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A comparative study of the centroid and ring-polymer molecular dynamics methods for approximating quantum time correlation functions from path integrals
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10.1063/1.3126950
/content/aip/journal/jcp/130/18/10.1063/1.3126950
http://aip.metastore.ingenta.com/content/aip/journal/jcp/130/18/10.1063/1.3126950

Figures

Image of FIG. 1.
FIG. 1.

Kubo-transformed position autocorrelation function for quartic potential , , and computed from RPMD in various coordinates: primitive (solid), NMs (dashes), and staging (dots). The minor discrepancies between curves are due to the sampling over many different initial conditions.

Image of FIG. 2.
FIG. 2.

Schematic representation of how to efficiently accumulate ergodic time correlation functions without distorting the dynamics during the observation stage.

Image of FIG. 3.
FIG. 3.

Relative error in the mean (defined in Ref. 58) of the virial estimator as a function of the block size for different sampling techniques in RPMD. Quartic potential , , and . Circles: sequential sampling (previous works); crosses: suggested parallel sampling scheme.

Image of FIG. 4.
FIG. 4.

Standard deviation of the virial estimator as a function of the block size for different sampling techniques in RPMD. Liquid para-hydrogen at and . The time step was 0.76 fs. Circles: sequential sampling (previous works); crosses: suggested parallel sampling scheme.

Image of FIG. 5.
FIG. 5.

Mean absolute error of the RPMD correlation functions for parallel and sequential sampling methods. is the fully converged Kubo-transformed position autocorrelation function. Above: quartic potential. Below: mildly anharmonic potential. A inverse temperature and Trotter number were used in both model potentials. The number of steps per trajectory was 60 000 for each sampling method.

Image of FIG. 6.
FIG. 6.

Mean absolute error of the RPMD correlation functions, , for parallel and sequential sampling methods. is the fully converged Kubo-transformed velocity autocorrelation function for the para-hydrogen at and obtained using RPMD. The number of steps per trajectory was 3500 and time step of 0.76 fs for each sampling method.

Image of FIG. 7.
FIG. 7.

Kubo-transformed position autocorrelation functions for the mildly anharmonic potential at two different temperatures.

Image of FIG. 8.
FIG. 8.

Comparison of the Kubo-transformed position autocorrelation function for the purely quartic potential at two different temperatures.

Image of FIG. 9.
FIG. 9.

Comparison of the Kubo-transformed position autocorrelation function for an asymmetric double well potential at and : exact (solid), RPMD (dots), and adiabatic CMD (dashes). The bare potential is shown in the inset.

Image of FIG. 10.
FIG. 10.

Kubo-transformed velocity autocorrelation function for para-hydrogen at as computed from adiabatic CMD (dashes) and RPMD (dots).

Image of FIG. 11.
FIG. 11.

Comparison of the ITMSD correlation function (ima, in the figure) as given by Eq. (16), and its reconstructed version (“rco” in the figure) from Eq. (17) for para-hydrogen at . The error as defined in Eq. (32) is shown for each method.

Image of FIG. 12.
FIG. 12.

Comparison of the ITVACF (“ITVACF,” in the figure) as given by Eq. (15), and its reconstructed version from Eq. (33) for para-hydrogen at . The error as defined in Eq. (32) is shown for each method.

Image of FIG. 13.
FIG. 13.

Comparison of the computed Fourier transform of the standard velocity autocorrelation function for para-hydrogen at . To facilitate comparisons, the same scale was used on the -axis as in Fig. 5 of Ref. 22. Only the real part is shown.

Tables

Generic image for table
Table I.

Parameters (in a.u.) in the SG potential, Eq. (34).

Generic image for table
Table II.

The Kubo-transformed velocity autocorrelation function at various selected points for para-hydrogen at with . The standard deviation in the last digit is given in parentheses.

Generic image for table
Table III.

Self-diffusion coefficient for para-hydrogen at with as computed from PI methods. The experimental value is from Ref. 64. The standard deviation in the last digit is given in parentheses.

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/content/aip/journal/jcp/130/18/10.1063/1.3126950
2009-05-12
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A comparative study of the centroid and ring-polymer molecular dynamics methods for approximating quantum time correlation functions from path integrals
http://aip.metastore.ingenta.com/content/aip/journal/jcp/130/18/10.1063/1.3126950
10.1063/1.3126950
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