Pairwise Additive Model for the He−MgO(100) Interaction - The

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Pairwise Additive Model for the He-MgO(100) Interaction Britta Johnson and Robert J. Hinde* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, United States ABSTRACT: We develop a model, based on pairwise additive He-Mg and He-O interactions, for the potential energy of He adsorbates above a rigid MgO(100) surface. The attractive longrange He-Mg and He-O interactions are assumed to have the form C6/r6, with the C6 coefficients determined from atomic data within the context of the Slater-Kirkwood approximation. The repulsive short-range He-Mg and He-O interactions are assumed to have the form Cp/rp, with the exponent p and the Cp coefficients taken as adjustable parameters. We find that for p = 9, the Cp coefficients can be chosen so that the laterally averaged He-MgO(100) pairwise additive interaction supports low-lying selective adsorption states, some of whose energies agree very well with the states’ apparent energies inferred from experimental measurements. However, for realistic values of the adjustable parameters that define our model, the lateral corrugation of the model pairwise additive He-MgO(100) potential energy surface far exceeds the corrugation that has been inferred both from experimental measurements and from density functional calculations of the short-range He-MgO(100) interaction.

1. INTRODUCTION The nature of the interaction between He atoms and the (100) surfaces of rocksalt-type ionic solids (such as NaCl and MgO) has attracted considerable attention over the past few decades.1-7 Although these solids have strong and highly nonuniform near-surface electric fields,8 which play an important role in molecular adsorption,9,10 these fields are less relevant for understanding the He-surface interaction both because the He atom’s polarizability is fairly small11 and because the He atom lacks any permanent electrostatic multipole moment. As a consequence, the He-surface interaction is dominated at short range by repulsive forces arising from the overlap of the closed electronic shells of the He atom and the ions that make up the substrate and at long range by attractive van der Waals forces arising from correlated fluctuations of these closed electronic shells. This makes interactions between He atoms and ionic solids good test cases for techniques12-14 that have been developed for performing post-Hartree-Fock electronic structure calculations on extended, nonconducting systems. Experimental information about the He-substrate interaction in these systems comes from angle- and energy-resolved studies of the inelastic scattering of He atoms off of the ionic solid’s (100) surface, studies that were originally designed to probe the dispersion curves of the solid’s surface phonons.1 Information about the corrugation of the surface can be extracted from an analysis of the Bragg diffraction of the incident He atoms,15-18 while the bound states of the laterally averaged He-substrate potential energy function can be inferred from observations of selective adsorption and desorption of the incident atoms.2,15,17 If the surface corrugation is substantial, however, some care must be used in deducing bound-state energies from selective adsorption experiments.19 In this situation, one begins with a realistic model He-surface interaction and computes the band structure r 2011 American Chemical Society

associated with the He atom’s lateral motion to determine the correction that must be applied to the apparent bound-state energies obtained from the selective adsorption resonances.20 This process requires as input a model for the He-surface interaction. The model interaction must represent both the short-range repulsive He-surface interaction that gives rise to the corrugation and the long-range attractive He-surface van der Waals interaction that is responsible for the bound states’ existence in the first place. In the present work, we develop such a model for the interaction between He adsorbates and the MgO(100) surface; our model is based on the assumption that the interaction can be written as a pairwise additive sum of He-Mg and He-O interactions. Although the short-range repulsive He-MgO(100) interaction has been studied using quantum chemical methods based on either Hartree-Fock theory or density functional theory,14,18 these methods do not quantitatively describe the attractive long-range He-MgO(100) van der Waals interaction and thus cannot be the basis for studies of the He adsorbate’s band structure. A recent study of the He-MgO(100) interaction within the framework of local second-order Møller-Plesset (MP2) perturbation theory14 shows that dispersion interactions give rise to a long-range attraction between the He adsorbate and the MgO(100) surface. However, a comparison of the HeMgO(100) binding energy estimated at the MP2 and coupledcluster levels of theory14 (employing a finite cluster model for the MgO surface) shows that the MP2 treatment of dispersion Special Issue: J. Peter Toennies Festschrift Received: December 31, 2010 Revised: February 13, 2011 Published: April 07, 2011 7112

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interactions is probably quantitatively inadequate. Consequently, no He-MgO(100) interaction that is derived solely from first principles is yet available for performing calculations of the band structure of He adsorbates on MgO(100) surfaces, and an empirical model for the He-MgO(100) interaction is required to study the effects of surface corrugation on the apparent He atom bound-state energies extracted from an analysis of selective adsorption experiments. Our goal in the present work is to investigate whether our pairwise additive model can provide a useful starting point for these studies. Our model He-MgO(100) potential energy surface incorporates three adjustable parameters, which control the short-range behavior of the potential energy surface. Two of the parameters determine the steepness of the short-range portion of the laterally averaged He-MgO(100) interaction; the third parameter determines the lateral corrugation of the He-MgO(100) interaction. The long-range portion of the potential energy surface is determined by the attractive van der Waals tails of the He-Mg and He-O atom pair interactions, which are determined here from atomic polarizabilities via the SlaterKirkwood approximation. We show that while our model can reproduce the apparent energies of some (not all) bound states of the laterally averaged He-MgO(100) interaction, its corrugation amplitude is much larger than the corrugation amplitude inferred from experiment, at least for realistic values of the third adjustable parameter. Further work must be done to refine the He-MgO(100) interaction presented here before it can be a useful starting point for modeling the band structure of He adsorbates on MgO(100) surfaces.

2. COMPUTATIONAL METHODS 2.1. MgO(100) Substrate. We consider a semi-infinite MgO crystal cleaved to expose the (100) surface, which is depicted in Figure 1. We define a Cartesian coordinate system oriented so that the surface layer of Mg and O ions lies in the z = 0 plane, the x and y axes are aligned with nearest-neighbor Mg-O ion pairs, and the positive z axis points away from the MgO(100) surface. We place one of the surface layer Mg ions at the origin (x,y,z) = (0,0,0). We take the nearest-neighbor Mg-O distance to be b = 2.11 Å and neglect any rumpling, buckling, or other relaxation of the ions in the (100) surface layer; theoretical studies21,22 of the MgO(100) surface indicate that surface rumpling or buckling is negligible in this system. As Figure 1 shows, the two-dimensional unit mesh, or surface net, associated with the MgO(100) surface is a square with edge length b(2)1/2 = 2.98 Å that is rotated by 45
with respect to the x and y axes. We define the coordinates u = (y þ x)/(2)1/2 and v = (y - x)/(2)1/2; the unit mesh associated with the Mg ion at the origin thus extends from u = -a to þa and from v = -a to þa, where a = b/(2)1/2 = 1.49 Å. 2.2. He-MgO Interaction. We let (x,y,z) denote the Cartesian coordinates of the He adsorbate atom, suspended above the MgO(100) surface. We assume that the interaction between the He adsorbate and the MgO(100) substrate can be written as the sum of pairwise additive He-Mg and He-O interactions with the Lennard-Jones (p,6) form

VHe-X ðrÞ ¼

Cp , X C6 , X - 6 rp r

ð1Þ

where X = Mg or O and r is the distance between the He adsorbate atom and a specific Mg or O ion in the MgO substrate. The

steepness of the repulsive portion of the He-X interaction is governed by the exponent p, which we take to be an integer and which we assume is the same for both He-Mg and He-O interactions. We restrict ourselves to exponents p > 6 so that the He-X interactions are attractive at large distances. This mathematical form for the He-MgO(100) interaction is chosen primarily for computational convenience because it leads to a simple form23 for the laterally averaged potential energy function Vlat(z) for the He atom ! ¥ Cp, Mg þ Cp, O C6, Mg þ C6, O 2π X Vlat ðzÞ ¼ 2 a n ¼ 0 ðp - 2Þðz þ nbÞp - 2 4ðz þ nbÞ4 ð2Þ where n indexes the layers of ions, beginning with n = 0 for the surface layer at z = 0. For large z, the short-range repulsive HeMg and He-O interactions can be neglected, and Vlat(z) has the form Vlat ðzÞ ¼ -

¥ C6, Mg þ C6, O π X 2 2a n ¼ 0 ðz þ nbÞ4

ð3Þ

which can be expressed in terms of the polygamma function ψ(3)(q) πðC6, Mg þ C6, O Þ ð3Þ Vlat ðzÞ ¼ ψ ðz=bÞ 12a2 b4

ð4Þ

Employing the asymptotic expansion of ψ(3)(q) for large values of q,24 we obtain ! πðC6, Mg þ C6, O Þ 3b b2 b4 Vlat ðzÞ ¼ 1 þ þ 2 - 4 þ ... 2z z 2z 6a2 bz3 ð5Þ We see that while the pairwise additive model for the HeMgO(100) interaction recovers the asymptotically correct C3/z3 form for the laterally averaged gas-surface interaction as z f ¥, higher-order contributions to this interaction that depend on 1/z4 and 1/z5 make up 10% of Vlat(z) at z = 30 Å and 5% of Vlat(z) at z = 60 Å. This indicates that the atomistic, or particulate, nature of the MgO crystal has a measurable influence on the HeMgO(100) interaction (evaluated within the pairwise additive model) even at relatively large values of z. We obtain the values of the C6,X coefficients from the SlaterKirkwood formula using both the static polarizabilities R listed in ref 25 for He and for the Mg2þ and O2- ions in MgO and also the values for the effective number of valence electrons NA given there, NA(He) = 1.430 and NA(Mg2þ) = NA(O2-) = 4.379. The Slater-Kirkwood formula gives C6,Mg = 0.7665 au and C6,O = 9.087 au. Inserting these values into eq 5 gives an effective C3 gas-surface interaction coefficient for He-MgO(100) of C3 = (π/6a2b)(C6,Mg þ C6,O) = 164.5 meV Å3. Fowler and Hutson25 point out that the actual He-MgO(100) C3 coefficient will be slightly smaller than the effective C3 coefficient obtained from a pairwise additive model for the HeMgO(100) interaction that employs accurate C6,X coefficients; this is primarily because the long-range attractive dispersion interactions between the He adsorbate and the Mg2þ and O2ions deep inside of the MgO substrate will be screened by the substrate’s dielectric properties. The actual He-MgO(100) C3 coefficient can be computed26,27 from the frequency-dependent 7113

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The Journal of Physical Chemistry A dipole polarizability of He at imaginary frequencies and the frequency-dependent dielectric function of the MgO substrate. The latter quantity can be obtained from a Kramers-Kr€onig inversion of experimental measurements of the frequencydependent reflectance of MgO crystals. On the basis of this approach, Chung and Cole4 computed C3 = 128 meV Å3. Using the same approach, but different experimental reflectance data, Nath et al.28 computed C3 = (151 ( 15) meV Å3, where the quoted uncertainty in the computed C3 value reflects the error possibly introduced into the calculation by an extrapolation of the reflectance measurements to infinite frequency. Our effective C3 coefficient is in rough, order-of-magnitude agreement with these two computations, which we take as an indication that our pairwise additive model for the He-MgO(100) interaction provides a reasonably realistic description of the attractive forces binding the He adsorbate to the MgO(100) surface. In principle, we could incorporate the dielectric screening effects just mentioned into our model by reducing the magnitude of the C6,X coefficients for Mg2þ and O2- ions in successively deeper layers of the MgO substrate. The additional effort involved in such an approach does not appear to be justified in the present study for three reasons. First, we note that the two values of C3 computed from experimental reflectance data differ from one another. This discrepancy would need to be resolved before we could use an “experimental” C3 value derived from reflectance data to choose a screening factor for the interior ions’ C6,X coefficients. Second, we have neglected the possibility that the C6,X coefficients for surface layer ions may differ from those for interior ions. The static polarizabilities of atomic anions in crystals increase with the anion-cation separation in the crystal,29 largely because the anions’ valence orbitals are compressed by overlap interactions with the surrounding cations. This compression reduces an anion’s polarizability from its gasphase value. Because the surface layer O2- ions in the MgO(100) surface layer are missing one Mg2þ neighbor, their static polarizabilities and C6 coefficients may be somewhat larger than those of the interior O2- ions (and may even be anisotropic). If the C6 coefficients of the surface layer O2- ions were larger than those of the interior ions, this would increase the dispersion forces binding the He adsorbate to the MgO(100) surface and would counteract, to some degree, the dielectric screening of the dispersion interactions involving interior O2- ions. Finally, the low-lying selective adsorption states that are supported by our model He-MgO(100) interaction have 99% or more of their probability density located at z values of z < 20 Å. At these heights, the laterally averaged He-MgO(100) interaction Vlat(z) has not yet reached its asymptotic C3/z3 limit, and therefore, it seems less critical that the asymptotic limit of Vlat(z) incorporate the effects of dielectric screening. We close this section by noting that we have ignored the contribution to the He-MgO(100) interaction that originates in polarization of the adsorbed He atom by the near-surface electric field of the MgO(100) substrate. This field decays exponentially with increasing z,8 and we expect it to play a role only for those He adsorbates that are in direct contact with the MgO(100) surface. Because the He polarizability is small, the polarization energy is a small perturbation to the pairwise additive model He-MgO(100) interaction; Guo and Bruch11 estimate that the polarization contribution to the total He-MgO(100) interaction is about 0.1 meV for He adsorbates in direct contact with the

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Figure 1. View from above the topmost layer of the MgO(100) surface. Smaller filled circles represent Mg2þ ions; larger open circles represent O2- ions. The dashed lines enclose the two-dimensional unit mesh associated with the (100) surface. The coordinates x and y are in units of Å.

MgO(100) surface. We will find that this is less than 1% of the well depth of the laterally averaged interaction Vlat(z).

3. RESULTS AND DISCUSSION 3.1. Bound States of the Laterally Averaged Potential Energy Surface. The pairwise additive model He-MgO(100)

interaction that we employ in this study has three adjustable parameters, the exponent p for the short-range repulsions and the coefficients Cp,Mg and Cp,O. However, if we define C6,sum = C6,Mg þ C6,O and Cp,sum = Cp,Mg þ Cp,O, we can write the laterally averaged He-MgO(100) interaction of eq 2 as ! ¥ Cp, sum C6, sum 2π X Vlat ðzÞ ¼ 2 ð6Þ a n ¼ 0 ðp - 2Þðz þ nbÞp - 2 4ðz þ nbÞ4 This makes it clear that Vlat(z) has only two adjustable parameters, namely, p and Cp,sum. If we rewrite the (p,6) pair interaction involving the summed coefficients Cp,sum and C6,sum in terms of an overall strength parameter V0 and a distance parameter σ, we have "   6 # Cp, sum C6, sum σ p σ V ðrÞ ¼ - 6 ¼ V0 ð7Þ p r r r r which shows that Cp,sum = C6,sumσp-6. Because σ is the distance at which V(r) = 0, it is a more physically intuitive parameter than is Cp,sum; we therefore treat p and σ as our adjustable parameters for Vlat(z) and compute Cp,sum from the relationship Cp,sum = C6,sumσp-6. We begin by choosing several values for the exponent p and, for each exponent, computing the energy E0 of the ground state of the laterally averaged interaction Vlat(z) as a function of σ. We 7114

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Figure 2. Dependence on p and σ of the ground-state energy E0 for the laterally averaged He-MgO(100) interaction Vlat(z). Horizontal dashed lines indicate the experimentally measured15 ground-state energy E0 = -(10.2 ( 0.1) meV.

perform this computation using the Numerov-Cooley method,30 terminating the sum of eq 6 at n = 1000. The results of these computations are shown in Figure 2. Several combinations of p and σ give ground-state energies in agreement with the apparent value of -(10.2 ( 0.1) meV inferred from experimental results.15 For each of these combinations of p and σ, we compute the energies of the excited states supported by Vlat(z). We find that at p = 9 and σ = 2.53 Å, where E0 = -10.23 meV, the first three excited states supported by Vlat(z) have energies in excellent agreement with the apparent excited-state energies inferred from experimental results;15 our computed energies are E1 = -5.29 meV, E2 = -2.41 meV, and E3 = -0.93 meV, while the experimental values are E1 = -(5.3 ( 0.1) meV, E2 = -(2.4 ( 0.1) meV, and E3 = -(0.9 ( 0.1) meV. The laterally averaged He-MgO(100) interaction and the probability densities for the five lowest-energy states are shown in Figure 3. Although for p = 9 and σ = 2.53 Å, Vlat(z) supports at least five excited states (in accord with experiment), the energies of the fourth and fifth excited states do not agree with the apparent excited-state energies inferred experimentally. The fourth and fifth excited states supported by Vlat(z) have energies E4 = -0.28 meV and E5 = -0.06 meV, but the corresponding experimentally inferred values15 are E4 = -(0.55 ( 0.1) meV and E5 = -(0.2 ( 0.1) eV. As Figure 3 shows, the fourth excited state has the bulk of its probability density at a substantial distance from the MgO(100) surface. The fifth excited state is dominated by even larger values of z. The disagreement between the computed values of E4 and E5 and those inferred from experiment might therefore, at first glance, reflect the increasing influence of the asymptotic (large z) portion of the laterally averaged potential energy function for the most weakly bound He-MgO(100) states; as we noted previously, the effective C3 coefficient derived from our pairwise additive model for the He-MgO(100) interaction is somewhat larger than the C3 coefficients derived from an analysis of the

frequency-dependent reflectance of MgO crystals. However, it seems intuitively reasonable that using a He-MgO(100) interaction whose effective C3 coefficient is too large would yield excited-state energies that are more negative than the true energies, not less negative, so that another explanation for the disagreement between theory and experiment is needed. It could be that the discrepancies between the computed and experimentally inferred values of E4 and E5 simply reflect shortcomings in our model He-MgO(100) interaction that are inherent in the rather simple, pairwise additive mathematical form that we have employed. The authors of ref 15 developed a laterally averaged HeMgO(100) potential energy function that supports six bound states, the energies of which are in good agreement with the experimentally inferred apparent ground-state energies. We denote this laterally averaged interaction as V*lat(z) to distinguish it from the function Vlat(z) obtained from our pairwise additive model He-MgO(100) interaction; it has the functional form 

Vlat ðzÞ ¼





C9 C3 z9 z3

ð8Þ

where C9* = 2.099  104 meV Å9 and C3* = 299.9 meV Å3. We compare Vlat(z) and V*lat(z) in Figure 4. For 1.8 e z e 5 Å, the two laterally averaged potential energy curves agree closely with * (z) is consistently one another. At larger values of z, however, Vlat more negative than Vlat(z), in accord with the fact that C3* is nearly twice as large as the effective C3 coefficient derived from our C6,Mg and C6,O values. One consequence of this is that the * (z) are more strongly high-lying excited states supported by Vlat bound than are the high-lying excited states supported by our laterally averaged pairwise additive model interaction. We note that the equilibrium He-MgO(100) distance pre* (z) is roughly 2.5 Å. A recent local dicted by both Vlat(z) and Vlat MP2 study14 of the He-MgO(100) interaction suggests that the equilibrium He-MgO(100) distance may be as much as 1 Å 7115

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Figure 3. (top) Laterally averaged He-MgO(100) interaction Vlat(z) for p = 9 and σ = 2.53 Å. The well depth of Vlat(z) is 13.75 meV. (bottom) Probability densities (in Å-1) for the five lowest-energy states for 4He in the potential energy function Vlat(z). The probability densities for the first, second, third, and fourth excited states are shifted up by 1.5, 2.5, 3.5, and 4.5 Å-1, respectively.

larger than this. However, the same study indicates that the equilibrium He-MgO(100) distance would shrink somewhat if a more sophisticated treatment of electron correlation could be used in computing the He-MgO(100) interaction. Unfortunately, no direct experimental information on the equilibrium atom-substrate distance is available to help benchmark either our model interaction or the results presented in ref 14. 3.2. Corrugation Amplitudes. Information about the lateral corrugation of the He-MgO(100) interaction can be obtained from an analysis of the Bragg diffraction of He atoms scattered off of the MgO(100) surface.15-18 To investigate the corrugation of our pairwise additive model He-MgO(100) interaction, we need to specify values for the two adjustable parameters C9,Mg

and C9,O, subject to the constraint that C9,Mg þ C9,O = C9,sum = C6,sum  (2.53 Å)3 = 95.347 eV Å9. As before, we introduce physically intuitive adjustable parameters σMg and σO, where C9,X = C6,XσX3; the pairwise interaction VHe-X(r) is zero at r = σX. Because C9,sum is fixed, only one of the two parameters σMg and σO is adjustable. Here, we choose σMg as our adjustable parameter and compute σO from C9,sum and σMg. The He-Mg2þ potential energy curve provides some guidance on the range of values that might be reasonable for σMg; the QCISD(T) He-Mg2þ potential energy curve of ref 31 suggests that σMg ≈ 1.55 Å. To extract information about the lateral corrugation of the He-MgO(100) interaction from the angular distribution of He 7116

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Figure 4. Laterally averaged He-MgO(100) interaction Vlat(z) for p = 9 and σ = 2.53 Å (solid line), compared with the laterally averaged interaction V*lat(z) described in ref 15 (dashed line). (top) The two potential energy curves are fairly similar at short range 1.8 e z e 5 Å. (bottom) At long range, however, Vlat * (z) is more strongly attractive than Vlat(z).

atoms scattered from the MgO(100) surface, a model for the lateral corrugation must be developed and used to compute the amplitudes of the Bragg peaks in the observed angular distribution.16 In refs 15 and 18, a simple one-parameter model for the lateral corrugation has been adopted ζðu, vÞ ¼ z0 - ðζ0 =2Þ½cosðπu=aÞ þ cosðπv=aÞ

ð9Þ

where ζ0 is the corrugation amplitude and z0 is the mean value of ζ(u,v). The corrugation function ζ(u,v) is interpreted as follows. For a specified positive interaction energy Vc, if V(u,v,z) = Vc, then ζ(u,v) = z. This indicates that ζ(u,v) depends implicitly on the value chosen for Vc, which is typically taken to be roughly the

incident kinetic energy of the He atoms being scattered from the MgO(100) surface. Here, we set Vc = 20 meV; the results that we present below are only very weakly dependent on our choice of Vc. Using the corrugation ansatz of eq 9, we see that when the He adsorbate sits above one of the Mg2þ ions, at (u=0,v=0), ζ = z0 ζ0, while when the He adsorbate sits above one of the O2- ions, at (u=a,v=a), ζ = z0 þ ζ0. At the points (u=(a,v=0) and (u=0,v= (a), which are midway between two nearest-neighbor Mg2þ ions (and also midway between two nearest-neighbor O2- ions), ζ = z0. If eq 9 faithfully represents the corrugation of the HeMgO(100) interaction, then the He adsorbate’s height above the O2- ion is 2ζ0 larger than the adsorbate’s height above the Mg2þ ion. The corrugation amplitude ζ0 can thus be computed from 7117

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Figure 5. Dependence on σMg on the height of the He adsorbate above Mg2þ ions (solid line) or above O2- ions (dashed line).

the difference between the He adsorption heights (when V = Vc) above the Mg2þ and O2- ions ζ0 ¼ ½ζðu ¼ a, v ¼ aÞ - ζðu ¼ 0, v ¼ 0Þ=2

ð10Þ

In addition, eq 9 implies that ζðu ¼ a, v ¼ 0Þ ¼ ½ζðu ¼ a, v ¼ aÞ þ ζðu ¼ 0, v ¼ 0Þ=2

ð11Þ

For He atom incident energies in the range of 13-60 meV, Benedek et al.15 estimated the corrugation amplitude ζ0 to be between 0.14 and 0.16 Å. In ref 18, electronic structure techniques based on density functional theory were used to evaluate the repulsive interaction between He adsorbates and the MgO(100) surface. These calculations showed that eq 9 was a good model for the lateral corrugation of the He-MgO(100) interaction and indicated that ζ0 ≈ 0.17 Å for Vc = 10 meV and ζ0 ≈ 0.19 Å for Vc = 80 meV, values that are broadly consistent with the experimentally inferred values15 for ζ0. The density functional theory results suggest that ζ0 does not depend strongly on Vc for Vc values between 10 and 80 meV. Figure 5 shows how the He adsorbate heights above Mg2þ and above O2- ions depend on σMg for our model He-MgO(100) interaction. The height ζ(u=0,v=0) above Mg2þ depends strongly on σMg and increases as σMg increases; on the other hand, the height ζ(u=a,v=a) above O2- is nearly independent of σMg. The gap between the two curves shown in Figure 5 is 2ζ0. Our model He-MgO(100) interaction has ζ0 values comparable to the experimentally inferred values at σMg ≈ 2.90 Å; this is almost twice as large as the value σMg ≈ 1.55 Å that we might deduce from the He-Mg2þ potential energy curve shown in ref 31. At σMg = 1.55 Å, our model He-MgO(100) interaction has ζ0 = 0.40 Å. Furthermore, if we set σMg = 2.90 Å to reproduce the experimentally inferred value of ζ0, the constraint C9,Mg þ C9,O = 95.347 eV Å9 gives σO = 2.49 Å, which is somewhat smaller than

σMg. This indicates that the hard-sphere radius of the He-O2interaction is slightly smaller than that of the He-Mg2þ interaction. Because the repulsive short-range He-X interaction arises in large part from overlap repulsion between the closedshell He and X species, we expect that the hard-sphere radius of the He-O2- interaction should be larger than that of the HeMg2þ interaction because the O2- electron cloud is larger. Finally, we note that for our pairwise additive model HeMgO(100) interaction, ζ(u=a,v=0) is independent of σMg. This originates in the functional form of our model interaction and the symmetry of the MgO(100) surface; when the He adsorbate is at (u=a,v=0), every He-Mg distance in the pairwise additive sum that defines the He-MgO(100) interaction is matched by an equal He-O distance. Consequently if C9,sum and C6,sum are held fixed, the He-MgO(100) interaction at (u=a,v=0) is independent of the way in which C9,sum is partitioned into C9,Mg and C9,O. For the C9,sum and C6,sum values that we employ, ζ(u=a,v=0) = 1.71 Å. According to eq 11, this value should be the average of ζ(u=0,v=0) and ζ(u=a,v=a). This is only true at σMg = 1.3 Å. For larger values of σMg, including both the physically reasonable value σMg = 1.55 Å and the value σMg = 2.90 Å that gives the correct lateral corrugation ζ0, the corrugation function of our pairwise additive model He-MgO(100) interaction deviates from the simple form given by eq 9. As σMg increases, this equation becomes an increasingly poor description of the corrugation of our pairwise additive model interaction, and the corrugation amplitude ζ0 computed from eq 10 no longer represents the actual lateral corrugation of the surface.

4. SUMMARY We have developed a simple model for the He-MgO(100) interaction based on a pairwise additive sum of Lennard-Jones (9,6) He-Mg and He-O potential energy functions. The laterally averaged He-MgO(100) interaction Vlat(z) derived from our pairwise additive model supports at least six bound states, in agreement with the results of selective adsorption 7118

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The Journal of Physical Chemistry A experiments15 for He adsorbates on MgO(100). However, only the four most strongly bound states supported by Vlat(z) have energies that agree with the apparent bound-state energies inferred from experiment. Furthermore, for realistic values of the parameter σMg that describes the repulsive core of the HeMg2þ potential energy curve, the lateral corrugation of our model interaction far exceeds both the corrugation inferred from an analysis of the Bragg diffraction of the incident He atoms15 and the corrugation of a potential energy surface for He adsorbates on MgO(100) computed using density functional theory.18 More work needs to be done to understand these discrepancies. One area where our model interaction could be improved is in the treatment of the long-range dispersion interactions between the He adsorbate and the surface layer O2- ions. As we mentioned previously, the O2- ions in the surface layer could have polarizabilities that differ substantially from the polarizability of O2- ions in deeper layers; the surface layer atoms could also have fairly anisotropic polarizabilities.22 It is not obvious how modifying the surface layer O2- polarizability tensors would affect either the bound-state energies of the laterally averaged interaction or the corrugation of the He-MgO(100) interaction; this is an area for future work. Additional information that could help us better understand the He-MgO(100) interaction would come from studies of the phonons of monolayers of He adsorbed on MgO(100) surfaces.32 Measurements of this type have already been made for H2 monolayers on MgO(100);33 if similar measurements were available for He monolayers on MgO(100), they would provide additional constraints on the form of the He-MgO(100) interaction.

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dx.doi.org/10.1021/jp1124316 |J. Phys. Chem. A 2011, 115, 7112–7119