Calculating the Cohesive Energy of Rare Gas Solids - American

Article pubs.acs.org/jchemeduc

From Dimer to Crystal: Calculating the Cohesive Energy of Rare Gas Solids Arthur M. Halpern* Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809, United States S Supporting Information *

ABSTRACT: An upper-level undergraduate project is described in which students perform high-level ab initio computational scans of the potential energy curves for Ne2 and Ar2 and obtain the respective Lennard-Jones (LJ) potential parameters σ and ε for the dimers. Using this information, along with the summation of pairwise interactions in the face-centered cubic structures of the crystalline solids, the students determine the cohesive energies of solid Ne and Ar. By applying the Debye theory of solids, they then adjust these values to take zero-point vibrational effects into account after obtaining the respective Debye temperatures from published low-temperature heat capacity data. The students also calculate the lattice parameters and solid densities. An assessment of their work is made by comparing their results with experimental data. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Computational Chemistry, Quantum Chemistry, Solids

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the sublimation energy of the solid, extrapolated to 0 K, and can be obtained experimentally from low-temperature heat capacity or vapor pressure measurements.5 The experimental approach for determining Ecoh from heat capacity data is summarized in the Supporting Information. Students calculate Ecoh for solid Ne and Ar from a computational chemistry approach and compare their results with experimental data. The project has several steps. First, the students obtain pair potentials of the respective dimers using high-level ab initio quantum chemistry calculations. Second, they apply these potentials to the solid state by using the summation of the twobody interaction energies in the crystal lattice. These two rare gas systems are chosen not only because they are amenable to appropriately high-level computational study, but also because students can see how well their modeling results compare with experimental data for two members of the rare gas family. Third, they discover the need to refine their model by taking into account the zero-point energies of the solids because the results obtained from the dimer pair potential extrapolations pertain to the energetics relative to the well bottoms, not the zero-point levels. These “quantum kinetic energy” effects must be considered to permit a more rigorous comparison of their results to be made with experimental data, especially for Ne2. To do this, students apply the Debye theory of solids and obtain the Debye temperatures of Ne and Ar from the respective low-temperature heat capacities. They also find the unit cell dimensions and calculate the respective solid densities. After performing this project, it is reasonable to expect students to have an understanding of the topics and tasks listed in Table 1.

he one-year physical chemistry curriculum, which traditionally spans a range of topics that includes thermodynamics and phase equilibrium, transport properties and chemical kinetics, quantum mechanics and quantum chemistry, and spectroscopy, does not usually include coverage of solids. In fact, only a few textbooks devote space to this topic.1 Many discuss solids in the context of other subjects such as quantum theory or low-temperature heat capacities.2 Most students, therefore, learn about condensed-phase properties and materials science in separate courses dedicated to these topics. This article describes a project for upper-level undergraduate students that engages them in tasks that will help them to bring together and apply ideas from such areas as quantum chemistry and solid-state physics. The project will also help them (one hopes) to appreciate the sweep of physical chemistry that ranges from rigorous treatments of microscopic systems (e.g., diatomic molecules) to fundamental considerations of the condensed phase (such as the Debye solid). A level of accountability, or outcomes assessment, is involved in the project because students will compare their results from the phases of the project with experimental data. The value of such an approach has been implicitly recognized in a recent article in this Journal dealing with the relationship between the bond energies of metals and their respective surface energies.3

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PROJECT STRATEGY AND EXPECTED LEARNING OUTCOMES The primary objective of the project is to determine the cohesive energies of two rare gas solids (neon and argon). Cohesive energy, Ecoh, is the energy needed to transform one mole of a (crystalline) solid at 0 K to isolated (gas-phase) atoms.4 Note that here Ecoh > 0. This quantity is equivalent to © 2012 American Chemical Society and Division of Chemical Education, Inc.

Published: January 27, 2012 592

dx.doi.org/10.1021/ed200348j | J. Chem. Educ. 2012, 89, 592−597

Journal of Chemical Education

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objectives of this project that they obtain this information themselves from ab initio computational methods. To this end, they calculate the energy of a pair of atoms at different internuclear separations (r values) in the region of the PE well minimum. In carrying out this first step of the project, students should be made aware that there are several options available in selecting the quantum computational method to be used, as well as the number of points in the PE scan. These choices greatly affect the time requirement, as well as the quality of the results obtained. Students should be reminded that bonding in the rare gas dimers arises from dispersion (induced dipole−induced dipole) interactions and is much weaker than in covalent or hydrogen bonding. For this reason, low levels of theory, such as the Hartree−Fock method, are inadequate. Although the Møller− Plesset perturbation method, which attempts to account for electron correlation effects, might be used for these dimers, students should be guided to use a higher level of theory, the coupled cluster with single, double, and perturbative triple excitations method, denoted as CCSD(T),7 along with a large basis setthe augmented correlation consistent polarized valence quintuple zeta (aug-cc-pV5Z with 254 basis functions).8 The CCSD(T) method, which accounts for many-body electron correlation, is capable of producing accurate results for atoms and small molecules in comparison with experimental data. It has become a standard in high-level ab initio calculations and is available in several computational chemistry packages. The use of the large basis sets reduces the need to obtain results in the complete basis set limit, which requires multiple calculations with different basis sets in the correlation consistent family, and greatly extends the computational and labor time without providing significant improvement in the outcome. These calculations are feasible for the neon dimer (20 electrons), but not for the argon dimer (36 electrons). In the latter case, the recommendation is to use the CCSD(T) method, but with a smaller basis set, one with quadruple-ζ functions (aug-cc-pVQZ with 168 basis functions). Students can readily carry out these calculations on a personal computer (PC) using the Gaussian suite of programs, available on the PC (Windows) platform.9 Details for running these calculations and obtaining the LJ parameters are presented in the Supporting Information. An important consideration in designing this part of the project is the decision about how many, and which, points in the LJ PE curve are to be calculated. Because each point in the Ne2 calculation takes about an hour (on a typical PC), students should give some thought as to how they set up their scan jobs. In this part of the project, the PE scan calculations can be distributed among several groups. First, they should estimate the position of the well minimum, re, where dV(r)/dr = 0. To do this, they could find the LJ σ value from their textbook or the literature. They should show that, from eq 1, re = 21/6σ. Another possibility is to estimate re from 2 times the respective atomic radii. Because the repulsive part of V(r) is very steep, only a few points are needed for r < re. To capture the characteristics of the bonding part of the potential, more points are needed for r > re. A suggestion (and the approach used in this article) is to perform the scan of the Ne2 PE curve between 2.7 and 4.0 Å in increments of 0.1 Å, for a total of 14 points. Students can use their judgment in choosing fewer or more points or a different range for their scan calculations. In the case of Ar2, a scan between 3.3 and 4.7 Å is recommended.

Table 1. Topics Covered and Tasks Performed in This Project Topic or Task •Perform ab intio quantum chemistry calculations; scan dimer potential energy (PE) curve •Obtain Lennard-Jones (LJ) potential parameters from computational results •Extend LJ interactions to fcc structure of solid •Calculate Ecoh values from lattice constants •Use the Debye theory of solids to obtain zero-point energies (ZPE) •Obtain Debye Temperatures from published experimental CV(T) data •Obtain lattice constants and solid densities •Compare all results with experimental data •Suggest refinements and improvements

In carrying out this project, students are made aware of the importance of the art of compromise. They will recognize the need to balance rigor with practicality and to make decisions about performing calculations that lead to acceptably accurate results. Another useful outcome would accrue by encouraging students to explore the additional cost (in time) of a calculation (e.g., the number of calculations performed) relative to the benefit of improved accuracy. This project is designed to keep each component to a modicum of simplicity and clarity, allowing students to better see their work from the perspective of achieving their goal of obtaining experimentally testable information about rare gas solids from computational quantum chemistry calculations of the dimers along with the application of the Debye theory of solids.6 The solids studied in this project are crystalline neon and argon. These rare gases are chosen to avoid complications that may arise with studies of more complex systems such as molecules. Although the interactions between the rare gas atoms are weak, often characterized as van der Waals, or dispersion interactions, the computational method described here is able to account very satisfactorily for these energetics. Although it may be argued that rare gas solids do not represent “real-life” examples, students nevertheless employ the same fundamental ideas and applications to pursue the project objectives as they would for more complex examples.

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THE PAIR POTENTIAL AND DIMER STRUCTURE To begin the project, students computationally obtain the parameters of the well-known Lennard-Jones (LJ) potential V(r), which is expressed as

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ V (r ) = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝ r ⎠ ⎥⎦ ⎢⎣⎝ r ⎠

(1)

where r is the internuclear distance, ε is the potential energy (PE) well depth, and σ is the value of r at which V = 0 [the potential changes from repulsive (V > 0) to attractive (V < 0)]. The first term in eq 1 accounts for repulsive interactions. The power of 12 is an arbitrary but convenient way to express these interactions. The second term in r−6 represents the attractive forces between the atoms that arise from induced dipole− induced dipole interactions. Students should be made aware that there are many other similar PE functions, many containing additional terms in r−n (e.g., n = 8, 10, 14, 16), to better represent pairwise interactions. The rationale for using the LJ potential in this project will be made clear later. Although students could obtain values of the LJ parameters, ε and σ, from the literature, it is in keeping with the learning 593



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The method for extracting the CCSD(T) results from the output file is described in the Supporting Information. The scan data can be imported into Excel, and a curve-fitting application capable of performing regression analysis with user-defined functions, such as SDAS,10 can be used to fit the data to eq 1. The results of these scans are presented in Table 2. For Table 2. Potential Energy Curve Parameters for Ne2 and Ar2 Ne2 a

Ar2. a

Parameter

This Work

Experiment

This Work

ε/cm−1 σ/Å De/cm−1 re/Å

29.4(3) 2.763(2) 30.15(4)b 3.097(1)b

24.3c 2.78c 29.40(12)d 3.094(1)d

93.8(6) 3.381(1) 95.73(4)b 3.801(1)b

Experiment 83.3c 3.405c 99.2(10)e 3.761(3)e

a

Ne2 and Ar2 parameters obtained from fitting the CCSD(T)/aug-ccpV5Z and CCSD(T)/aug-cc-pVQZ scan data, respectively, to eq 1. b Obtained from a sixth order polynomial fit described in ref 11. c Obtained from B2(T) data as cited in ref 12. dData from ref 13. eData from ref 14.

comparative purposes, this table also contains the well depth, De, and the equilibrium internuclear distance, re, obtained from a sixth-order polynomial fit to the scan data.11 The comparison between the LJ parameters, obtained from the ab initio scans, and those found experimentally from gas imperfection studies, for example, the temperature dependence of the second virial coefficients, B2(T), illustrates an important fact regarding the relationship between eq 1 and the “true” potential curves, and it is worthwhile to emphasize the following points to students: (1) the experimental LJ well depths ε, shown in Table 2, are found from analyzing the B2(T) data of the gases with respect to eq 1 (see ref 12 for a discussion of this procedure); (2) values of ε and σ depend on the method used to obtain them, for example, gas viscosity and other properties; (3) the experimental values of the well depths, De, are obtained from high-resolution spectroscopic studies of the rare gas clusters produced by in a supersonic expansion; (4) ε values for the rare gases (Ne−Xe), found from B2(T) data, are systematically smaller than those obtained from direct spectroscopic studies of the dimer by factors ranging from 0.92 (Ne) to 0.88 (Xe); and (5) high-level ab initio calculations can provide well depths for the rare gas dimers (Ne2−Xe2) that accurately reproduce the experimental values. The scans that students obtain in this project yield results that are respectably close to those obtained from even higher-level quantum chemical studies.15 Because the values of Ecoh obtained in this project depend on the well depths of the dimer potentials, the question arises, which values should be used, ε from gas imperfection studies or ε from the ab initio potential energy scans? The answer to this question is avoided by using both sets of values. Students can see for themselves how well these two approaches work when they compare their Ecoh results with the experimental data. The results of the PE scans for Ne2 and Ar2 along with the regression fits of the data to eq 1 are shown in Figure 1.

Figure 1. (Top) Regression fit of the CCSD(T)/aug-cc-pV5Z scan for Ne2 to the Lennard-Jones potential, eq 1. The parameters are σ = 2.763(2) Å and ε = 29.4(3) cm−1. (Bottom) Regression fit of the CCSD(T)/aug-cc-pVQZ scan for Ar2 to the Lennard-Jones potential, eq 1. The parameters are σ = 3.383(1) Å and ε = 93.8(6) cm−1.

themselves) that the rare gas solids form face-centered cubic (fcc) solids. This is the case even though calculations have shown that the energy of the body-centered structure (bcc) is near that of the fcc form.16 At this point it is a good idea to review the three cubic structural forms, fcc, bcc, and hexagonal close-packed. The idea is to imagine that a given atom, j, is surrounded by a vast number of identical atoms, i, in a fcc arrangement, and then to calculate the pairwise interaction energies between atom j and all other i atoms using eq 1. As one would expect, nearest neighbors will contribute the most (and equally), and those atoms situated in farther-lying arrangements will add smaller and smaller amounts to the stabilization energy. The total interaction energy of all i atoms in the lattice with atom j is expressed by the summation

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Vj ,tot =

12 ⎡ 1 ⎢ ⎛σ⎞ 4ε ∑ ⎜ ⎟ − 2 ⎢⎣ ⎝ Sir ⎠ i=1

PAIRWISE INTERACTIONS IN THE SOLID AND THE COHESIVE ENERGY In the next part of the project, students use their LJ parameters for the dimers to estimate the energetic of the (pairwise) interactions between atoms in the crystalline solids. At this point, they are told (or otherwise directed to find out for

⎛ σ ⎞6 ⎤ ⎟ ⎥ Sir ⎠ ⎥⎦ ⎝ i=1

∑⎜

(2)

where Si ir is the distance between the atom j and the ith atom in the lattice. The factor of 1/2 has been included to avoid the double counting of atom pairs. Fortunately, sums of 1/xn (analogous to Madelung constants) have been tabulated for several cubic lattice structures. For the fcc arrangement and n = 594



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12 and 6, those sums are 12.1318 and 14.4539, respectively.17 Note that these values are close to 12, the number of nearest neighbors in the fcc lattice. Multiplying eq 2 by Avogadro’s number, NA, gives the total interaction energy of a mole of rare gas atoms in an fcc structure. One should realize that vibrational characteristics of the dimer are not accounted for in this model, and thus, the energy represented in eq 2, similar to that in eq 1, ignores zero-point energy. Using this information, eq 2 can be applied to the solid-state potential, Vs(R), and rewritten as

⎡ ⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ Vs(R ) = 2NA ε⎢12.1318⎜ ⎟ − 14.4539⎜ ⎟ ⎥ ⎝R⎠ ⎝ R ⎠ ⎥⎦ ⎢⎣

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ZERO-POINT ENERGY AND THE DEBYE SOLID Focusing attention on Ne and Ar, both calculated values of Ecoh are larger that the respective experimental ones. Qualitatively, these differences can be readily explained in terms of zero-point vibrational effects. Students should understand the difference between the well depth of a dimer, that is, De, as obtained from the PE scans in the study described here, and the dimer dissociation energy D0, the energy difference between the dimer zero-point level, and the separated atoms. The experimental value obtained for Ecoh represents, of course, the energy of the transition of atoms in the zero-point levels in the solid to isolated gas phase atoms (at 0 K). Thus, relative to the expression for Ecoh in eq 4, one must subtract the zero-point energy of the solid to obtain a more realistic estimate of the experimental value. The concept of calculating the zero-point vibrational energy of a solid, Ezpe, appears straightforward: the vibrational zeropoint energies of the atoms in the solid are summed. Assuming the harmonic oscillator model for atomic vibrations and assigning a vibrational frequency νi to an atom imbedded in the solid, one can write

(3)

where R denotes the interatomic distance in the solid. With this seemingly straightforward simplification, students realize that they can use eq 3 to find (the negative of) Ecoh by fixing the value of R so that it is equal to the minimum, Re, of the multipairwise interaction potential, eq 3. To find Re, they set dVs(R)/ dr = 0, solve for σ/Re, and then find that σ/Re = 0.9173. Using this result, they see that from its definition Ecoh = −V(Re) and thus from eq 3

Ezpe =

Ecoh = 8.610NA ε (4) This model predicts that eq 4 can be used to obtain Ecoh for any fcc solid for which the isolated dimer interaction follows eq 1. This simple expression follows from the choice of a two-term (6-12) expression for V(r), for example, the LJ potential, eq 1. Some students might be interested in fitting their scan data to PE functions containing one or more additional r−n terms. In these cases, however, although the 1/lir−n sums are available,17 the calculation needed to express σ/Re must be performed numerically. The Ecoh values are presented in Table 3; they are obtained from eq 4 for Ne and Ar based on ε values obtained from both

i

g (ν) =

Ecoh/(kJ mol−1) Rare Gas Solid

From PE Calculationsb

Experimental Valuec

Ne Ar Kr Xe

2.49 8.58 12.2 15.8

3.03d 9.66d 14.2e 18.0f

1.92 7.74 11.2 15.9

1 hνi 2

(5)

where h is Planck’s constant. The task now is to assign values to νi. At this point, students should be introduced to the Debye theory of solids.1,6 For the purpose of this project, students should understand that, according to the Debye model, atomic vibrations in a solid (phonons)21 are distributed throughout its entire volume; that is, all atoms participate in vibrational motion, and there is a distribution of vibrational frequencies whose waveforms accommodate the particular interatomic spacing in the solid. The Debye treatment (which is analogous to Plank’s consideration of photons in a blackbody radiator) gives a distribution function of frequencies1

Table 3. Cohesive Energies of Rare Gas Solids From B2(T)a

∑

9NA ν 2 ν 3max

(6)

where νmax is the maximum frequency, which depends on the physical characteristics of the solid such as the speed at which sound is propagated in it. The energy of this maximum vibration can be expressed as the Debye temperature ΘD through the relationship

Calculated from eq 4 based on ε values from B2(T) data from ref 12.. From ab initio calculations, this work. cFrom ref 20. dThis study. e From ref 18. fFrom ref 19. a b

ΘD =

hνmax k

(7) 22

where k is Boltzmann’s constant. Using eqs 5 and 6, and assuming a high density of vibrational states (so the sum in eq 5 can be replaced by the integral), Ezpe can be expressed as

gas imperfection studies and from the ab initio potential energy scans described above. In addition, for comparative purposes, the results are included for two other rare gas solids, Kr and Xe, for which ε values from B2(T) data and from high-level ab initio calculations are used. The data in Table 3 are reported in kJ mol−1 (1 kJ mol−1 = 83.5935 cm−1). From the results in Table 3, it is apparent that the simple model used to calculate the cohesive energy of a solid from gasphase dimer properties works qualitatively well. One also notes a range of values for each solid depending on which method is used to obtain ε, with those from gas imperfection studies always smaller than those from computational or spectroscopic studies (because ε is always smaller than De).

Ezpe =

1 h 2

∫0

νmax

νg (v) dν

(8)

Finally, after expressing g(ν) from eq 6 in eq 8 and integrating, the zero-point energy in terms of the Debye temperature is

Ezpe =

9R ΘD 8

(9)

The derivation of the distribution function in eq 6 is not straightforward and interested students should be directed to ref 23 for details. 595



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Although it is possible to calculate values of ΘD from measurable properties of a solid, it is more practical for the purposes of this project, and more instructionally valuable, to ask students to obtain ΘD using experimental low-temperature heat capacity data and the T3 dependence of CV (a well-known outcome of the Debye theory of solids). That relationship24 is 3 12π R ⎛ T ⎞ CV = ⎟ ⎜ 5 ⎝ ΘD ⎠

and Ar obtained from this study, along with experimental values are shown in Table 5. Table 5. Solid-State Density, d, and the fcc Lattice Parameter, a0, of Ne and Ar Ne

4

(10)

Accepted values of ΘD for Ne and Ar are 75 and 92 K, respectively.25 Analysis of C V(T) data reported for these solids using eq 10 gives values of 73 and 92 K, respectively. For Ne, results from ref 26 between 1 and 3 K were used, and for Ar, data between 1 and 2 K were analyzed.27 These data sets are in the Supporting Information. The zero-point-corrected cohesive energies of the rare gas solids E′coh (Ecoh − Ezpe), based on ε obtained from B2(T) and PE data, are presented in Table 4 along with the experimental

a

Ezpea/ (kJ mol−1)

E′cohb/ (kJ mol−1)

E′cohc/ (kJ mol−1)

Experimental Valued/ (kJ mol−1)

Ezpe/ Ecoh,exp

Ne Ar Kr Xe

0.68e 0.86f 0.67g 0.60g

1.81 7.72 11.5 15.2

2.35 8.80 13.5 17.4

1.92 7.74 11.2 15.9

0.35 0.11 0.060 0.038

From eq 9. bBased on ε from B2(T) data. cBased on dimer PE calculations. dFrom ref 20. eUsing ΘD from analysis of data from ref 26 as described earlier. fUsing ΘD from analysis of data from ref 27 as described earlier. gBased on ΘD from ref 25. a

values. The ratio of zero-point energies to experimental cohesive energies is also shown and these indicate the relative importance of zero-point effects for the lighter-element solids and hence the particular need to include the zpe contribution for Ne.

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LATTICE CONSTANTS AND SOLID DENSITIES Having obtained the equilibrium spacing between Ne and Ar atoms in the fcc solid, students are now in a position to obtain two other experimental quantities, the lattice constants and densities (in the 0 K limit). By analyzing a fcc model, they can readily determine that the lattice parameter, a0, which is the length of the unit cell, is simply √2Re, and therefore the mass density is d=

3 4M ⎛ 1 ⎞ ⎜ ⎟ NA ⎝ a0 ⎠

3 4· 2−3/2M ⎛ 1 ⎞ = ⎜ ⎟ NA ⎝ Re ⎠

=

2M ⎛ 1 ⎞ ⎜ ⎟ NA ⎝ R e ⎠

Parameter

This Study

Experiment

This Study

Experiment

Rea/Å a0/Å d/(g cm−1)

3.012 4.260 1.734

− 4.429b 1.543b

3.686 5.213 1.880

− 5.256b 1.827b

Re = σ/0.9173 and σ from Table 2. bValues at 4.2 K, from ref 29.

The agreement between the calculated and experimental values is surprisingly good. One notices, however, that the a0 values obtained from the model are somewhat smaller than the experimental ones, more so for Ne than for Ar and the correspondingly opposite can be seen for the densities. If asked to analyze these observations, students should, in light of the earlier discussion about the Ecoh values, realize that zero-point effects again come into play here. The experimental lattice parameters and densities listed in Table 5, obtained at 4.2 K (kT = 2.9 cm−1), reflect properties of the solids in the zeropoint levels; hence, the relevant nearest-neighbor distance is not Re, but R0, the most probable distance between nearest neighbor atoms in the n = 0 vibrational level. This adjustment can be approximated by using the rare gas dimer PE functions obtained in the ab initio scans to find the value of r, that is, r0, at which Ψ02(r), the probability density function for the n = 0 vibrational state, is a maximum. It is possible to estimate values of r0 by solving the (onedimensional) Schrödinger equation using the PE scans for the dimers obtained in this study (recognizing that these scans consist of a relatively small number of points). For this purpose one can use FINDIF, which produces the eigenfunctions and eigenvalues associated with a one-dimensional potential.28 Details are provided in the Supporting Information. From such an analysis, r0 values of 3.231 and 3.843 Å are found for Ne2 and Ar2, respectively. The results also indicate a zero-point energy of 15.2 cm−1 for Ne2 (and only one bound vibrational state) and a fundamental vibrational frequency of 25.1 cm−1 for Ar2. The latter value compares surprisingly well with the experimental value of 25.6 cm−1.14 Assuming that one can scale r0 to R0 using the same factor discussed above for converting σ to Re, one can obtain the corresponding values of a0 and d for the solids. After this project is completed, students should be encouraged to write or present an overview of their work in which they summarize their strategy and methods for obtaining measurable (and verifiable) information about crystalline neon and argon. In addition to their assessment of how well their results compare with experimental data, they might also be engaged in a discussion about intermolecular potentials. In particular, they should contemplate why the well depths of the rare gases, expressed in the Lennard-Jones potential, that is, ε, obtained from B2(T) data, are systematically smaller than the actual well depths, De, found from experiment or high level ab initio calculations. Instructors might also lead a discussion about value and limitations of the pair-potential model and the neglect of three-body (and higher) interactions.16 They should also emphasize the relative importance of zero-point vibrational

Table 4. The Zero-Point and Calculated “Corrected” Cohesive Energies of the Rare Gas Solids and the Experimental Values Rare Gas Solid

Ar

(11)

where M and NA are the molar mass and Avogadro’s number, respectively. The factor of 4 represents the number of atoms per unit cell. The values of d (in g cm−3) and a0 (in Å) for Ne 596



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(23) Kestin, J.; Dorfman, J. R. A Course in Statistical Mechanics; Academic Press: New York, 1971; pp 388−394. (24) Levine, I. N. Physical Chemistry, 6th ed.; McGraw Hill Higher Education: Boston, MA, 2009; p 945. (25) Kittel, C. Introduction to Solid State Physics, 6th ed.; John Wiley & Sons: New York, 1986; p 110. (26) Fenichel, H.; Serin, B. Phys. Rev. 1966, 142, 490−495. (27) Finegold, L.; Phillips, N. E. Phys. Rev. 1969, 177, 1383−1391. (28) FINDIF Home Page. http://carbon.indstate.edu/FINDIF (accessed Jan 2012). (29) Henshaw, D. G. Phys. Rev. 1958, 111, 1470−1475.

effects on the properties of solid Ne as compared with the other rare gas solids.

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ASSOCIATED CONTENT

S Supporting Information *

Experimental approach for determining Ecoh from heat capacity data; details for running these calculations and obtaining the LJ parameters; the method for extracting the CCSD(T) results from the output file; data sets; use of FINDIF. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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REFERENCES

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