Monte Carlo data

Bennett C H 1976 Efficient estimation of free energy differences from Monte Carlo data J. Comput. Phys. 22 245-68... 

Ferrenberg A M and Swendsen R H 1989 Optimized Monte Carlo data analysis Phys. Rev.L 63 1195-8... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

In Fig. 10(b) one can see the density profiles calculated for the system with /kgT = 5 and at a high bulk density, p = 0.9038. The relevant computer simulation data can be found in Fig. 5(c) of Ref. 38. It is evident that the theory of Segura et al, shghtly underestimates the multilayer structure of the film. The results of the modified Meister-Kroll-Groot theory [145] are more consistent with the Monte Carlo data (not shown in our... 

FiG. 10 Normalized density profiles p z)/for the associating fluid at a hard wall. The association energy is jk T — 7 and the bulk density is p = 0.2098 (a), e ykgT = 5 and the bulk density equals 0.9038 (b). The solid and dashed lines denote the results of the modified Meister-Kroll theory and the theory of Segura et al., respectively. The Monte Carlo data in (a) are marked as points. (From Ref. 145.)... 

FIG. 21 Dependence of the average density on the configurational chemical potential. The solid line denotes the grand canonical Monte Carlo data, the long dashed fine corresponds to the osmotic Monte Carlo results for ZL = 40, and the dotted line for ZL = 80. (From Ref. 172.)... 

Ferrenberg, A. M. Swendsen, R. H., Optimized Monte Carlo data analysis, Phys. Rev. Lett. 1989, 63,1195-1198... 

Monte Carlo data for y were generated according to with mean x, to simulate process sampling data. A window size of 25 was used here and to demonstrate the performance of the Bayesian approach. 

In obtaining Monte Carlo data such as shown in Figs. 2, 3, 5, it is also necessary to understand the statistical errors that are present because the number of states M — Mq over which we average (Eq. (24)) is finite. If the averages m, E, i/ are calculated from a subset of n uncorrelated observations m(Xy), E(Xy), ilf Xy), Standard error analysis applies and yields estimates for the expected mean square deviations, for n- cx),... 

In the previous subsections, only a small number of typical applications of the Monte Carlo study of adsorbed layers was treated in detail to show how one proceeds with the analysis of Monte Carlo data, and to illustrate the type of questions that can be answered. In the present subsection, we give a brief... 

The correlation energy is known analytically in the high-and low-density limits. For typical valence electron densities (1 < r, < 10) and lower densities (r, > 10), it is known numerically from release-node Diffusion Monte Carlo studies [33]. Various parametrizations have been developed to interpolate between the known limits while fitting the Monte Carlo data. The first, simplest and most transparent is that of Perdew and Zunger (PZ) [34] ... 

Table IV compares Eqs. (18)-(21) (using PW92 input) with Monte Carlo data [54] as fitted by Gori-Giorgi, Sachetti and Bachelet (GSB) [55, 56], which we take to be the standard of accuracy here. (The GSB data were kindly provided by Dr. Gori-Giorgi.) In the range 1 < r, < 10, the Stoll decomposition (18) and Eq. (19) seem inaccurate in comparison with the SKTP of Eq. (20), as pointed out by GSB, and so does Eq. (21).

For the total correlation energy (r Q, so much is known about the r, — 0 and r, — 00 no limits that accurate values for all r, and can be found by interpolation [58], without ever using the Monte Carlo or other data. For the spin resolution e r, Q/e (r, Q, however, so little is known about these limits that we must and do rely on the Monte Carlo data. The spin resolution of Eq. (24) has recently been generalized to all [59]. 

By integrating Equations (3) and (4), neglecting the A term, with random initial conditions, mounds similar to those of the simulation can be obtained (Fig. 3). These mounds also coarsen in time. However, there has not been direct test of this equation as a description of multilayer growth. In particular, Eq. (4) was derived by fitting to Monte-Carlo data in the submonolayer regime. In this paper we show that certain aspects of multilayer growth by the Monte-Carlo model are well represented by Eq. (3) and (4). 

I am grateful to Dr. M. F. Sykes and Dr. D. L. Hunter for providing data on which the plots in Figures 1 and 2 are based, and to Dr. J. Mazur for communicating his Monte Carlo data prior to publication. 

A. Scemama, P. Chaquin, M. Caffarel, Electron pair localization function A practical tool to visualize electron localization in molecules from quantum Monte Carlo data. J. Chem. Phys. 121, 1725-1735 (2004)... 

Kincaid and Weis [ 29] have proposed the following analytical expression for the radial distribution function by fitting the Monte Carlo data ... 

The application of this relation to the Monte Carlo data is also illustrated in Fig. 3 by the large black circles, which lead to the parameter values (L2yojn = 0.60 0.15 and /8/c2 = 2.6 1.0, and finally cmin = 0.8. It is seen that (L2 jn is seriously underestimated1. 

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