Quantum mechanics: a new tool for engineering thermodynamics [PDF]

Fluid Phase Equilibria 210 (2003) 147–160

Quantum mechanics: a new tool for engineering thermodynamics Stanley I. Sandler∗ Department of Chemical Engineering, Center for Molecular and Engineering Thermodynamics, University of Delaware, Newark, DE 19716, USA Received 16 August 2002; accepted 14 November 2002

Abstract Computational quantum mechanics is leading to new, theoretically based methods for the prediction of thermodynamic properties and phase behavior of interest to engineers. Three such methods we have been working on are reviewed here. In the most direct and computational intensive form, computational quantum mechanics is used to obtain information on the multidimensional potential energy surface between molecules, which is then used in computer simulation to predict thermodynamic properties and phase equilibria. At present, this method is limited to the study of small molecules due to the computational resources available. The second method is much less computationally intensive and provides a way to improve group-contribution methods by introducing corrections based on the charge and dipole moment of each functional group that is unique to the molecule in which it appears. The final method we consider is based on the polarizable continuum model, in which the free energy of transferring a molecule from an ideal gas to a liquid solution is computed, leading directly to values of activity coefficients and phase equilibrium calculations. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Quantum mechanics; Engineering thermodynamics; Polarizable continuum model

1. Introduction Quantum chemistry calculations, computer simulation and theory have now developed to the point that they are useful tools for predicting thermodynamic properties and phase behavior of some substances to an accuracy useful in engineering calculations. Here, I review three different methods that we have been developing in my research group to use quantum mechanics as a basis for making such predictions. The three methods are as follows: (1) use quantum mechanics to obtain the intermolecular potential energy surface for the interaction between a pair of molecules (as a function of intermolecular separation and relative orientation), and ∗

Tel.: +1-302-831-2945; fax: +1-302-831-3266. E-mail address: [email protected] (S.I. Sandler). 0378-3812/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-3812(03)00176-6

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use this potential to calculate second virial coefficients and, in simulation, to determine vapor–liquid phase behavior; (2) use quantum mechanics to improve current group-contribution methods; (3) use quantum mechanics-based continuum solvation models to make predictions of excess Gibbs energies, activity coefficients and phase behavior. The first of the methods is very computationally intensive, so that we have limited our attention to small molecules largely to answer two questions: (a) Can we obtain two-body intermolecular potential functions from quantum mechanics that are of sufficient accuracy to make useful first-principles estimates of thermodynamic properties? (b) Are two-body potentials and the assumption of pairwise additivity sufficient for the accurate prediction of thermodynamic properties? Studies of the second and third methods were undertaken to determine whether accurate prediction methods could be based on quantum-mechanical methods that were not computationally intensive, and could be used by the non-specialist in routine engineering calculations. This paper reviews the recent progress of my research group in each of these areas. As a review paper, ideas and results will be presented, but few equations or details. This information is provided in the papers to which we refer.

2. Quantum-mechanical calculation of the intermolecular interaction energies A straightforward, but computationally intense use of quantum mechanics is to determine the interaction energy landscape between two molecules. We first determine the optimum (minimum energy) structure of an isolated molecule, and then calculate the interaction energy of two molecules for a large collection (hundreds) of intermolecular separations and relative orientations. There are many subtleties in such calculations, such as the quantum-mechanical description used (which takes electron correlation into account), the size of the basis set, whether or not one keeps the molecular structure fixed in the optimum geometry or allows it relax, how one takes into account that a larger basis set is used for a pair of molecules than for a single molecule (basis set superposition error), etc. The specific techniques we used are discussed in [1–8]. Once energies have been computed at a large number of points, the next step is to use this information to obtain an analytical representation of the multidimensional potential surface. As the functional form of the interaction potential is not known, the simplest procedure is to fit parameters in site–site potentials, with sites located on each of the atoms, and in some cases at additional off-atom sites to improve the fit. In addition, the electrostatic charge on each of the atoms is computed for the isolated molecule, and these are used in a damped Coulomb potential. We have tried numerous site–site potential functions including the simple Lennard-Jones 6-12 and the exponential-n potentials, to the more complicated BSC potential [1,2]. Our general conclusions from such studies [6] are: (1) that the Lennard-Jones form does not provide a sufficiently accurate description; and (2) that it is difficult to obtain a globally optimum fit with potentials that contain very many parameters (number of sites times the number of parameters per site–site interactions). Consequently, we now use potentials of the exponential-n form. However, even with such potentials, different methods of fitting the potential parameters lead to different predictions

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of bulk properties. From a study of the use of interaction potentials for atoms [6], we have found the following: (1) the liquid phase radial distribution function is sensitive mainly to the well region of the potential; (2) the second virial coefficient is very sensitive to the potential well region at low temperatures, to the long-range part of the potential at moderate temperatures, and to the short-range repulsion region only at very high temperatures; (3) the predicted phase behavior (saturation densities and vapor pressures) is sensitive to both the well and long-range parts of the potential; (4) the predicted critical temperature depends almost equally on the well and the long-range regions of the potential, while the critical density is only slightly dependent on these; (5) all phase behavior and critical properties are very sensitive to the location of the point at which the potential changes from repulsive to attractive (that is, the crossing point). Clearly the situation is much more complicated when one is dealing with molecules, and multiple site–site interactions. However, the point of this discussion is that even if one has computed interaction energies very accurately, how these are fit to an analytic function, and particularly the weighting functions used, can greatly affect the predicted properties. 3. Second virial coefficient from intermolecular potentials generated from quantum mechanics We first consider the second virial coefficient, which results from only two-body interactions. The results for several fluids are shown in Figs. 1–4. It is interesting that the predicted second virial coefficient is quite good for methyl fluoride and hydrogen chloride. However, for methanol and acetonitrile, the agreement is only qualitatively, not quantitatively, good. There can be several reasons for this. One is that these latter fluids were the first we studied, and since then we have learned how to better fit the energy

Fig. 1. The predicted second virial coefficient for methyl fluoride compared to experimental data.

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Fig. 2. The predicted second virial coefficient for hydrogen chloride compared to experimental data.

landscape. However, and more importantly, we have found from studies of hydrogen-bonding fluids, such as HF and methanol, that for polar molecules, a higher level of theory and larger basis set may be needed to obtain accurate interaction energies. (It might seem surprising that the same is not true for CH3 F and HCl; however, we shall present a plausible argument later that these are not strongly hydrogen-bonding fluids.) One way to correct for possible inadequacies of the level of theory and/or incomplete basis set in the quantum chemical calculations is to assume that the shape of the interaction energy landscape is correct,

Fig. 3. The predicted second virial coefficient for methanol with various scaling parameters compared to experimental data.

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Fig. 4. The predicted second virial coefficient for acetonitrile with various scaling parameters compared to experimental data.

but that the absolute values of the energies are not, and therefore to multiply the potential obtained by a scaling factor. This factor can be obtained in one of two ways. One is, at the potential energy minimum, to do a calculation at a still higher level of theory and with a very large basis set, a calculation that is very time consuming, and thus impractical for the hundreds of points we use to determine the potential surface, and use the ratio of this energy to the one computed earlier as the scaling factor. The second procedure is, by trial and error, to choose a scale factor so that the measured and calculated second virial coefficients agree. We have examined both ways, and have found that while the scale factors obtained are in general agreement, obtaining the factor from matching the second virial coefficient is much easier (see Fig. 4).

4. Phase behavior based on intermolecular potentials generated from quantum mechanics Vapor–liquid equilibrium can be obtained from the intermolecular potentials by Monte Carlo or molecular dynamics computer simulation. Most computer simulation results reported in the literature are based on the assumption of pairwise additivity; that is, only two-body interactions are considered. A more accurate way to proceed would be at each step in a simulation to quantum-mechanically calculate, for the assembly of all molecules, the total system energy after each move (Monte Carlo) or force on each molecule due to all others at each time step (molecular dynamics) without the pairwise additivity assumption. Unfortunately, such a calculation is not possible for most molecules with currently available computer resources. Therefore, instead we have done Gibbs ensemble Monte Carlo (GEMC) calculations using the quantum-mechanically based pair potentials and the assumption of pairwise additivity. These results are shown in Figs. 5–9. We see that for some fluids, such as hydrogen chloride and methyl fluoride, the completely first-principles predictions of the phase boundaries and vapor pressures are in remarkably good agreement with experiment, especially when one considers that there are no adjustable parameters in the prediction. For

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Fig. 5. Vapor pressure (a) and vapor–liquid coexistence diagram (b) for methyl fluoride from GEMC NVT simulations. The solid line is experimental data and the symbols are simulation results based on the ab initio pair potential. The closed symbol is the estimated critical point.

acetonitrile the predictions are somewhat satisfactory, while the predictions for methanol and especially hydrogen fluoride are not in good agreement with the experimental data. We can obtain some insight into why this is so by looking at the bulk thermodynamic properties of the fluids shown in Table 1. There we see that HF has a much higher critical temperature and much lower critical compressibility than would be expected for a molecule of its size, indicating strong association in the vapor and liquid phases. Similar behavior is not seen for HCl or methyl fluoride, so we conclude that hydrogen bonding is much less

Fig. 6. Vapor pressure (a) and vapor–liquid phase boundary (b) of hydrogen chloride as a function of temperature from experiment (line or 䊏) and simulation (䉱).

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Fig. 7. Vapor pressure (a) and vapor–liquid phase boundary (b) for acetonitrile as a function of temperature from experiment (line or 䊏) and simulation (䉱).

important for these fluids. We believe that some of this inaccuracy results from the fact that a higher level of theory and larger basis set is needed when computing the interaction energies of hydrogen-bonding fluids. However, there is also another reason that the phase behavior predictions can be inaccurate for these fluids. Hydrogen-bonding molecules are in close proximity, and the strongly polar nature and polarizability

Fig. 8. Vapor pressure (a) and vapor–liquid phase boundary (b) for methanol as a function of temperature from experiment (lines) and simulation from simulation with combinations of scaling and polarization.

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Fig. 9. Vapor pressure (a) and vapor–liquid phase boundary (b) for hydrogen fluoride as a function of temperature from experiment (lines) and simulation from simulation with combinations of scaling and polarization.

of the molecules suggest that the pairwise additivity assumption is not valid, and multi-body effects need be considered. The obvious way to add multi-body effects is to use quantum chemical methods to compute the energy of the whole assembly of molecules, rather than summing the interactions of pairs of molecules. However, this is computationally prohibitive at the present time. A less rigorous method is to use quantum mechanics to compute three- and higher-body interaction energies, and then fit these to complicated potential functions that can be used in simulations. Three-body potentials have been proposed for water [9] and some noble gases [10–12], and their use in simulation [13] and virial coefficient calculations [14] has been shown to improve gas phase properties and vapor–liquid equilibrium predictions. However, the quantum chemical calculations involved, the fitting of these non-additivities, and then their use in simulation is also currently computationally prohibitive for molecules and mixtures of engineering interest. Table 1 Critical properties of the fluids for which we have made quantum-mechanical calculations, and some related fluids (CH3 CH3 , CH3 SH and H2 O) Species

Tc (K)

Pc (bar)

Zc

ω

HCl CH3 F HF CH3 OH CH3 CN CH3 SH CH3 CH3 H2 O

324.7 315.0 461 512.6 545.5 470.0 305.4 647.3

83.1 56 64.8 80.9 48.3 72.3 48.8 221.2

0.249 0.240 0.117 0.224 0.184 0.268 0.285 0.235

0.133 0.187 0.329 0.556 0.327 0.153 0.099 0.344

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A less computationally demanding, but less accurate procedure is the use of classical polarizable potentials that approximately account for multi-body induction interactions by adding a contribution to the total interaction energy of the system due to the interaction of each permanent dipole with the induced dipoles in all the other molecules. This polarization energy is a multi-body effect since the induced dipole in each molecule is a result of the electric field that depends on the electrostatics of all other molecules in the system. This effect has to be recalculated at each Monte Carlo step, i.e. each time a trial move is performed. Therefore, Monte Carlo simulations with a polarizable potential are computationally more demanding by approximately an order of magnitude than those with only pairwise additive interactions. We have done such studies using a single polarizable site in each molecule [1,8], which is the simplest description of polarization. There are two reasons for our choice of this model. First, the incorporation of a more complicated polarization correction would be computationally too expensive. Second, while not completely accurate, even this simple polarization model should exhibit the correct qualitative behavior, and indicate the importance and magnitude of the polarization. From our calculations, we find that polarization results in a greater attraction between the molecules (i.e. a more negative interaction energy), as evidenced by the resulting lower vapor pressure, an increase in the critical temperature, and greater association in the vapor phase. Because of the computational cost of simulations including polarization, we have not done calculations for all the molecules we have studied. For those that we have studied, we find the effect of polarization is small for CH3 F and acetonitrile (Figs. 6 and 7), and very significant for methanol and hydrogen fluoride (Figs. 8 and 9). In particular, the inclusion of polarization improves the agreement between simulation results and experiment, though for hydrogen fluoride both scaling and polarization are needed to obtain vapor pressures that agree with experiment. (As these calculations are so time consuming, calculations have so far been done at only a few points.) As a final example of using the combination of ab initio quantum mechanics and simulation, we consider the vapor–liquid equilibrium of the methanol + acetonitrile mixture. To do such a calculation, we used the

Fig. 10. Vapor pressure (a) and vapor–liquid phase boundary (b) for hydrogen fluoride as a function of temperature from experiment (lines) and simulation from simulation with combinations of scaling and polarization.

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acetonitrile potential developed earlier, the methanol potential scaled to give approximately the correct pure component vapor pressure, and developed a methanol–acetonitrile potential from ab initio quantum mechanics calculations [2]. The GEMC results we obtained are shown in Fig. 10. What is remarkable about these results is that the azeotropic behavior is correctly predicted based on quantum mechanics, even though no mixture data have been used. The only adjustable parameter in this otherwise first-principles calculation is the scaling parameter used to obtain the correct pure component methanol vapor pressure. Our results suggest that the combination of ab initio quantum mechanics calculations with computer simulation has developed to the point where it can be useful in making predictions of thermodynamic properties and vapor–liquid equilibria. However, a computational limitation at present is the size of molecules that can be studied since ab initio calculations scale approximately from the 4th to the 7th power of the number of basis sets used, depending on the extent of electron correlation used. As computational power improves, this will be become less of a limitation. The computational method discussed above, of first using ab initio calculations to determine the intermolecular potential function followed by Monte Carlo simulation, is sufficiently computationally intensive that it cannot presently be used for routine engineering calculations. Therefore, we next turn our attention to quantum chemistry-based methods that can now be used in this manner.

5. The use of quantum mechanics to improve group-contribution methods An assumption central to all group-contribution methods is that a functional group behaves the same (that is, contributes an identical amount to a thermodynamic property) independent of the molecule on which it is located. Empirically, this assumption has been found to be incorrect when two strong (that is, non-alkyl) functional groups are located close to each other on the same molecule. An example of such a “proximity effect” failure of group-contribution method is shown in Fig. 11 for octanol–water partition coefficients [15]. The obvious reason for the failure is seen when computing the charges on each atom and

Fig. 11. Predictions from the GCSKOW model without and with multipole corrections for 204 monofunctional and 246 multifunctional compounds.

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then each functional group in a molecule. For example, below we show the charges and dipole moments on functional groups in the molecules n-propanol and hydroxyacetonitrile. n-Propanol

Charge (e) Dipole moment (d)

Hydroxyacetonitrile

CH3 –

CH2 –

CH2 –

OH

NC–

CH2 –

OH

−0.01 0.14

+0.02 0.16

+0.30 0.06

−0.31 0.35

−0.15 0.42

+0.41 0.13

−0.26 0.36

For comparison, the charges on (and dipole moments of) the CH3 and CH2 groups in an alkane are approximately zero. Thus, we see that a CH2 group has a zero charge in an alkane, 0.41e in hydroxyacetonitrile, and +0.02e or +0.30e in propanol, depending on where in the molecule it is located. Since interaction energies are a result of electrostatics we see from these examples that functional groups generally do not have the same properties independent of the molecule on which they are located, thus violating the underlying group-contribution assumption. There have been several proposals to correct group-contribution methods for these proximity effects. One has been to add corrections to the properties of a functional group depending on its first (and sometimes second) intramolecular nearest neighbors [16]. To do so requires a great deal of high quality data, and sophisticated data regression methods. Another suggestion has been to redefine groups so that each is approximately electrically neutral [17]. Thus, for example, the CH2 and OH groups in propanol would be combined into a single CH2 OH group. The problem with this suggestion is that it leads to increasing numbers of larger functional groups. For example, the only way this could be accomplished for hydroxyacetonitrile is by defining the whole molecule to be a functional group. However, quantum mechanics calculations provide a path forward in that we can keep the commonly defined small functional groups, but correct their properties for the charges and dipole moments of each group that will change depending on the molecule, or more specifically the other functional groups in the molecule and their distance away from the functional group of interest [18]. Thus, the contribution that a functional group makes to the thermodynamic property Θ could be written as Θ = a + be2 + cd2 or 2 Θ = Θref + b(e2 − e2ref ) + c(d 2 − dref )

where ref indicates the property of the functional group when it is in a reference state as the sole non-alkyl functional group in an alkane molecule, e is the charge and d is the dipole moment. The form of the correction is based on a theory of Kirkwood for the free energy of adding a charge or dipole into a cavity in a dielectric continuum [19], and the parameters b and c depend on the size of the functional group and the dielectric properties of the solvent. The success of this hybrid method of using a quantum-mechanically corrected group contribution is illustrated in Fig. 11 for the octanol–water partition coefficient. The beauty of the method described here is that it improves upon the well-known concept of group contributions by requiring only a single molecule ideal gas quantum mechanics calculation that is easily done.

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6. Polarizable continuum models The final class of quantum mechanics models considered here is based on calculating the Gibbs energy change on transferring a molecule from the ideal gas into a solvent that is treated as a polarizable continuum. While there is a long history of such models, it is only recently, based on the very clever pioneering work of Klamt and co-workers [20–22], and then slightly modified by us [23], that the COSMO class of models have been developed that are of sufficient applicability, generality and accuracy to be of engineering use. Indeed, I believe that as this model is developed further, it will replace current predictive models based on the group-contribution concept such as UNIFAC [24,25]. The basic idea of this class of methods is that a molecule, rather than being separated into a set of functional groups as in group-contribution methods, is deconstructed into a collection of very small surface elements, and the charge density on each surface element is computed using a quantum electrostatic calculation. The unique characteristic of each molecule is its sigma profile that is a representation of charge density versus likelihood of occurrence. From the sigma profiles of each molecule and a statistical-mechanical analysis, the excess Gibbs energy at any composition can be computed. There are several interesting features of this class of models. First, as each molecule has a unique sigma profile, the method does not suffer from the problem of the proximity of other strong functional groups, or from structural effects (the inability to distinguish between isomers) that are shortcomings of group-contribution methods. Second, the method does not use a pre-specified form for the excess Gibbs energy, as is the case for UNIFAC, which is built upon UNIQUAC [26]. Finally, there are only a few adjustable parameters in the model; the diameter of each type of atom, several parameters related to hydrogen bonding, and the area of a surface element used in computing the sigma profile. Once these parameters have been fixed, the properties of any mixture containing these atoms can be computed. Consequently, the method

Fig. 12. Comparison of vapor–liquid equilibrium predictions from COSMO-SAC, UNIFAC, and modified UNIFAC models for water (1)/1,4-dioxane (2) at temperatures 308.15 and 323.15 K.

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Fig. 13. Vapor–liquid equilibrium prediction from COSMO-SAC for benzene (1)/n-methylformamide (2) at temperatures 318.15 and 328.15 K.

does not require the very large databases needed to determine the much larger number of parameters inherent in group-contribution methods. Also, unlike group-contribution methods, this class of model does not require the introduction of new functional groups to accommodate new types of compounds or to resolve unsatisfactory predictions. New parameters are needed only if a new atom is introduced (the current implementation provides parameters for H, C, Cl, F, O, N and S). However, what is required is a one-time, easy and quick, quantum mechanic electrostatic calculation for each molecule using density functional theory. Predictions based on the COSMO-SAC model are shown in Figs. 12 and 13. A limitation of the model is that, like UNIFAC, it is applicable only to a dense liquid, as in the theory each surface element must interact with another; vacancies or unpaired surface elements are not permitted.

7. Conclusions I believe that the development of computational quantum mechanics is leading to a new paradigm in the prediction of thermodynamic properties and phase behavior. In the most direct and computationally intense form, computational quantum mechanics can provide the information needed to construct the multidimensional interaction potential surface between molecules, and this can be used in computer simulation to predict thermodynamic properties and phase equilibria. At present, this method is limited to the study of small molecules due to the computational resources available. However, there are much less computationally intensive ways that quantum chemical methods can be used, which opens up their applicability to very much larger molecules. The electrostatic correction method for group-contribution models, and the polarizable continuum method, both very briefly discussed here are two examples.

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Acknowledgements Financial support for this research was provided by the National Science Foundation (CTS-0083709 CHE-9982134) and the Department of Energy (DE-FG02-85ER13436). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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