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Thermodynamics of liquids: standard molar entropies and heat capacities of common solvents from 2PT molecular dynamicsw
Tod A. Pascal,ab Shiang-Tai Linc and William A. Goddard III*ab
Received 19th August 2010, Accepted 7th October 2010 DOI: 10.1039/c0cp01549k
We validate here the Two-Phase Thermodynamics (2PT) method for calculating the standard molar entropies and heat capacities of common liquids. In 2PT, the thermodynamics of the system is related to the total density of states (DoS), obtained from the Fourier Transform of the velocity autocorrelation function. For liquids this DoS is partitioned into a diffusional component modeled as diffusion of a hard sphere gas plus a solid component for which the DoS(u) - 0as u - 0 as for a Debye solid. Thermodynamic observables are obtained by integrating the DoS with the appropriate weighting functions. In the 2PT method, two parameters are extracted from the DoS self-consistently to describe diffusional contributions: the fraction of diffusional modes, f, and DoS(0). This allows 2PT to be applied consistently and without re-parameterization to simulations of arbitrary liquids. We find that the absolute entropy of the liquid can be determined accurately from a single short MD trajectory (20 ps) after the system is equilibrated, making it orders of magnitude more efficient than commonly used perturbation and umbrella sampling methods. Here, we present the predicted standard molar entropies for fifteen common solvents evaluated from molecular dynamics simulations using the AMBER, GAFF, OPLS AA/L and Dreiding II forcefields. Overall, we find that all forcefields lead to good agreement with experimental and previous theoretical values for the entropy and very good agreement in the heat capacities. These results validate 2PT as a robust and efficient method for evaluating the thermodynamics of liquid phase systems. Indeed 2PT might provide a practical scheme to improve the intermolecular terms in forcefields by comparing directly to thermodynamic properties.
I. Introduction Alternatively, the free energy can be obtained from potential of mean-force simulations,7 which are markedly simpler, but 8 Downloaded by California Institute of Technology on 29 November 2010 Modern quantum mechanics (QM) methods provide powerful not as accurate. Widom particle insertion schemes yield the Published on 23 November 2010 http://pubs.rsc.org | doi:10.1039/C0CP01549K means for predicting the energetics and enthalpies of molecules chemical potential but require extensive sampling for all but at low temperatures, including accurate estimates for solvation 1–3 the simplest of systems. Alternatively, Jorgensen and others energies. However, neither QM nor molecular dynamics have shown9–13 that Monte Carlo (MC) methods14 coupled (MD) using forcefields (FF) have proved feasible for predicting with intermolecular forcefields can lead to accurate free accurate free energies from practical first principles calculations, energies of solution, but again this usually involves very long primarily due to uncertainty in calculating entropy. Numerous simulations to reduce the statistical uncertainty. Indeed, methods have thus been proposed to calculate accurate except in the context of thermodynamic integration using entropies, although there is usually a tradeoff between accuracy umbrella sampling, MD has not generally been useful for and efficiency. Most perturbation MD methods,4 based on predicting the free energy or entropy of complex molecular Kirkwood–Zwanzig thermodynamic integration,5,6 have systems. shown to be very accurate for a range of systems and can in It would be quite useful to have efficient ways to estimate principle lead to accurate free energy change from a reference the entropy directly from MD (thereby preserving the system A to the target system B. However, complexities related dynamical information lost from MC techniques). Indeed, to the choice of appropriate approximation formalism limit entropy is expected to be the driving force behind most their straightforward application. biochemical processes, ranging from protein folding and 15–17 a Materials and Process Simulation Center, California Institute of ligand/protein binding, to DNA transformations and 18 19,20 Technology, Pasadena, CA 91125, USA. recognition and hydrophobic effects. In particular, E-mail: [email protected] solubility and therefore miscibility of molecules in organic b Graduate School of EEWS, Korea Advanced Institute of Science and liquids may be dominated by changes in entropy,21–23 making Technology, Daejeon, Korea 305-701 c Department of Chemical Engineering, National Taiwan University, accurate measures of the standard molar entropy critical to Taipei 10617, Taiwan understanding solvation phenomena. w Electronic supplementary information (ESI) available: Description We propose here a practical approach to obtain accurate of forcefield parameters, heats of vaporization, coefficient of thermal expansions and isothermal compressibilities. See DOI: 10.1039/ thermodynamics from short MD trajectories, which we c0cp01549k validate by predicting entropies and specific heats of 15 standard
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solvents. Our approach is based on the 2PT method24 for Amber Force Field (GAFF),32 OPLS AA/L11,33 and extracting absolute standard molar entropies and free energies Dreiding II.34 Overall, we find good agreement with from classical MD trajectories. In the 2PT paradigm, the entropy experiment from all these forcefields, but OPLS AA/L, derived of the system is derived from the atomic velocity autocorrelation to fit MC simulations of liquids, was the best. Perhaps more function C(t) extracted from a 20 ps trajectory. The process is importantly, we find that after equilibration 20 ps of MD leads to first to calculate the total density of states (DoS), from a Fast deviations in the standard molar entropy of 0.25 cal mol 1 K 1 Fourier Transform of C(t) to obtain DoS(u). Obtaining the (or 0.6%). thermodynamic properties from the DoS(u)ofaliquidis In the 2PT framework, the heat capacity is calculated complicated by diffusional effects (a finite DoS at u =0),which (as any other thermodynamic quantity) directly from the would lead to infinite values for the entropy for the standard DoS by applying the appropriate weighting function. In this harmonic oscillator partition function. paper we show that the calculated molar heat capacities are in In the 2PT model, we assume that the DoS can be very good agreement with experiment and in excellent
partitioned into a solid like component, DoS(u)solid, and a agreement with other theoretical methods, further validating diffusional component, DoS(u)diff. This DoS(u)diff component the 2PT method. is modeled in terms of a gas of hard spheres, completely Finally, we present the partition of the total entropy of described by DoS(u=0), the zero frequency intensity of the these solvents into the rotational, translational and internal total DoS and f, the fraction of the 3N degrees of freedom vibrational components. We find that 46% of the entropy of (dof) that are diffusional. These parameters DoS(u) and f are the liquids arises from translation, 37% from rotation and extracted from the total DoS, leading to the vibrational 17% from intra-molecular vibrations, with large variations
DoS(u)solid that can be analyzed using the standard harmonic depending on the nature of the liquid. This is the first time that oscillator partition function to account for the vibrational and such a theoretical component analysis has been reported for librational components. these organic liquids. We expect that such analyses may be The original validation of the 2PT method was for a useful in formulating macroscale methods of estimating Lennard-Jones fluid, where very extensive MC calculations entropies of solvation. were available for the free energies for all regimes of the phase Taken together, we demonstrate that the 2PT method can diagram, including solid, liquid, gas, supercritical, metastable, be applied without reparameterization to study liquids, and unstable. Lin et al.24 showed that 2PT from short 10 ps producing precise results from standard molecular dynamics simulations gave excellent agreement with the MC for the forcefields and simulation codes. The remainder of the paper is entropies and free energies for all phases, including metastable organized as follows: Section II presents the major results with and unstable regions. discussions. Section III presents the theoretical background of The 2PT method has since been applied to several complex, the 2PT method and details for the application to the systems condensed phase systems. For example, Lin et al.25 showed considered here. that the water molecules in a full solvent simulation (B40 000 water molecules) of a generation 4 PAMAM Dendrimer Downloaded by California Institute of Technology on 29 November 2010
Published on 23 November 2010 http://pubs.rsc.org | doi:10.1039/C0CP01549K (one molecule with 2244 atoms) leads to dramatically different II. Results and discussion entropies for the inner hydrophobic region, compared to the II.a Applicability of forcefields for liquid simulations surface and bulk waters. Similarly, Jana et al.26 used 2PT to show that the entropies of water molecules in the grooves of Table 1 compares the computed static dielectric constants DNA are significantly lower than in bulk water. More (see Appendix A.II.3 of ESIw) of the 15 liquids in this study. recently, Pascal et al.27 used 2PT to examine the entropic We find that all forcefields (FF) underestimate the dielectric effects in binding a disaccharide (chitobiose) to the outer constants, with the magnitude of the error depending on the membrane protein A on Escherichia coli (a system involving nature of the solvent. This is expected since these FF all use 2589 atoms in the protein, 114 in the ligand and 42 600 solvent/ fixed charges and hence cannot account for the molecular lipid atoms). Here it was shown that various mutations led to a polarizability contribution to the dielectric constant. We plan significant change in the contribution of entropy to the binding, later to use the QEq method35 to include such polarization so that enthalpies alone did not correlate well with effects. experimental invasion rates, but that the 2PT derived free For non-polar solvents, the OPLS AA/L FF has the best energies led to an excellent correlation. This work demonstrated performance, with a mean absolute deviation—MAD—error the feasibility of calculating accurate ligand binding free of 0.078 (2.5%) and a root mean squared—RMS—error energies and entropies on proteins embedded in a phospholipid (a measurement of the variance) of 0.103. The next best is membrane with explicit water molecules and salt. GAFF (3.6% error), the AMBER 2003 FF (15.5%) and While 2PT has been applied successfully to these and other finally the Dreiding FF (20%). The excellent performance of complex systems,28,29 its performance in computing the absolute OPLS AA/L might be expected since the charges of benzene,10 entropy and free energy of liquids other than water has not chloroform,10 furan36 and toluene10 were explicitly parameterized been demonstrated. In this paper, we use the 2PT method to to reproduce the experimental dielectric properties. calculate standard molar entropies and molar heat capacities The gas-phase RESP37 charges in GAFF generally are more of 15 common organic liquids. Of course the 2PT analysis polar than the solution phase fitted charges in OPLS cannot be better than the forcefield used, so we test the (Table S1, ESIw). Perhaps unsurprisingly, the calculated gas accuracy of several forcefields: AMBER-2003,30,31 General phase dipole moments (Table 2) are closer to the experimental
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Table 1 Comparison of the calculated static dielectric constants (e0) of all 15 organic liquids to experiments. Effects due to charge polarization are not included
Expa AMBER 2003b,c,d Dreidingc,e GAFFb,c,d OPLS AA/Lf Non-polar Benzene 2.283 1.031 0.000 1.018 0.000 1.014 0.000 1.027 0.000 Chloroform 4.807 2.136 0.000 1.738 0.000 3.236 0.001 3.636 0.001 1,4-Dioxane 2.219 1.082 0.000 1.079 0.000 1.055 0.000 1.057 0.000 furan 2.940 1.232 0.001 1.160 0.000 1.668 0.000 1.534 0.000 Toluene 2.379 1.049 0.000 1.182 0.000 1.039 0.000 1.204 0.000 M.A.D. — 0.456 — 0.587 — 0.106 — 0.078 0.456 RMS error — 0.626 — 0.812 — 0.152 — 0.103 0.626 Polar-aprotic Acetonitrile 36.640 18.843 0.126 17.908 0.113 22.429 0.186 14.468 0.069 Acetone 21.100 12.319 0.047 19.535 0.138 12.343 0.048 14.417 0.069 DMSO 47.240 53.590 1.205 45.723 0.865 59.147 1.500 49.221 0.975 THF 7.520 10.263 0.030 16.701 0.095 9.487 0.023 5.340 0.005 M.A.D. — 8.918 — 7.787 — 9.211 — 7.454 8.918 RMS error — 11.034 — 11.547 — 11.599 — 10.550 11.034 Polar protic Acetic acid 6.200 — — 4.623 0.001 — — 4.311 0.002 Ethanol 25.300 16.020 0.083 7.796 0.015 16.184 0.072 19.075 0.120 Ethylene glycol 41.400 37.212 0.514 13.284 0.042 11.708 0.013 30.389 0.350 Methanol 33.000 24.678 0.228 14.965 0.074 33.399 0.423 25.540 0.249 NMA 179.000 85.662 0.836 69.445 1.794 99.101 1.748 62.611 1.599 TFE 27.680 13.437 0.054 33.408 0.422 14.294 0.064 23.931 0.201 M.A.D. — 26.986 — 27.126 — 22.765 — 30.645 26.986 RMS error — 37.883 — 41.704 — 31.856 — 45.148 37.883 a Ref. 64. b Undefined valence and vdW parameters obtained from Antechamber78 atom typing program. c Structures optimized HF/6-31G* level using Jaguar 7.079 electronic structure package. d Charges from RESP37 charge fitting scheme. e Charges from Mulliken population analysis.80 f Parameters and charges from MacroModel 9.7program.81
Table 2 Comparison of the calculated gas-phase dipole moments The agreement with experimental dielectric constants (Debye) vs. experiment. Each liquid is first minimized in the appro- deteriorates considerably for polar solvents. We find MAD priate forcefield errors of 8.92, 7.79, 9.21 and 7.5 (33%, 28%, 32% and 27%) Expa AMBER 03 Dreiding GAFF OPLS AA/L for the aprotic solvents for the AMBER, Dreiding, GAFF and OPLS, respectively, and significantly worse agreement for the
Downloaded by California Institute of Technology on 29 November 2010 Acetic acid 1.70 1.678 2.307 1.506 1.542
Published on 23 November 2010 http://pubs.rsc.org | doi:10.1039/C0CP01549K Acetone 2.88 3.226 3.608 3.194 3.111 protic solvents, with errors greater than 50% for all but the Acetonitrile 3.92 4.035 4.681 4.030 4.129 GAFF forcefield (44%). The largest errors are observed for Benzene 0.00 0.000 0.000 0.000 0.000 NMA (the most polar solvent), where the dielectric constant is Chloroform 1.04 1.085 0.934 1.083 1.462 1,4-Dioxane 0.45 0.000 0.000 0.000 0.000 underestimated by 93.4, 109.6, 80.0 and 116.4 respectively DMSO 3.96 4.933 5.450 4.711 4.700 (cf. the experimental value of 179.0). We again attribute these Ethanol 1.69 1.876 1.921 1.900 2.323 large errors to the lack of charge polarization in these force- Ethylene glycol — 0.000 0.000 0.000 0.000 fields. Indeed Anisimov et al.38 showed that the dielectric Furan — 0.868 0.429 0.843 0.732 Methanol 1.70 2.150 2.060 2.177 2.274 constants of common alcohols are underestimated by 36% NMA 4.04 4.414 4.881 4.378 4.030 using non-polorizable forcefields compared to polarizable THF 1.75 2.621 3.617 2.690 1.972 forcefields, and that polarizable forcefields show significantly Toluene 0.36 0.211 0.606 0.221 0.566 better agreement to experiments. TFE — 3.563 4.640 3.624 4.087 The AMBER, GAFF and OPLS forcefields slightly a Ref. 82. underestimate the liquid densities and molar volumes, with average errors of 1.4%, 2.2% and 1.6%, respectively, gas phase values. This relatively good performance of GAFF across all liquids (Table 3). Since the density of the liquid is a indicates that these gas phase static charges are transferable parameter used in fitting these forcefields, the better for simulations of non-polar liquids, greatly simplifying FF performance compared to the dielectric constant (usually not development. The relatively poor performance of the AMBER a fitting parameter except the cases of OPLS outlined above) is 2003 FF compared to the GAFF is also not surprising, since to be expected. This indicates that the vdW parameters,
the AMBER forcefield was not optimized for simulations of and specifically the equilibrium distance R0, are tuned to liquids. We find that the Mulliken charges used with the compensate for inaccuracies in the electrostatics. The Dreiding generic Dreiding FF are more polar than the RESP charges underestimates the densities by 10%, which might be of GAFF, leading Dreiding to overestimate the gas phase expected due to the generic nature of this FF. The results dipole moments, relative to GAFF and charges that are not obtained here could be used to optimize the Dreiding vdW transferable to the condensed phase. parameters.
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Table 3 Comparison of experimental and predicted densities and molar volumes of organic liquids. Calculated values obtained from statistical averaging over 2.5 ns MD, sampled every 100 ps. Numbers in parentheses indicate the uncertaintya
Density hri/g cm 3 Molar volumeb hVi/cm3 Expc AMBER 2003 Dreiding GAFF OPLS AA/L Expc AMBER 2003 Dreiding GAFF OPLS AA/L Acetic acid 1.045 1.020 (99) 1.047 (16) 95.45 97.75 95.19 Acetone 0.785 0.757 (19) 0.697 (32) 0.764 (31) 0.777 (26) 122.9 127.33 138.41 126.25 124.11 Acetonitrile 0.786 0.729 (18) 0.730 (18) 0.705 (27) 0.711 (71) 86.75 93.53 93.37 96.62 95.92 Benzene 0.877 0.827 (31) 0.802 (54) 0.796 (62) 0.913 (26) 147.96 156.82 161.77 162.90 141.98 Chloroform 1.479 1.380 (34) 1.331 (55) 1.412 (34) 1.447 (82) 134.03 143.59 148.96 140.40 136.96 1,4-Dioxane 1.034 1.088 (26) 0.871 (27) 1.068 (37) 0.987 (24) 141.51 134.49 167.89 137.03 148.15 DMSO 1.101 1.096 (28) 1.089 (15) 1.103 (35) 1.095 (23) 117.82 118.38 119.09 117.59 118.42 Ethanol 0.789 0.790 (27) 0.642 (34) 0.785 (33) 0.783 (18) 96.90 96.83 119.11 97.38 97.66 Ethylene glycol 1.114 1.166 (12) 0.997 (44) 1.115 (50) 1.064 (24) 92.55 88.38 103.31 92.38 96.83 Furan 0.951 0.959 (29) 0.852 (32) 0.950 (32) 0.972 (37) 118.80 117.86 132.64 118.96 116.25 Methanol 0.791 0.801 (27) 0.644 (29) 0.799 (27) 0.766 (24) 67.22 66.42 82.62 66.57 69.49 NMA 0.937 0.954 (18) 0.847 (24) 0.952 (21) 0.952 (20) 129.50 127.14 143.32 127.47 127.54 THF 0.883 0.909 (32) 0.826 (19) 0.901 (43) 0.786 (24) 135.53 131.68 144.98 132.84 152.25 Toluene 0.862 0.816 (23) 0.782 (10) 0.822 (21) 0.900 (33) 177.41 187.47 195.57 186.09 170.06 TFE 1.384 1.300 (20) 1.354 (52) 119.99 127.79 122.68 a Estimations of the statistical uncertainties are obtained by fluctuation auto-correlation analysis via the estimation of correlation times t.83 b Calculated from molecular weight. c NIST Reference Database Number 69.64
II.b Convergence, efficiency and precision of the 2PT method calculated every 100 ps, showing that accurate thermo- dynamics can be obtained from uncorrelated or correlated Lin et al.24 showed that the entropies predicted with 2PT for a trajectories. LJ gas converge for just 10 ps of dynamics. To validate the In all our simulations, we chose not to constrain the motion time needed for convergence, we calculated the properties of of the hydrogen atoms by the SHAKE40 algorithm, as is benzene using OPLS AA/L for 5 independent 1 ns trajectories, commonly the practice in the AMBER/GAFF and OPLS calculating the properties for various length trajectories: 1 ps, forcefields. While these constraints would presumably not 4 ps, 10 ps, 20 ps, 40 ps, 100 ps and 200 ps (Fig. 3). We find affect the dynamics,41 the calculated thermodynamics depends that by 20 ps the entropy and heat capacity are converged, on integrating over the entire DoS, thus SHAKE might affect while the self-diffusivity took 50 to 100 ps to converge the thermodynamics. Conversely, the high frequency of the (Fig. S1, ESIw). This convergence in the thermodynamic quantities vibrations may render any effect due to SHAKE minimal, as is consistent with a recent study of Lin et al.39 that found that the high frequency modes contribute exponentially less to the entropy of liquid water convergesafter10to50ps.
Downloaded by California Institute of Technology on 29 November 2010 thermodynamics than low frequency modes. We note however Due to the short 20 ps trajectories required and the Published on 23 November 2010 http://pubs.rsc.org | doi:10.1039/C0CP01549K that the 2PT formalism allows for accurate calculation of efficiency of FFTs, the 2PT calculations presented here require thermodynamic quantities regardless of external constraints, only a trivial increase in additional computation time. This by accounting for the removed degrees of freedom (Dof): the allows one to calculate the system thermodynamics on-the-fly Dof is used to calculate the system’s temperature from the during dynamics. Such calculations provide a rigorous check atomic velocities. of numerical stability and precision of the method. Fig. 4 reports the standard molar entropies and heat II.c Comparison of standard molar entropies vs. experiment capacities for acetic acid, benzene and DMSO with OPLS AA/L. Here, we used the last 20 ps of dynamics to evaluate the Fig. 5 and Table 4 present the standard molar entropies S0. thermodynamic properties every 100 ps during the 2.5 ns Contributions due to configurational entropy are included by dynamics, for a total of 25 data points. Convergence is statistical averaging over 5 discrete and uncorrelated observed after only 300 ps of equilibration, with fluctuations microstates, obtained from 20 ps trajectories every 500 ps of of 0.36 cal mol 1 K 1 in specific heat (0.6%). This indicates the 2.5 ns dynamics, as described previously. Effects due to that 2PT gives robust and precise thermodynamic quantities configurational changes are captured in the diffusive from short MD trajectories. The additional simulation and component and explicitly included in our model. computational time is also minimal, with the trajectories All FF underestimate S0, with average errors of 4.13, generated automatically during regular dynamics and the post 2.12, 6.36, 2.97 cal mol 1 K 1 for AMBER 2003, Dreiding, trajectory analysis taking less than 2% of the total simulation Gaff and OPLS AA/L, respectively, or approximately 5 to 15%. time. For example, the total time to simulate 512 molecules of As was the case with the dielectric constant, the largest benzene for 2.5 ns with LAMMPS took approximately discrepancy occurs in the polar solvents, in particular ethylene 110 CPU hours on a 3.2 GHz Intel Xenon processor, while glycol (average of 16% error) and DMSO (average of 15% the additional analysis of the 25 NVT trajectories to obtain the error). This may again point to the deficiency of using a fixed 2PT prediction took an additional 16 minutes (0.2%). Further, charge model, since the diffusion constant of other liquids42 is the average values of the entropy and heat capacity calculated known to be also affected by the lack of polarization. Self- every 500 ps (5 trajectories) are within 0.1% of the average diffusion and other low frequency librational modes contribute
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Table 4 Comparison of average standard molar entropy S0 (cal mol 1 K 1) for the 15 liquids and 4 different forcefields in this study. Entropies evaluated last 20 ps every 500 ps of 2.5 ns MD simulation. Average fluctuations of 0.31 kcal mol 1 molecule 1 is observed over all forcefields. Overall, the OPLS AA/L forcefield is the best performer, with a mean absolute error (M.A.D) of 1.47 cal mol 1 molecule 1, an average error of 5.85 cal mol 1 molecule 1 and a R2 correlation coefficient of 92%
AMBER 03 Dreiding GAFF OPLS AA/L Solvent Expa Best estimatec Avg Avg Avg Avg Acetic acid 37.76 32.23 0.25 37.79 2.35 30.67 0.17 35.13 0.22 Acetone 47.90 44.72 0.15 44.26 0.26 44.79 0.21 47.27 0.08 Acetonitrile 35.76 38.06 0.15 34.08 0.21 34.75 0.28 33.99 0.22 Benzene 41.41 40.58 0.22 39.01 0.25 38.37 0.52 41.19 0.16 1,4-Dioxane 46.99 39.47 0.22 41.71 0.27 38.00 0.20 42.82 0.26 DMSO 45.12 39.57 0.25 38.71 0.13 37.99 0.24 39.10 0.12 Ethanol 38.21 33.92 0.33 41.97 0.13 30.36 0.13 33.63 0.16 Ethylene glycol 39.89 30.43 0.05 40.13 0.31 28.95 0.18 33.62 0.16 Furan 42.22 39.09 0.00 38.88 0.38 37.60 0.25 40.03 0.23 Methanol 30.40 28.61 0.10 25.87 0.26 26.13 0.12 29.12 0.08 THF 48.71 41.89 0.22 48.45 0.12 38.01 0.18 46.96 0.28 Toluene 52.81 48.98 0.23 45.91 0.24 45.30 0.19 48.67 0.23 M.A.D.b 2.39 2.48 2.62 1.72 Avg. error 4.13 2.53 6.36 2.97 RMS error 3.17 3.12 3.15 2.03 R2 0.75 0.74 0.76 0.90 Chloroform 43.01 47.47 0.10 47.44 0.30 53.88 0.22 45.98 0.31 NMA 40.23 41.86 0.29 33.51 0.24 40.18 0.14 43.20 0.07 TFE 43.54 48.54 0.30 46.07 0.08 44.30 0.23 46.51 0.22 a NIST Reference Database Number 69. 64 b Mean absolute deviation. c No experimental values are available for chloroform, NMA and TFE. We estimate their values here but subtracting the average error from the calculated OPLS AA/L values.
most to the entropy calculated from approaches that rely on the where ap is the coefficient of thermal expansion (Table S2, ESIw) DoS such as 2PT. and kT is the isothermal compressibility (Table S3, ESIw). 0 While none of the forcefields reproduce the experimental S , We find that the corrections to the Cv are all less than the OPLS forcefield shows the best overall correlation with 0.25 cal mol 1 K 1 (Table S4, ESIw) (Fig. 6). experiment, with a 1.72 cal mol 1 K 1 MAD and a 90% Overall, all forcefields reproduce the experimental heat correlation. Particularly exciting here is the performance of the capacities to within 5%. More importantly, the values Dreiding forcefield (2.5 cal mol 1 K 1 MAD and 74% calculated using the 2PT approach show a 96% correlation correlation), since no parameters related to the thermodynamics to the approach used by Jorgensen and coworkers9,10,12,13,36,43 of liquids were used in determining the forcefield parameters. with the OPLS forcefield, which was based on extensive Downloaded by California Institute of Technology on 29 November 2010
Published on 23 November 2010 http://pubs.rsc.org | doi:10.1039/C0CP01549K The AMBER and GAFF forcefields have performance similar Monte Carlo sampling. This validates that 2PT can capture to Dreiding: 2.39 and 2.67 cal mol 1 K 1 MAD and 75% and the essential physics in these systems from short 20 ps MD 76% correlation respectively. We find that 2PT predicts trajectories. Further, the statistical deviations in our calculated standard molar entropies of these pure liquids to within heat capacities are 0.2 cal mol 1 K 1, or 0.5% (Table 5). 0.25 cal mol 1 K 1 (0.6%) standard deviation over all forcefields. Since there are no experimental standard molar entropy II.e Components of liquid entropy values for chloroform, NMA and TFE, we provide here a priori predictions based on the OPLS AA/L forcefield An attractive feature of 2PT is the facility to separate the average error of 2.97 cal mol 1 K 1: 43.01 for chloroform, individual components of the entropy as detailed in 40.23 for NMA and 43.54 cal mol 1 K 1 for TFE. Section III.c.ii. We performed this decomposition for all the liquids, with the OPLS AA/L forcefield (Table 6). Here we find II.d Comparison of molar heat capacities vs. experiment the ratio of the contributions to the entropy of 2 : 4 : 5 for valence vibrations : rotation : translation across all molecules, In 2PT we prefer to keep the volume constant (NVT MD) leading to a non-negligible contribution of 17% to the entropy leading to C and Helmholtz free energies, because we consider v from the internal vibrations. As expected, the vibrational this to be the least ambiguous framework for describing the entropy is greatest for the large flexible solvents (31%, DoS. However experiments are generally carried out under 24% and 21% for NMA, TFE and ethylene glycol respectively) conditions of NPT, leading to C and Gibbs free energies. To p and least for the small rigid solvents (3% and 6% for compare the C from 2PT to the C from experiment, we apply v p acetonitrile and methanol respectively). Since the vibrational a correction: component of the total DoS is analogous to the experimental @H IR and Raman spectra, forcefields that more closely reproduce Cp ¼ ¼ Cv þ DCv;p the experimental vibrational frequencies should lead to @T p 2 improved entropies. For illustrative purposes, we show the @p @V ap ¼ Cv þ T ¼ Cv þ VT ð1Þ vibrational DoS for chloroform using the OPLS AA/L force- @T N;V @T N;P kT field in Fig. 2b. The vibrational frequencies are on average
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