In the Classroom
Dispersion Interactions between Rare Gas Atoms: Testing the London Equation Using ab Initio Methods Arthur M. Halpern Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809, United States
[email protected] From their early experiences in learning chemistry, students are introduced to the concept of intermolecular interactions and the hierarchy of forces that leads to liquifaction and solid formation (1). Of the three main types of interactions, dipoledipole (or ion-dipole) types are easiest for students to visualize because of the explicit electrostatic nature of these forces. Whereas the covalent interactions, and the formation of the covalent bond, are difficult for students to conceptualize (without, or even with, knowledge of quantum chemistry), they readily grasp the profound importance of compound formation and usually adapt easily to the working concept of electron pairing. But it is with dispersion interactions that students seem to have the greatest challenge. Not only is the term dispersion likely to be new to them, it does not represent or convey a mechanism that necessarily results in an attractive force. Furthermore, students are told that these dispersion interactions are universal, that they contribute to all interatomic or intermolecular attractive forces, and in some cases they represent the only attractive forces between species. As far as offering a “mechanistic” interpretation of dispersion forces is concerned, textbooks point out to students that the distribution of electrons in an atom or molecule is described by statistical fluctuations, which give rise to an “instantaneous dipole” that can set up an induced dipole in a neighboring atom or molecule (2). The electrostatic association between these fleeting, fluctuating polar species is, then, viewed as the origin of dispersion, or London forces. Fritz London described the first theoretical treatment of the attractive interactions between nonpolar molecules and atoms in 1930 (3). He accounted for these forces, now called, appropriately, London forces, in terms of the correlation of electron movement in the two species. The basis of London's approach is the representation of the electrostatic potential energy of the neutral system in a multipole series (dipole, quadrupole, etc.) in which the leading term, dipole-dipole interactions, was dominant. In a later paper (4), London referred to the statistical ensemble of neutral species as consisting of an “orchestra of periodic dipoles”. These oscillators represent the motion of electric charge within the atomic or molecular system. In its most simplified form, the potential energy, V, describing the interaction between two such particles with separation R, which London called the “dispersion effect,” is the familiar expression in 1/R6, V ðRÞ ¼ -
C6 R6
ð1Þ
For identical particles, London's treatment represents C6 as (4) C6 ¼
174
3 hνR2 4
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ð2Þ
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where ν represents a single oscillator frequency of the particle and R is its polarizability volume. London empirically assigned the oscillator energy, hν, to the ionization energy of the interacting species. Thus, out of the “orchestra” of oscillators, London essentially selected but one instrument.1 With this ad hoc assignment of hν in eq 2, C6 can be calculated from experimental quantities.2 In the following discussion, we denote this attractive, “dispersion” interaction energy as a positive quantity, that is, the London energy, EL. Thus, EL =-V. The Project Considering the fundamental importance of dispersion interactions and the apparent simplicity of the London equation (eqs 1 and 2), it is reasonable to invite students to test how well this relationship works for a series of weakly interacting particles. A logical choice for this project is the family of dimers formed between rare gas atoms. In the plan described here, students carry out high-level ab initio quantum mechanical calculations of the interaction energies of the 10 rare gas pairs comprising He, Ne, Ar, and Kr. These pairs consist of the four homonuclear dimers and the six heterodimers, NeHe, ArHe, KrHe, ArNe, KrNe, and KrAr. Given the opportunity to vary the scope of the project, students can work individually, or in groups, where each team works on one or more components of the project. To complete the project, the groups can meet, discuss, and exchange their results, as appropriate. Students first consider the methodology of choosing interatomic distances for which repulsive interactions may be neglected relative to the attractive or “dispersion” interactions expressed in eq 1. To begin, it may be helpful for the students to visualize this issue by examining the Lennard-Jones (6-12) pair potential, VLJ, that is, " 6 # σ 12 σ ð3Þ VLJ ðRÞ ¼ 4ε R R in which R is the internuclear separation, and ε and σ are the well depth and the R value, respectively, at which VLJ = 0. The attraction between the particles is accounted for by the 1/R6 term, whereas the repulsion is accounted for (arbitrarily) by the 1/R12 term, which represents repulsive interactions by a steep wall. [Many other pair potentials exist. For example, the Buckingham potential represents the repulsive wall by an exponential function in R, that is, exp(aR), where a is a parameter.] The point is that the second term in eq 3 represents “dispersion” interactions; many simple pair potentials contain such a term. One simple way to probe only the attractive contributions to molecular interactions, to a satisfactory level of approximation, is
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In the Classroom Table 1. Internuclear Separations of the Rare Gas Dimers Determined from van der Waals radii, RW, and Obtained from CCSD(T)/avqz optimizations, Rea
Table 2. Properties of the Rare Atoms, Ground State Electronic Energies, E, obtained from CCSD(T)/avqz Calculations and Experimental Polarizability Volumes, R, and Ionization Energies, I
Dimer RW/Å Re/Å De/cm-1 Dimer RW/Å Re/Å De/cm-1
Rare Gas
R /Å3 b
E /Eh a
I /105 cm-1 c
He2
2.80
2.98
7.05
ArHe
3.28
3.49
21.19
He
-2.902533601
0.205
1.98307
Ne2
3.08
3.08
33.27
KrHe
3.42
3.72
20.53
Ne
-128.8474595
0.3955
1.73926
Ar2
3.76
3.80
95.76
ArNe
3.42
3.50
48.71
Ar
-527.0750553
1.642
1.27107
Kr2
4.04
4.08
128.20
KrNe
3.56
3.68
50.53
Kr
-2752.2734834
2.488
1.12912
NeHe
2.94
3.00
16.91
KrAr
3.90
3.94
106.28
a
The well depths, De, obtained from this method are also listed.
to perform calculations at intermolecular separations that are two times larger than the sum of the van der Waals radii of the species. For example, using the Lennard-Jones potential as a model, students can readily show that when R = 2σ the ratio of the repulsive-to-attractive energies is 2-6 or 0.0156. This might be a good opportunity to ask students to think about the concept of “atomic size”. Because the position of the electron in an atom is not an eigenvalue of the Hamiltonian, it is not a precisely measurable quantity; this is a central point of London's analysis, namely, that it is the distribution of electron position probabilities that gives rise to instantaneous dipoles in a species. The van der Waals atomic radius, rW, here is the nonbonded contact distance between atoms, as determined, for example, from the crystal structure. One advantage of the students' basing their calculations on the rare gas rW values is that they do not have to determine the minimum-energy internuclear separations of the dimers, Re, from ab initio calculations. Although doing so is a viable approach and a logical one from the point of view of consistency, it presents unnecessary difficult challenges because optimization methods implemented in different computational applications give different results. Such differences arise because of the shallow nature of the (light) rare gas potentials and thus the flatness of these potentials near the minima, and the possibility that different minimization algorithms are employed in the applications. In implementing the use of rW data for the rare gas atoms studied in this project for determining the internuclear separations at which dispersion interaction energies are calculated, students employ the following simple combining rule, RWij ¼ rWi þ rWj ,
ð4Þ
where RWij is internuclear separation of the dimer of rare gas atoms i and j, and rWi and rWj are the respective van der Waals atomic radii. Table 1 shows the RWij values of the rare gases based on eq 4 and rW assignments of 1.40, 1.54, 1.88, and 2.02 Å for He, Ne, Ar, and Kr, respectively (6). For comparative purposes, this table also lists the well depths, De, and minima, Re, obtained from the computational method described below. Computational Method Students calculate the interaction energy between two rare gas atoms using a high-level ab initio method known as the coupled cluster with single, double, and perturbative triple excitations, CCSD(T) (7). This method accounts for electron correlation effects, which are important in describing the type of nonbonded interactions involved in this project. The basis set is the augmented correlation-consistent polarized valence quadruple-ζ
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a
Obtained using G03W. See the supporting information for details. b Data obtained from ref 11. c Data obtained from ref 12.
set (aug-cc-pVQZ, or avqz) (8). This approach is a good compromise between quantum chemical rigor and computational feasibility. In fact, given the large number of electrons in Kr, this approach is close to the limit that may be reasonably expected for calculations involving the family of rare gas elements He through Kr. These calculations can be readily performed using applications available to undergraduates on a PC [e.g., Gaussian03W (9)] or on a Unix platform. Details of performing these calculations, along with results, are described in the supporting information. The London interaction energies of the 10 rare gas dimers, ELij, at internuclear separations of 2RWij are calculated from ELij ð2RWij Þ ¼ Eij ð2RWij Þ - ðEi þ Ej Þ
ð5Þ
where ELij(2RWij) is the CCSD(T)/avqz energy of the ij dimer with internuclear separation 2RWij, and Ei and Ej are the CCSD(T)/ avqz energies of the two rare gas atoms i and j. Setting R in eq 1 equal to 2RWij, and recalling that EL = -V, we obtain ELij ð2RWij Þ ¼
C6ij ð2RWij Þ6
ð6Þ
The left-hand side of eq 6 is obtained from quantum chemical calculations, and the C6 coefficient on the right-hand side can be calculated, according to London's treatment of dispersion interactions, from experimental quantities (10), that is, C6ij ¼
3 Ri Rj Ii Ij 2 Ii þ Ij
ð7Þ
where Ri and Rj, and Ii and Ij are the respective polarizability volumes and ionization energies of rare atoms i and j. Students can readily confirm that eq 7 reduces to eq 2 for homonuclear dimers (and if hν is associated with the ionization energy, I). To show more clearly the relationship between calculated and experimental quantities, eqs 6 and 7 may be combined to ELij ð2RWij Þ ¼
3 Ri Rj Ii Ij ð2RWij Þ - 6 2 Ii þ Ij
ð8Þ
Thus, one can test the London approach by plotting the calculated interaction energy of a pair of rare gas atoms at an internuclear distance of twice the sum of their van der Waals radii (the left-hand side of eq 8) versus the right-hand side of eq 8, which consists of experimental quantities and the respective van der Waals radii and expect to obtain a slope of one and an intercept of zero. Table 2 contains the CCSD(T)/avqz energies of the rare gas atoms, their experimental polarizability volumes, and ionization energies. Table 3 lists the CCSD(T)/avqz energies of the dimers, along with the dimer interaction energies, at
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In the Classroom Table 3. Electronic Energies of the Rare Gas Dimers at Internuclear Distances 2RW, the Respective Interaction Energies, EL, and the C6 and C8 Coefficients Dimer He2 Ne2
E(2RW)/Eh a
EL(2RW)/cm-1 b
C6/cm-1 Å6 c
C8/cm-1 Å8 d
-5.8050683571
0.254
6.25 103
1.09 104
0.639
2.04 10
4
6.04 104
5
2.13 106
-257.69492199
Ar2
-1054.1501202
2.10
2.57 10
Kr2
-5504.5469794
2.77
5.24 105
6.34 106
1.13 10
4
2.67 104
3.91 10
4
2.21 105
4
4.10 105
NeHe ArHe
-131.47999503
0.416
-529.97759208
0.693
KrHe
-2755.7760205
0.768
5.50 10
ArHe
-655.92251986
1.10
7.15 104
4.40 105
1.24
1.01 10
5
7.97 105
3.66 10
5
3.87 106
KrNe KrAr a
-2881.1209486 -3279.3485497
2.41 b
c
d
Obtained using G03W. See the supporting information for details. Obtained from eq 5. Obtained from eq 7. Obtained from eq 10.
The coefficient C8 is given by (10) C8ij
45Ri Rj Ii Ij Ri Ii Rj Ij ¼ þ 2 8e 2Ii þ Ij Ii þ 2Ij
! ð10Þ
where R and I have the same meaning as in eq 7 and e is the elementary charge. (The use of e in cm-1 and Å units is described in the supporting information.) With the inclusion of the C8 term, the London equation is tested by extending eq 6 to read C6ij C8ij þ ð11Þ ELij ð2RWij Þ ¼ 8 6 ð2RWij Þ ð2RWij Þ Figure 1. Plot of eq 8 for the 10are gas dimers: homodimers (filled circles) and heterodimers (open circles). The regression value of the slope is 1.48(1).
internuclear distances of twice their respective van der Waals separations. Table 3 also shows the C6 values, calculated from eq 7. A plot of eq 8 based on the data presented in Tables 2 and 3 is shown in Figure 1. Although students are impressed that the 10 points fall close to the linear relationship predicted by eq 8, they notice that the slope of the regression line (the fitting function is y=mx) is 1.49 (standard deviation of regression 5.2 10-2; r2 = 0.9967), rather than 1.0, as predicted by eq 8. Students may be dismayed by this discrepancy and wonder how well high-level ab initio methods account for these “dispersion” interactions. They should be reminded that the London equation represents a simple electrostatic model that should not be expected to account precisely for these interactions. Also, they should be reassured that rigorous quantum mechanical methods, when applied to rare gas dimers, for example, Ar2, provide values of detailed structural and spectroscopic properties that are very close to the experimental values (13). It is worthwhile to introduce students at this point to a refinement of the London equation that takes into account higher-order, instantaneous dipole-induced quadrupole interactions, and to see if this enhanced model better accounts for the data. In accounting for these interactions, eq 1 is modified to include a 1/R8 term. The London energy is now expressed as EL ¼
176
C6 C8 þ R6 R8
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ð9Þ
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Values of C8 for the 10 dimers are listed in Table 3. Linear regression of the data according to eq 11 produces a slope of 1.27(2) (standard deviation of regression 7.0 10-2, r2 = 0.9957). As might be expected, the slope is smaller than the value obtained from the fit to eq 8, showing that the inclusion of the R-8 term is an improvement to the London model. From these observations,students may logically posit that including the next higher multipole term, in R-10, further reduces the value of the corresponding slope. And, in fact, it does, but only slightly. Adding the term C10ij/(2RWij)10 to the right-hand side of eq 11 reduces the slope to 1.26(2). Those who want to test this result will find the expression for C10 in the supporting information. Argon Dimer Potential Students can further examine the ability of the London “dispersion effect”, viz. eq 1 to account for interatomic attractions by calculating the interaction energy between two rare gas atoms over a range of internuclear distances such that repulsion between the two atoms may be neglected (as in the calculations described earlier). In this part of the project they test the R dependence of eq 1. A good choice for this study is the argon dimer because the interaction energy is large enough at extended internuclear separations while the number of electrons, 36, does not pose inordinate computing resource problems. The approach is to perform a scan of the potential energy surface of the argon dimer for R between 2RW and 3RW, that is, ca. 7.5 and 11 Å. A step size of 0.25 Å furnishes 15 points in the scan. The same computational method as described earlier should be used, that is, CCSD(T)/avqz. Details describing these calculations, as well as the energies of the 15 points, are contained in the supporting information. The interaction energy, V(R),
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r 2010 American Chemical Society and Division of Chemical Education, Inc.
In the Classroom
their calculated values of the rare gas dimer interaction energies, they recognize the ability of the simple London equation to account qualitatively well for the (R-6 and R-8)-dependence of the interaction energy. These considerations enforce their understanding of the mechanism of these nonbonding associations in terms of instantaneous dipole-induced dipole interactions, which, from classical electrostatics, is expressed as 1/R6 (and higher terms). Acknowledgment It is a pleasure to acknowledge many helpful and stimulating conversations with Eric. D. Glendening. A reviewer provided helpful suggestions. Orienting experiments were performed on the Indiana State University High Performance Computing facility. Notes 1. London's treatment of intermolecular forces is semiclassical and treats atoms as quantized harmonic oscillators. It should be noted that this work was published just several years after the advent of quantum mechanics. 2. Other investigators have sought to refine the assignment of the hν quantity, incorporating such concepts as the number of valence electrons and the oscillator strength of electronic transitions. See the review article by Pitzer (5).
Figure 2. Calculated interaction energy of Ar2 (eq 12). Fit to eq 1 (upper graph). Fit to eq 12 (lower graph). Insets show the residuals of the respective fits.
here expressed as a negative quantity, is obtained from V ðRÞ ¼ EðRÞ - 2EAr
ð12Þ
where E(R) is the CCSD(T)/avqz energy of the argon dimer with internuclear separation R and EAr is energy of the Ar atom. Figure 2 (upper panel) displays the results of the scan, along with a one-parameter regression fit of the 15 points to eq 1. This analysis returns a value of C6 of 3.63(4) 105 cm-1 Å6, which compares with 2.57 105 cm-1 Å6 calculated from eq 7 (see Table 3), and is, in fact, higher by a factor similar to the slope of the plot in Figure 1, showing consistency between the test of eq 6, applied to the series of 10 rare gas dimers and the scan of the argon dimer attractive potential energy surface. Students now find it logical to employ the C8 term in analyzing the argon dimer scan results and thus fit their data to V ðRÞ ¼ -
C6 C8 R6 R8
ð13Þ
From this analysis, shown at the bottom half of Figure 2, they obtain regression values of C6 and C8 of 2.71(7)105 cm-1 Å6 and 5.9(4)106 cm-1 Å8, respectively, and see, as well, a better fit to the calculated points. They, no doubt, note the good agreement between this value of C6 and that obtained from eq 7 and the less satisfactory comparison between C8 and its calculated value from eq 10 (see Table 3). Conclusion What can students learn from carrying out this project? They are probably impressed by the trend illustrated in Figure 1 for the 10 rare gas dimers that span a large range of polarizability volumes and ionization energies. If they accept the validity of
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Literature Cited 1. Chang, R. Chemistry, 10th ed.; McGrawHill Higher Education: Boston, MA, 2010; pp 464-466. 2. Atkins, P.; de Paula, J. Physical Chemistry, 9th ed.; W. H. Freeman and Co.: New York, 2010; pp 636-637. 3. London, F. Z. Phys. 1930, 63, 245–279. 4. London, F. Trans. Faraday Soc. 1937, 33, 8–26. 5. Pitzer, K. S. Adv. Chem. Phys. 1959, 2, 59–83. 6. Bondi, A. J. Phys. Chem. 1964, 68, 441–451. 7. Levine, I. N. Quantum Chemistry, 5th ed.; Prentice Hall: Upper Saddle River, NJ, 2000; 568-574. 8. (a) Levine, I. N. Quantum Chemistry, 5th ed.; Prentice Hall: Upper Saddle River, NJ, 2000; pp 492-494. (b) Wilson, A. K.; Woon, D. E.; Peterson, K. A. J. Chem. Phys. 1999, 110, 7667–7676. 9. Frisch, M. J. et al. Gaussian03W, revision D.01; Gaussian, Inc.: Wallingford, CT, 2004. 10. (a) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc.: New York, 1954; pp 964-965. (b) Atkins, P.; de Paula, J. Atkins' Physical Chemistry, 8th ed.; W. H. Freeman and Company: New York, 2006; pp 633-634. 11. Lupinetti, C.; Thakkar, A. J. J. Chem. Phys. 2005, 122, 044301– 1 - 7. 12. NIST: Ground Levels and Ionization Energies. http://physics.nist. gov/PhysRefData/IonEnergy/tblNew.html (accessed Nov 2010). 13. Slavícek, P.; Kalus, R.; Paska, P; Odvarkova, I.; Hobza, P; Malijevsky, A. J. Chem. Phys. 2003, 119, 2102–2119.
Supporting Information Available Details of performing these calculations, along with results. This material is available via the Internet at http://pubs.acs.org.
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