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Efficient methods and practical guidelines for simulating isotope effects
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10.1063/1.4772676
/content/aip/journal/jcp/138/1/10.1063/1.4772676
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/1/10.1063/1.4772676
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Panel (a) shows the integrand of Eq. (4) , as computed for a P = 64 PIMD simulation of liquid water with one hydrogen atom being transformed into deuterium. Panel (b) shows that the integrand becomes nearly constant when a transformed integration variable is used, as in Eq. (7) . Lines are just guides for the eye, and statistical error bars are reported for each point. The dashed line shows the estimate from the two end points. Note the difference in the scale: in panel (a) the integrand varies by a factor of 2, whereas in panel (b) it varies by less than 3%.

Image of FIG. 2.
FIG. 2.

The figure shows the integrand of the transformed thermodynamic integration (7) as a function of the mass of the isotope. The results from fully-converged PIMD (P = 64, blue lines) are compared with those from a harmonic/quasi-harmonic approximation (red lines). Curves are shown for both liquid water (full lines) and an isolated molecule in the gas phase (dashed lines). The harmonic approximation for the gas phase is based on static calculations and therefore has no error bars. For all other calculation lines are just guides for the eye, and statistical error bars are reported for each data point.

Image of FIG. 3.
FIG. 3.

(a) Expectation values for the centroid-virial kinetic energy estimator for hydrogen in liquid water as a function of the number of beads. Results are shown for both PIMD and PIGLET simulations, and for the tagged particle having mass set to that of hydrogen (m H ) or deuterium (m D ). (b) Convergence of the isotope fractionation ratio α lv between the liquid and the gas phase of water for the H/D substitution. Results are reported as a function of the number of beads, using conventional path integral MD (blue circles) and PIGLET (red lozenges). Lines are just guides for the eye, and statistical error bars are smaller than the size of the points.

Image of FIG. 4.
FIG. 4.

Average kinetic energy (blue circles) computed from simulations in which n D hydrogen atoms have been substituted with deuterium, in a simulation box containing 64 water molecules. At most one atom per molecule was substituted.

Image of FIG. 5.
FIG. 5.

Cumulative average of , as computed for hydrogen isotopes in a single PIMD trajectory of liquid water, with P = 32. Lines, from top to bottom, correspond to different mass ratios ranging from μ/m H = 1 to μ/m H = 2.

Image of FIG. 6.
FIG. 6.

Contour plot of the variance of the difference Hamiltonian for the thermodynamic free energy perturbation h TD, as a function of the number of beads P and the mass ratio α = μ/m H . For σ2(h TD) ≳ 1 computing becomes impractical.

Image of FIG. 7.
FIG. 7.

Convergence of the isotope fractionation ratio α lv between the liquid and the gas phases of water at ambient conditions for the 16O/18O substitution. Upper panel depicts the convergence of the 18O kinetic energy in the two phases, as computed from thermodynamic free energy perturbation, while the lower panel reports the fractionation ratio. Lines are just guides for the eye.

Image of FIG. 8.
FIG. 8.

The figure demonstrates the performance of the TD-FEP estimator used together with PIGLET. Panel (a) represents as a function of the number of beads. PIMD results are shown for comparison, and the black line indicates the reference value (direct substitution, PIMD, P = 64). In Panel (b) the isotope fractionation ratio between liquid and vapor is shown, for water at room temperature, as estimated from TD-FEP and PIGLET. The black line indicates the reference value (direct substitution, PIMD, P = 64).

Image of FIG. 9.
FIG. 9.

Contour plot of the variance of the difference Hamiltonian for the coordinate-scaling free energy perturbation h SC, as a function of the number of beads P and the mass ratio α = μ/m H . Throughout the range we considered, σ2(h SC) < 1.

Image of FIG. 10.
FIG. 10.

The figure demonstrates the poor convergence of the SC-FEP estimator when used together with PIGLET. (a) as a function of the number of beads. PIMD results are shown for comparison, and the black line indicates the reference value (direct substitution, PIMD, P = 64). (b) Isotope fractionation ratio between liquid and vapor for water at room temperature, estimated from SC-FEP and PIGLET. The black line indicates the reference value (direct substitution, PIMD, P = 64).

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/content/aip/journal/jcp/138/1/10.1063/1.4772676
2013-01-07
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Efficient methods and practical guidelines for simulating isotope effects
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/1/10.1063/1.4772676
10.1063/1.4772676
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