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2007, 111, 5825-5829 Published on Web 03/30/2007
Physisorption and Diffusion of Hydrogen Atoms on Graphite from Correlated Calculations on the H-Coronene Model System Matteo Bonfanti,† Rocco Martinazzo,† Gian Franco Tantardini,†,‡ and Alessandro Ponti*,‡ Dipartimento di Chimica Fisica ed Elettrochimica, UniVersita` degli Studi di Milano, V. Golgi 19, 20133 Milano, Italy, and Istituto di Scienze e Tecnologie Molecolari, Consiglio Nazionale delle Ricerche, V. Golgi 19, 20133 Milano, Italy ReceiVed: January 24, 2007; In Final Form: March 4, 2007
Correlated, counterpoise corrected wave function calculations on the hydrogen-coronene system are used to investigate the energy landscape and the dynamic behavior of hydrogen atoms physisorbed on graphite. The adopted MP2 correlation level, employing the aug-cc-pVDZ basis set augmented with bond functions, has been selected after extensive investigation on the smaller hydrogen-benzene system. The computed physisorption energy (39.7 meV) is in excellent agreement with the existing experimental value of (39.2 ( 0.5) meV for a graphite single layer (Ghio, E.; Mattera, L.; Salvo, C.; Tommasini, F.; Valbusa, U. J. Chem. Phys. 1980, 73, 557) and makes one confident of the computed barriers to diffusion. A simple, analytical expression of the corrugated potential energy surface fitted to the calculated energy values is then used in 3D quantum dynamical calculations of the tunneling contribution to the diffusion coefficient. Results show that hydrogen atoms physisorbed on graphite are highly mobile on the surface even at T ) 0 K. This suggests that hydrogen formation in cold, interstellar clouds can indeed occur down to very low temperatures through recombination of hydrogen atoms previously physisorbed on the surface of dust grains.
Introduction Recent years have witnessed an ever growing interest in studying hydrogen recombination on graphite surfaces. This interest is largely due to the relevance of hydrogen-graphite systems for understanding hydrogen formation in the interstellar medium (ISM). Hydrogen is the most abundant element in the universe (>90% in number1) and mainly occurs in atomic and molecular (H2) form. It follows that molecular hydrogen is the most abundant molecule in several interstellar structures, ranging from dense to diffuse clouds and from photon dominated regions to regions with shocked gases, even though hydrogen molecules are continuously dissociated by stellar UV radiation and cosmic rays. Thus, an efficient formation reaction from atomic species (recombination) is needed, and it is now generally accepted that it basically occurs on the surface of interstellar dust grains.2-4 Gas-phase chemical routes (such as the so-called H-/H+ routes)5,6 are deemed to be important for H2 formation in the early universe only, when dust was absent and surface-catalyzed reactions could not take place. Recent observations suggest that cosmic dust is composed of very small particles of different sizes and nature.7-9 In diffuse clouds, where the intense stellar radiation heats the gas, the largest particles are composed of a silicate core covered by an “organic refractory” mantle, whereas smaller particles are entirely carbonaceous, being even simple * To whom correspondence should be addressed. Tel: +39(0)250314280. Fax: +39(0)250314300. E-mail:
[email protected]. † Universita ` degli Studi di Milano. ‡ Consiglio Nazionale delle Ricerche.
10.1021/jp070616b CCC: $37.00
polycyclic aromatic hydrocarbons. Hydrogen formation in these regions of interstellar space may thus occur on graphitic surfaces, and hydrogen-graphite has become the prototypical system for studying hydrogen formation in the ISM. Depending on the physical conditions of interest and the actual morphology of the surface, a number of formation processes are possible, and only an accurate knowledge of adsorption, diffusion and recombinative elementary acts allows us to ascertain with confidence the role of each given pathway and to estimate the corresponding rate constant. Atom adsorption is the first step of any molecular formation mechanism. Hydrogen atoms may adsorb on graphite surfaces either chemically or physically. Several experimental10,11 and theoretical12-17 studies agree on the fact that hydrogen chemisorption on the regular (0001) surface occurs on top of a carbon atom and requires substantial lattice reconstruction. The carbon atom involved in forming the chemical bond rehybridizes its valence orbitals and puckers out of the surface plane by about 0.4 Å. As a consequence, a barrier to chemisorption of about 0.2 eV appears and essentially prevents (direct) hydrogen sticking in the chemisorption well at temperatures typical of the ISM. However, hydrogen chemisorption is needed to explain molecular formation in photon-dominated and shocked regions,18,19 where the temperature is much higher than the desorbing temperature of physisorbed species. This problem has been recently solved by the direct observation of clustering of hydrogen atoms on graphite surfaces which has been shown to occur through barrierless sticking of a H atom close to a previously adsorbed one.20,21 In this way, molecular formation at “high” temperature may follow recombination of hydrogen © 2007 American Chemical Society
5826 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Letters TABLE 1: Counterpoise Corrected, MP2 Physisorption Energies of Atomic Hydrogen on the Hollow Site of Benzene (3.11 Å Above the Molecular Plane) for Several Dunning-Type Basis Setsa
Figure 1. Coronene molecular model for the (0001) graphite surface, with a Cartesian coordinate system. H, B, and A stand for hollow, bridge, and atop adsorption sites, respectively.
pairs in the para position of a hexagonal ring,22,23 in addition to direct (Eley-Rideal) abstraction.13,24-31 At low temperature, however, formation of hydrogen molecules from chemisorbed species cannot occur. Even if a hydrogen atom chemisorbed on the surface, direct Eley-Rideal recombination would hardly occur in the temperature range 1-100 K.32,33 Thus, H2 formation in diffuse clouds must involve physisorbed species, and may occur either through a LangmuirHinshelwood, an Eley-Rideal, a hot-atom mechanism, or a combination of them. Eley-Rideal formation out of a physisorbed species is efficient down to very low temperatures;33 Langmuir-Hinshelwood recombination between neighboring atoms has been shown to be efficient,34,35 too, but a knowledge of hydrogen mobility under these conditions is required to definitely assess the feasibility of this process. We focus in this paper on the physisorption tail of the hydrogen-graphite potential. Current DFT methods are known to be unreliable in describing such weak dispersion forces and have indeed provided variable results for the physisorption binding energy, ranging from 8 to 74 meV.12,13 We therefore resorted to standard quantum chemical methods on a hydrogencoronene model system in order to accurately determine the physisorption binding energies over the high-symmetry sites, and the values of the barriers to diffusion. We have used these computed energy values to build a simple, corrugated model for the potential energy surface (PES) of a hydrogen atom physisorbed on graphite, which in turn has been employed in quantum dynamical calculations of the site-to-site tunneling time. The nice agreement between our energy results and the experimentally derived physisorption binding energy36 makes us confident of the dynamical results and suggests that physisorbed hydrogen atoms are highly mobile on graphite down to very low temperatures. Calculations and Results Calculations were carried out with the help of the Gaussian 0337 and GAMESS-US38 program suites. The coronene molecule was kept fixed at planar geometry with a C-C distance of 1.421 Å and a C-H distance of 1.07 Å; these values correspond to the experimental C-C and C-H bond lengths in graphite and benzene, respectively.39 The physisorption potential energy surface was sampled over three high symmetry points (see Figure 1): the hollow (H), bridge (B), and atop (A) sites. For each site, calculations were performed for several H-coronene distances, z, ranging from 2.7 to 7.5 Å. The physisorption energy Eph was computed as
Eph ) E(H-coronene) - E(coronene) - E(H)
(1)
where the full basis set was employed consistently for each term on the right-hand side, according to the counterpoise correction (CP)40 of the basis set superposition error.
aug-cc-pVDZ d-aug-cc-pVDZ b.aug-cc-pVDZ bb.aug-cc-pVDZ bbb.aug-cc-pVDZ aug-cc-pVTZ d-aug-cc-pVTZ b.aug-cc-pVTZ bb.aug-cc-pVTZ bbb.aug-cc-pVTZ aug-cc-pVQZ b.aug-cc-pVQZ
basis set size
Eph/meV
201 283 223 245 267 437 596 459 481 503 802 824
-18.8 -19.3 -24.0 -25.1 -25.3 -23.8 -24.6 -25.4 -25.5 -25.6 -25.3 -25.8
a The number of prepended b denotes the number of bond function groups present in the basis set.
TABLE 2: Counterpoise Corrected, MP2 and Coupled-Cluster Physisorption Energies Eph (meV) for Atomic Hydrogen on the Hollow Site of Benzene (3.11 Å above the Molecular Plane), for Several Double-ζ Basis Sets aug-cc-pVDZ b.aug-cc-pVDZ bb.aug-cc-pVDZ bbb.aug-cc-pVDZ
MP2
CCSD(T)a
-18.8 -24.0 -25.1 -25.3
-18.7 -23.6 -24.7 -24.8
a The physisorption energy does not depend on the excitations included in the calculation: Eph(CCD/aug-cc-pVDZ) ) -18.7 meV, Eph(CCD/b.aug-cc-pVDZ) ) -23.7 meV, and Eph(CCSD/aug-cc-pVDZ) ) -18.7 meV.
Preliminary tests on the H-coronene system (109 active electrons in the frozen core approximation) showed that at most we could perform second-order Møller-Plesset (MP2) calculation with a double-ζ basis set. Therefore, before attacking the problem of computing the physisorption energy of atomic hydrogen on coronene, we used the smaller H-benzene system to check whether MP2/double-ζ calculations were sufficient to achieve an energy accuracy of ∼1 meV. Test calculations on H-benzene were performed with the H atom above the center of the benzene molecule (hollow site) at a distance of 3.11 Å. First, several Dunning-type correlation-consistent polarizedvalence n-uple-ζ basis sets (cc-pVnZ)41,42 augmented with one (aug-cc-pVnZ) or two (d-aug-cc-pVnZ) sets of diffuse functions and with additional groups of bond functions43 (evenly spaced along the H-benzene axis) were checked at the MP2 level (see Table 1). The results show that (i) diffuse functions are not sufficient to properly describe the physisorption interaction, particularly with the double-ζ basis set [They are however necessary since the physisorption interaction is repulsive when using the cc-pVDZ basis set.]; (ii) bond functions are very effective in describing physisorption: the bbb.aug-cc-pVDZ set (267 functions) performs as well as the much larger aug-ccpVQZ set (802 functions); (iii) three groups of bond functions are sufficient: indeed the physisorption energy lowers by less than 0.2 meV on passing from the bb.aug-cc-pVDZ to the bbb.aug-cc-pVDZ set. Next, in order to find out whether MP2 is sufficiently accurate, counterpoise corrected coupled-cluster calculations with different truncation [CCD, CCSD, and CCSD(T)] were performed on the same H-benzene system with several double-ζ basis sets (Table 2). We found that all coupledcluster physisorption energies are very close to the MP2 ones, thereby making us confident of the MP2 correlation level for the H-coronene system.
Letters
J. Phys. Chem. C, Vol. 111, No. 16, 2007 5827 TABLE 3: Best-Fit Morse Potential Parameters for the MP2/bbb.(aug)-cc-pVDZ Physisorption Energies of Atomic Hydrogen on Coronenea site
zeq/Å
Deq/meV
β/Å-1
barrier/meV
atop bridge hollow
3.07 3.03 2.93
34.5 35.5 39.5
1.42 1.35 1.29
5.0 4.0
a Energy Barriers refer to the transfer of a hydrogen atom between hollow sites in two adjacent rings.
Figure 2. CP-corrected MP2/bbb.(aug)-cc-pVDZ physisorption energies of atomic hydrogen on the hollow (circles), bridge (squares), and atop (diamonds) sites defined on Figure 1. Dashed lines are best-fitting Morse curves (see Table 3). Full line is the best fitting modified Morse potential, including the long-range contribution (see text for details).
Calculations on the H-coronene system were thus attempted at the MP2 correlation level with the bbb.aug-cc-pVDZ set (735 basis functions) but extensive linear dependence, even without bond functions, prevented us from getting converged SCF wave functions, despite many efforts. We were thus forced to delete a number of diffuse functions, namely those located on the peripheral carbon and hydrogen atoms of the coronene molecule, which were deemed to be of little relevance for computing the interaction energy of the hydrogen atom on the sites defined in Figure 1. The resulting basis set is denoted as bbb.(aug)-ccpVDZ and comprises 525 basis functions. The CP-corrected MP2/bbb.(aug)-cc-pVDZ physisorption energies of atomic hydrogen on the three coronene sites considered in Figure 1 are shown in Figure 2, as a function of the distance z of the H atom from the molecular plane. We found a steep repulsive wall at distances below 2.9 Å; when the H atom is on the atop site, this repulsive interaction represents the known barrier to chemisorption, whose accurate description would require (at least) relaxation of the vertical position of the binding carbon atom. Physisorption binding energies range between 34 and 40 meV at distances close to 3 Å. The latter value is much shorter than the DFT predicted ones.13 For z g 4 Å, the three adsorption curves are almost superimposed, indicating little corrugation of the PES at large distances. The important result is that the (computed) physisorption binding energy at the stable adsorption site (hollow site, z ) 2.9 Å) turns out to be 39.7 meV. This value is in excellent agreement with that derived from analysis of the selective adsorption resonances in H-graphite scattering experiments,36 (43.3 ( 0.5) meV. The agreement is even better when this value is compared with that estimated for adsorption on a single C layer,36 (39.2 ( 0.5) meV. In order to check the reliability of our results, we also performed MP2 calculations employing the Pople-type 6-31++G(d,p) basis set. The resulting physisorption energies and geometries (data not shown) are very close to the MP2/bbb.(aug)-cc-pVDZ, thus ensuring that our results and conclusions are not biased by a particular choice of the basis set. Apart from the long-range region, the physisorption energy results reported in Figure 2 are nicely fit by a Morse potential
V(z,R) ) Deq(R)[e-β(R)(z-zeq(R)) - 1]2
(2)
where the dependence of the Morse parameters Deq, zeq, and β
on the position R in the molecular plane has been explicitly indicated. The best-fitting parameters are reported in Table 3. Note that the difference between the binding energy at the top and that at the bridge site defines the lowest energy barrier (Eb) to diffusion. The latter turns out to be 4.0 meV, only 1 meV lower than the energy barrier for passing over the atop site. Inclusion of the long-range part of the form VLR(z))-C4/z4 is possible with the help of a smooth “switching function” f(z)
V(z) ) VSR(z)(1 - f(z)) + VLR(z)f(z)
(3)
where VSR(z) is the above Morse potential and f(z) ) 0 for z less than a cutoff distance zc. When using the switching function adopted in ref 44 (with n ) 2) to fit the data on the hollow site, we find that that the best-fitting Morse parameters are essentially the same as those reported in Table 3, zc ) 3.51 Å and C4 ) 4.48 eV/ Å4. The latter value is in reasonably good agreement with the experimental one,36 C4 ) (5.2 ( 0.3) eV/ Å4, if the finite size of the our graphite model is taken into account. The above functional form may be used to build an analytic representation of the 3D physisorption PES by interpolating the Morse parameters over the molecular plane. Restricting the analysis to the sites of Figure 1, the following functional form is obtained by imposing the correct (x,y) translation symmetry
f(x,y) ) g(r) + g(r) )
h(r) - g(r) (1 + cos(6φ)) 2
(4)
RH + RB RH - RB 2πr + cos + 3 2 3d RH + 3RB - 4RA 4πr (5) cos 6 3d
( )
( )
h(r) )
( )
RH + RB RA - RB 2πr + cos 2 2 x3d
(6)
where R is one of the Morse parameters (Deq, zeq, and β), H, B and A stand for the hollow, bridge and atop sites, respectively, (r, φ) are the standard polar coordinates for the reference axes of Figure 1, and d is the C-C bond length. A cut of the resulting PES at z ) 2.9 Å is shown as a contour plot in Figure 3. The resulting potential has been employed to study diffusion of hydrogen atoms physisorbed on graphite. The theory of activated surface diffusion, including multiple hopping and quantum effects, is well developed (for recent, comprehensive reviews see refs 45 and 46). The theory, however, cannot be applied to the present case since the basic requirement of a large reduced barrier height (Eb/kBT . 1) is never satisfied, except in the deep quantum regime. Therefore, we limit ourselves to the T ) 0 K case, considering nearest-neighbor transitions only, and computing the tunneling escape rate by 3D wavepacket
5828 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Letters
Figure 4. Survival probability for a wavepacket initially localized in an adsorption site (bold line). For comparison, results for the same initial wave packet on a flat surface are shown as a thin line.
Figure 3. Section of the PES for atomic hydrogen physisorption on graphite obtained by interpolating MP2 physisorption energy values for the H-coronene system. Contour map of the physisorption potential as a function of the H position on the surface plane, with the origin of the Cartesian axes at a hollow site, and the height of the hydrogen atom kept fixed at z ) 2.9 Å. Energy levels are 0.7 meV spaced starting from the lowest energy value (dark region).
dynamics. The diffusion coefficient D for the present surface geometry is then given by46
1 D ) a02Γh 4
(7)
where a0 is the “jump” length (i.e., the site-to-site distance, dx3) and Γh is the total escape rate. It is worth noting at this point that dissipation is known to increase the tunneling time,47 and therefore the results presented below do not represent strict lower bounds to hydrogen mobility. Even though it is expected to be of minor importance in this physisorption regime, it might be considered in the near future, once friction coefficients become available, thanks to recent developments in real time quantum dynamics for large systems.48 Time-dependent wavepacket calculations employed the Colbert-Miller discrete-variable representation states49 for representing the wavefunction along the three Cartesian degrees of freedom, and the standard Split-Operator scheme50 for the shorttime approximation of the time-evolution operator. The grid was chosen to be sufficiently large on the surface plane that no reflection from the grid boundaries could be observed in the relevant time scale. Tests calculations showed that more than 3 adsorption sites along the x axis on each side of the initially occupied well were necessary, and therefore, a 21 Å × 21 Å grid was adopted on the surface plane. A much smaller grid of 4 Å was sufficient for the z axis. The initial wavepacket was placed at the hollow adsorption site at the center of the grid, and the tunneling time was determined as the minimum of the “survival probability” in the initial state, i.e., of the squared autocorrelation function shown in Figure 4. For comparison, in that figure, the same quantity for the same initial wavepacket but on a flat surface is also reported. Although the shape of the autocorrelation function depends on the initial state, the tunneling time does not, as tests calculations confirmed. When the resulting tunneling time, Γh ≈ 0.9 ps, is used in eq 7 we get D ) 1.7 × 10-4 cm2 s-1. The limiting coefficient for H-atom diffusion on metal surfaces (where the barrier is ∼0.2 eV) is known to be in the range of 10-13-10-10 cm2 s-1 (ref 46) (i.e., orders of magnitude lower than in the present case, as expected).
We note, however, that the value given above is only indicative of the hydrogen mobility on graphite. Diffusion becomes (almost) free already at low temperatures, since Eb ) 4 meV ≈ 46 K. Even though a quantum description is always mandatory for hydrogen motion, in this regime, one can reasonably use the classical estimate D ) kBT/mγ where γ is the (unknown) friction coefficient. In this “high” temperature regime, diffusion dynamics competes with thermal desorption, whose rate can be estimated to be kdes ≈ (ω/2π) exp(-Deq/kBT), where ω ≈ 3 × 1013 s-1 and Deq ) 460 K in the present case (see Table 3). In addition metastable hot-atoms with long lifetimes may form and travel for long distances across the surface thanks to the small surface corrugation.51,52 This might contribute to explain the large abstraction cross-sections observed by Zecho et al.26 Conclusions We used accurate, correlated ab initio calculations on the H-coronene system to investigate physisorption and diffusion of hydrogen atoms on graphite. The resulting physisorption binding energy is in nice agreement with available experimental results and makes us confident of the computed barriers to diffusion. Our results suggest that hydrogen atoms are extremely mobile on the surface down to very low temperatures, thereby making the Langmuir-Hinshelwood mechanism a possible efficient route for hydrogen formation in the ISM cold clouds.34,35 Eley-Rideal recombination is known to be efficient too,33 and possible HA scenarios could also be envisaged if thermalization of hydrogen atoms required long times. The relative importance of the possible mechanisms remains to be assessed on the basis of relevant surface coverage conditions. Concluding, it is worth noting that, when physisorbed species are involved, there is little interaction between the hydrogen electron and the substrate electrons and therefore molecular hydrogen formation is expected to occur only in singlet-state H atom pairs, i.e., 1/4 of the possible electronic spin combinations. This statistical factor would reduce both the LH and ER rates involving physisorbed H atoms and has already been taken into account by Morriset et al.35 in their calculation of LH crosssections. References and Notes (1) Herbst, E. J. Phys. Chem. A. 2005, 109, 4017. (2) Gould, R. J.; Salpeter, E. E. Astrophys. J. 1963, 138, 393. (3) Hollenbach, D. J.; Salpeter, E. E. J. Chem. Phys. 1970, 53, 79. (4) Hollenbach, D. J.; Salpeter, E. E. Astrophys. J. 1971, 163, 155. (5) Lepp, S.; Stancil, P. C. In The Molecular Astrophysics of Stars and Galaxies; Hartquist, T. W., Williams, D. A., Eds.; Clarendon Press: Oxford, 1998.
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