δ13CH3D and ±1.61). ‰forΔCH2D2, 1𝜎). PIMC calculations provide a way to calculate stable isotope fractionation factors independent of the traditionally used BMU model, and are more rigorous and accurate (assuming a high quality PES and a sufficiently large number of beads and number of samples for the systematic and statistical error convergence, respectively ) .
Summary
Using the minimization scheme described above (Fig. 5.3), the DBO-corrected PIMC calculations are offset by 0.49‰ from the experimental data. We also note that in some cases the influence of the DBO correction is less noticeable due to (at least partial) cancellation with the anharmonic effects discussed in the following section, while in other cases (notably site-specific isotope effect) the two effects reinforce each other.
Abstract
Here we provide a calibration of the hydrogen isotopic fractionation equilibrium factor for CH4 and hydrogen gas (H2) (𝐷𝛼CH. Additionally, we provide new theoretical estimates of the hydrogen isotopic equilibrium between CH4(𝑔), H2(𝑔) and H2O(𝑔) and the carbon isotopic equilibrium between CH4(𝑔 ) and CO2(𝑔) using Path Integral Monte Carlo (PIMC) calcns.
Introduction
This proposal is based on the observation that the measured temperatures of methane accumulated in the isotope are similar to the temperatures calculated based on the assumption of isotopic equilibrium of hydrogen between methane and other alkanes.155. Carbon isotope equilibrium between CH4 and CO2 and hydrogen isotope equilibrium between CH4 and liquid water.
Methods
Here, we provide an experiment-based calibration of the equilibrium hydrogen isotope fractionation factor for methane and H2 from 3 to 200°C. Non-Born-Oppenheimer effects (ie, inaccuracies associated with the use of the Born-Oppenheimer approximation) are usually negligible compared to those inherent to PES.
Results
To use the experimental data as a constraint on the temperature dependence of 1000×ln𝐷𝛼H. This is done to guarantee the theoretically expected form of the temperature dependence for 1000×ln𝐷𝛼H.
Discussion
We interpret convergence of harmonic RPFRs with respect to the level of theory or basis set size to indicate sufficient accuracy of the electronic structure methods. We plot 𝛿rpfr on the y-axis, where 𝛿rpfr = 1000 × ln (𝑅 𝑃 𝐹 𝑅𝑋/𝑅 𝑃 𝐹 𝑅 𝑅reference) and is reported in ‰ .𝑋enotes the value of the variable tested and the reference the RPFR was calculated using harmonic frequencies from the same PES as used in the PIMC calculations (the so-called reference harmonic lines; . Section 5.3).
Summary
For the clumped heavy isotope effects, we list only the polynomial fits for D+D clumping in dihydrogen and water in Table 6.9; the PIMC-based values are in Table 6.10. Manolopoulos, "Comparison of path integral molecular dynamics methods for the infrared absorption spectrum of liquid water", The Journal of Chemical Physics.
𝐷 and 13 𝐶 isotopic equilibria in alkanes
Abstract
On the other hand, bulk isotope effects are much more forgiving and can be obtained with reasonable accuracy using less expensive DFT/B3LYP methods. Clustered isotope effects tend to be less affected by anharmonic effects as well.
Introduction
We analyze approximations involved in calculating heavy isotope balances in and between alkanes (methane, ethane, and propane), water, and hydrogen gas, including heavy isotope fractionation, lumped isotope effects, and site-specific isotope effects. Finally, we relate the calculated equilibrium isotopic effects 𝛼𝑒 𝑞 and Δ𝑒 𝑞 to the corresponding experimental quantities 𝛼 and Δ, respectively, and discuss complications in describing isotope pairing effects using existing nomenclature arising from the presence of different rotamers of ethane and propane and from different isomers of doubly substituted isotopologues, from each of which has a different preference for combining.
Methods
We have observed this for the harmonic RPFRs both in this study and in [99]—the RPFRs calculated with different methods and basis set sizes vary by 10s to 100s per mil. Since molecular hydrogen is a diatomic molecule, its potential energy depends on only one coordinate - the distance between two hydrogen atoms; thus, we fit the PES for dihydrogen using a 1-dimensional spline interpolation of 76 F12/ATZ calculations for interatomic distances between 0.5 and 6 Bohr (equilibrium geometry is at 1.402 Bohr).
Results
Position-specific isotope effect in propane (Fig. 6.4) shows similar trends to the fractionation presented in Fig. Position-specific isotope effect in propane is affected by both the BOD correction and the anharmonic effects, and the effects coincidentally align, reinforcing each other relative to the harmonic result.
Discussion
This is particularly pronounced for the MP2 method (green), as the relative importance of the error cancellation is greatest for this method. The strength of the clumped isotope effect also depends strongly on the proximity of the heavy isotopes, falling below 1‰ for the deuterium atoms bonded to different carbon atoms (panel b) and for 13C+D clumping when the two heavy atoms are not directly associated with each other (panel d).
Summary
Marx, "On the applicability of centroid and ring polymer path integral molecular dynamics to vibrational spectroscopy", The Journal of Chemical Physics. An extended Gaussian-type basis for molecular-orbital studies of organic molecules", The Journal of Chemical Physics.
ORCA calculations
It is worth noting that hydrogen atoms bonded to one heavy atom (oxygen or carbon) are structurally equivalent for the PIP surfaces described in section 3.3.2. So replacing one of the two hydrogen atoms in water or one of the four hydrogen atoms in methane gives exactly the same harmonic frequencies for the PIP surfaces.
Heavy isotope fractionations involving propane
We used the experimentally available structures as an estimate for the cheapest RHF/DZ calculations of each molecule and the final optimized geometry served as an initial estimate for the next (more expensive) method. This is not the case for the calculations typically performed with a standard software package (Gaussian, ORCA or MolPro), unless molecular symmetry is enforced.
Approximating isotope fractionations by considering singly substi-
Clumped heavy isotope effect equilibrium reactions
Relationship between Δ and Δ 𝑒 𝑞
As a result, the Δvalues for clumping at the individual sites of propane represent a combination of the clumped and position-specific heavy isotope effects. To be more specific, the Δ value for clumping in the methylene group of propane is significantly larger than expected based on clumped heavy isotope effect considerations alone, while the same value for the methyl group is negative.
Averaging of clumped heavy isotope effect due to different isotopo-
Bulk fractionations
Clumped isotope effects
