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Interaction of atomic hydrogen with lithium metal clusters: Breakdown of the adiabatic approximation
111
Surface Science 121 (1982) 111-122 North-Holland Publishing Company
INTERACTION OF ATOMIC HYDROGEN WITH LITHIUM METAL CLUSTERS: BREAKDOWN OF THE ADIABATIC APPROXIMATION J. VOJTiK J. Heyrovsk+ Institute of Physical Chemistty and Electrochemistry, of Czechoslovakia, 12138 Prague, Czechoslovakia
Academy
of Sciences
and J. FISER Departmetit Received
of Physical Chemistry
Charles University,
12840 Prague, Czechoslovakia
15 May 1982
Adsorption of hydrogen atom on the (100) surface.of Li, clusters is studied using the diatom&.-in-molecules method. Special attention is paid to crossing and/or avoided crossing of the ground and excited state potential energy curves for normal approach of H to some surface sites. The most obvious effects of the non-adiabatic behaviour of the system on the course of the adsorption process are considered and the basic aspects of the breakdown of the Born-Oppenheimer approximation are discussed.
1. Introduction In recent years, there have been a number of theoretical studies of chemisorption on metal surfaces simulated by atomic clusters. Most studies have been concerned with the calculation of interaction energies for selected sets of important configurations of adsorbate-substrate systems, with the aim of providing data either for qualitative interpretation of the basic features of the chemisorption processes [ 1,2] or for the construction of model potential energy hypersurfaces to be used in dynamical studies [3-71. Because of the complexity of these systems, the quantum chemical methods used to produce the necessary interactions have generally been less accurate than those employed in treating simple chemical reactions in the gas phase. In theoretical studies of the adsorption of hydrogen on Li clusters, for example, the calculations have been done by the ab initio Hartree-Fock method [8- lo], the CNDO scheme [ 11,121 and semiempirical crystal orbital technique [ 121. The potential energy curves (PEC’s) produced in these works for selected idealized approaches of H to Li cluster surfaces were then used to obtain 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
112
J. Vojrik, 3. FiSer / Interaction
of H
with Li metal dusters
information on equilibrium properties of the adsorption process, namely the adsorption energies and the equilibrium adatom-surface distances. The above-mentioned methods are all based on a singfe-determinant wave function and provide an approximation to the ground state energy of the system. It is partly due to this inherent feature of these methods that the problem of whether the excited states can affect the adsorption process has not been considered. In a recent communication [13] we reported the calculation of the interaction of atomic hydrogen with the (4? f, 4) lithium cluster performed by the semiempirical valence bond technique of diatom&-in-molecules (DIM) ] f4161,which has turned out to be quite successful in the theoretical description of the adiabatic and non-adiabatic processes in the gas phase [ 17,181. Ground and excited state PEC’s obtained for normal approach of H to some surface sites were found to intersect or pseudointersect, thus reveahng the possible nonadiabatic character of the adsorption process. The importance of non-adiabatic effects in adsorption was already stressed at, e.g., the Gstaad meeting [20,2t]. Recently, several authors have treated in detail various aspects of non-adiabatic behaviour in adsorption on semi-infinite metals [22-251. The aim of the present paper is to extend our previous study [13] to the (5,4,0) and (4,5,0) lithium clusters and to discuss some common features and possible consequences of the non-adiabatic behaviour of the Li,H systems.
2. The computational method The goal of the DIM method is to compute a polyatomic potential energy surface in terms of more accessible energies of the constituent atomic and diatomic fragments. The method is based on the valence bond type wave function but the role of the basis is rather formal. Actual analytic expressions for the atomic functions from which the basis is constructed are nat required. Essential are their symmetry properties which are used to set up the DIM approximation to the Hamiltonian matrix [ 14- 16], A sir&et wave function of the Li,H system is taken as a linear combination of polyatomic basis functions (PBF’s) which are antisymmetrized products of atomic (ground state) ‘S states. A PBF may be written as
Here, B is the ~~sy~et~~ation operator. AL:: is the ‘S wake function of atom K with the spin projection M,~. A product of these functions, +,, is called a primitive function. Since msK can take on two values, l/2 and - l/2, there are 252 PBF’s with m,, -I-M,~ + . . * +m,,,= 0. The calculations are done in a spin adapted basis (&?‘+m}constructed by the direct diagonalization of the total S2 operator in the subspaces of the [+,_] space, pre-arranged so as
J. Vojtik, J. FiSer / Inleraciion
of H with Li metal clusters
113
to be invariant under the diatomic fragment spin operators S&, S$,,..., S;,, [26] and time reversal [27]. The approximations to the true singlet potential energy hypersurfaces of the system are obtained by the diagonalization of the S-‘H matrix of the nonHermitean version of the DIM method [28]
K=l
L>K
K=l
(2)
Here S and H are, respectively, overlap and Hamiltonian matrices, /rcK) and AcKL) are (diagonal) matrices of the accurate atomic and diatomic fragment energy levels, respectively. Each of the yKLj matrices represents the KL diatomic fragment states in the basis of (properly antisymmetrized) spinadapted primitive functions. As our basis is composed of *S states, each diatomic fragment can occur in a ‘Z+ or ‘Z+ state. Another consequence of the present choice of the basis is .of the system and that the yKLj matrices do not depend on the configuration are deducible from the spin properties of the basis. Specifically, due to the way of constructing the {‘$I~} basis we have TKLj = 1 (unit matrix) for KLE{ 12,34 ,..., 910}, while the other matrices are obtained as representatives in {‘$,,} of the operators permuting appropriately the atomic fragments. These matrices are all found to be unitary, and it makes no difference for our choice of the basis whether one exactly includes the overlap through the application of the non-Hermitean version of the scheme or uses the standard version of DIM with the neglect-of-overlap assumption invoked. Adopting the latter point of view we can write for the Hamiltonian matrix
In evaluating the diagonal elements of the diatomic fragment matrices, the ‘2+ and ‘Xl potential energy curves (PEC’s) of Li, are taken from the work bysPickup [29], while the ‘2+ and 3Z+ PEC’s of LiH are based on the multiconfiguration SCF calculations by Docken and Hinze [30]. The monatomic contributions to the polyatomic Hamiltonian matrix are obtained under the usual assumption that the primitive functions are eigenstates of atomic Hamiltonians. A comment concerning the basis appears to be warranted. Due to the fact that Li,H is the largest system to which the DIM method has been applied, it seems natural, in view of the programming and computer time difficulties, to start with the minimum basis calculations. Though the quality of the resulting DIM model of the potential energy hypersurface is difficult to asses, the situation is certainly not so disastrous as if the ab initio valence bond calculations with the same basis were contemplated. The main reasons are the indirect role played by the basis within the DIM scheme and the use of very
accurate PEc”s for diatomic fragments Li, and LiH. It should perhaps be remarked in this connection that if PEC’s of comparabie quality were to be obtained by the (ab initio) valence bond technique, one should take a basis consisting of a great many structures, including ionized ones. Thus, while the present basis can be considered to be not sufficiently flexible, one cannot claim that inclusion of the ionized structures i3If into the DIM model of L&H will lead to a better approximation to the true potential energy hypersurfaces.
Three different Li, clusters were emptoyed (fig. I), namely (4,1,4), (4,5,0) and (5,4,0). PEC’s were obtained for normal approach of the hydrogen atom to the (100) lithium cluster surface for three adsorption sites: atop (A), bridge (B) and centred (C) (fig. 1). Ground and excited state PEC’s together with their symmetry notation for each cluster and site are shown in figs. 2-4. The most important point to notice in these figures concerns the mutual arrangement of the ground and excited state singlet PEC’s. In most cases studied crossing and/or avoided crossing between the PEC’s occurs, indicating that in some regions of the configuration space the Born-Oppenheimer approximation breaks down, and the electronic motion cannot be separated from the nuclear one. In these regions, the relative adatom-substrate motion can produce transitions between the potential energy hypersurfaces. 3.1. Model For gas phase processes, the ‘~surfa~-Hopkins-trajecto~ method” has been developed [173, which takes into account non-adiabatic transitions. This technique is based on the semiclassical theory of non-adiabatic behaviour and requires a thorough treatment of the dynamics of the elementary process. A
Fig. I. The (4, I, 4), (4,5,0) and (5,4,0) modeIs of bee I_i(lOO) surface C. The cube edge is taken to be 3.491 A.
and adsorption
sites A, B,
J. Vojtik, J. FiSer / Interaction
115
of H with Li metal clusters
-7
-0
3 iii -9
-10
I
I
I
I
1
2
3
4
r(A)
c
4
I
I
I
I
I
I
I
I
1
2
3
4
1
2
3
4
c(A)
r(A)
Fig. 2. PEC’s for the A-site approach of the atomic hydrogen to the lithium clusters. (a) (4, 1,4), (b) (4, 5,0), (c) (5,4,0) cluster.
necessary prerequisite for such an approach is the knowledge of the adiabatic potential energy hypersurfaces and non-adiabatic coupling among them in the whole configuration space relevant to the model. At present, such information on the L&H system is not available and we have to combine the “surface-hopping-trajectory” idea with the limited information on the interactions and non-adiabatic coupling provided by PEC’s relevant to a normal approach of H to our selected surface sites. In doing so, we follow the approach usually adopted in cases where only information on the ground state hypersurface represented by a few PEC’s is available. In this latter case a surface site is assigned adsorption energy (and equilibrium distance) according to the PEC corresponding to the site considered. In this way, even those adsorption energies can be arrived at which, if changes of the potential energy with respect to the lateral change of the adatom-cluster configuration were known, might turn out not to correspond to a local minimum on the potential energy hypersurface. We adopt here a similar viewpoint and assign to, a surface site every adsorption energy (and equilibrium adatom-surface distance) which corresponds to a minimum on either ground and excited state PEC provided, of course, that both the electronic structure and nuclear configuration corresponding to this minimum can be acquired by the system during the adsorption process. In order to make a rough estimate of the probability with which sudden changes in the electronic structure of the system occur, we have to make an assumption about the relative adatom-cluster motion in the non-
J. Vojtik, J. FiSer / Interaction
116
of H with Li metal clusters
-1
-8
3J
-9
Ill
-10
-11
-2
-1
0
3
-1
0
r/A)
1
3
4
-2
I
I
-1
0
I
iA)
I
I
2
3
1 4
-7
-0
2
-9
w
-10
-11
-2
1
2
4
G, Fig. 3. (a), (b) PEC’s for the B-site approach of the H atom to the Li, clusters: (a) (4, 1,4), (b) (4, 5,0) cluster. (c) PEC’s for the B-site approach of the H atom to the (5,4,0) lithium cluster.
adiabatic regions. Because of the limited information on the potential energy hypersurfaces we have, and in view of the approach discussed above, we restrict ourselves to the normal component of the H-cluster surface motion.
117
J. Vojtik, J. FiSer / Interaction of Ii with Li metal clusters
-7
t -8
9 2% -9 u
-10
-11
I
-1
0
1
2
3
4
-1
0
1
2
3
4
6)
-7
-8
7 s-9 u
-10 I-
-11
Fig. 4. (a), (b) PEC’s for the C-site approach of the H atom to the Li, clusters: (a) (4, 1,4), (b) (4,5,0) cluster. (c) PEC’s for the C-site approach of the H atom to the (5.4.0) cluster. Dashed line represents the non-adiabatic coupling matrix element (+21a/arl+l) = c2, of eq. (5) as a function of r.
118
J. Vojtik, J. F&r
3.2. Non-adiabatic
/ Interaction
of H with Li metal clusters
transitions
A rough estimate of the transition probabilityp,; j can be obtained from the expression [32,33].
from the state i to the state
p,;-exp(-ICE,-E,)/v,.d,,(). Here, vi is the nuclear velocity vector appropriate and d,, is the non-adiabatic coupling vector d,,=
-i~(ll;lVlG,>.
Ei and 4, are the electronic According estimate the batic coupling dj,“=
to the adiabatic
surface
E,
(5)
the corresponding eigenvalue and eigenfunction, respectively, of Hamiltonian of the system. to our model, we need normal components of u, and d,, to transition probability. The normal component of the non-adiaelement is, as usual in the DIM scheme [ 181, approximated by
-iA[~i(MI/i3r)C;]/(Ej-E,),
(6)
where r is the distance between the hydrogen atom and the cluster surface, aH/ar is obtained by differentiating each of the matrix elements of eq. (3) and C, is the column matrix of the expansion coefficients of qi in the PBF basis. In order to facilitate the discussion of the qualitative features of the adsorption process for individual binding sites, it is convenient to distinguish the following three types of the mutual arrangement of the PEC’s. In the first type (cf. figs. 2a-2c and 4b), the ground state PEC is neither crossed nor pseudocrossed by the excited state PEC’s, so that to each of these sites a single adsorption energy and equilibrium distance corresponding to the ground state PEC must be assigned. The second type (figs. 3b, 3c and 4c) is characterized by the presence of an avoided crossing between the ground state PEC and an excited one. For the B-site approach (both clusters), the transition probability is practically negligible, because the curves do not get close enough (cf. eqs. (4) and (5)) to make transition possible. As a consequence, adsorption energies corresponding to the ground state PEC’s are to be assigned to these binding sites. In the (5,4,0) C-site case, however, the situation is different and even for thermal adatomsurface relative velocities (about a few times lop4 au), the transition probability can be expected to be large. Thus, in this case, besides the ground state energy minimum, the other one, corresponding to the excited A” state will apply. It should be noted in this connection that our simple model does not provide a mechanism for the system to get to the excited A’ state. This is due to the fact that for the normal component of the vector (5) we have, on symmetry grounds, dj,n)= -ift(~,(A’)(a/arl~,(A”))=O.
(8)
J. Yojtik* J. FiSer / Interaction of H with Li metal clusters
119
A thorough inspection of the situation [ 13,171 then reveals that the transition probability between the states under question vanishes. In the third type (figs. 3a and 4a), the PEC’s that are most energetically favourable in the asymptotic region are intersected by the curves corresponding to states of different symmetry. At crossing points, the normal component of the coupling element (5) is in both cases zero, so that within our model, a transition between the PEC’s cannot occur. The system will not acquire the electronic structure corresponding to the A’ and A, states. On the contrary, the adsorption process is pictured as governed solely by the A” and B, curves, respectively. Thus the adsorption energies corresponding to these curves, not the overal minima on the A’ and A, PEC’s, are to be assigned to these adsorption sites. 3.3. Equilibrium properties for different adsorption sites For the A-site approach, the arrangement of the PEC’s belongs, for all three clusters, to the first type. The adsorption energies and equilibrium distances are similar for all the clusters (cf. table 1) and are in good agreement with those obtained by other authors [8- 1I]. We now move our attention to the B-site .adsorption (figs. 3a-3c). In the case of the (4, 1,4) cluster, the two lowest A’ and A” PEC’s cross in the vicinity of the minimum of the A” curve. However, in accordance with the above general remarks, the system will be governed by the A” PEC leading to the equlibrium distance of 0.28 A and the adsorption energy of -2.39 eV. For the (4,5,0) and (5,4,0) clusters, the non-adiabatic effects are negligible so that the
Table 1 ~uilib~um
properties of the Li,H systems for different clusters and adsorption sites
Site
Cluster
A
(4, 174) (49% 0) (%4> 0)
1.61 1.61 1.59
B
(4,I,4) (4, 5, 0) (5,4,0)
0.28 - 1.39 -0.36
-2.39 -2.81 -2.81
C
(4,1.4) (42% 0) (5,4.0)
0.30 -0.06 -0.11 0.40
- 1.80 -3.03 - 2.33 -1.90
r, ‘) (A)
&is (eV) - 1.89 - 1.64 - 1.74
‘) r. is the equilibrium distance of the H atom to the cluster surface.
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J. Vojtik, 3. Fi3er / Interaction
of H with Li metal clusters
adsorption process is essentially controlled by the lowest PEC’s which are attractive inside the clusters. The subsurface minimum for this site was reported by Kunz et al. [S] at r, = -0.5 A for the (4, 1,4) cluster, by Fantucci et al. [lo] at r, = -0.53 A for the (4,1,0) cluster, and by Lavery and Hillier [ 121 for the (4,1,4) cluster at r, = - 1.75 A. Companion [ 1l] found a periodic attractive potential inside the lithium lattice, with the first minimum at about - 1.0 A. As far as the adsorption energy is concerned, our values are lower in magnitude than the results of the above-mentioned calculations: -3.0 eV [S], -2.97 eV [Ii] and -3.48 eV 1121. The other minima reported lie above the surface. The CNDO calculations with the Pople parametrization on the (4, 1,4) cluster [ 121 predict a second minimum at 1.12 A (-4.12 eV), while the Sichel parametrization leads to the values -2.68 eV at r, = 1.0 A [ 121. The crystal orbital technique gives 0.4 A (-3.60 eV) [12]. In the C-site case the non-adiabatic effects are found to play an important role. For the (5,4,0) cluster, the present model leads to two energy minima (cf. table l), one lying below, and the other above the surface. It should be remarked that the latter minimum is accessible thanks to the non-adiabatic behaviour of the system. With the (4,1,4) cluster, the situation is even more flagrant. It is perhaps interesting to note that in both cases these effects are accompanied by a decrease of the adsorption energy. Another consequence of the non-adiabatic behaviour of the L&H system in this case is the occurrence of relatively large differences in the equilibrium properties obtained for different clusters. However, the present results do not contradict those obtained by other authors for this adsorption site. A subsurface minimum was found by Kunz et al. [8] and by Companion [I 11.The adsorption energies obtained here are all negative, which is in accord with all the reported calculations but that of Fantucci et al. [lo]. In concluding this section, we would like to mention that with the exception of the (4,5,0) cluster, the chemisorption of a H atom in the B position is favoured by our calculations to that in the C position. This result is in agreement with the ab initio calculations [lo] on the (4,1,0) cluster but contradicts the CNDO results [ 121. It is interesting to note that the basic trends obtained by Hjelmberg in his self-consistent study of a hydrogen atom interacting with a jellium model of the Na(lOO) surface [34] are similar to the present ones.
4. Concluding remarks The present study of the interaction of. atomic hydrogen with Li, clusters by the DIM method also includes excited states. Ground and excited state potential energy curves obtained for normal approach of the H atom to some
J. Vojtik, J. F&r / Interaction of H with Li metal clwters
121
adsorption sites are found to intersect or to avoid crossing. Even a qualitative analysis of the situation reveals that the adsorption process cannot be pictured as taking place on one energy hypersurface. Each of the configurations studied here is characterized by non-trivial spatial symmetry and leads to PEC’s which, when having different symmetry, can cross. Thus these configurations in a sense represent limiting cases qualitatively different from general binding positions where the PEC’s can at most avoid crossing. In this connection the question arises as to how the present picture of the adsorption process would be changed if, instead of taking the binding sites examined in the previous section, general sites, lying in their vicinity were studied. In view of the remarks made previously, a change of this kind in the binding site will affect the arrangement of the PEC’s in all the cases but the (4,5,0) A-site one, causing the actual crossing to become avoided. In accordance with the model used here, changes in the non-adiabatic behaviour of the Li,H system will occur with the (4, 1,4) B-site, (4, 1,4) C-site and (5,4,0) C-site cases. In the first two cases, the model used here will lead to an additional pair of equilibrium properties. According to this new pair of data, the hydrogen atom, besides being adsorbed, is predicted to be also absorbed and the over-all picture of the chemisorption is modified in favour of the penetration of H through the cluster surface. The probability with which a resulting new pair applies will be connected, through the quantities occurring in eq. (4) with the change in the binding site. Mainly due to the dependence on the energy difference between the PEC’s (cf. eq. (4)) one can expect that the probabilities will change more dramatically than the PEC’s themselves. Our sample calculation on the (4, 1,4) cluster [ 131 seems to confirm this point. We can therefore infer that when non-adiabatic effects occur, equilibrium characteristics of chemisorption deduced from a few PEC’s corresponding to configurations of nontrivial spatial symmetry can be expected to be less reliable than the information obtained in an analogous way for processes involving only ground state hypersurfaces. This conclusion may be of value in those situations where cluster calculations are done with the aim of generating data for the construction of model potential energy hypersurfaces to be used in dynamical studies of the gas-surface processes [3-71. In concluding this communication, we would like to remark that we should have felt much safer if we could have based our discussion on results obtained for larger clusters and with a more flexible basis than the minimum one. That is why our results should be taken with a reserve. However, due to the conceptual background of the DIM method, one is inclined to believe that the present picture of the chemisorption process allows for that part of what is actually going on which cannot be taken into account by the SCF MO cluster approaches.
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of H with Li metal clusters
References [1] [2] [3] [4] [S] [6] [7] [8] [9] [lo] [ 1 l] [12] [13] [ 141 [15] [16] [17] [18] [ 191 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
For a review, see R.C. Baetzold, Advan. Catalysis 25 (1976) 1. For a review, see M. Simonetta and A. Gavezotti, Advan. Quantum Chem. 12 (1980) 103. J.H. McCreery and G. Wolken, Jr., J. Chem. Phys. 63 (1975) 2340; 65 (1976) 1310. A.B. Elkowitz, J.H. McCreery and G. Wolken, Jr., Chem. Phys. 17 (1976) 423. G. Wolken, Jr. and J.H. McCreery, Chem. Phys. Letters 54 (1978) 35. G.F. Tantardini and M. Simonetta, Surface Sci. 105 (1981) 517. J.C. Tully, J. Chem. Phys. 73 (1980) 6333. A.B. Kunz, D.J. Mickish and P.W. Deutsch, Solid State Commun. 13 (1973) 35. H. Stoll and H. Preuss, Surface Sci. 65 (1977) 229. P. Fantucci, G. Pacchioni and J. Fernandez-Rico, J. Mol. Catalysis 12 (1981) 245. A.L. Companion, Chem. Phys. 14 (1976) 1. R. Lavery and I.H. Hillier, J. Mol. Catalysis 4 (1978) 9. J. Vojtik, J. FiSer and R. PO& Chem. Phys. Letters 80 (1981) 569. F.O. Ellison, J. Am. Chem. Sot. 85 (1963) 3540. J.C. Tully, J. Chem. Phys. 58 (1973) 1396. E. Steiner, P.R. Certain and P.J. Kuntz, J. Chem. Phys. 59 (1973) 47. J.C. Tully and R.K. Preston, J. Chem. Phys. 55 (1971) 562. R.K. Preston and J.C. Tully, J. Chem. Phys. 54 (1971) 4297. E. Drauglis and R.L. Jaffe, Eds., Physical Basis for Heterogeneous Catalysis (1975). J.R. Schrieffer, in ref. [19], p. 576. M. Boudart, T.B. Grimley and D.M. Newns, in ref. [ 191, p. 576. G.P. Brivio and T.B. Grimley, J. Phys. Cl0 (1977) 2351; Surface Sci. 89 (1979) 226. B.I. Lundqvist, 0. Gunnarsson, H. Hjelmberg and J.K. Ndrskov, Surface Sci 89 (1979) 196. K. Schonhammer, 0. Gunnarsson, Phys. Rev. B22 (1980) 1629. R. Brako and D.M. Newns, Solid State Commun. 33 (1980) 713. WI. Salmon, K. Ruedenberg and L.M. Cheung, J. Chem. Phys. 57 (1972) 2787. J. Vojtik and J. FiSer, Theoret. Chim. Acta 45 (1977) 301. J.C. Tully and C.M. Truesdale, J. Chem. Phys. 65 (1976) 1002. B.T. Pickup, Proc. Roy. Sot. (London) A333 (1973) 69. K.K. Docken and J. Hinze, J. Chem. Phys. 57 (1972) 4298. A.C. Roach and P.J. Kuntz, J. Chem. Phys. 74 (1981) 3435. J.C. Tully, J. Chem. Phys. 59 (1973) 5122. M.B. Faist and J.T. Muckerman, J. Chem. Phys. 71 (1979) 233. H. Hjelmberg, Surface Sci. 81 (1979) 539.
Surface Science 121 (1982) 111-122 North-Holland Publishing Company
INTERACTION OF ATOMIC HYDROGEN WITH LITHIUM METAL CLUSTERS: BREAKDOWN OF THE ADIABATIC APPROXIMATION J. VOJTiK J. Heyrovsk+ Institute of Physical Chemistty and Electrochemistry, of Czechoslovakia, 12138 Prague, Czechoslovakia
Academy
of Sciences
and J. FISER Departmetit Received
of Physical Chemistry
Charles University,
12840 Prague, Czechoslovakia
15 May 1982
Adsorption of hydrogen atom on the (100) surface.of Li, clusters is studied using the diatom&.-in-molecules method. Special attention is paid to crossing and/or avoided crossing of the ground and excited state potential energy curves for normal approach of H to some surface sites. The most obvious effects of the non-adiabatic behaviour of the system on the course of the adsorption process are considered and the basic aspects of the breakdown of the Born-Oppenheimer approximation are discussed.
1. Introduction In recent years, there have been a number of theoretical studies of chemisorption on metal surfaces simulated by atomic clusters. Most studies have been concerned with the calculation of interaction energies for selected sets of important configurations of adsorbate-substrate systems, with the aim of providing data either for qualitative interpretation of the basic features of the chemisorption processes [ 1,2] or for the construction of model potential energy hypersurfaces to be used in dynamical studies [3-71. Because of the complexity of these systems, the quantum chemical methods used to produce the necessary interactions have generally been less accurate than those employed in treating simple chemical reactions in the gas phase. In theoretical studies of the adsorption of hydrogen on Li clusters, for example, the calculations have been done by the ab initio Hartree-Fock method [8- lo], the CNDO scheme [ 11,121 and semiempirical crystal orbital technique [ 121. The potential energy curves (PEC’s) produced in these works for selected idealized approaches of H to Li cluster surfaces were then used to obtain 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
112
J. Vojrik, 3. FiSer / Interaction
of H
with Li metal dusters
information on equilibrium properties of the adsorption process, namely the adsorption energies and the equilibrium adatom-surface distances. The above-mentioned methods are all based on a singfe-determinant wave function and provide an approximation to the ground state energy of the system. It is partly due to this inherent feature of these methods that the problem of whether the excited states can affect the adsorption process has not been considered. In a recent communication [13] we reported the calculation of the interaction of atomic hydrogen with the (4? f, 4) lithium cluster performed by the semiempirical valence bond technique of diatom&-in-molecules (DIM) ] f4161,which has turned out to be quite successful in the theoretical description of the adiabatic and non-adiabatic processes in the gas phase [ 17,181. Ground and excited state PEC’s obtained for normal approach of H to some surface sites were found to intersect or pseudointersect, thus reveahng the possible nonadiabatic character of the adsorption process. The importance of non-adiabatic effects in adsorption was already stressed at, e.g., the Gstaad meeting [20,2t]. Recently, several authors have treated in detail various aspects of non-adiabatic behaviour in adsorption on semi-infinite metals [22-251. The aim of the present paper is to extend our previous study [13] to the (5,4,0) and (4,5,0) lithium clusters and to discuss some common features and possible consequences of the non-adiabatic behaviour of the Li,H systems.
2. The computational method The goal of the DIM method is to compute a polyatomic potential energy surface in terms of more accessible energies of the constituent atomic and diatomic fragments. The method is based on the valence bond type wave function but the role of the basis is rather formal. Actual analytic expressions for the atomic functions from which the basis is constructed are nat required. Essential are their symmetry properties which are used to set up the DIM approximation to the Hamiltonian matrix [ 14- 16], A sir&et wave function of the Li,H system is taken as a linear combination of polyatomic basis functions (PBF’s) which are antisymmetrized products of atomic (ground state) ‘S states. A PBF may be written as
Here, B is the ~~sy~et~~ation operator. AL:: is the ‘S wake function of atom K with the spin projection M,~. A product of these functions, +,, is called a primitive function. Since msK can take on two values, l/2 and - l/2, there are 252 PBF’s with m,, -I-M,~ + . . * +m,,,= 0. The calculations are done in a spin adapted basis (&?‘+m}constructed by the direct diagonalization of the total S2 operator in the subspaces of the [+,_] space, pre-arranged so as
J. Vojtik, J. FiSer / Inleraciion
of H with Li metal clusters
113
to be invariant under the diatomic fragment spin operators S&, S$,,..., S;,, [26] and time reversal [27]. The approximations to the true singlet potential energy hypersurfaces of the system are obtained by the diagonalization of the S-‘H matrix of the nonHermitean version of the DIM method [28]
K=l
L>K
K=l
(2)
Here S and H are, respectively, overlap and Hamiltonian matrices, /rcK) and AcKL) are (diagonal) matrices of the accurate atomic and diatomic fragment energy levels, respectively. Each of the yKLj matrices represents the KL diatomic fragment states in the basis of (properly antisymmetrized) spinadapted primitive functions. As our basis is composed of *S states, each diatomic fragment can occur in a ‘Z+ or ‘Z+ state. Another consequence of the present choice of the basis is .of the system and that the yKLj matrices do not depend on the configuration are deducible from the spin properties of the basis. Specifically, due to the way of constructing the {‘$I~} basis we have TKLj = 1 (unit matrix) for KLE{ 12,34 ,..., 910}, while the other matrices are obtained as representatives in {‘$,,} of the operators permuting appropriately the atomic fragments. These matrices are all found to be unitary, and it makes no difference for our choice of the basis whether one exactly includes the overlap through the application of the non-Hermitean version of the scheme or uses the standard version of DIM with the neglect-of-overlap assumption invoked. Adopting the latter point of view we can write for the Hamiltonian matrix
In evaluating the diagonal elements of the diatomic fragment matrices, the ‘2+ and ‘Xl potential energy curves (PEC’s) of Li, are taken from the work bysPickup [29], while the ‘2+ and 3Z+ PEC’s of LiH are based on the multiconfiguration SCF calculations by Docken and Hinze [30]. The monatomic contributions to the polyatomic Hamiltonian matrix are obtained under the usual assumption that the primitive functions are eigenstates of atomic Hamiltonians. A comment concerning the basis appears to be warranted. Due to the fact that Li,H is the largest system to which the DIM method has been applied, it seems natural, in view of the programming and computer time difficulties, to start with the minimum basis calculations. Though the quality of the resulting DIM model of the potential energy hypersurface is difficult to asses, the situation is certainly not so disastrous as if the ab initio valence bond calculations with the same basis were contemplated. The main reasons are the indirect role played by the basis within the DIM scheme and the use of very
accurate PEc”s for diatomic fragments Li, and LiH. It should perhaps be remarked in this connection that if PEC’s of comparabie quality were to be obtained by the (ab initio) valence bond technique, one should take a basis consisting of a great many structures, including ionized ones. Thus, while the present basis can be considered to be not sufficiently flexible, one cannot claim that inclusion of the ionized structures i3If into the DIM model of L&H will lead to a better approximation to the true potential energy hypersurfaces.
Three different Li, clusters were emptoyed (fig. I), namely (4,1,4), (4,5,0) and (5,4,0). PEC’s were obtained for normal approach of the hydrogen atom to the (100) lithium cluster surface for three adsorption sites: atop (A), bridge (B) and centred (C) (fig. 1). Ground and excited state PEC’s together with their symmetry notation for each cluster and site are shown in figs. 2-4. The most important point to notice in these figures concerns the mutual arrangement of the ground and excited state singlet PEC’s. In most cases studied crossing and/or avoided crossing between the PEC’s occurs, indicating that in some regions of the configuration space the Born-Oppenheimer approximation breaks down, and the electronic motion cannot be separated from the nuclear one. In these regions, the relative adatom-substrate motion can produce transitions between the potential energy hypersurfaces. 3.1. Model For gas phase processes, the ‘~surfa~-Hopkins-trajecto~ method” has been developed [173, which takes into account non-adiabatic transitions. This technique is based on the semiclassical theory of non-adiabatic behaviour and requires a thorough treatment of the dynamics of the elementary process. A
Fig. I. The (4, I, 4), (4,5,0) and (5,4,0) modeIs of bee I_i(lOO) surface C. The cube edge is taken to be 3.491 A.
and adsorption
sites A, B,
J. Vojtik, J. FiSer / Interaction
115
of H with Li metal clusters
-7
-0
3 iii -9
-10
I
I
I
I
1
2
3
4
r(A)
c
4
I
I
I
I
I
I
I
I
1
2
3
4
1
2
3
4
c(A)
r(A)
Fig. 2. PEC’s for the A-site approach of the atomic hydrogen to the lithium clusters. (a) (4, 1,4), (b) (4, 5,0), (c) (5,4,0) cluster.
necessary prerequisite for such an approach is the knowledge of the adiabatic potential energy hypersurfaces and non-adiabatic coupling among them in the whole configuration space relevant to the model. At present, such information on the L&H system is not available and we have to combine the “surface-hopping-trajectory” idea with the limited information on the interactions and non-adiabatic coupling provided by PEC’s relevant to a normal approach of H to our selected surface sites. In doing so, we follow the approach usually adopted in cases where only information on the ground state hypersurface represented by a few PEC’s is available. In this latter case a surface site is assigned adsorption energy (and equilibrium distance) according to the PEC corresponding to the site considered. In this way, even those adsorption energies can be arrived at which, if changes of the potential energy with respect to the lateral change of the adatom-cluster configuration were known, might turn out not to correspond to a local minimum on the potential energy hypersurface. We adopt here a similar viewpoint and assign to, a surface site every adsorption energy (and equilibrium adatom-surface distance) which corresponds to a minimum on either ground and excited state PEC provided, of course, that both the electronic structure and nuclear configuration corresponding to this minimum can be acquired by the system during the adsorption process. In order to make a rough estimate of the probability with which sudden changes in the electronic structure of the system occur, we have to make an assumption about the relative adatom-cluster motion in the non-
J. Vojtik, J. FiSer / Interaction
116
of H with Li metal clusters
-1
-8
3J
-9
Ill
-10
-11
-2
-1
0
3
-1
0
r/A)
1
3
4
-2
I
I
-1
0
I
iA)
I
I
2
3
1 4
-7
-0
2
-9
w
-10
-11
-2
1
2
4
G, Fig. 3. (a), (b) PEC’s for the B-site approach of the H atom to the Li, clusters: (a) (4, 1,4), (b) (4, 5,0) cluster. (c) PEC’s for the B-site approach of the H atom to the (5,4,0) lithium cluster.
adiabatic regions. Because of the limited information on the potential energy hypersurfaces we have, and in view of the approach discussed above, we restrict ourselves to the normal component of the H-cluster surface motion.
117
J. Vojtik, J. FiSer / Interaction of Ii with Li metal clusters
-7
t -8
9 2% -9 u
-10
-11
I
-1
0
1
2
3
4
-1
0
1
2
3
4
6)
-7
-8
7 s-9 u
-10 I-
-11
Fig. 4. (a), (b) PEC’s for the C-site approach of the H atom to the Li, clusters: (a) (4, 1,4), (b) (4,5,0) cluster. (c) PEC’s for the C-site approach of the H atom to the (5.4.0) cluster. Dashed line represents the non-adiabatic coupling matrix element (+21a/arl+l) = c2, of eq. (5) as a function of r.
118
J. Vojtik, J. F&r
3.2. Non-adiabatic
/ Interaction
of H with Li metal clusters
transitions
A rough estimate of the transition probabilityp,; j can be obtained from the expression [32,33].
from the state i to the state
p,;-exp(-ICE,-E,)/v,.d,,(). Here, vi is the nuclear velocity vector appropriate and d,, is the non-adiabatic coupling vector d,,=
-i~(ll;lVlG,>.
Ei and 4, are the electronic According estimate the batic coupling dj,“=
to the adiabatic
surface
E,
(5)
the corresponding eigenvalue and eigenfunction, respectively, of Hamiltonian of the system. to our model, we need normal components of u, and d,, to transition probability. The normal component of the non-adiaelement is, as usual in the DIM scheme [ 181, approximated by
-iA[~i(MI/i3r)C;]/(Ej-E,),
(6)
where r is the distance between the hydrogen atom and the cluster surface, aH/ar is obtained by differentiating each of the matrix elements of eq. (3) and C, is the column matrix of the expansion coefficients of qi in the PBF basis. In order to facilitate the discussion of the qualitative features of the adsorption process for individual binding sites, it is convenient to distinguish the following three types of the mutual arrangement of the PEC’s. In the first type (cf. figs. 2a-2c and 4b), the ground state PEC is neither crossed nor pseudocrossed by the excited state PEC’s, so that to each of these sites a single adsorption energy and equilibrium distance corresponding to the ground state PEC must be assigned. The second type (figs. 3b, 3c and 4c) is characterized by the presence of an avoided crossing between the ground state PEC and an excited one. For the B-site approach (both clusters), the transition probability is practically negligible, because the curves do not get close enough (cf. eqs. (4) and (5)) to make transition possible. As a consequence, adsorption energies corresponding to the ground state PEC’s are to be assigned to these binding sites. In the (5,4,0) C-site case, however, the situation is different and even for thermal adatomsurface relative velocities (about a few times lop4 au), the transition probability can be expected to be large. Thus, in this case, besides the ground state energy minimum, the other one, corresponding to the excited A” state will apply. It should be noted in this connection that our simple model does not provide a mechanism for the system to get to the excited A’ state. This is due to the fact that for the normal component of the vector (5) we have, on symmetry grounds, dj,n)= -ift(~,(A’)(a/arl~,(A”))=O.
(8)
J. Yojtik* J. FiSer / Interaction of H with Li metal clusters
119
A thorough inspection of the situation [ 13,171 then reveals that the transition probability between the states under question vanishes. In the third type (figs. 3a and 4a), the PEC’s that are most energetically favourable in the asymptotic region are intersected by the curves corresponding to states of different symmetry. At crossing points, the normal component of the coupling element (5) is in both cases zero, so that within our model, a transition between the PEC’s cannot occur. The system will not acquire the electronic structure corresponding to the A’ and A, states. On the contrary, the adsorption process is pictured as governed solely by the A” and B, curves, respectively. Thus the adsorption energies corresponding to these curves, not the overal minima on the A’ and A, PEC’s, are to be assigned to these adsorption sites. 3.3. Equilibrium properties for different adsorption sites For the A-site approach, the arrangement of the PEC’s belongs, for all three clusters, to the first type. The adsorption energies and equilibrium distances are similar for all the clusters (cf. table 1) and are in good agreement with those obtained by other authors [8- 1I]. We now move our attention to the B-site .adsorption (figs. 3a-3c). In the case of the (4, 1,4) cluster, the two lowest A’ and A” PEC’s cross in the vicinity of the minimum of the A” curve. However, in accordance with the above general remarks, the system will be governed by the A” PEC leading to the equlibrium distance of 0.28 A and the adsorption energy of -2.39 eV. For the (4,5,0) and (5,4,0) clusters, the non-adiabatic effects are negligible so that the
Table 1 ~uilib~um
properties of the Li,H systems for different clusters and adsorption sites
Site
Cluster
A
(4, 174) (49% 0) (%4> 0)
1.61 1.61 1.59
B
(4,I,4) (4, 5, 0) (5,4,0)
0.28 - 1.39 -0.36
-2.39 -2.81 -2.81
C
(4,1.4) (42% 0) (5,4.0)
0.30 -0.06 -0.11 0.40
- 1.80 -3.03 - 2.33 -1.90
r, ‘) (A)
&is (eV) - 1.89 - 1.64 - 1.74
‘) r. is the equilibrium distance of the H atom to the cluster surface.
120
J. Vojtik, 3. Fi3er / Interaction
of H with Li metal clusters
adsorption process is essentially controlled by the lowest PEC’s which are attractive inside the clusters. The subsurface minimum for this site was reported by Kunz et al. [S] at r, = -0.5 A for the (4, 1,4) cluster, by Fantucci et al. [lo] at r, = -0.53 A for the (4,1,0) cluster, and by Lavery and Hillier [ 121 for the (4,1,4) cluster at r, = - 1.75 A. Companion [ 1l] found a periodic attractive potential inside the lithium lattice, with the first minimum at about - 1.0 A. As far as the adsorption energy is concerned, our values are lower in magnitude than the results of the above-mentioned calculations: -3.0 eV [S], -2.97 eV [Ii] and -3.48 eV 1121. The other minima reported lie above the surface. The CNDO calculations with the Pople parametrization on the (4, 1,4) cluster [ 121 predict a second minimum at 1.12 A (-4.12 eV), while the Sichel parametrization leads to the values -2.68 eV at r, = 1.0 A [ 121. The crystal orbital technique gives 0.4 A (-3.60 eV) [12]. In the C-site case the non-adiabatic effects are found to play an important role. For the (5,4,0) cluster, the present model leads to two energy minima (cf. table l), one lying below, and the other above the surface. It should be remarked that the latter minimum is accessible thanks to the non-adiabatic behaviour of the system. With the (4,1,4) cluster, the situation is even more flagrant. It is perhaps interesting to note that in both cases these effects are accompanied by a decrease of the adsorption energy. Another consequence of the non-adiabatic behaviour of the L&H system in this case is the occurrence of relatively large differences in the equilibrium properties obtained for different clusters. However, the present results do not contradict those obtained by other authors for this adsorption site. A subsurface minimum was found by Kunz et al. [8] and by Companion [I 11.The adsorption energies obtained here are all negative, which is in accord with all the reported calculations but that of Fantucci et al. [lo]. In concluding this section, we would like to mention that with the exception of the (4,5,0) cluster, the chemisorption of a H atom in the B position is favoured by our calculations to that in the C position. This result is in agreement with the ab initio calculations [lo] on the (4,1,0) cluster but contradicts the CNDO results [ 121. It is interesting to note that the basic trends obtained by Hjelmberg in his self-consistent study of a hydrogen atom interacting with a jellium model of the Na(lOO) surface [34] are similar to the present ones.
4. Concluding remarks The present study of the interaction of. atomic hydrogen with Li, clusters by the DIM method also includes excited states. Ground and excited state potential energy curves obtained for normal approach of the H atom to some
J. Vojtik, J. F&r / Interaction of H with Li metal clwters
121
adsorption sites are found to intersect or to avoid crossing. Even a qualitative analysis of the situation reveals that the adsorption process cannot be pictured as taking place on one energy hypersurface. Each of the configurations studied here is characterized by non-trivial spatial symmetry and leads to PEC’s which, when having different symmetry, can cross. Thus these configurations in a sense represent limiting cases qualitatively different from general binding positions where the PEC’s can at most avoid crossing. In this connection the question arises as to how the present picture of the adsorption process would be changed if, instead of taking the binding sites examined in the previous section, general sites, lying in their vicinity were studied. In view of the remarks made previously, a change of this kind in the binding site will affect the arrangement of the PEC’s in all the cases but the (4,5,0) A-site one, causing the actual crossing to become avoided. In accordance with the model used here, changes in the non-adiabatic behaviour of the Li,H system will occur with the (4, 1,4) B-site, (4, 1,4) C-site and (5,4,0) C-site cases. In the first two cases, the model used here will lead to an additional pair of equilibrium properties. According to this new pair of data, the hydrogen atom, besides being adsorbed, is predicted to be also absorbed and the over-all picture of the chemisorption is modified in favour of the penetration of H through the cluster surface. The probability with which a resulting new pair applies will be connected, through the quantities occurring in eq. (4) with the change in the binding site. Mainly due to the dependence on the energy difference between the PEC’s (cf. eq. (4)) one can expect that the probabilities will change more dramatically than the PEC’s themselves. Our sample calculation on the (4, 1,4) cluster [ 131 seems to confirm this point. We can therefore infer that when non-adiabatic effects occur, equilibrium characteristics of chemisorption deduced from a few PEC’s corresponding to configurations of nontrivial spatial symmetry can be expected to be less reliable than the information obtained in an analogous way for processes involving only ground state hypersurfaces. This conclusion may be of value in those situations where cluster calculations are done with the aim of generating data for the construction of model potential energy hypersurfaces to be used in dynamical studies of the gas-surface processes [3-71. In concluding this communication, we would like to remark that we should have felt much safer if we could have based our discussion on results obtained for larger clusters and with a more flexible basis than the minimum one. That is why our results should be taken with a reserve. However, due to the conceptual background of the DIM method, one is inclined to believe that the present picture of the chemisorption process allows for that part of what is actually going on which cannot be taken into account by the SCF MO cluster approaches.
122
J. Vojtik, J. FiSer / Interaction
of H with Li metal clusters
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