Langmuir 1989, 5, 550-557
550
Art ic 1es Calculation of Physisorption Potentials in the Electron Gas Approximation+ G. Ihm* and Milton W. Cole 104 Davey Laboratory, Department of Physics, The Pennsylvania State University,
University Park, Pennsylvania 16802 Received October 27, 1988. I n Final Form: January 18, 1989
The laterally averaged physisorption potentials of He and Ne on noble metal, graphite, and MgO surfaces are calculated with a modified Gordon-Kim ( G K ) method. Each of the energy terms in the electron gas formula is slightly modified to include the inhomogeneity effects of the system. Overall, our results show this to be a simple way of predicting potentials that are reasonably consistent with experiments. The orientationally anisotropic H,/metal physisorption potential is also investigated and compared with experiments.
I. Introduction There has been great interest in obtaining an accurate description of gas-surface interaction potentials for a wide variety of systems. The importance of knowing this potential arises because it is a prerequisite for the understanding of various kinds of kinetic and equilibrium properties of particles near the surface. Theoretical understanding of the physisorption potentials is not, however, totally satisfactory as yet. While the long-rangetail of the interaction, i.e., the dispersion interaction, is rather well-known, the repulsive part of the physisorption potential is not. It has been understood, for example, that the repulsive part of the helium potential at a surface is approximately proportional to the local (unperturbed) surface electron density, according to both Hartree-Fock14 and effective medium calculations.&' However, discrepancies in the value of the proportionality constant between these theories are substantial, differing by factor of 3 from each other. The basic idea of these different theoretical approaches is summarized in a paper by Karikorpi et a1.8 Recently, Takada and Kohng calculated the repulsive interaction between a He atom and a metal surface using an S-matrix scattering theory. They showed that the proportionality constant depends on the work function CP; their value is about factor of 2 higher than the one predicted by the effective medium theory for a typical work function CP 4-5 eV. The situation is even less clear for the repulsive part of the physisorption potentials of other gases than He because of the assumption made in the theory that the adsorbate should be small and inert. One potential of particular interest is that of H2because of the role of the internal and rotational degrees of freedom. This system has been explored extensively with the help of diffractive selective adsorption (DSA) scattering resonan~es~"'~ and rotationally mediated selective adsorption (RMSA) scattering resonances.1° The HREELS (high-resolution electron energy loss spectroscopy) spectra for H2-Ag14 and Hz-Cu16 physisorbed systems reveal roF;:
t Presented a t the symposium on "Adsorption on Solid Surfaces", 62nd Colloid and Surface Science Symposium, Pennsylvania State University, State College, PA, June 19-22, 1988; W. A. Steele, Chairman.
0743-7463/89/2405-0550$01.50/0
tational excitations that show small shifts relative to gas-phase values, implying that the molecule behaves like an essentially free 3D rotor on these surfaces. Comparative DSA studies'O utilizing rotationally state-selected H, and D2molecules have revealed small level shifts and line width changes which are the result of an average over individual J- and m-dependent level shifts, where J and m are rotational quantum numbers. The first experimental evidence for the anisotropy of the molecule-surface potential was obtained by Chiesa et al.l' by measuring the sublevel splitting due to the different values of J and m (order of 1-meV level splitting is observed for the free hydrogen molecule). However, only one pertinent theoretical study has been done, that by Nordlander et a1.16 The theory is adapted from a model due to Zaremba and Kohn,' in which the repulsive interaction is formed by the overlap of the closed shell adsorbate with the substrate conduction electrons. They showed that difficulties arise because of the lack of spherical symmetry, the delicate partial cancellation of repulsive and attractive branches, and that there exists extreme sensitivity of the net potential to the imperfectly known van der Waals parameters. In this paper, we present physisorption interaction potentials for several systems (He, Ne, and H2 on metals, graphite, and MgO) by applying a recently formulated (1) Zaremba, E.; Kohn, E. Phys. Reu. B 1977, 15, 1769. (2) Harris, J.; Liebsch, A. J. Phys. C 1982, 15, 2275. (3) Nordlander, P. Surf. Sci. 1983, 126, 675. (4) Nordlander, P.; Harris, J. J. Phys. C 1984, 17, 1141. (5) Esbjerg, N.; Horskov, J. K. Phys. Reu. Lett. 1980, 45, 807. (6) Lang, N. D.; Norskov, J. K. Phys. Rev. B 1983, 27,4612. (7) Manninen, M.; Norskov, J. K.;Puska, M. J.; Umrigar, C. Phys. Reu. 1984,29, 2314. (8)Karikorpi, M.; Manninen, M.; Umrigar, C. Surf. Sci. 1986,169,299. (9) Takada, Y.; Kohn, W. Phys. Reu. B 1988,37,826. (10) Yu, C.; Whaley, K. B.; Hogg, C. S.; Sibener, S. J. Phys. Rev. Lett. 1983,51, 2210. (11) Chiesa, M.; Mattera, L.; Musenich, R.; Salvo, C. Surf. Sci. 1985, 151, L145. (12) (a) Yu, C.; Whaley, K. B.; Hogg, C. S.; Sibener, S. J. J. Chem. Phys. 1986,83, 4235; (b) 1985,83, 4235. (13) Andersson, S.; Wilzen, L.; Persson, M., preprint. (14) Avouris, Ph.; Schmeisser, D.; Demuth, J. E. Phys. Reu. Lett. 1982, 48, 199. (15) Andersson, S.; Harris, J. Phys. Reu. Lett. 1982, 48, 545. (16) Nordlander, P.; Holmberg, C.; Harris, J. Surf. Sci. 1985, 152/153, 702.
1989 American Chemical Society
Langmuir, Vol. 5, No. 3, 1989 551
Calculation of Physisorption Potentials electron gas approximation (modified Gordon-Kim method)I7 in the calculation of the repulsive part of the gas-surface interaction potential. Our main motive for these studies is to give a better understanding of these systems from a theoretical point of view and so explain a rather abundant set of experimental results for these systems. The essence of the electron gas approach, proposed initially by Gordon and Kim ( G K henceforth),18is that each system is represented by its electronic charge density, and in the overlap region the densities are assumed to be additive. This method enables us to determine the interaction energy from knowledge of only the charge density of each species and the energy functional. It does not require that the surface charge density be slowly varying on the scale of an atom (so it works potentially for big atoms as well as He). In further developing the G K method, we assumed that most of the error originates from the incorrect energy functionals, while the basic idea of density additivity holds in van der Waals systems. Some justification for this exists. First, the rearrangement of electron density for van der Waals systems is quite small. For example, for two He atoms, the total molecular density is everywhere within r 1 0 % of the sum of the atomic densities, even when the internuclear separation is as small as one-half of the equilibrium ~eparation.'~Even in bimetallic interfaces, self-consistently calculated electron density distributions are very similar to the simple charge superposition resultam Second, Heller et alF1 showed that the extra interaction energy due to the difference between the true density and the simply added density, say A(r), is of the order of A(@. Finally, later variants of the G K were successful in predicting intermolecular forces between other closed-shell atoms, ions, and molecules. In another paper1' we developed our modified G K method. Improvement of the energy functional includes a correction to remove the self-exchange term= from the exchange energy and adjustments to take into account the nonuniformity of the system. Detailed derivation of our method can be found in this paper,17where we succeasfully tested it on a prototype model problem: the calculation of the immersion energy of an atom into jellium. The original G-K method was used previously by Freeman%and by Van Himbergen and SilbeyaS in studying physical adsorption. Their results were qualitatively reasonable. It turned out in the present work that our method is both simple and reasonably accurate in calculating physisorption problems. The method is explained in section 11, and the results are analyzed and compared with experiments in section 111. Finally, conclusions are provided in section IV.
11. Model We study here only the laterally averaged potential (17) Ihm, G.;Cole, M. W., to be published. (18) Gordon. R. G.: Kim. Y. S. J. Chem. Phvs. 1972.56. 3122. (19) (a) Gordon, M: D.; Secrest, D.; Llaguno,-C.J.Chem.'Phys. 1971, 55,1046. (b) Jorgenson, W. L.; Allen, L. C. J. Am. Chem. SOC.1971,93, 567. (c) Bader, R. F. W.; Preston, H. J. T. Znt. J. Quantum Chem. 1969,
-. (20) Ferrante, J.; Smith, J. R. Phys. Reu. E 1985, 31, 3427.
3. 927.
(21) Heller, D. F.; Harris, R. A.; Gelbart, W. M. J. Chem. Phys. 1975,
62. - -, 1947. - - - ..
(22) Clugston, M. J. Ado. Phys. 1978,27,893. (23) Lloyd, J.; Pugh, D. Chem. Phys. Lett. 1974, 26, 282. (24) (a) Waldman, M.; Gordon, R. G.J. Chem. Phys. 1979,1325,1340. (b) LeSar, R. J.Phys. Chem. 1984,88,4272. (25) (a) Freeman, D. L. J. Chem. Phys. 1975,62,941. (b) Van Himbergen, J. E.; Silbey, R. Solid State Commun. 1977,23,623. (26) We actually ignore any lateral variation of the solid's input functions (e.g., charge density). If that were included, it would in principle affect the laterally averaged potential somewhat.
1
V(z,8) = -Jd2r A
V(r',e)
where A is the surface area, z is the vertical coordinate, and 8 is the polar angle of the molecular axis with respect to the surface normal. The laterally averaged He,Ne/ surface interaction potential depends only on z, while the H2/surface interaction potential depends on both z and 8. We can write the total interaction potential as a sum of the repulsive part, VR, and the attractive part, VA: v(z,e) = vR(z,e) + vA(z,e) (1) where VA is the dispersion energy. For an ionic crystal like MgO or an alkali halide, we have also the induced dipole energy, so that the interaction can be written v(z) = VR + v, - y z a p (2) 'where a is the static polarizability of the adatom and E(r) is the ionic electric field. Higher order polarization terms (induced quadrupole, etc.) are assumed to be negligible. Repulsive Part of Interaction Potential. The repulsive part of the interaction energy is obtained with a modified G K method without the correlation energy (which is included in the attractive part of the potential as the van der Waals interaction) vR(z,e) = v,,,(z,e) + Jd3r [ ( P a + Ps)EG(Pa + P s ) - P J G ( P a ) - P$G(PI)I (3) where pa and ps are the atomic and solid densities, respectively. The electron-gas functional (in hartrees) is Ec(p) Sk(3/10)(3g)2/3p2/3- S,(3/4)(3/~)'/~p'/' (4) where densities are in atomic units (1 au = 1b0hr3 = 6.76 A-7. Sk and S, are scaling parameters which depend on the adsorbed atom; they were one in the original G K theory. Here they are evaluated from" Sk = KHF/KED (5)
S, = Y(XHF/XEG)
(6)
where the Sk correction factor is the Hartree-Fock kinetic energy relative to the electron gas approximation for an atom; the S, ratio is a similar ratio for exchange. This nonuniformity correction is guided by previous ~ o r k ~ , ~ ~ and a recent paper of Lee and Parrn; the latter developed a Gaussian ansatz for the single particle density matrix which leads to the same functional form as the uniform electron gas but with a different coefficient. y represents the self-energy correction factor, which depends on the number of valence electrons of the adsorbate. We modified Lloyd and Pugh's correction facto$ for molecular systems, yip, because one component of the system (surface) is infinitely extended while the other (adsorbate) is finite. We simply take the geometric mean rule, i.e. Ygas-surf
=
(YgaB-gasYsurf-sur31'z
= Ylp 1/2
(7)
where ysurf-surf = 1. The s k (S,) values are 1.1175 (0.4217), 1.092 (0.692), and 1.1678 (0.422) for He, Ne, and Hz, respectively. The first term, Vmd(z,8), represents the Coulomb energy, which can be written as V,,,(z,8)
= 1 d 3 r ' p,(r")@(z+z')
(8)
Here @ ( z )is the electrostatic potential due to the surface charge. We neglect any lateral dependence. For the H2 molecule, we use prolate spheroidal coordinates (e, Q, and (27) Lee, C.; Parr, R. G. Phys. Reu. A 1987,35, 2377.
552 Langmuir, Vol. 5, No. 3, 1989
Ihm and Cole
4) with R as the distance between the two nuclei in the hydrogen molecule. Then z' = ( 1 / 2 ) R cosh u cos u = (1/2)67for the J = 1, m = 0 state (8 = OO), z ' = ( 1 / 2 ) Rsinh u sin u cos 4 for the J = 1, m = 1 state (8 = goo); and d3r' = ( R / 2 I 3d4 dE dv (t2- v2). Here J = 1 corresponds to ortho hydrogen and J = 0 corresponds to para hydrogen. In our calculation, Roothaan-Hartree-Fock atomic wave functions28are used for He and Ne while Kolos and Root h a n ' s electron densitymVmis used for H2. For the metal surface charge density and electrostatic potential, Lang and Kohn's self-consistent calculation^^^ are used (r, = 3.02 for silver and r, = 2.67 for copper), except that the longrange tail of the electron density is fitted to the following form to give the asymptotic behavior suggested by recent calculation^,^^^^^ depending on the face of the metal:
+ E F ) ) 1 /exp(-2kWz) 2 ( 1 + kFZ/9)'
p0(2W / (W p(z) =
(9)
Here pot EF, and kF are the electron density, the Fermi energy, and the Fermi wave vector of the bulk jellium, respectively, while W is the work function; k , = ( 2 r n W Y 2 /h . Attractive Part of Interaction Potential. We consider first metal surfaces. For H2, the attractive part of the potential, VA, consists of dipole and quadrupole parts: c3
vA(z,8) = - - f ( k $ ) ( l
c5
0.06Pz(c0s 8)) - -g(k&)
x3
(10)
x5
where x = z - zo and zo is the reference plane position for the van der Waals interaction. The origin of z coordinates for metals is taken as the jellium edge; for graphite and MgO it is the top ionic plane. The two terms of this equation represent the dipolar and quadrupolar dispersive interactions, respectively. The angular term in the dipolar interaction is due to the anisotropy of the molecular polarizability; the factor 0.06 multiplying the Legendre polynomial is proportional to the difference between the polarizabilities perpendicular and parallel to the bond axis of hydrogen.34 C3 and C5are the dipolar and quadrupolar van der Waals coefficients. The functions f ( x ) (ref 16) and g ( x ) describe corrections to the asymptotic van der Waals interaction, and k, is a cutoff wave vector, which is the order of inverse of the size of the molecule. f ( x ) and g ( x ) are defined as
+ 11 exp(-2x) g ( x ) = 1 - [ 2 x ( 1 + x + Y3x2 + Y3x3) + 11 exp(-2x) f ( x ) = 1 - [2x(1 + x )
- -
--
(11)
Table I. Interaction Potential Parameters U s e d i n Eq 10 a n d 15.' system C3, meV.AS C5, meV-A5e 23, aut k,, au He-Cu 235' 0.324 1.0 Ne-Cu 488' 0.300 1.0 HZ-CU 675' 952 0.409 0.46 He-Ag 249, 0.285 1.0 Ne-Ag 520d 0.266 1.0 H2-Ag 714b 1013 0.376 0.46 H2-Ge 550,8 714" 700 He-MgO 151' 73.2 Ne-MgO 325' 202
'
Reference 84. Reference 85. Reference 86. Reference 87. e S e e text eq 13 and ref 37. 'Reference 36. BReference 64. "Reference 62. 'Reference 79. jC, and C5 are the dipolar and quadrupolar van der Waals coefficients. z, is the reference plane position for the van der Waals interaction, measured from the jellium edge. k, is a wave vector cutoff introduced in eq 10-12.
fit to experimental data. However, their value of zo was incorrect (by 0.4 and they did not include the quadrupole term, which gives about 10% contribution of the dipole term near the equilibrium position of the interaction potential. We used Liebsch's improved values% for zo, calculated from the dynamical image-plane position of a metal surface for external fields that vary exponentially in time. The quadrupolar van der Waals coefficient, C5,can be written in a simple form3' in the approximation of a local dielectric response of the substrate: goY&,E, c, = W E , + E,) (13) where yo is the static quadrupole polarizability of the hydrogen molecule.38 E , and E, are characteristic excitation energies for quadrupole excitation of the molecule and of the solid. go depends on the substrate. We have neglected the interaction due to the permanent quadrupole moment (Qzz = 0.484 of H2 because it is small. By use of the perfect image method* with the erect orientation of Hz (maximum response), it is estimated to be -C/z5, where C = ( Q Z z 2 ) / 9 = 6 2.75 meVA5, which is negligible compared to C,. All the values we used for the interaction parameters are given in Table I. For He and Ne interactions, we neglect the quadrupolar dispersive interaction because it has a neglegible effect compared to the dipolar term (less than 5% of the latter); hence
(12)
which have the following behavior. As x m, both become 1. As x 0, f / x 3 4 / 3 and g l x 5 4/15. We used k , = 0.46 au (winverse of size of H2).In a similar study (H2/Ag),Nordlander et al.16 regarded k, as a free parameter because of the extreme sensitivity of the results to the van der Waals corrections and got k , = 0.4 au based on a (28) Clementi, E.; Roetti, C. At. Data N u l . Data Tables 1974,14, no. 3-4. (29) Koloe, W.; Roothaan, C. C. J. Reu. M.Phys. 1962, 32, 219. (30) Stewart, R. F.; Davidson, E. R.; Simuson, W. T. J . Chem. Phvs. 1965,42, 3175. (31) Lang, N. D.; Kohn, W. Phys. Reu. B 1970, 1, 4555. (32) Dondi, M. G.; Mattera, L.; Terreni, S.; Tommasini, F.; Linke, U. Phys. Reu. B 1986,34, 5897. (33) Gupta, A. K.; Singwi, K. S. Phys. Reu. B 1977,15, 1801. (34) The factor 0.05 is introduced bv J. Harris and P. Feibelman (Surf. Sci. 1982, 115, 133). We used 0.06, which is more accurate. We'todk values (in au) from: Kolos, W.; Wolniewicz, L. J . Chem. Phys. 1967,46, = 5.8, DIL- DIT = 1.8). 1426 (CUL= 6.38, CUT= 4.578, ck = ( 1 / 3 ) ( a ~+ 2~~77) Rychlewski, J. Mol. Phys. 1980, 41, 833 (CYL = 6.387, DIT = 4.579, 5 = ( 1 / 3 ) ( @+~ 2 a ~ =) 5.18, DIL- DIT = 1.8).
k, = 1 au is used for He and Ne adsorption on the basis of the inverse square root of (9)of He and Ne (0.92 and 1.09, respectively). In contrast to the H2 adsorption, adsorption potentials for He and Ne turned out to be quite insensitive to the choice of k,; less than 1% changes were observed for the variation of Ak, = f0.2. We consider next the cases of graphite and MgO. The attractive van der Waals interaction is written41
~~
(35) Persson, B. N. J.; Zaremba, E. Phys. Rev. B 1985, 32, 6916. (36) Liebsch, A. Phys. Rev. B 1986, 33, 7249. (37) Jiang, X.; Toigo, F.; Cole, M. W. Surf. Sci. 1984, 145, 281. (38) Langhoff, P. W. J. Chem. Phys. 1971,55, 2126. (39) Radzig, A. A.; Smirnov, B. M. In Reference data on atoms, molecules and ions; Springer: Berlin, 1985. (40) Jiang, X. P.; Toigo, F.; Cole, M. W. Chem. Phys. Lett. 1983, 101, 159.
Langmuir, Vol. 5, No. 3, 1989 553
Calculation of Physisorption Potentials
(1lO)O
Eo El
E2 E3
E4 D zeq
6.00 3.15 1.57 0.92 0.29 7.83 5.2
( l l O ) b exptl
Table 11. Results for He-Agh ( l l O ) c exptl
4.48 f 0.05 2.31 f 0.06 1.04 f 0.07 0.43 f 0.08 6.0 f 0.1
4.44 f 0.02 2.18 f 0.02 0.92 f 0.02 0.31 f 0.02 0.095 f 0.015 6.0
(111)
theord
theore
4.3 2.2 0.96 0.23
3.53
5.7 5.7
4.95
9.3' 10.99
"This work. bReference45. cReference 32. dReference 16. eReference 1. 'Reference 46. BReference 47.
'k
Energies ( E )in meV, distances
(D)(measured from the jellium edge) in au. For distances, 1 au = 0.529 A. Bound-state energies, E,, well depth, D, and minimum location, zw, are presented and compared with previous experiments and theories.
EO El
E2 E3
E4 D zeq
(100)" 5.40 2.60 1.19 0.48 0.17 7.14 5.6
(110)' 5.01 2.49 1.16 0.48 0.17 6.70 5.6
Table 111. Results for He-Cut (111)' (113Ib 6.79 3.89 2.10 1.05 0.48 8.85 5.0
4.6 f 0.1 2.2 f 0.1 1 0.1
(115Ib 4.45 f 0.15 2.1 f 0.1 0.9 f 0.1 0.35 f 0.15
6.35 f 0.05b3e
6.35 f 0.05b2e
*
(lO0)C
theord 4.2 2.0 0.80 0.13
9.8
5.6 5.7
"This work. bReference 42. CReference44. dReference 16. eReference 43. 'Energies ( E )in meV; distances (D)in au. Other notation same as in Table 11.
where the summation over j represents the consecutive planar contributions and cl is the interplanar distance: 3.37 A for graphite and 2.103 A for MgO. 111. Results He-Metal. The potential parameters used in our calculations are given in Table I. Results are given in Tables I1 and I11 along with experimental and other theoretical results obtained by numerical solution of the Schrodinger equation. Our results for He-Cu (100, 110, 111)interactions show that the face dependence of the interaction is closely related to the work function difference between faces. Adsorption on the face with a larger work function produces a larger well depth because of the shorter decay length of the surface electron density (i.e., closer approach to the surface). These differences have not been used previously in potential energy calculations. Our values for Eo and D are about 0.5 meV larger than values obtained from bound-state resonances in scattering experiment^.^^,^ Specifically, Perreau and L a p ~ j o u l a d emeasured ~~ the bound-state energies for selective adsorption on Cu(113) and Cu(115). They found Eo = 4.6 f 0.1 meV for Cu(113) and Eo = 4.45 f 0.15 meV for Cu(115). Lapujoulade et al.'s measurementsu of the thermal attenuation of elastic helium scattering from Cu(100) gave D = 9.8 meV, which is higher than expected from the other data. The calculated values of Zaremba and Kohn' (E, = 3.55 meV) and Nordlander and Harris4 ( E , = 4.2 meV) are lower than experimental values. Our results for He-Ag listed in Table I1 are similar to those for He-Cu surfaces. We get Eo = 6.00 meV, which is higher than the selective adsorption measurement (about 4.5 meV in ref 32 and 45). Earlier results for well depth (9-11 meV), based on diffraction intensities, are presumably too high.46,47 (41) See: Toigo, F.; Cole, M. W. Phys. Reu. E 1985, 32, 6989. It concerns the omission of damping; this approach is nearly universal for these systems. (42) Perreau, J.; Lapujoulade, J. Surf. Sci. 1982, 119, L292. (43) Gorse, D.; Salanon, B.; Fabre, F.; Kara, A.; Perreau, J.; Armand, G.; Lapujoulade, J. Surf. Sci. 1984, 147, 611. (44) Lapujoulade, J.; Lejay, Y.; Armand, G. Surf. Sci. 1980, 95, 107. (45) Luntz, A.; Mattera, L.; Rocca, M.; Tommasini, F.; Valbusa, U. Surf. Sci. 1980, 120, L447.
H2-Ag. The laterally averaged interaction potential V(z,O) can be expanded in a Legendre series with coefficient V,(z);only even terms in 1 contribute for homonuclear molecules. Approximately, we have v(z,e) = v,(z) + v,(z)P,(~os01 (16) First-order perturbation theory48 in the second term gives EnJ'"' = E ,
+ J(J+ l ) B + AEnJJ"
(17)
where B is the rotational constant (about 86 K) and E , is the eigenvalue for motion in the laterally averaged well. The shifts for molecules with J # 0, m # 0 in a bound state n of the isotropic potential V,(z) are then expressed as AEnJJ"= GV,(J,mlP,(cos O)lJ,m)
(18)
where SV, = (9,~V2(z)~9,). are the wave functions corresponding to E,,. IJ,m) denotes a spherical harmonic. Finally we have48 AE,J'"' =
-(
36V, $-m2 ___ ( 2 J + 3) 2 5 - 1
;)
(19)
If the repulsive term (attractive term) dominates in V2(z), 6 V, is positive (negative). We calculated the interaction potential of ortho hydrogen ( J = 1) adsorbed on silver (110), which is expected to be rather smooth. The J = 1, m = f l states correspond to O = 90°, while the J = 1, m = 0 state corresponds to O = 0'. The orientation dependence of V, i.e., the V, term, is a compromise between the dispersion interaction, which prefers erect orientation (see eq lo), and a repulsion, which favors flat orientation. The balance is a delicate one in the H2case. Our results (Figure 1)show that the interaction potential is almost isotropic and thus is consistent with experimental results.'lJ4 Well depths of 31.1 (28.4) meV of the H2/Ag interaction potential are estimated for the J = l , m = 0 (J= l, m = f l ) states. Figure 2 shows each contribution to the total interaction potential. The estimated quadrupolar contribution is about 10% of the dipolar van der Waals con(46) Horne, J. M.; Miller, D. R. Surf. Sci. 1977, 66, 365. (47) Weinberg, W. H. J. Chem. Phys. 1972,57, 5463. (48) Harris, J.; Feibelman, P. J. Surf. Sci. 1982, 115, 133.
554 Langmuir, Vol. 5, No. 3, 1989
Ihm and Cole Table IV. Results for H2-Ag' (110)* exptl
(110)O J=l,m=fl 23.5
EO
J=l,m=O 26.0
E,
15.1
17.1
E2
8.8
10.1
E3
4.3 2.20 1.04 0.44 28.4 4.8
4.9 2.55 1.23 0.53 31.1 4.8
E4 E6 E6
D Zeq
(111)' exptl
(1ll)d
n
P
n
P
P
26.0 f 0.4 24.5 f 0.4 17.1 f 0.1 16.1 f 0.1 10.1 f 0.2 9.5 f 0.2 5.3 f 0.2 2.5 f 0.2 1.1 f 0.3
25.9 f 0.4
25.53 f 0.2
25.74 f 0.7
23.8
16.4 f 0.2
16.55 f 0.1
16.87 f 0.7
15.5
10.0 f 0.2
9.91 f 0.1
10.11 f 0.1
9.9
5.4 f 0.3 2.7 f 0.3
5.35 f 0.1 2.52 f 0.07 1.12 f 0.05
31.7 f 0.2
5.47 f 0.1 2.61 f 0.13 1.21 f 0.06
32.46
5.1 2.5 1.1 0.4
32.46
OThis work. *Reference 11. cReference 10. dReference 16. e J and m are rotational quantum numbers. n and p are natural and para hydrogen, respectively. Other notation same as Table 11. 50
0
25
I' !
-
0
E i
q:
>
-20
0
W
-25
0
o
~
o
o
?
0 '
e
.
ob ~
A d.
'
-40 3
b .
4
5
6
7
8
I
9
z ( a.u. )
Figure 1. H2-Ag(l10) interaction potential versus distance from the jellum edge. Open and solid squares correspond to J = 1, m = 0 and J = 1, m = h1 rotational states of hydrogen, Le., erect and flat orientations, respectively.
tribution a t the well minimum; hence it affects the well depth by -20%. We find that the interaction potential is quite sensitive to the choice of k, (in contrast to the case of He or Ne/Ag); AEo 2 meV for Akc = 0.3 au. This means that this poorly known quantity is the major source of uncertainty in the theory. The eigenvalues and well depth of the interaction potential are presented in Table IV,along with experimentalresults. The ground-state-level splitting is here calculated as AEtio - AEo"*' = -2.5 meV. In previous work16 it was assumed that v(z,e) = v,(z) + V,(Z)P,(COS e) (20) vR(z,e)= vr0w(i+ hp2(c0se)) (21) vA(z,e) = vAo(z)(l + ppZ(cOs 0)) (22) v2(z) = Xvro(z) + @vAo(z) (23) Thus the eigenvalue shifts corresponding to different J, m states are determined by two anisotropy parameters, X and p. Nordlander et ala's calculations give X = 0.18 and P = 0.05 so that SVo = 0.9 meV, AEolto = 0.36 meV, AEO1,*' = -0.18 meV, and AEo'so- AEO1**l= 0.54 meV. Schinke et al.49did close-coupling calculations to fit Chiesa et al.'s
-
(49) Schinke, R.; Engel, V.; Voges, H. Chem. Phys. Lett. 1984,104,279.
-50:
5
7
z ( a.u. )
Figure 2. H2-Ag(ll0) interaction potential for the J = 1, m = 0 rotational state by components: open squarea for the total,solid squares for the attraction, solid diamonds for the repulsion, open triangles for the dipole, and open diamonds for the quadrupole dispersion interactions, respectively.
experimental results'l and got similar values for these parameters, i.e., X = 0.2 and @ = 0.05. They determined the level splitting hEol~o - AEO1~*'= 1.5 meV. Yu et al.so assumed X = p in their analysis of rotationally mediated selective adsorption resonances and achieved best agreement between theory and experiment for X = p = 0.05. Whaley et al.51 also did a detailed investigation of this problem. They performed close coupling scattering calculations to obtain transition probabilities and resonance shifts in agreement with experiment. Their best fitting values for X and p are X = 0.09 and p = 0.06, giving AEo1p0 - hEO1,*l = -0.323 meV; the values X = 0.2 and 0 = 0.0516v49 did not work well in their analysis. This analysis shows that the lowest state is the J = 1, m = 0 state, in agreement with our work, and that the overall anisotropy is apparently even smaller than our potentials indicate. It has been found e ~ p e r i m e n t a l l f lthat ~ ~ ~the ~ ~intensities of diffracted beams for H2 scattering from Ag(1ll) (50) Yu, C. F.; Hogg, C. S.;Cowin, J. P.; Whaley, K. B.; Light, J. S.; Sibener, S. J. Zsr. J . Chem. 1982, 22, 305. (51) Whaley, K. B.; Yu, C.; Hogg, _ _ C. S.;Light, J. C.; Sibener, S.J. J.
Chem. Phys. 1985,83,4235. (52) Boato, G.; Cantini, P.; Tatarek, R. J. Phys. 1976, F6, L237. (53) Horne, J. M.; Yerkes, S.C.; Miller, D. R. Surf. Sci. 1980, 93,47.
Calculation of Physisorption Potentials
EO El E2 E3 E4 E6
D %
'This work.
Langmuir, Vol. 5, No. 3, 1989 555 Table V. Results for H2-Cuf (110)' J=l,p=fl J=l,pO (loo)*exptl 18.16 20.21 25.5 f 0.2 11.21 12.40 16.9 f 0.2 6.68 7.43 10.2 f 0.2 3.79 4.25 5.8 f 0.2 2.03 2.29 2.8 f 0.2 1.01 1.15 22.60 24.83 30.9 5.2 5.4
(lOO)n J = 1,m=f1 J=l,p=O 19.13 21.26 11.72 13.34 6.97 8.17 3.95 4.79 2.11 2.66 1.05 1.39 23.66 26.01 5.2 5.2
(110)~ exptl 17.7 11.0 6.0 3.5 1.9 0.6 22"
(110)d 19.1 11.9 6.8 3.5 1.6 0.6
Reference 13. Reference 56. Reference 16. " Reference 55. Relerence 57. 'Notation same as Table IV. 20 0
0
.
'0-
0
0-
0
:
> -401 I
...* .... 4
5
...e'
-20.
6
7
8
-30;
l
..... .I
"
4
"
5
z ( a.u. )
Figure 3. He, Hz,and N e A g ( l l 0 ) interaction potentials; notation is same as Figure 1,except open squares for H e A g , solid squares for Hz-Ag, and solid diamonds for Ne-Ag.
were about 1order of magnitude larger than the intensities for He. We find this plausible because the equilibrium position of the potential well for Hz/Ag(llO) is -0.2 A closer to the surface than for He-Ag(ll0); this suggests that the hydrogen molecule experiences a stronger corrugation of the surface than helium does in the scattering process. Thus the higher diffraction intensities for Hz/ Ag(ll1) scattering can be understood. This description needs to be supplemented by actual calculations of the Fourier components of the potential and of the scattering intensities. As it stands, the results are similar to those of Liebsch and Harris.64 Hz-Cu. We calculated the ortho hydrogen interaction with Cu (100) and (110) faces. Our results are given in Table V. The interaction potential is rather isotropic, as for Ag. The well depth for the (100) face is about 1 meV deeper than for the (110) face, so the facet dependence of the interaction potential is rather weak. This small dependence is related to the facet-dependent work function of the metal. Our calculated well depths in this work are lower than Andemson's values13but higher than the values of others. Glachant and BardiS6did a thermodynamic study for the H2/Cu(l10) system and deduced a well depth D = 22 meV for this system. This depth is comparable to that found in the scattering experiments; Lapujoulade and Perreauss obtained a ground-state energy of 17.7 meV (54)Liebsch, A.; Harris, J. Surf. Sci. 1983,130, L349. (55)Glachant, A.; Bardi, U. Surf. Sci. 1979,87,187. (56)Lapujoulade, J.; Perreau, J. Phys. Scr. 1983, T4, 138.
"
6
"
7
"
0
z ( a.u. )
Figure 4. He, Hz, and Ne-Cu(ll0) interaction potentials in comparison; notation same as Figure 3.
EO
E; E2 E3 E4
E5 E6
D ZWl
Table VI. Results for Ne-Ag, Cud Ag(ll0)' Cu(ll0)' Cu(llO)* Cu(llO)c 23.12 16.00 10.8 20.12 13.53 8.34 17.40 11.33 6.74 15.02 9.47 5.36 12.93 7.87 4.16 11.08 6.51 3.17 9.46 5.34 2.47 24.72 17.34 12.0 33.2 4.6 5.2
'This work. *Reference 59. Reference 58. Notation same as Table 11. for H2/Cu(l10),and Stiles and Wilkins6' determined D == 22.3 meV for Hz-Cu(lOO). Nordlander et al.I6 calculated Eo= 19.1 meV for the H2/Cu(l10) interaction potential. Recent BSR measurements for H2/Cu(l10) by Andersson et al.I3 deviate a lot, however, from the previous estimation of Eoand D. They found Eo = 25.5 f 0.2 meV and D = 30.9 meV. Furthermore, their data strikingly resemble those reported for the H2/Ag system. Thus these most recent data are semiquantitatively consistent with our findings. Ne-Ag,Cu. There has been little work done for these systems, either theoretically or experimentally. Our results are listed in Table VI. We find that the classical turning point of the Ne-metal interaction potential is closer to the surface than that of He-metal, suggesting that the non(57) Stiles, M. D.; Wilkins, J.
s.,preprint.
556 Langmuir, Vol. 5, No. 3, 1989
Ihm and Cole
specular intensity for Ne-metal should be higher than those for He-metal. Again, calculations of the lateral dependence of V are needed to confirm this. Our potential for Ag is nearly 40% deeper than that for Cu. The well depths for Ne are about a factor of 3 larger than the well depths for He but 25% smaller than those for H2. Boheim and Brenig5*studied the Debye-Waller factor of Ne-Cu from inelastic atom-surface scattering and deduced D = 33.2 meV, which is almost factor of 2 higher than our value. S a l a n ~ studied n ~ ~ the diffraction from the (110) face of copper. He derived Eo = 10.8 meV from the selective adsorption measurements and estimated that D = 12 meV. These latter values are about 30% lower than our values. They found the potential to be more corrugated and steeper in its repulsive part than in the HeCu(l10) case, consistent with our calculation; the derived corrugation amplitude of the isopotentials for Ne-Cu(ll0) was 0.23 and 0.13 8, for He-Cu(ll0). Unfortunately, there are no more reliable experiments or theories to explain these discrepancies. We tried to test this by using semiempirical correlations. Ihm et alewfound that there is a strong correlation between the parameters in physisorption potentials (C3/D)1/3 = 1.2(zeq- z,)
(24)
The correlation coefficient of the above fit was 0.94. If we test it with the Ne-Cu(ll0) system, we get zw - z, = (C3/D)ll3/1.2= 2.79 and 2.47 8, using D = 12 and 17.3 meV, respectively. Our actual value (from Tables I and VI) for zeq - z, is 2.49 A, which agrees well with the prediction of eq 24 obtained by using our value of D = 17.3 meV. A conceivable, but very speculative, resolution of the discrepancy is the possibility of missing deeper levels in selective adsorption experiments; without the two lowest bound states, Salanon’s values are comparable to ours. H2-Graphite. Mattera et a1.61determined the boundstate spectrum for this system by selective adsorption measurements. Their values are Eo = 41.6 f 0.25 meV and D = 51.7 f 0.5 meV. A previously used semiempirical approach is the summation of pairwise potentials, by Crowell and Brown,62based on parameters derived from likeatom interactions and including the anisotropy of C atoms in graphite. We estimated the quadrupolar coefficient C5 as 700 meV.A5,63so the corresponding energy is about 15% of the dipole interaction around the equilibrium position; it is totally neglected in that previous approach. The difference is characteristic of gas-phase modeling in which Lennard-Jones functions subsume all dispersion terms into an effective r4 potential energy. Our results are shown in Figure 5 , where two different potentials correspond to different values of C3.62964The graphite charge density is from Posternak et al.,65which can be written in the vicinity of the minimum (z L 1.7 A) as ps = 1.053 exp(-1.941z) in au. The result is D = 29.0 meV, which is much smaller than the experimental value. An increase in C3 to the value62714 meV.A3 would yield consistency with experiment, but there is no known jus(58)Boheim, J.; Brenig, W. Z.Phys. B 1981,41,243. (59)Salanon, B. J . Phys. (Les Ulis,Fr.) 1984,45,1373. (60)Ihm, G.;Cole, M. W.; Toigo, F.; Scoles, G. J . Chem. Phys. 1987, 87,3995. (61)Mattera, L.;Rosatelli, F.; Salvo, C.; Tommasini, F.; Valbusa, U.; Vidali, G.Surf. Sci.1980,93,515. (62) Crowell, A. D.; Brown, J. S. Surf. Sci.1982,123,296. (63)We used eq 13 with go = 0.619,yo = 20,E, = 0.667,E, = 0.993 au. (64)Vidali, G.;Cole, M. W. Surf. Sci. 1981,110,10. (65)Posternak, M.; Baldereschi, A.; Freeman, A. J.; Wimmer, E. Phys. Reu. Lett. 1984,52,863. Private communication with Baldereschi, A.
0
0-.
- I >
-50
3.6
3.2
4.0
z(A)
Figure 5. H,-graphite interaction potential; potential, V, in meV and distance, z, from the top layer of nuclei in A. Open and solid squares correspond to two different choices for CB,i.e., 550 (ref 64) and 714 (ref 62) meV.A3,respectively.
tification for this. We should note that our method is extremely sensitive to the surface charge density a t large z, where it is decaying rapidly and is most uncertain. He,Ne-MgO. MgO is of great interest because of the possibility of large reconstruction66and the existence of a large set of data for p h y s i s o r p t i ~ n . ~ ~The + ~ electronic properties and the MgO surface have been studied by ab initio self-consistent calculations.70~71 Experimentally, there are several contradictory sets of He scattering s t ~ d i e P J ~ -of’ ~the He-MgO interaction potential. Brusdeylins et a1.66measured bound-state energies of 10.2,6.0,2.6, and 1.2 meV and deduced a well depth of 12.5 meV, while Mahgerefteh et al.76 recently measured bound-state energies of 5.52,2.57, 1.16,0.54, and 0.26 meV and obtained a well depth of 7.5 meV. The latter ground state is rather close to the adsorption isotherm results77 for He on MgO smoke (ground-state energy of 4.8 meV) and to the value (well depth of 8 meV) extracted from scattering intensity of He beam by Cantini and C e ~ a s c o . ~ ~ We constructed the laterally averaged density by summing contributions from individual surface ions; the charge density of the 0- ion is slightly different from its crystal value due to the competition between the anisotropic electrostatic and overlap interaction^.^^ The net laterally (66)Brusdeylins, G.; Do&, R. B.; Skofronick, J. G.; Toennies, J. P. Surf. Sci. 1983,128,191. (67)Coulomb, J. P.; Sullivan, T. S.; Vilches, 0. E. Phys. Reu. B 1984, 30,4753. (68)Bienfait, M.; Coulomb, J. P.; Palmari, J. P. Surf. Sci.1987,182, 5.57.
(69)Meichel, T.;Girard, C.; Suzanne, J.; Girardet, C., preprint (sub. . mitted to Phys. Rev. B ) . (70)Fowler, P. W.; Tole, P. Surf. Sci. 1988,197,457. (71)Causa, M.; Dovesi, R.; Pisani, C.; Roetti, C. Surf. Sci. 1986,175, ?C
