Thermodynamic Properties of Molecular Crystals Calculated within the

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Thermodynamic Properties of Molecular Crystals Calculated within the Quasi-Harmonic Approximation Ctirad Cervinka, Michal Fulem, Ralf Peter Stoffel, and Richard Dronskowski J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b00401 • Publication Date (Web): 09 Mar 2016 Downloaded from http://pubs.acs.org on March 14, 2016

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The Journal of Physical Chemistry

Thermodynamic Properties of Molecular Crystals Calculated within the Quasi-Harmonic Approximation Ctirad Červinka †,*, Michal Fulem †, Ralf Peter Stoffel ‡, Richard Dronskowski ‡ †

Department of Physical Chemistry, University of Chemistry and Technology Prague,

Technická 5, CZ-166 28 Prague 6, Czech Republic ‡

Institute of Inorganic Chemistry and Jülich-Aachen Research Alliance (JARA-HPC),

RWTH Aachen University, Landoltweg 1, D-52056 Aachen, Germany

*Corresponding author: [email protected]

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ABSTRACT. A computational study of the possibilities of contemporary theoretical chemistry as regards calculated thermodynamic properties for molecular crystals from first principles is presented. The study is performed for a testing set of 22 low-temperature crystalline phases whose properties such as densities of phonon states, isobaric heat capacities, and densities are computed as functions of temperature within the quasi-harmonic approximation. Electronic structure and lattice dynamics are treated by plane-wave based calculations with optPBE-vdW functional. Comparison of calculated results with reliable critically assessed experimental data is especially emphasized.

1. INTRODUCTION The thermodynamic properties of the crystalline phase represent data necessary for modeling technological processes such as solid-state chemical reactions, adsorption on solid surfaces, or separation processes. Also, such data are required for solving the problems connected to polymorphism of crystalline phases and phase equilibrium with liquid and vapor phases which is the main reason of our interest in the thermodynamic properties of solids. Since most organic compounds and many inorganic species exist in the form of molecular crystals at low temperatures, including species such as water ice, carbon dioxide, hydrocarbons, or numerous drugs, explosives, fertilizers, etc., they represent an important and large group of chemical compounds whose properties are required in many applications. The thermodynamic properties of solids have been accessible experimentally down to temperatures of liquid helium for decades of years with a relatively low uncertainty. Nonetheless, even for some simple molecules such as formaldehyde, there is a lack of thermodynamic data at low temperatures which usually holds even more so for more complex molecules. Therefore, a reliable computational methodology enabling to predict the thermodynamic properties of the solid phase would be useful to help generating data in cases 2 Environment ACS Paragon Plus

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of missing reference data or newly synthesized compounds. Such a methodology needs to be verified by carrying out a thorough comparative study of calculated and experimental data which is the main goal of this work. A testing set of 22 molecular crystals has been selected – neon, argon, ethane, ethene, ethyne, propane, butane, nitrogen, fluorine, carbon dioxide, ammonia, hydrogen peroxide, hydrogen fluoride,

methanol,

formaldehyde,

aminomethane,

dimethylether,

hydrazine,

methylhydrazine, formic acid, formamide, and acetic acid; all molecular structures are illustrated in Figure 1. This set of species contains various substances of different properties and structures (polar and non-polar molecules, monoatomic and polyatomic molecules forming orthogonal or non-orthogonal unit cells, or molecules capable of forming hydrogen bonds) so that a methodology being able to treat all such different species is expected to be general enough and in principle to be able to describe any molecular crystal. For compounds exhibiting polymorphic behavior, the calculations were performed for the stable lowtemperature phases. When the dominant effect of the cohesive energy is taken into account, the vibrational characteristics of the solid phase are responsible for a rather minor contribution to quantities as the sublimation enthalpy. However, the temperature dependence of the enthalpy of a molecular crystal is governed by the vibrational degrees of freedom, so that the quantification of the density of phonon states is a necessary task to be carried out. For this purpose, phonon computations based on plane-wave quantum-chemical calculations with periodic boundary conditions can be employed1-2 whose strategy fully exploits the perfect periodicity of the crystal environment. The goal of this work is to use the plane-wave based quantum calculations within the DFT formalism to calculate the thermodynamic properties, above all the temperature-dependent densities and isobaric heat capacities for the molecular crystals. The density of phonon states is calculated for the testing molecular set and, based on it, the thermodynamic properties of



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the crystalline phase are evaluated within the quasi-harmonic approximation which has become a frequently used tool for evaluating complex thermodynamic data for solids3-6, although it has rather been applied for metallic systems or semiconductors so far. Its application for simple molecular crystals studied in this work is still not sufficiently explored in the literature, although several works aiming to calculate the thermal contribution to the enthalpy of molecular crystals have been published.7-8 Some other works assume only the harmonic approximation or use experimental heat capacity data.9-10 Recently, several groups have studied the possibilities of the quasi-harmonic approximation coupled with the calculations from first principles to predict the relative stability of different polymorph modifications of particular molecular crystals11-13 and to calculate their temperature and pressure dependent thermodynamic properties.14-16 These works are driven by the motivation to construct the phase diagrams of molecular crystals ab initio.17-18 A broad study, evaluating the uncertainties of calculations of thermodynamic properties of molecular crystals at finite temperatures is still missing. A few benchmark studies comparing the performance of various DFT functionals accounting for dispersion interactions can be found in the literature,9,

19-20

taking into account their

capability to predict unit-cell geometries and/or cohesive energies. In this work, we briefly tested some of the recommended functionals, and due to the high computational cost of the phonon calculations for our testing set of molecular crystals, even in the case of using empirically corrected DFT, only one level of theory was used to calculate the density of phonon states (DPS), as is described below. The calculated vibrational frequencies of the solid phase are compared with available literature experimental data obtained by infrared or Raman spectroscopy or inelastic neutron scattering. Based on this comparison, some generalized statements regarding the accuracy of phonon calculations for molecular crystals are concluded. Based on the volume-dependent density of phonon states and electronic energy of the unit cell, the thermal expansion of the


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crystals is studied and the isobaric heat capacities are calculated in the temperature interval from 0 K to the lowest temperature of a phase transition (either crystal − liquid or crystal − crystal). All calculated data are compared with critically assessed experimental data, and statistical post-processing of the calculated results is performed to analyze the computational uncertainties.

2. COMPUTATIONAL METHODS The quantum-chemical calculations with periodic boundary conditions were used to compute selected static and dynamic properties for the set of 22 molecular crystals. The density functional theory with a non-local functional optPBE-vdW21, accounting for the dispersion interactions in a semi-empirical way, and the projector augmented wave (PAW) method22 as implemented in program package VASP 5.3.523 were used in general for all the calculations. At first, the parameters of the plane-wave calculations were subjected to a preliminary analysis which is described below and which resulted into the following setting which was used throughout this work. The k-space was sampled using the Monkhorst-Pack method24 with a mesh centered at the Γ-point. The numbers of subdivision along each of the reciprocal space vectors were set by formula Ni =·K / ai, where K ≈ 60 and ai stands for the direct unitcell vector lengths. The so-called “hard” PAW potentials were used in this work as long as they are reported to yield more accurate description of all properties of solids than the standard PAW potentials.25 The use of the “hard” PAW potentials caused need for enlarging the plane wave cut-off parameter which was set to 1000 eV (except neon and argon, for which no “hard” PAW potentials were available, so the standard potentials along with a 700 eV cutoff were used). After having found the optimal computational parameters, unit cells of all 22 studied crystals were optimized using the conjugate-gradient approach26 until the criterion (fairly strict) that forces acting on all atoms had to be lower than 10−6 eV · Å -1 was met. The criterion for the 5 Environment ACS Paragon Plus


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electronic iteration loop was set to 10−10 eV per cell. The optimization procedure started from the experimentally determined unit-cell parameters which are summarized in the Supporting information in Table S1. During the course of optimization, unit-cell volume and shape as well as the atomic positions were allowed to relax, only keeping the total space group symmetry fixed. The obtained unit-cell geometry was considered to be the equilibrium structure at 0 K neglecting the ZPE effect. Then, the Eel(V) curve around the equilibrium volume was constructed. The equilibrium unit-cell vectors were gradually scaled by factors 0.94 to 1.06 with a step of 0.01 so that 13 more structure optimizations were carried out; this time with fixed volume and with the same 10−6 eV · Å −1 force criterion. When a volume point possessing a lower energy than the former minimum was found, the procedure was repeated starting with a relaxed volume optimization and following the Eel(V) curve calculation around the new minimum found. This issue occurred mostly in cases when a lower cut-off value than 1000 eV was used. This fact supports the selection of the 1000 eV cut-off as the optimum. Having the Eel(V) curve complete, the force constants needed for the computation of the vibrational modes of the crystal structure were calculated. Following the quasi-harmonic approximation,27 the same finite-displacement procedure28 was performed in VASP for 5 optimized unit cells obtained by scaling by factors 0.96 to 1.00 with a step 0.01. The unit cell was replicated to form a super-cell so that its dimensions exceed 10 Å in each direction. The k-space grid was made sparser accordingly. The program Phonopy29 was then used to determine the symmetry-inequivalent displacements of all symmetry-inequivalent atoms from the super-cell. All these displacements were performed by 0.01 Å in both positive and negative sense which usually led to several tens of super-cells distorted from its equilibrium geometry. Every such distorted super-cell was then treated by a single-point computation (no further optimization) on the basis of which the Hellmann-Feynman forces acting on all atoms were evaluated. Taking into account the large number of particular super-cell computations


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for each crystal, this represents the most expensive step from the computational time point of view. The description of the thermal expansion is based on DPS of the compressed supercells since the expanded supercells usually exhibit imaginary vibrational frequencies due to the circumstance that the calculations are related to zero temperature and the expanded unit cells would correspond to unphysical negative pressure. The Hellmann-Feynman forces were processed by the Phonopy code29 which produces the dynamical matrix, searches for its eigenvalues (vibrational frequencies) and evaluates the density of phonon states, based on which the vibrational Helmholtz energy Avib can be evaluated using the harmonic approximation: ∞

(

)

− hν hν   Avib = RT ∫ ln 1 − e kT + DPS (ν ) dν , 2kT  0 

(1)

In this way, Avib was evaluated using the DPS calculated for the 5 values of unit-cell volume and these values were then fitted using a linear function Avib = a V + b, where a and b were the temperature dependent fitting parameters. Such approach massively saves the computational cost without any significant loss of accuracy, although it has to be verified whether the Avib really exhibits a nearly linear volume-dependence. A detailed statistics of the linear fits can be found in the SI in Table S3. The total Helmholtz energy of the crystal was evaluated as follows:

Acrcalc (T,V ) = Eel (V ) + Avib (T,V ) .

(2)

The isobaric properties were extracted from the constant volume phonon calculations after fitting the Helmholtz energy of the crystal by the Murnaghan equation of state30:   V  B0  0   BV   B0V   V  A(V ) = A(V0 ) + ' + 1 − '0 0 , ' B0  B0 − 1  B0 − 1     '


(3)


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containing four parameters, namely the equilibrium volume V0, the Helmholtz energy at the equilibrium volume A(V0), the ground state bulk modulus B0, and the dimensionless volume '

derivative of the bulk modulus (evaluated at V0) B0 . In addition, the Murnaghan equation provides an analytic expression for the volume derivative of the Helmholtz energy so that, the Gibbs energy can be determined using the equation:

 ∂A  G(T , p) = A(T ,V ) + pV = A(T ,V ) − V   .  ∂V T

(4)

3. THE REFERENCE SET OF EXPERIMENTAL DATA Establishing a consistent reliable database of experimental data including their uncertainties is key to the testing and evaluating the performance of computational methodologies. However, evaluating the uncertainty of reference experimental data is not an easy task as the experimental uncertainties in the original papers are presented in a non-uniform and mostly insufficient manner – often the way the uncertainties were derived is not specified, the reported estimates of uncertainties are the standard deviations of the mean reflecting only random not systematic errors (this results in too optimistic estimates of reported uncertainties) or no uncertainties are given at all. In this work, three classes of experimental data were used for comparison: spectroscopic data, density and isobaric heat capacities for the crystalline phases. The experimental data on vibrational frequencies for the crystalline phases were mostly obtained by IR and/or Raman spectroscopy. Both techniques usually provided the frequencies not differing by more than a few units of cm‒1 (this is also the estimated uncertainty of these measurements). In rare cases, when several contradicting values for a given frequency were found in the literature, the most recent experimental data were used for the comparison with calculated values. Experimental crystalline phase densities exhibited a relatively high scatter in cases when more data sets were available, although the authors of X-ray or neutron scattering experiments 8 Environment ACS Paragon Plus

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claim that the uncertainty of their determinations of unit-cell volumes are in the interval 0.01 ‒ 0.5 %. Based on the scatter of available literature data, we conservatively estimated the uncertainty of experimental densities typically to a few units of percent (Table 2). For the statistics, the average or interpolated values of the reported densities were used while obviously outlying experimental values were omitted. The experimental data on heat capacities of the crystalline phase were primarily searched for using the NIST ThermoData Engine (TDE31-32). All the reference experimental heat capacity data were measured by adiabatic calorimetry which is the principal method for low temperature calorimetric measurements concerned in this work and which belongs among the most accurate calorimetric techniques with typical uncertainties ranging from 0.2 to 0.5 %. At temperatures close to 0 K and phase temperature transitions the uncertainties is slightly higher (typically 2 – 3 % close to 0 K and 0.5 – 1 % close to the phase transition temperatures). The combined expanded (0.95 level of confidence) uncertainties given in Table 3 were estimated based on i) on experimental uncertainties reported in the original publications, ii) the consistency and mutual agreement of data from various literature sources and their number (however often only one literature source was available for the studied compounds), iii) our subjective view on the reliability of the experimental data from individual sources based on our experience with certain laboratories and uncertainties of their measurements (also for compounds for which more data sets on heat capacities are available) considered when developing recommended thermophysical data (see for example33-34). In the comparisons of calculated to experimental values given throughout this paper, the mean percentage deviations are defined as

100 N X icalc − X iexp σX = ∑ X exp , N i =1 i

(5)

and the mean absolute percentage deviations as

σ Xabs =

calc exp 100 N X i − X i ∑ X exp , N i =1 i


(6)


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where X is the property compared (X = vibrational frequencies ν, crystal phase densities ρcr, cr and heat capacities Cp ). Detailed comparisons of calculated to experimental data is given in

the sections 4.2 (vibrational frequencies), 4.3 (crystal phase densities), and 4.4 (crystal phase heat capacities).

4. RESULTS AND DISCUSSION 4.1 Verification of computational parameters At first, a brief comparison of various methods of treatment of the dispersion interactions was carried out on crystals of formic acid and ethane, exemplarily for one polar and one non-polar compound. The empirical corrections35 developed by Grimme and co-workers, namely DFTD2,36 DFT-D3,37 and DFT-D3 with the Becke-Johnson dumping function,38 were tested together with the method suggested by Tkatchenko and Scheffler.39 These methods were combined with the PBE functional.40 Also, several non-local correlation functionals specially optimized to account for the dispersion interactions within the DFT41-42 (implemented in VASP by Klimeš and co-workers43) were included in this preliminary test. Namely, the following functionals were tested: optB88-vdW,21 optPBE-vdW,21 optB86b-vdW,43 and vdWDF2.44 In total, 8 approaches how to account for the dispersion were compared. The unit cells corresponding to the experimental structures were optimized using these 8 methods such as to compare the lattice parameters and unit-cell volumes with the original experimental values. Based on this test, the DFT-D3 and optPBE-vdW techniques were selected for further computations as they provided the results in the closest agreement with the experimental parameters. The vdW-DF2 functional, which is often used by other authors,9, 19 gave results of almost a comparable accuracy as the optPBE-vdW functional. The differences in the optimized unit-cell volumes obtained by the examined methods are below 10 % for formic acid and below 25 % for ethane, see Table S2 in SI for particular values. However, it should be noted that a larger set of molecules would be required to compare the performance of mentioned techniques reliably. Still, as a clue which option to choose for further calculations, 10 Environment ACS Paragon Plus

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such a preliminary test seems sufficient. Due to the very high computational costs of the desired phonon calculations, all calculations were further carried out only at the optPBE-vdW level of theory. The effect of the density of the grid sampling the k-space was studied by a similar test of optimizing the experimental crystal structures of formic acid and ethane. From this analysis, it was concluded that the effect of the K value on the optimized geometry is negligible when it holds K > 40 so that in most cases approximately K ≈ 60 was used to get integer values of Ni divisions, e. g. a lattice vector of 10 Å gives 6 k-points in the same direction. For non-cubic unit cells, the Ni parameters were set to sample all directions in the k-space in a balanced way. Much stronger influence on the resulting unit-cell geometry was found to originate from the plane wave cut-off. A preliminary test of the cut-off influence was carried out for the whole testing set of studied 22 molecules. The experimental geometries were optimized using cutoff values of 700 eV, 800 eV, 900 eV and 1000 eV, respectively. The highest assumed cut-off of 1000 eV appeared to provide the closest unit-cell structures to the experiment so that it was used for the whole set (except neon and argon). Such a high cut-off value should also prevent the volume optimizations from suffering from the errors caused by the basis set incompleteness. The unit-cell parameters resulting from a fully performed structural optimization with the symmetry constraint are given in Table 1 along with the deviation of the calculated volume from the experimental values. We note that this comparison is rather preliminary since the calculated volumes correspond to 0 K temperature and do not include vibrational contributions while the experimental volumes correspond to finite temperatures. The mean absolute percentage deviation of the optimized unit-cell volumes from the experimental data amounts to 3 % which can be regarded as a satisfactory agreement. The optimized unit-cell volumes do not exhibit any significant bias compared to the experimental data. The largest relative deviation was observed for neon which is in part caused by its reported large zero-



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point motion.45 The testing set includes 5 crystals whose lattice is not orthogonal so that the β angle was allowed to vary during the optimization and its average change in the absolute value amounts to 9 °. The most significant deformation of the unit cell during its optimization was observed for ethane where the β angle changed from an obtuse angle to an acute angle. Since the phonon calculations require a structure corresponding to an energetic minimum related to the theoretical framework, even such modified unit cells were used for further calculations. However, this comparison of unit-cell volumes does not include the vibrational effects nor the temperature dependence of the volume. Since such a proper comparison requires performing all the calculations of the quasi-harmonic approximation, it is discussed below.

4.2 Lattice dynamics The densities of phonon states were calculated for the whole set of crystals using the finite displacement method28 as implemented in the program Phonopy.29 This quantity represents the basis for the evaluation of the temperature-dependent vibrational Helmholtz energy Avib(T,V) via the methodology of statistical thermodynamics. The results of these calculations are summarized in SI in the form of plots of the densities of phonon states as functions of the phonon wavenumber. It can be seen that the distribution of the low-frequency vibrations which belong to the lattice (intermolecular) modes is close to the Debye model while the intramolecular vibrations remind the Einstein model and are usually represented in the DPS function by isolated peaks of a well-defined frequency. In total, 9 molecular crystals do not exhibit any imaginary vibrational frequencies which confirms their dynamical stability. Concerning the remaining 13 crystals, only minor fractions of the DPS are located in the imaginary region. The mean percentage integral contribution of modes of imaginary frequencies to the DPS amounts to 0.35 % while the largest imaginary part of the DPS was observed for fluorine (1.29 %). Such small values suggest that the spurious imaginary frequencies represent rather a computational artifact caused by the finite size of the treated


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super-cells than dynamic instability of given crystals. The imaginary frequencies are expected to vanish with increasing the super-cell size, as was discussed in earlier works.5, 32 The complete density of phonon states usually cannot be directly compared to experimental data since complete inelastic neutron or X-ray scattering measurements of phonon dispersion properties are rather limited to simpler (covalent or metallic) crystals. However, vibrational frequencies measured experimentally by methods of infrared or Raman spectroscopy are abundantly accessible and can serve as a benchmark for comparison of calculated dynamical properties of the crystal lattice. For a direct comparison, the calculated frequencies need to be evaluated at the Γ-point. Concerning the assignment of particular frequencies to fundamental vibrational modes, this procedure is based on group-theoretical considerations combined with the knowledge of the eigenvectors of vibrational modes. This task is simpler for the intramolecular modes where one can benefit from the knowledge of an assignment of an isolated molecule, and the vibrational modes are usually distinguishable for smaller molecules. A unit cell containing Z N-atomic molecules possesses 3ZN−3 optical vibrational modes, out of which 3ZN−6Z modes belong to the intramolecular vibrations and the remaining 6Z−3 modes represent the lattice vibrations (for linear molecules, these numbers are 3ZN−5Z and 5Z−3, respectively). Figure 2 shows a plot of relative deviations of calculated wavenumbers of intramolecular vibrational modes from the experimental values. In total, it was possible to compare the wavenumbers assigned to 891 out of 992 intramolecular modes overall possessed by 22 studied crystals. The experimental data on crystal lattice dynamics were culled from the following literature: neon,46 argon,47 ethane,48-49 ethene,50-52 ethyne,53 propane,48, peroxide,61

54

butane,54-56 nitrogen,57 fluorine,58-59 carbon dioxide,57 ammonia,60 hydrogen hydrogen

fluoride,62

methanol,63

formaldehyde,64-65

methaneamine,66

dimethylether,67-68 and hydrazine,69 formic acid,70-72 formamide,73 and acetic acid.74 The deviation plot seems almost unbiased, namely the mean percentage deviation σν, defined according to equation (5), amounts to +0.21 % which corresponds to a hypothetical scale 13 Environment ACS Paragon Plus


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factor 0.9979. This could demonstrate that the plane-wave based calculations of intramolecular vibrational frequencies of molecular crystals with use of the optPBE-vdW functional are more accurate than common DFT calculation of vibrational frequencies of isolated molecules. Recommended scale factors75 for isolated molecules vary between 0.987 and 0.995 for the PBE functional and range from 0.961 to 0.969 for the B3-LYP functional, depending on the size of used basis set. However, the very low absolute value of σν is caused by compensation of positive and negative deviations. The mean absolute percentage deviation of calculated vibrational frequencies

σνabs , defined according to equation (6), amounts to 2 %

which points to a comparable accuracy as for isolated molecules. Still, the deviations

σνi for

particular crystals exhibit a significant scatter (standard deviation 4 %) and some calculated wavenumbers of the crystal differ by 10 % from the experiment. Moreover, Figure 2 shows that the calculated low or middle wavenumbers below 2500 cm−1, belonging mostly to deformation vibrations of the molecular skeleton, exhibit a different mean deviation from the experimental values than the high wavenumbers from the region around 3000 cm−1 reflecting stretching modes of single bonds to hydrogen atoms. For isolated molecules, it proved to be useful to distinguish the scale factors for low and high frequencies.76-77 Performing an analogous statistical analysis for the vibrational frequencies obtained in this work by the plane-wave based calculations leads to the scale factors 1.0032 and 0.9861 for the lowfrequency and high-frequency region, respectively. In the language of mean deviations of calculated frequencies, the mean low-frequency deviation amounts to −0.32 % and the highfrequency value to 1.41 %. In the case of the lattice, or intermolecular vibration modes, a group-theory based procedure can be again used to assign the frequencies to particular vibrational modes. However, the situation here is not so optimistic. Inelastic neutron scattering experiments are quite rare for the type of molecular crystals studied in this work. Next, the modes of very low wavenumbers close to 0 cm−1 are usually not detectable by common spectrometers or the intensities of such 14 Environment ACS Paragon Plus

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lattice modes are very low in the absorption spectrum. Also, the frequencies of the lattice modes may vary with temperature significantly due to the changes in the vibrational potential energy caused by the thermal expansion of the crystal. Therefore, the complete vibrational assignment of the lattice modes is available in the literature only in some model cases of crystals whose unit cells contain a few molecules. Typically, only some of the lattice modes are recorded in the spectrum whose further assignment is only tentative. Figure 3 represents a plot of relative deviations of calculated wavenumbers of the 227 lattice modes (out of 442 total lattice modes belonging to the subset of 22 studied crystals). As long as the computed and experimental values for a significant part of these wavenumbers were coupled rather speculatively without any knowledge about the assignment of the experimental data, this figure should be taken only as illustrative, not allowing to draw a quantitative conclusion. The mean absolute percentage deviation of the calculated frequencies from the experiment

σνabs

amounts to 15 % ( σν = 8 %) and the standard deviation of this difference is 32 % which is in accord with the fact that calculated frequencies differing from the experimental values by a factor of 2 were often observed. Such a high scatter is due to the nature of the lattice modes, closely linked to intermolecular (noncovalent) interactions which would require an unaffordably high level ab initio description to obtain more accurate data on lattice dynamics of molecular crystals. It should be pointed out that for non-polar molecules, the mean deviation of the calculated lattice frequencies from the experiment

σν amounts to 18 % while

for polar molecules, it is below 4 %. Such a result, being still only a clue, is not surprising because the description of the interactions of non-polar molecules is of semi-empirical character in the optPBE-vdW functional. Much better accuracy of calculated frequencies for polar molecules reflects significant electrostatic interactions which can be described better within the DFT formalism. Since the low-frequency modes contribute most to the thermodynamic properties, lower accuracy of calculated solid phase heat capacities should be expected, especially for non-polar molecules on this level. 15 Environment ACS Paragon Plus


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Based on the density of phonon states (DPS) calculated at the optimized unit-cell volume, the crystal isochoric heat capacity CVcr was computed. Subsequently, the DPS was computed at four smaller unit-cell volumes. Example of the dependence of the DPS for hydrogen peroxide on the volume can be seen in Figure 4. The plot contains three DPS functions − at optimized unit-cell volumes multiplied by factors 1.00, 0.94, and 0.88, respectively. Evidently, the lattice modes (region below 500 cm−1) as well as the internal rotation modes (around 700 cm−1) increase their frequency as the crystal is compressed. It happens because the compression of the crystal decreases the interatomic distances which makes the mutual vibration movement of the molecules more energy-intensive, as reflected in the frequency increase. On the other hand, the intramolecular O−H bond stretching modes exhibit a significant frequency decrease during the crystal compression. This is a typical behavior for the hydrogen bonds forming species and happens due to the strengthening of the hydrogen bond invoked by a shortening of its length during the crystal compression. The covalent bond to the hydrogen atom donor is simultaneously weakened which is reflected in the frequency decrease of the stretching modes. The evidence that this phenomenon concerns only species capable of forming hydrogen bonds can be found in Figure 5 which represents a similar plot of volume-dependent DPS functions for crystalline butane. All peaks are shifted towards higher wavenumbers in this case, also those corresponding to C−H stretching modes.

4.3 Density of the crystalline phase The knowledge of the DPS at different volumes finally enables to fit the total Helmholtz energy of the crystal by the Murnaghan equation of state, for details see equations (1) and (2). Examples of the Acr(V) profiles for ammonia at several temperatures are given in Figure 6. Following the quasi-harmonic approximation, the Acr(V) curves reach their minima at higher molar volumes as the temperature increases. This is a significant success of this approach capturing the major part of the anharmonicity of the crystal, although the molar volumes for crystalline ammonia are overestimated by roughly 10 %, as can be seen in Figure 6. In the 16 Environment ACS Paragon Plus

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case of molecular crystals, the minima of the Acr(V) curves are rather broad and insignificant as the volume expansion of such crystals easily leads to destroying the crystal lattice and formation of unbound molecules. This may be an issue for practical calculations since it requires the computation of some additional points forming the Eel(V) curve at higher volume values to ensure that a minimum on the Acr(V) curve is found. In this work, it happened in the cases of ethane, and hydrazine for which the energies corresponding up to 1.33 V0 had to be included in the Eel(V) curve construction. The molar volumes can be recalculated to densities which are accessible experimentally. Figure 7 represents a plot of relative deviations σρ, defined according to equation (5). A reduced temperature Tr = T/Ttp, where Ttp is the lowest lying triple point temperature (corresponding either to the point of coexistence of two crystalline and a vapor phase, or coexistence of a crystalline, a liquid and a vapor phase), was chosen as the independent variable for this plot to get a generally well-arranged temperature scale, not affected by large differences in Ttp among the studied crystals. Particular experimental density data were culled from the references listed in Table 2, based on X-ray or neutron diffraction studies or eventually direct densimetric determinations. The Table 2 contains the comparison of ρcrcalc and ρcrexp for particular crystals in the form of absolute percentage deviations σ ρabs (see eqution (6)), as well as the estimated expanded uncertainties U c ( ρ crexp ) (coverage factor k = 2). The calculated densities are underestimated in the most of the cases (the only observed exceptions are argon at higher temperatures, and neon and nitrogen in the whole temperature range) and their mean percentage deviation amounts to −5 %. The polar molecules exhibit the same mean deviation of −7 % ± 2 % (coverage factor k = 1) while the non-polar species exhibit surprisingly smaller mean deviation of −3 % ± 9 %. As can be seen in Figure 7, ρcrcalc for nonpolar species such as neon and ethane exhibit the largest deviation from the experiment and cause a high scatter of σρ in this case, although they coincidentally compensate which leads to lower σρ for non-polar molecules than for polar molecules. The mean absolute percentage 17 Environment ACS Paragon Plus


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deviation σ ρabs amounts to 8 %, and to 7 % for non-polar, and polar compounds, respectively. For most crystals where temperature-dependent ρ crexp data were available, the σρ do not exhibit significant temperature dependence, as is indicated in Figure 7 by the dashed and dotted lines connecting points representing some examples of the same crystals. Therefore, it can be concluded that the temperature trends of ρcrcalc and the thermal expansion of the crystals are captured satisfactorily within the quasi-harmonic approximation. When the mean σ ρabs value of 7 % is compared to the mean absolute percentage deviations of unit-cell volumes σV discussed in section 4.1 (based on electonic energy minimization only, and amounting to 3 % on average), it can be concluded that a partial compensation of errors occurs, namely of the errors caused by i) not including the zero-point vibrational contributions to unit-cell parameters in section 4.1, and by ii) performing the unit-cell structure optimization at a given level of theory. Moreover, it holds for 18 out of 22 studied crystals that | σV | < σ ρabs , which means that a proper inclusion of the vibrational-based contributions does not increase the accuracy of ρcrcalc due to this compensation of errors.

4.4 Isobaric heat capacity of the crystalline phase Since the Gibbs energy equals the Helmholtz energy at the minima of the Acr(V) curve at zero pressure, it is a straightforward task to evaluate the Gibbs energy followed by the evaluation of the isobaric heat capacity C crp of the crystals as functions of temperature. The plots of calculated C pcr, calc for individual species can be found in the SI. For propane, butane, carbon dioxide, and hydrazine, a perfect agreement of C pcr, calc and C pcr, exp was recorded as the percentage deviation does not exceed 5 % at Tr = 0.5. On the other hand, the largest discrepancies between the theory and experiment were observed for neon and fluorine. In these cases, the percentage deviation of C pcr, calc from C pcr, exp exceeds 50 %. Figure 8 illustrates the global comparison of C pcr, calc to C pcr, exp , again in the form of relative deviations σc (see 18 Environment ACS Paragon Plus

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equation (5)) being a function of the reduced temperature Tr. Experimental heat capacity data were culled from the references listed in Table 3 which contains the comparison of C pcr, calc and

C pcr, exp for individual crystals in the form of absolute percentage deviations σ Cabs (see equation (6)) along with the estimated expanded uncertainties U c (C pcr, exp ) (coverage factor k = 2). At temperatures close to Tr = 0, the average value σ Cabs is relatively large, namely 18 % which is partly caused by the fact that C crp amounts to low values at low temperatures and the absolute difference C pcr, calc – C pcr, exp usually does not exceed a few units of J mol−1K−1. At temperatures close to Tr = 1, that means in the vicinity of the triple points, the mean of σc amounts to 15 %. Such results seem to be in contrast to the assumed worse performance of the quasi-harmonic approximation when approaching a temperature of a phase transition (the extent of anharmonicity increases). However, it is more likely caused by the high uncertainty of the calculated lattice vibrational frequencies whose error compensates with the missing explicit anharmonic terms at temperatures close to a phase transition. Generally, C pcr, calc are underestimated in 85 % of cases while for non-polar molecules, σc exhibits a large scatter and the accuracy and reliability of C pcr, calc is much higher for polar molecules. Although relative deviations of C pcr, calc from C pcr, exp are considerable, the uncertainty in C pcr, calc may play only a minor role when some integral thermodynamic properties, such as enthalpy or entropy differences, are evaluated. To illustrate this phenomenon, integral quantities ∆Habs defined as: Tmax

∆H abs =

∫

(C pcr,calc − C pcr,exp ) dT ,

(7)

Tmin

and ∆Sabs defined as:  C pcr,calc − C pcr,exp ∫  T Tmin 

Tmax

∆ S abs =

  dT , 


(8)


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where the integration runs over the interval of known C pcr, exp data, were evaluated for all studied crystals. The mean value of ∆Habs is 0.2 kJ mol−1 and ∆Sabs amounts 0.7 J K−1 mol−1 on average. This means that replacing the experimental heat capacity of the crystal with its calculated equivalent leads only to a minor uncertainty compared i. e. to the uncertainty arising from the cohesive energy calculations. To express the relative error of such integral quantities, ∆Habs and ∆Sabs were normalized using the experimental enthalpy and entropy change over given temperature intervals. In this case the mean relative errors in both enthalpy (σH) and entropy (σS) changes are 14 % (23 % for non-polar molecules and 7 % for polar molecules). See Table S3 in the SI for σH and σS evaluated for particular molecular crystals.

5. CONCLUSIONS This work thoroughly tested the calculations of the thermodynamic properties of molecular crystals from first principles. For a testing set of 22 molecular crystals, plane-wave based calculations of the lattice dynamics, using the optPBE-vdW functional and resulting into the density of phonon states within the harmonic approximation, were performed and their results thoroughly analyzed and compared to the experimental spectroscopic data. The mean absolute percentage deviation of intramolecular vibrational frequencies amounts to 2 % which can be regarded as a satisfying accuracy. However, the same deviation for the lattice vibrational modes amounts to 15 % which is the reason for higher uncertainty of the calculated thermodynamic properties. The phonon densities of states were calculated at 5 different unitcell volumes for each crystal and the thermodynamic properties of the crystalline phase were calculated within the quasi-harmonic approximation. The calculated densities of the crystalline phases are underestimated by 5 % on average when compared to the experimental data, whereas the thermal expansion of the molecular crystals is satisfactorily captured within the quasi-harmonic approximation. On average, the calculated isobaric heat capacities differ by less than 20 % from the experimental values for the whole molecular set (8 % for polar molecules) which leads to a mean relative integral error of 14 % in both enthalpy and entropy 20 Environment ACS Paragon Plus

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changes over the temperature interval of existence of a given phase. The overall accuracy is encouraging and suggests application in the field of predictions of the sublimation equilibrium. On the other hand, the C pcr, calc obtained in this work exhibits significant uncertainty so that other alternatives of the treatment of the dispersion interactions within DFT or fully ab initio should be searched and tested.

ASSOCIATED CONTENT

Supporting Information The supporting information is available free of charge on the ACS Publications website at DOI: 1. Experimental unit-cell parameters of studied molecular crystals. 2. Comparison of various methods for evaluation of the dispersion corrections. 3. Calculated densities of phonon states. 4. Comparison of calculated and experimental isobaric heat capacities. 5. Statistics of the linear fits of the vibrational Helmholtz energy.

AUTHOR INFORMATION

Corresponding Author * E-mail: [email protected]

ACKNOWLEDGMENT C. Č and M. F. acknowledge financial support from specific university research (MSMT no. 20/2015) and Czech Science Foundation (GACR no. 15-07912S). The access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum, provided under the program "Projects of Large Infrastructure for Research, Development, and Innovations" (LM2010005) and the CERIT-SC under the program Centre CERIT Scientific Cloud, part of the Operational Program Research and Development for Innovations, Reg. no. CZ.1.05/3.2.00/08.0144.is highly appreciated. 21 Environment ACS Paragon Plus


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REFERENCES (1) Blaha, P.; Schwarz, K.; Sorantin, P.; Trickey, S. B. Full-Potential, Linearized Augmented Plane-Wave Programs for Crystalline Systems. Comput. Phys. Commun. 1990, 59, 399-415. (2) Kresse, G.; Furthmuller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169-11186. (3) Deringer, V. L.; Lumeij, M.; Stoffel, R. P.; Dronskowski, R. Ab Initio Study of the High-Temperature Phase Transition in Crystalline GeO2. J. Comput.Chem. 2013, 34, 23202326. (4) Deringer, V. L.; Stoffel, R. P.; Dronskowski, R. Vibrational and Thermodynamic Properties of Gese in the Quasiharmonic Approximation. Phys. Rev. B 2014, 89, 094303. (5) Deringer, V. L.; Stoffel, R. P.; Dronskowski, R. Thermochemical Ranking and Dynamic Stability of TeO2 Polymorphs from Ab Initio Theory. Cryst. Growth Des. 2014, 14, 871-878. (6) Stoffel, R. P.; Dronskowski, R. First-Principles Investigations of the Structural, Vibrational and Thermochemical Properties of Barium Cerate - Another Test Case for Density-Functional Theory. Z. Anorg. Allg. Chem. 2013, 639, 1227-1231. (7) Pamuk, B.; Soler, J. M.; Ramirez, R.; Herrero, C. P.; Stephens, P. W.; Allen, P. B.; Fernandez-Serra, M. V. Anomalous Nuclear Quantum Effects in Ice. Phys. Rev. Lett. 2012, 108. (8) Ramirez, R.; Neuerburg, N.; Herrero, C. P. The Phase Diagram of Ice: A QuasiHarmonic Study Based on a Flexible Water Model. J. Chem. Phys. 2013, 139. (9) Otero-de-la-Roza, A.; Johnson, E. R. A Benchmark for Non-Covalent Interactions in Solids. J. Chem. Phys. 2012, 137, 054103. (10) Reilly, A. M.; Tkatchenko, A. Understanding the Role of Vibrations, Exact Exchange, and Many-Body Van Der Waals Interactions in the Cohesive Properties of Molecular Crystals. J. Chem. Phys. 2013, 139, 024705. (11) Li, J.; Sode, O.; Voth, G. A.; Hirata, S. A solid-solid phase transition in carbon dioxide at high pressures and intermediate temperatures. Nat. Commun. 2013, 4. (12) Reilly, A. M.; Tkatchenko, A. Role of Dispersion Interactions in the Polymorphism and Entropic Stabilization of the Aspirin Crystal. Phys. Rev. Lett. 2014, 113. (13) Rivera, S. A.; Allis, D. G.; Hudson, B. S. Importance of Vibrational Zero-Point Energy Contribution to the Relative Polymorph Energies of Hydrogen-Bonded Species. Cryst. Growth Des. 2008, 8, 3905-3907. (14) He, X.; Sode, O.; Xantheas, S. S.; Hirata, S. Second-Order Many-Body Perturbation Study of Ice Ih. J. Chem. Phys. 2012, 137, 204505. (15) Gilliard, K.; Sode, O.; Hirata, S. Second-Order Many-Body Perturbation and CoupledCluster Singles and Doubles Study of Ice VIII. J. Chem. Phys. 2014, 140, 174507. (16) Li, J.; Sode, O.; Hirata, S. Second-Order Many-Body Perturbation Study on Thermal Expansion of Solid Carbon Dioxide. J. Chem. Theory Comput. 2015, 11, 224-229. (17) Heit, Y. N.; Nanda, K. D.; Beran, G. J. O. Predicting Finite-Temperature Properties of Crystalline Carbon Dioxide from First Principles with Quantitative Accuracy. Chem. Sci. 2016, 7, 246-255. (18) Hirata, S.; Gilliard, K.; He, X.; Li, J.; Sode, O. Ab Initio Molecular Crystal Structures, Spectra, and Phase Diagrams. Acc. Chem. Res. 2014, 47, 2721-2730. (19) Carter, D. J.; Rohl, A. L. Benchmarking Calculated Lattice Parameters and Energies of Molecular Crystals Using van der Waals Density Functionals. J. Chem. Theory Comput. 2014, 10, 3423-3437. (20) Reilly, A. M.; Tkatchenko, A. Seamless and Accurate Modeling of Organic Molecular Materials. J. Phys. Chem. Lett. 2013, 4, 1028-1033. (21) Klimeš, J.; Bowler, D. R.; Michaelides, A. Chemical Accuracy for the Van Der Waals Density Functional. J. Phys. Condens. Mat. 2010, 22, 022201. 22 Environment ACS Paragon Plus

Page 22 of 45

Page 23 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(22) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 1795317979. (23) Hafner, J. K., G; Vogtenhuber, D.; Marsman, M. Vienna Ab-initio Simulation Package, 5.3.5; 2014. (24) Monkhorst, H. J.; Pack, J. D. Special Points for Brillouin-Zone Integrations. Phys. Rev. B 1976, 13, 5188-5192. (25) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector AugmentedWave Method. Phys. Rev. B 1999, 59, 1758-1775. (26) Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D. Iterative Minimization Techniques for Abinitio Total-Energy Calculations - Molecular-Dynamics and Conjugate Gradients. Rev. Modern Phys. 1992, 64, 1045-1097. (27) Stoffel, R. P.; Wessel, C.; Lumey, M.-W.; Dronskowski, R. Ab Initio Thermochemistry of Solid-State Materials. Angew. Chem. Int. Edit. 2010, 49, 5242-5266. (28) Parlinski, K.; Li, Z. Q.; Kawazoe, Y. First-Principles Determination of the Soft Mode in Cubic ZrO2. Phys. Rev. Lett. 1997, 78, 4063-4066. (29) Togo, A. Phonopy, 1.3; 2009. (30) Murnaghan, F. D. The Compressibility of Media under Extreme Pressures. Proc. Natl. Acad. Sci. U.S.A. 1944, 30, 244-247. (31) Frenkel, M.; Chirico, R. D.; Diky, V.; Kroenlein, K.; Muzny, C. D.; Kazakov, A. F.; Magee, J. W. A., I. M. Lemmon, E. W. ThermoData Engine (TDE), NIST Standard Reference Database 103b - Pure Compounds, Binary Mixtures, Ternary Mixtures and Chemical Reactions, 2.7; Thermodynamics Research Center: Boulder CO, 2013. (32) Frenkel, M.; Chirico, R. D.; Diky, V.; Yan, X.; Dong, Q.; Muzny, C. ThermoData Engine (TDE): Software Implementation of the Dynamic Data Evaluation Concept. J. Chem. Inf. Model. 2005, 45, 816-838. (33) Fulem, M.; Růžička, K.; Červinka, C.; Rocha, M. A. A.; Santos, L. M. N. B. F.; Berg, R. F. Reccomended Vapor Pressure and Thermophysical Data for Ferrocene. J. Chem. Thermodyn. 2013, 57, 530–540. (34) Růžička, K.; Fulem, M.; Červinka, C. Recommended Sublimation Pressure and Enthalpy of Benzene. J. Chem. Thermodyn. 2014, 68, 40-47. (35) Bucko, T.; Hafner, J.; Lebegue, S.; Angyan, J. G. Improved Description of the Structure of Molecular and Layered Crystals: Ab Initio DFT Calculations with van der Waals Corrections. J. Phys. Chem. A 2010, 114, 11814-11824. (36) Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a LongRange Dispersion Correction. J. Comput. Chem. 2006, 27, 1787-1799. (37) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements HPu. J. Chem. Phys. 2010, 132, 154104. (38) Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the Damping Function in Dispersion Corrected Density Functional Theory. J. Comput. Chem. 2011, 32, 1456-1465. (39) Tkatchenko, A.; Scheffler, M. Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data. Phys. Rev. Lett. 2009, 102, 073005. (40) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. (41) Dion, M.; Rydberg, H.; Schroder, E.; Langreth, D. C.; Lundqvist, B. I. Van Der Waals Density Functional for General Geometries. Phys. Rev. Lett. 2004, 92, 246401. (42) Roman-Perez, G.; Soler, J. M. Efficient Implementation of a van der Waals Density Functional: Application to Double-Wall Carbon Nanotubes. Phys. Rev. Lett. 2009, 103, 096102. (43) Klimes, J.; Bowler, D. R.; Michaelides, A. Van Der Waals Density Functionals Applied to Solids. Phys. Rev. B 2011, 83, 195131. 23 Environment ACS Paragon Plus


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(44) Lee, K.; Murray, E. D.; Kong, L.; Lundqvist, B. I.; Langreth, D. C. Higher Accuracy van der Waals Density Functional. Phys. Rev. B 2010, 82, 081101. (45) Herrero, C. P. Isotope Effects in Structural and Thermodynamic Properties of Solid Neon. Phys. Rev. B 2002, 65. (46) Endoh, Y.; Shirane, G.; Skalyo, J. Lattice-Dynamics of Solid Neon at 6.5 and 23.7 K. Phys. Rev. B 1975, 11, 1681-1688. (47) Randolph, P. D. Slow Neutron Scattering Studies of Liquid and Solid Argon. Idaho Operations Office, U.S. Atomic Energy Commission: 1965. (48) Nelligan, W. B.; Lepoire, D. J.; Brun, T. O.; Kleb, R. Inelastic Neutron-Scattering Study of the Torsional and CCC Bend Frequencies in the Solid Normal-Alkanes, Ethane Hexane. J. Chem. Phys. 1987, 87, 2447-2456. (49) Wisnosky, M. G.; Eggers, D. F.; Fredrickson, L. R.; Decius, J. C. The VibrationalSpectra of Solid-II Ethane and Ethane-d6. J. Chem. Phys. 1983, 79, 3505-3512. (50) Brith, M.; Ron, A. Far-Infrared Spectra of Crystalline Ethylene C2H4 and C2D4. J. Chem. Phys. 1969, 50, 3053-3056. (51) Elliott, G. R.; Leroi, G. E. Raman Study of Crystalline Ethylenes and Structure Determination through Model Calculations of Lattice Spectra. J. Chem. Phys. 1973, 59, 12171227. (52) Zhao, G.; Ospina, M. J.; Khanna, R. K. Infrared Intensities and Optical-Constants of Crystalline C2H4 and C2D4. Spectrochim. Acta A 1988, 44, 27-31. (53) Binbrek, O. S.; Anderson, A. Lattice-Dynamics of Acetylene. Phys. Status Solidi B 1992, 173, 561-568. (54) Schachtschneider, J. H.; Snyder, R. G. Vibrational Analysis of the n-Paraffins. 2. Normal Coordinate Calculations. Spectrochim. Acta 1963, 19, 117-168. (55) Cangeloni, M. L.; Schettino, V. Infrared and Raman-Spectra and Polymorphism in Crystal Normal-Butane. Mol. Cryst. Liq. Cryst. 1975, 31, 219-231. (56) Snyder, R. G.; Schachtschneider, J. H. Vibrational Analysis of the n-Paraffins. 1. Assignments of Infrared Bands in the Spectra of C3H8 through n-C19H40. Spectrochim. Acta 1963, 19, 85-116. (57) Binbrek, O. S.; Higgs, J. F.; Anderson, A. Lattice-Dynamics of Nitrogen and CarbonDioxide. Phys. Status Solidi B 1989, 155, 427-436. (58) Kobashi, K.; Klein, M. L. Lattice-Vibrations of Solid Alpha-F2. Mol. Phys. 1980, 41, 679-688. (59) Niemczyk, T. M.; Getty, R. R.; Leroi, G. E. Vibrational-Spectra of Solid Fluorine. J. Chem. Phys. 1973, 59, 5600-5604. (60) Zeng, W. Y.; Anderson, A. Lattice-Dynamics of Ammonia. Phys. Status Solidi B 1990, 162, 111-117. (61) Arnau, J. L.; Giguere, P. A.; Abe, M.; Taylor, R. C. Vibrational-Spectra and Normal Coordinate Analysis of Crystalline H2O2, D2O2 and HDO2. Spectrochim. Acta A 1974, A 30, 777-796. (62) Higgs, J. F.; Zeng, W. Y.; Anderson, A. Dynamic-Model for the Vibrations of the Hydrogen Halide Crystals. Phys. Status Solidi B 1986, 133, 475-482. (63) Weng, S. X.; Anderson, A. Lattice-Dynamics of Methanol. Phys. Status Solidi B 1992, 172, 545-555. (64) Weng, S. X.; Anderson, A. Lattice-Dynamics of Formaldehyde. Phys. Status Solidi B 1991, 166, 359-368. (65) Weng, S. X.; Anderson, A.; Torrie, B. H. Raman and Far-Infrared Spectra of Crystalline Formaldehyde, H2CO and D2CO. J. Raman Spectrosc. 1989, 20, 789-794. (66) Durig, J. R.; Bush, S. F.; Baglin, F. G. Infrared and Raman Investigation of Condensed Phases of Methylamine and Its Deuterium Derivatives. J. Chem. Phys. 1968, 49, 2106-2117. (67) Allan, A.; McKean, D. C.; Perchard, J. P.; Josien, M. L. Vibrational Spectra of Crystalline Dimethyl Ethers. Spectrochim. Acta A 1971, A 27, 1409-1437. 24 Environment ACS Paragon Plus

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(68) Levin, I. W.; Pearce, R. A. R.; Spiker, R. C. Vibrational Raman-Spectra and IntraMolecular Potential Function of Solid-Solutions of Dimethyl Ether-d0 and Dimethyl Etherd6. J. Chem. Phys. 1978, 68, 3471-3480. (69) Durig, J. R.; Griffin, M. G.; Macnamee, R. W. Raman-Spectra of Gases. 15. Hydrazine and Hydrazine-d4. J. Raman Spectrosc. 1975, 3, 133-141. (70) Blumenfeld, S.; Fast, H. Low Frequency Raman Spectra of Solid and Liquid Formic Acid. Spectrochim. Acta A 1968, 24, 1449-1459. (71) Mikawa, Y.; Brasch, J. W.; Jakobsen, R. J. Infrared Spectra and Normal Coordinate Calculation of Crystalline Formic acid. J. Mol. Spectrosc. 1967, 24, 314-329. (72) Zelsmann, H. R.; Marechal, Y.; Chosson, A.; Faure, P. Raman and IR-Spectra of HCOOH and DCCOD Crystals at Low-Temperatures. J. Mol. Struct. 1975, 29, 357-368. (73) Torrie, B. H.; Brown, B. A. Raman and Far-Infrared Spectra of Formamide at Temperatures down to 20 K. J. Raman Spectrosc. 1994, 25, 183-187. (74) Haurie, M.; Novak, A. Spectres de Vibration des Molecules CH3COOH CH3COOD CD3COOH et CD3COOD. 3. Spectres Infrarouges des Cristaux. Spectrochim. Acta 1965, 21, 1217-1228. (75) Merrick, J. P.; Moran, D.; Radom, L. An Evaluation Of Harmonic Vibrational Frequency Scale Factors. J. Phys. Chem. A 2007, 111, 11683-11700. (76) Červinka, C.; Fulem, M.; Růžička, K. Evaluation of Accuracy of Ideal-Gas Heat Capacity and Entropy Calculations by Density Functional Theory (DFT) for Rigid Molecules. Journal of Chemical and Engineering Data 2012, 57, 227-232. (77) Červinka, C.; Fulem, M.; Růžička, K. Evaluation of Uncertainty of Ideal-Gas Entropy and Heat Capacity Calculations by Density Functional Theory (DFT) for Molecules Containing Symmetrical Internal Rotors. J. Chem. Eng. Data 2013, 58, 1382-1390. (78) Batchelder, D. N.; Losee, D. L.; Simmons, R. O. Measurements of Lattice Constant Thermal Expansion and Isothermal Compressibility of Neon Single Crystals. Phys. Rev. 1967, 162, 767-775. (79) Holste, J. C.; Swenson, C. A. Experimental Thermal Expansions for Solid Neon, 2-14 K. J. Low Temp. Phys. 1975, 18, 477-485. (80) Abrahams, S. C.; Collin, R. L.; Lipscomb, W. N. The Crystal Structure of Hydrogen Peroxide. Acta Crystallogr. 1951, 4, 15-20. (81) Prince, E.; Trevino, S. F.; Choi, C. S.; Farr, M. K. Refinement of Structure of Deuterium Peroxide. J. Chem. Phys. 1975, 63, 2620-2624. (82) Savariault, J. M.; Lehmann, M. S. Experimental-Determination of the Deformation Electron-Density in Hydrogen-Peroxide by Combination of X-Ray and Neutron-Diffraction Measurements. J. Am. Chem. Soc. 1980, 102, 1298-1303. (83) Barrett, C. S.; Meyer, L. X-Ray Diffraction Study of Solid Argon. J.Chem. Phys. 1964, 41, 1078-1081. (84) Dobbs, E. R.; Figgins, B. F.; Jones, G. O.; Piercey, D. C.; Riley, D. P. Density and Expansivity of Solid Argon. Nature 1956, 178, 483-483. (85) Atoji, M.; Lipscomb, W. N. The Crystal Structure of Hydrogen Fluoride. Acta Crystallogr. 1954, 7, 173-175. (86) Johnson, M. W.; Sandor, E.; Arzi, E. Crystal-Structure of Deuterium Fluoride. Acta Crystallogr. B 1975, 31, 1998-2003. (87) Le Boucher, L.; Fischer, W.; Biltz, W. Über Molekular- und Atomvolumina 41. Tieftemperaturdichten Kristallisierten Fluorwasserstoffs und einiger Kristallisierter Fluoride. Z. anorg. Chem. 1932, 207, 61-72. (88) Heuse, W. The Molar Volume of Hydrocarbons and Several Other Compounds at Low Temperature. Z. Phys. Chem., Abt. A 1930, 147, 266-274. (89) Klimenko, N. A.; Gal'tsov, N. N.; Prokhvatilov, A. I. Structure, Phase Transitions, and Thermal Expansion of Ethane C2H6. Low Temp. Phys. 2008, 34, 1038-1043.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(90) Mark, H.; Pohland, E. Über die Gitterstruktur des Äthans und des Diborans. Z. Kristallogr. 1925, 62, 103-112. (91) Stewart, J. W.; Larock, R. I. Compression and Densities of Four Solidified Hydrocarbons and Carbon Tetrafluoride at 77-Degrees-K. J. Chem. Phys. 1958, 28, 425-427. (92) van Nes, G. J. H.; Vos, A. Single-Crystal Structures and Electron-Density Distributions of Ethane, Ethylene and Acetylene. 1. Single-Crystal X-Ray Structure Determinations of 2 Modifications of Ethane. Acta Crystallogr. B 1978, 34, 1947-1956. (93) Biltz, W.; Fischer, W.; Wunnenberg, E. Molecular and Atomic Volumes. The Volume Requirements of Crystalline Organic Compounds and Low Temperatures. Z. Phys. Chem., Abt. A 1930, 151, 13-55. (94) Riembauer, M.; Schulte, L.; Wurflinger, A. PVT Data of Liquid and Solid-Phases of Methanol, Cyclohexanol, and 2,3-Dimethylbutane up to 300 MPa. Z. Phys. Chem. Neue Fol. 1990, 166, 53-61. (95) Tauer, K. J.; Lipscomb, W. N. On the Crystal Structures, Residual Entropy and Dielectric Anomaly of Methanol. Acta Crystallogr. 1952, 5, 606-&. (96) Torrie, B. H.; Binbrek, O. S.; Strauss, M.; Swainson, I. P. Phase Transitions in Solid Methanol. J. Solid State Chem. 2002, 166, 415-420. (97) Torrie, B. H.; Weng, S. X.; Powell, B. M. Structure of the Alpha-Phase of Solid Methanol. Mol. Phys. 1989, 67, 575-581. (98) van Nes, G. J. H.; Vos, A. Single-Crystal Structures and Electron-Density Distributions of Ethane, Ethylene and Acetylene. 3. Single-Crystal X-Ray Structure Determination of Ethylene at 85-K. Acta Crystallogr. B 1979, 35, 2593-2601. (99) Thakur, T. S.; Kirchner, M. T.; Blaeser, D.; Boese, R.; Desiraju, G. R. Nature and Strength of C-H Center Dot Center Dot Center Dot O Interactions Involving Formyl Hydrogen Atoms: Computational and Experimental Studies of Small Aldehydes. Phys. Chem. Chem. Phys. 2011, 13, 14076-14091. (100) Weng, S. X.; Torrie, B. H.; Powell, B. M. The Crystal-Structure of Formaldehyde. Mol. Phys. 1989, 68, 25-31. (101) Koski, H. K.; Sandor, E. Neutron Powder Diffraction Study of Low-Temperature Phase of Solid Acetylene-d2. Acta Crystallogr. B 1975, 31, 350-353. (102) McMullan, R. K.; Kvick, A.; Popelier, P. Structures of Cubic and Orthorhombic Phases of Acetylene by Single-Crystal Neutron-Diffraction. Acta Crystallogr. B 1992, 48, 726-731. (103) van Nes, G. J. H.; Vanbolhuis, F. Single-Crystal Structures and Electron-Density Distributions of Ethane, Ethylene and Acetylene. 2. Single-Crystal X-Ray Structure Determination of Acetylene at 141-K. Acta Crystallogr. B 1979, 35, 2580-2593. (104) Atoji, M.; Lipscomb, W. N. On the Crystal Structures of Methylamine. Acta Crystallogr. 1953, 6, 770-774. (105) Boese, R.; Weiss, H. C.; Blaser, D. The Melting Point Alternation in the Short-Chain n-Alkanes: Single-Crystal X-Ray Analyses of Propane at 30 K and of n-Butane to n-Nonane at 90 K. Angew. Chem. Int. Edit. 1999, 38, 988-992. (106) Vojinovic, K.; Losehand, U.; Mitzel, N. W. Dichlorosilane-Dimethyl Ether Aggregation: A New Motif in Halosilane Adduct Formation. Dalton Trans. 2004, 2578-2581. (107) Refson, K.; Pawley, G. The Structure and Orientational Disorder in Solid NormalButane by Neutron Powder Diffraction. Acta Crystallogr. B 1986, 42, 402-410. (108) Collin, R. L.; Lipscomb, W. N. The Crystal Structure of Hydrazine. Acta Crystallogr. 1951, 4, 10-14. (109) Bolz, L. H.; Boyd, M. E.; Mauer, F. A.; Peiser, H. S. A Re-Examination of the Crystal Structures of Alpha-Nitrogen and Beta-Nitrogen. Acta Crystallogr. 1959, 12, 247-248. (110) Hoerl, E. M.; Marton, L. Electron Diffraction Studies on Solid Alpha Nitrogen. Acta Crystallogr. 1961, 14, 11-19.


Page 26 of 45

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(111) Tolkachev, A. M.; Manzheliev, V. G. Density of Solidified Gases. Sov. Phys. Solid State 1966, 7, 1711. (112) Foulon, M.; Lebrun, N.; Muller, M.; Amazzal, A.; Cohenadad, M. T. Structure of the Stable Phase of Methylhydrazine - 1st Observations of Phase-Transitions. Acta Crystallogr. B 1994, 50, 472-479. (113) Meyer, L.; Barrett, C. S.; Greer, S. C. Crystal Structure of Alpha-Fluorine. J. Chem. Phys. 1968, 49, 1902-1907. (114) Holtzberg, F.; Post, B.; Fankuchen, I. The Crystal Structure of Formic Acid. Acta Crystallogr. 1953, 6, 127-130. (115) Nahringbauer, I. Hydrogen-Bond Studies. 127. Reinvestigation of Structure of Formic-Acid (At 98K). Acta Crystallogr. B 1978, 34, 315-318. (116) Curzon, A. E. A Comment on the Lattice Parameter of Solid Carbon Dioxide at 190°C. Physica 1972, 59, 733. (117) Keesom, W. H.; Köhler, J. W. L. The Lattice Constant And Expansion Coefficient Of Solid Carbon Dioxide. Physica 1934, 1, 655-658. (118) Kitaura, R.; Matsuda, R.; Kubota, Y.; Kitagawa, S.; Takata, M.; Kobayashi, T. C.; Suzuki, M. Formation and Characterization of Crystalline Molecular Arrays of Gas Molecules in a 1-Dimensional Ultramicropore of a Porous Copper Coordination Polymer. J. Phys. Chem. B 2005, 109, 23378-23385. (119) Simon, A.; Peters, K. Single-Crystal Refinement of the Structure of Carbon-Dioxide. Acta Crystallogr. B 1980, 36, 2750-2751. (120) Ladell, J.; Post, B. The Crystal Structure of Formamide. Acta Crystallogr. 1954, 7, 559-564. (121) Ottersen, T. Structure of Peptide Linkage - Structures of Formamide and Acetamide at -165 Degreesc and An Ab Initio Study of Formamide, Acetamide, and N-Methylformamide. Acta Chemica Scand. A 1975, 29, 939-944. (122) Stevens, E. D. Low-Temperature Experimental Electron-Density Distribution of Formamide. Acta Crystallogr. B 1978, 34, 544-551. (123) Torrie, B. H.; Odonovan, C.; Powell, B. M. Structure of Solid Formamide at 7-K. Mol. Phys. 1994, 82, 643-649. (124) Boese, R.; Niederprum, N.; Blaser, D.; Maulitz, A.; Antipin, M. Y.; Mallinson, P. R. Single-Crystal Structure and Electron Density Distribution of Ammonia at 160 K on the Basis of X-Ray Diffraction Data. J. Phys. Chem. B 1997, 101, 5794-5799. (125) Olovsson, I.; Templeton, D. H. X-Ray Study of Solid Ammonia. Acta Crystallogr. 1959, 12, 832-836. (126) Albinati, A.; Rouse, K. D.; Thomas, M. W. Neutron Powder Diffraction Analysis of Hydrogen-Bonded Solids .1. Refinement of Structure of Deuterated Acetic-Acid at 4.2 and 12.5 K. Acta Crystallogr. B 1978, 34, 2184-2187. (127) Boese, R.; Blaser, D.; Latz, R.; Baumen, A. Acetic Acid at 40K. Acta Crystallogr. C 1999, 55, IUC9900001. (128) Jones, R. E.; Templeton, D. H. The Crystal Structure of Acetic Acid. Acta Crystallogr. 1958, 11, 484-487. (129) Jonsson, P. G. Hydrogen Bond Studies. 44. Neutron Diffraction Study of Acetic Acid. Acta Crystallogr. B 1971, B 27, 893-898. (130) Nahringbauer, I. Hydrogen Bond Studies. 39. Reinvestigation of Crystal Structure of Acetic Acid (At +5 Degrees C and -190 Degrees C). Acta Chem. Scand. 1970, 24, 453-462. (131) Martin, J. F.; Andon, R. J. L. Thermodynamic Properties of Organic Oxygen Compounds. 52. Molar Heat-Capacity of Ethanoic, Propanoic, and Butanoic Acids. J. Chem. Thermodyn. 1982, 14, 679-688. (132) Fenichel, H.; Serin, B. Low-Temperature Specific Heats of Solid Neon and Solid Xenon. Phys. Rev. 1966, 142, 490-495.



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(133) Finegold, L.; Phillips, N. E. Low-Temperature Heat Capacities of Solid Argon and Krypton. Phys. Rev. 1969, 177, 1383-1391. (134) Flubacher, P.; Morrison, J. A.; Leadbetter, A. J. A Low Temperature Adiabatic Calorimeter for Condensed Substances - Thermodynamic Properties of Argon. Proceedings of the Physical Society of London 1961, 78, 1449-&. (135) Atake, T.; Chihara, H. Calorimetric Study of Phase-Changes in Solid Ethane. Chem. Lett. 1976, 7, 683-688. (136) Egan, C. J.; Kemp, J. D. Ethylene. The Heat Capacity from 15°K to the Boiling Point. The Heats of Fusion and Vaporization. The Vapor Pressure of the Liquid. The Entropy from Thermal Measurements Compared with the Entropy from Spectroscopic Data. J. Am. Chem. Soc. 1937, 59, 1264-1268. (137) Kemp, J. D.; Egan, C. J. Hindered Rotation of the Methyl Groups in Propane. The Heat Capacity, Vapor Pressure, Heats of Fusion and Vaporization of Propane. Entropy and Density of the Gas. J. Am. Chem. Soc. 1938, 60, 1521-1525. (138) Aston, J. G.; Messerly, G. H. The Heat Capacity and Entropy, Heats of Fusion and Vaporization and the Vapor Pressure of n-Butane. J. Am. Chem. Soc. 1940, 62, 1917-1923. (139) Giauque, W. F.; Clayton, J. O. The Heat Capacity and Entropy of Nitrogen. Heat of Vaporization. Vapor Pressures of Solid and Liquid. The Reaction 1/2 N2 + 1/2 O2 = NO from Spectroscopic Data. J. Am. Chem. Soc. 1933, 55, 4875-4889. (140) Hu, J. H.; White, D.; Johnston, H. L. Condensed Gas Calorimetry. 5. Heat Capacities, Latent Heats and Entropies of Fluorine from 13-Degrees-K to 85-Degrees-K - Heats of Transition, Fusion, Vaporization and Vapor Pressures of the Liquid. J. Am. Chem. Soc. 1953, 75, 5642-5645. (141) Giauque, W. F.; Egan, C. J. Carbon Dioxide. The Heat Capacity and Vapor Pressure of the Solid. The Heat of Sublimation. Thermodynamic and Spectroscopic Values of the Entropy. J. Chem. Phys. 1937, 5, 45-54. (142) Overstreet, R.; Giauque, W. F. Ammonia. The Heat Capacity and Vapor Pressure of Solid and Liquid. Heat of Vaporization. The Entropy Values from Thermal and Spectroscopic Data. J. Am. Chem. Soc. 1937, 59, 254-259. (143) Giguere, P. A.; Liu, I. D.; Dugdale, J. S.; Morrison, J. A. Hydrogen Peroxide - the Low Temperature Heat Capacity of the Solid and the 3rd Law Entropy. Can. J. Chem.-Rev. Can. Chim. 1954, 32, 117-128. (144) Hu, J. H.; White, D.; Johnston, H. L. The Heat Capacity, Heat of Fusion and Heat of Vaporization of Hydrogen Fluoride. J. Am. Chem. Soc. 1953, 75, 1232-1236. (145) Carlson, H. G.; Westrum, E. F. Methanol: Heat Capacity, Enthalpies of Transition and Melting, and Thermodynamic Properties from 5–300°K. J. Chem. Phys. 1971, 54, 1464-1471. (146) Aston, J. G.; Siller, C. W.; Messerly, G. H. Heat Capacities and Entropies of Organic Compounds. III. Methylamine from 11.5°K. to the Boiling Point. Heat of Vaporization and Vapor Pressure. The Entropy from Molecular Data. J. Am. Chem. Soc. 1937, 59, 1743-1751. (147) Kennedy, R. M.; Sagenkahn, M.; Aston, J. G. The Heat Capacity and Entropy, Heats of Fusion and Vaporization, and the Vapor Pressure of Dimethyl Ether. The Density of Gaseous Dimethyl Ether. J. Am. Chem. Soc. 1941, 63, 2267-2272. (148) Scott, D. W.; Oliver, G. D.; Gross, M. E.; Hubbard, W. N.; Huffman, H. M. Hydrazine: Heat Capacity, Heats of Fusion and Vaporization, Vapor Pressure, Entropy and Thermodynamic Functions. J. Am. Chem. Soc. 1949, 71, 2293-2297. (149) Aston, J. G.; Fink, H. L.; Janz, G. J.; Russell, K. E. The Heat Capacity, Heats of Fusion and Vaporization, Vapor Pressures, Entropy and Thermodynamic Functions of Methylhydrazine. J. Am. Chem. Soc. 1951, 73, 1939-1943. (150) Stout, J. W.; Fisher, L. H. The Entropy of Formic Acid. The Heat Capacity from 15 to 300°K. Heats of Fusion and Vaporization. J. Chem. Phys. 1941, 9, 163-168. (151) De Wit, H. G. M.; De Kruif, C. G.; Van Miltenburg, J. C. Thermodynamic Properties of Molecular Organic Crystals Containing Nitrogen, Oxygen, and Sulfur Ii. Molar Heat 28 Environment ACS Paragon Plus

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Capacities of Eight Compounds by Adiabatic Calorimetry. J. Chem. Thermodyn. 1983, 15, 891-902.

TABLE 1 Optimized unit-cell parameters a, b, c (Å) and the angle β (°), unit-cell volumes V0 (Å3) and its deviations σV (%) from the experimental values. All data were calculated using the optPBE-vdW functional. Molecule

Space group

a

b

c

βa

V0

σV b

Neon

Fm 3m

4.198

4.198

4.198

90.0

73.96

-14.35

Argon

Fm 3m

5.308

5.308

5.308

90.0

149.53

0.01

Ethane

P21/n

4.004

6.975

5.182

70.5

136.37

-2.70

Ethene

P21/n

4.488

6.628

4.026

94.7

119.35

-3.89

Ethyne

Acam

6.220

6.034

5.529

90.0

207.51

0.72

Propane

P21/n

4.059

11.65

7.906

95.9

372.09

1.97

Butane

P21/c

5.633

5.991

8.079

122.9

229.00

2.85

Nitrogen

P213

5.544

5.544

5.544

90.0

170.35

-6.15

Fluorine

C2/m

5.425

3.133

7.994

116.9

121.14

-5.69

Carbon dioxide

Pa3

5.644

5.644

5.644

90.0

179.75

1.05

Ammonia

P213

5.076

5.076

5.076

90.0

130.76

1.65

Hydrogen peroxide

P41212

4.075

4.075

7.813

90.0

129.71

-0.04

Hydrogen fluoride

Cmc21

3.272

4.223

5.414

90.0

74.80

1.62

Methanol

P212121

4.952

4.584

9.004

90.0

204.36

1.91

Formaldehyde

P421c

8.157

8.157

4.617

90.0

307.15

-4.27

Aminomethane

Pcab

5.725

6.043 13.607

90.0

470.74

-0.31

Dimethylether

P42/n

11.24

11.24

4.973

90.0

628.24

0.09

Hydrazine

P21/m

3.403

5.715

4.856

112.7

87.15

-0.82

Methylhydrazine

P21/c

10.08

3.911

7.661

106.9

289.09

0.14



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Page 30 of 45

Formic acid

Pna21

10.304 3.566

5.434

90.0

199.79

2.78

Formamide

P21/n

3.590

9.067

8.536

125.5

226.10

0.32

Acetic acid

Pna21

13.293 3.918

5.782

90.0

301.18

4.15

a

The remaining unit-cell angles α and γ are equal to 90 °.

b

Deviation defined as σ V = 100 (V calc − V exp ) / V exp .

TABLE 2 Comparison of calculated densities ρ crcalc with experimental values ρ crexp at finite temperatures and normal pressure (p = 101.325 kPa). Molecule

σ ρabs /

U c ( ρ crexp ) /

σ ρabs /

U c ( ρ crexp )

%a

%b

%a

/% b

Neon

11.2

2

78-79

Hydrogen peroxide

5.6

3

80-82

Argon

1.8

0.3

83-84

Hydrogen fluoride

6.0

2

85-87

Ethane

11.1

3

88-92

Methanol

8.7

1

93-97

Ethene

8.4

3

88, 91, 98

Formaldehyde

6.4

0.1d

99-100

Ethyne

3.1

1

101-103

Aminomethane

6.0

0.6d

104

Propane

11.3

2

88, 91, 105

Dimethylether

10.5

0.1d

106

Butane

10.6

7

105, 107

Hydrazine

12.4

1d

108

Nitrogen

1.2

1

109-111

Methylhydrazine

7.0

0.5d

112

Fluorine

0.4

1d

113

Formic acid

6.3

2

93, 114-115

8.7

2

116-119

Formamide

7.9

2

120-123

7.8

1

111, 124-125

Acetic acid

6.6

2

93, 126-131

Ref. c

Molecule

Ref. c

Carbon dioxide Ammonia

ρ crexp

a

Values averaged over available experimental data on

b

Estimated percentage relative combined expanded uncertainty (0.95 level of confidence) of

c

Literature source of

d

Estimated only from one experimental determination

ρ crexp


ρ crexp

Page 31 of 45

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TABLE 3 Comparison of calculated isobaric heat capacities C pcr, calc with experimental values C pcr, exp .a

Molecule

Tmin/K Tmax/K

Ref. b

U c (C pcr, exp )

σ Cabs

/%c

/

∆Habs/ kJ mol−1 e

|σH| f

−0.1

62

∆Sabs/ J K−1 mol−1 e −8.4

|σS| f

Neon

1

24.69

132

2

%d 64

Argon

2

83.81

133-134

0.2-2

32

−0.3

21

−8.8

24

Ethane

50

89.81

135

0.2

12

−0.3

14

−3.7

14

Ethene

16

103.97

136

0.5

18

−0.5

16

−8.4

17

Propane

15

85.53

137

0.5

8

−0.1

2

−0.3

1

Butane

12

107.55

138

0.5

6

0.0

0

1.3

2

Nitrogen

15

35.61

139

0.5

35

−0.2

35

−7.5

34

Fluorine

14

45.55

140

0.5

50

−0.3

44

−11.4

46

Carbon dioxide

15

216.59

141

0.5

3

0.1

1

0.8

1

Ammonia

15

195.48

142

0.5

12

−0.7

14

−5.5

13

Hydrogen peroxide

13

272.74

143

0.5

4

−0.3

4

−1.9

3

Hydrogen fluoride

15

189.79

144

0.5

16

−0.2

5

−3.1

9

Methanol

5

157.34

145

0.2

18

−0.8

16

−10.0

16

Aminomethane

13

101.52

146

0.5

10

−0.2

10

−3.4

10

Dimethylether

13

131.66

147

0.5

8

−0.4

8

−5.1

8

Hydrazine

12

274.69

148

0.3

4

0.2

2

−0.3

1

Methylhydrazine

15

220.79

149

0.2-2.5

4

−0.4

4

−2.9

4

Formic acid

15

281.40

150

0.5

6

−0.6

5

−3.8

6

Formamide

85

238.25

151

0.3

8

−0.6

8

−3.8

8

Acetic acid

10

289.69

131

0.2

7

0.6

4

0.1

0

a

cr, exp

Ethyne and formaldehyde are not listed in this table since no experimental isobaric heat capacity data C p

were found in the literature. b

cr, exp

Literature source of C p

.


64


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c

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cr, exp

Estimated percentage relative combined expanded uncertainty (0.95 level of confidence) of C p

. The

uncertainty is typically higher than that reported in the table at temperatures close to 0 K (typically 2-3 %) or phase transitions (typically 0.5 – 1 %). d

Relative absolute percentage deviations σ C

abs

cr, exp

values C p

cr, calc

of calculated isobaric heat capacities C p

defined by equation (6). The σ C

abs

cr, exp

are averaged over all selected C p

from experimental

from the temperature

interval from Tmin to Tmax. e

Absolute ∆Habs and ∆Sabs deviations over the temperature range from Tmin to Tmax defined by equations (7‒8)

f

Absolute percentage deviations σH and σS of the calculated enthalpy and entropy changes from the experimental

counterparts over the temperature range from Tmin to Tmax.

FIGURE 1. The molecules of 22 molecular crystals studied in this work. Atoms in grey: carbon, white: hydrogen, red: oxygen, dark blue: nitrogen, and light blue: fluorine.

FIGURE 2. The relative percentage deviations of calculated intramolecular vibrational frequencies (using the optPBE-vdw functional, evaluated at the Γ-point) from corresponding experimental data. Data points represent the fundamental vibration modes for the set of 22 molecular crystals.


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FIGURE 3. The relative percentage deviations of calculated lattice vibrational frequencies (using the optPBE-vdw functional, evaluated at the Γ -point) from corresponding experimental data. Data points represent the fundamental vibration modes for the set of 22 molecular crystals.

FIGURE 4. The density of phonon states DPS of hydrogen peroxide calculated at three different unit-cell volumes.

FIGURE 5. The density of phonon states DPS of butane calculated at three different unit cellvolumes. 33 Environment ACS Paragon Plus


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FIGURE 6. The Helmholtz energy Acr of the NH3 crystal as a function of volume at several temperatures. The calculated Vcrcalc and experimental Vcrexp molar volumes at pressure p = 101.325 kPa are included for comparison.

FIGURE 7. The relative percentage deviations of the calculated densities ρcrcalc from the experimental data ρcrexp at pressure p = 101.325 kPa as a function of reduced temperature Tr = T/Ttp for the testing set of 22 molecular crystals. Blue dashed line connects points representing neon, blue dotted line argon, blue dash-dotted line ethane, red dashed line carbon dioxide, and red dotted line acetic acid.


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FIGURE 8. The relative percentage deviations of the calculated isobaric heat capacities

Cpcr, calc from the experimental data at pressure p = 101.325 kPa as a function of reduced temperature Tr = T/Ttp for the testing set of 22 molecular crystals.



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TABLE OF CONTENTS IMAGE


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FIGURE 1. The molecules of 22 molecular crystals studied in this work. Atoms in grey: carbon, white: hydrogen, red: oxygen, dark blue: nitrogen, and light blue: fluorine. 43x22mm (600 x 600 DPI)

ACS Paragon Plus Environment


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FIGURE 2. The relative percentage deviations of calculated intramolecular vibrational frequencies (using the optPBE-vdw functional, evaluated at the Γ-point) from corresponding experimental data. Data points represent the fundamental vibration modes for the set of 22 molecular crystals. 57x40mm (300 x 300 DPI)


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FIGURE 3. The relative percentage deviations of calculated lattice vibrational frequencies (using the optPBEvdw functional, evaluated at the Γ -point) from corresponding experimental data. Data points represent the fundamental vibration modes for the set of 22 molecular crystals. 57x40mm (300 x 300 DPI)



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FIGURE 4. The density of phonon states DPS of hydrogen peroxide calculated at three different unit-cell volumes. 58x19mm (300 x 300 DPI)


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FIGURE 5. The density of phonon states DPS of butane calculated at three different unit cell-volumes. 58x19mm (300 x 300 DPI)



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FIGURE 6. The Helmholtz energy Acr of the NH3 crystal as a function of volume at several temperatures. The calculated Vcrcalc and experimental Vcrexp molar volumes at pressure p = 101.325 kPa are included for comparison. 57x40mm (300 x 300 DPI)


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FIGURE 7. The relative percentage deviations of the calculated densities ρcrcalc from the experimental data ρcrexp at pressure p = 101.325 kPa as a function of reduced temperature Tr = T/Ttp for the testing set of 22 molecular crystals. Blue dashed line connects points representing neon, blue dotted line argon, blue dashdotted line ethane, red dashed line carbon dioxide, and red dotted line acetic acid. 57x40mm (300 x 300 DPI)



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FIGURE 8. The relative percentage deviations of the calculated isobaric heat capacities Cpcr, calc from the experimental data as a function of reduced temperature Tr = T/Ttp for the testing set of 22 molecular crystals. 57x40mm (300 x 300 DPI)


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TABLE OF CONTENTS IMAGE 47x26mm (300 x 300 DPI)


Thermodynamic Properties of Molecular Crystals Calculated within the

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