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Surface crossing in interaction of atomic hydrogen with a lithium metal cluster
CHEMICAL. PHYSICS LJDTJZRS
Volume 80, number 3
SURFACE CROSSING IN INTERACTION
15 June 1981
OF ATOMIC HYDROGEN
WITH A LITHIUM METAL CLUSTER J. VOJT&, J. FIgER * and R POLk J. Heyrovskj, Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, 22138 Prague, Czechoslovakia Received 3 March 1981
Interaction of atomic hydrogen with a (4,1,4) lithium cluster, simulating the (100) metal surface, is studied using the diatom&in-molecules method.Ground-and excited&ate potential curves for normal approach of H to some attack positions on the surface intersect or pseudo-intersect. The results reveal possiile nonadiabatic character of the absorption process.
1. Introduction
2.
Model and method
Recently much interest has been directed toward the theoretical study of chemisorption on metal surfaces modelled by atomic clusters. These systems are rather complicated, so the quantum-chemical methods employed have generally been less sophisticated than those applied to treating simple chemical reactions in the gas phase. For example, in theoretical studies of the interaction of hydrogen with Li clusters, the calculations have been performed by the ab initio SCF method [l-3], semiempirical crystal orbital techniques [4] and the CNDO scheme ES]. Due to the inherent features of these methods, the studies have been confined to the ground-state energy hypersurface and the problem of whether excited states play a role in the adsorption process has largely been ignored. We have carried out calculations of the interaction of atomic hydrogen with the (4,1,4) lithium metal cluster, including excited states. The method used is the diatomics-in-molecules (DIM) approach [6---81 which has turned out to be quite successful in the theoretical interpretation of gas-phase reactions, including those where excited states and their coupling with the ground-state energy hypersurfaces play an important role.
The (100) face of lithium metal, assumed to have a perfect bee lattice, is modelled by the (4,1,4) cluster (fig. 1). Because of the Lig cluster size, we were forced to limit ourselves to the minimum basis DIM calculation, including only 2S states on each atom. A singlet wavefunction of the LigH system is expanded in terms of 252 polyatomic basis functions (PBFs) with &fs = 0. With this basis, each diatomic fragment can occur in a singlet or triplet state, so that the spin parts of the spin-adapted singlet PBF can be taken as those of the Serber type 191. Specifically, the basis is constructed in two steps [ 10,l l]. First, the space formed by the primitive (unantisymrnetrized) basis functions is split into the eigens aces of the diatomic f fragment spin operators Sfz: S34, .... SG 10 and of time reversal. The 42 singlet (see branching diagram)
* Present address: Department of Physical Chemistry, Charles University,128 40 Prague,Czechoslovakia.
Fig. 1. The (4,1,4)
0 009-2614/81/0000-OOOO/%
lithium clusterand adsorption sites: X, cross bond; S, supraatomic; H, hole; N, “near hole”,
02.50 0 North-Holland Publishing Company
569
Volume 80, number 3
primitive basis functions, denoted by lQln, are then obtained by the direct diagonalization of the total S* operator in these subspaces. The S-l H matrix of the non-hermitean DIM formulation f13-] for the Ll,H system is 9 &‘-lH
-7
10
= KG1 LqKTcKL
15 June 1981
CXEMICAL PHYSICS LETTERS
$z(~~)T;;KL
-8
)-- 8 5 hcK) .(I) K=l
As our basis includes only S-type states, the TcKt> matrices do not depend on the geometry of the system. Furthermore, due to the method of construction of the 1’4, Ibasis, we have TtKL) = 1 (identity matrix) for KL E { 12,34, . .. . 9 lo} . Each of the other T[KL) matrices relates the basis { 1 Qpn} to a new one which (after proper antisymmetrhation) is symmetry adapted with respect to the constants of motion of the corresponding fragment diatomic and is obtained as a representative in {I@, } of an operator permuting appropriately the fragment atoms. The diatomic fragment matrices /lcKL) are found to be diagonal so that with our minimum basis, the non-hermitean DIM formulation, which exactly excIudes overIap from the scheme, happens to lead to a hermitean matrix and is equivalent to the hermitean version of the DIM method with the neglect of overlap [ 13]_ For the evaluation of the diagonal elements of the fragment matrices, 1 q and 3 Zz potential energy curves (PECs) of Li, are taken from the work by Pickup [13], while z Zlf and 3 Cc PECs of LiH are based on the MC SCF results of Docken and Hinze [ 143 _ In evaluating the atomic parts of the total hamiltonian, the primitive functions ‘9, are assumed to be eigenfunctions of the atomic hamiltonians [6] _
3 z -9
-10 I
I
I
I
1
2
3
4
r(A)
Fig. 2. Interaction energy curves for the supra&omicapproach of the atomic hydrogen to the lithmm metal cluster.
system has C, symmetry. According to fig. 2, the system 1~11 follow the lowest (A’) PEC, leading to an equilibrium distance (I-61 A) and interaction energy (-1.89 eV) in reasonable accord with other calcula-
3. Results The cube edge in the (4,1,4) cluster is taken to be 3.491 a. Interaction energy curves obtained for normal approach of atomic hydrogen to the metal surface for four adsorption sites - supra-atomic, cross bond, hole, and “near hole” (fig_ l), together with their symmetry notation, are shown in figs. 24. In the asymptotic region, the differences between the PECs correspond to the relevant excitation energies of the Lig cluster. For the supra-atomic approach (fig. Z), the L.ig H 570
-11
I
-2
1
-1
I
1
I
I
0
1
2
3
L
4
AA)
Fig. 3. PECs for the cross-bond site approach of the atomic hydrogen to the lithium metal cIuster.
CHEMICAL PHYSICS LElTERS
Volume 80, number 3 -7
-
A¶ a)
15 June 1981
5
b,
E
.4
A, -8
2
2
-
2
Bl
1
-9 -
-10
-
bE:;;
2
81
1
A, -11
’ -1
I 0
1 1
I
2
1
3
t
4 6)
1 -1
I 0
20
L :: II 11C,, :: :! :: ‘1’I_
,OG u’ I
2
1
3
’
4
0
Fig. 4. PECs for the normal approach of the atomic hydrogen to (a) the hole site and (b) the poution lying at a distance of 0.1 A from the hole site. The latter case is supplemented by the non-adiabatic coupling element Cl2 calculated according to ref. [ 151.
tions for normal supra-atomic approach [2,4,5]. Unlike the previous case, in the hole and cross-bond site attacks (figs_ 4a and 3) the PECs that are the most energetically favourable in the asymptotic region are intersected at distances of x0.9 and 0.2 A, respectively. According to the general rules about the possible arrangement of two PECs [ 161, such an intersection can occur only between the PECs of different symmetry. The PECs in fig. 4b are obtained for the normal approach to the attack possition which lies 0.1 a from the hole site and is chosen so that the L&-H system has no spatial symmetry. According to the abovementioned rules, the PECs avoid crossing, indicating that even a very simple interpretation of the (elementary) adsorption process should include corrections to the Born-Oppenheimer approximation.
4. Discussion and conclusions First we shall briefly mention the most obvious effects on the course of the normal approach of H to the surface, caused by the potential curve crossing and pseudo-crossing. According to the semiclassical theory of non-adiabatic behaviour 1171, the transition probability be-
tween two adiabatic electronic
states \k, (R) and \k2(R) (R stands for a set of nuclear coordinates) is COMeCted with the scalar product of the coupling matrix element
and the classical nuclear velocity u of the system. If the scalar product Cl2 -u vanishes as the system passes through a certain region of configuration space, no transition is induced [17]. This situation is encountered in the normal approach of H to the hole site of the cluster surface, governed in the asymptotic region by the PEC of BI symmetry (cf. fig. 4a), provided we take \kl(R) as corresponding to this B, PEC ’ and denote by e2(R) the state closest in energy. Indeed, as the C,, symmetry of the system is conserved during the normal approach, the component of Cl2 which is relevant to the motion considered will, on symmetry grounds, vanish regardless of whether the lowest PECs get close or even cross_ We can therefore conclude that during this idealized normal approach the system will not reach the over-all minimum on the Al PEC. On the contrary, the system wilI follow the Bi PEC leading to the equilibrium distance of 0.4 A and interaction energy -1.77 eV. For a norma! 571
Volume 80, number 3
CHEMICAL PHYSICS LETTERS
approach of H to the cross-bond site (fig. 3), anab gous reasoning leads to the values re = 0.25 A and Eint = -2.42 eV. Now let us turn to the attack position which lies in the vicinity of the hole site. According to fig. 4b, the Li9H system ~111 follow the lowest curve until it reaches the non-adiabatic region characterized by a large magnitude of the coupling matrix element C12_ The motion of the system through this narrow region will cause a transition from the ground to the excited state. In order to get a rough estimate of the probability PI, of transition from the adiabatic state *I(R) to the state e,(R), resulting from a single passing of the coupling region, we use the Landau-Zener model [ 1 S] _ Aswming the normal component of the hydrogen atom velocity at the pseudo-crossing point to be 7 X 10d4 au (corresponding to the mean-square speed at room temperature), and taking the other parameters required by the model from the curves in fig. 4b, we arrive at PI2 = 0.6. If we ignore other possible transitions, we find that the application of this simple model to the normal approach of H to the attack posirion near the hole site leads to two pairs of adsorption characteristics. The values re = -0-2 & E lnt = -2.83 eV apply with probability ~0.4, while the other pair, r, = 0.4 A, Eint = -1.77 eV, applies with probability m-6. Thus, analogous to the hole and cross-bond attack, we arrive at the result which is in deep contrast to that which can be deduced when only one PEC is available. It indicates that - besides the ground state the excited state should be taken into consideration when interpreting experimental adsorption data. Of course, a more sophisticated treatment of the non-adiabatic behaviour of the system during the adsorption process would require a thorough investigation of its kinematics, including an appropriate averaging of the calculated data for specific initial conditions, e.g. adsorbing particle velocities, sites and angles of attack (cf. surface hopping trajectory method [ 191). A necessary prerequisite for such a treatment of the problem would be knowledge of the interaction energy hypersurfaces and non-adiabatic coupling between them in the configuration space relevant to the model. Concerning the interpretation of non-adiabatic behaviour of the system, it should be noted that the DIM calculation was carried out using a minimum 572
15 June 1981
basis set and consequently no charge transfer could occur. Also, other common causes of non-adiabatic behaviour (quenching, radiationless transitions, etc.) cannot count with our model. In an attempt to shed some light on the cause of non-adiabatic behaviour, we have evaluated those parts of the total energy in the lowest states which correspond to the singlet and triplet contributions to the interaction of atomic hydrogen with individual lithium atoms. While the subsequent analysis has not revealed any apparent relation between the changes in these quantities and the non-adiabatic behaviour, the results seem to indicate that the crossing might be connected With changes in that part of the energy and/or wavefunction corresponding to the (interacting) metal cluster. At present we are not able to specify whether these changes are confined ro a certain bond of the cluster or whether they are of a more collective nature. These calculations represent an application of the DIM method to a system with the largest number of atoms. However, it should be noted that we should have preferred to base our discussion on results obtained for a larger Li cluster and with a more flexible basis than the minimum one. So our results should be viewed with caution. Clearly, caloulations at a relatively very high level of sophistication are necessary to draw definite conclusions about the possible role of non-adiabatic effects on the course of adsorption. Though calculations of such a kind on clusters of sufficient size would be very expensive, our results indicate that they are justifiable.
References [ 11 A-B. Kum, D-J. Mickishand P.W. Deutsch, SoIid State Commun. 13 (1973) 35. 121 H. StoII and H. Preuss, Surface Sci. 65 (1977) 229. 131 J-B. Moffat, Surface Sci. 84 (1979) 65. [4] R. Lavery and 1-H. HiIIier, J_ Mol. CataI. 4 (1978) 9. [S] A-L. Companion, Chem. Phys. 14 (1976) I. [61 F-0. EIIison, J- Am. Chem. Sot. 85 (1963) 3540. 171 J-C- Tully, J. Chem. Phys. 58 (1973) 1396. [81 E. Steiner, P-R Ceitain and P-J_ Kuntz, J. Chem. Phys. 59 (1973) 47_ [9] R. Serber, Phys. Rev. 45 (1934) 461.
[IO1 W-I. Salmon, K. Ruedenbergand L.M. Cheung, J. Chem.
Phys. 57 (1972) 2787. 1111 J. Vojtik and J. Fser, Theoret. Chim. Acta 45 (1977) 301.
Volume 80, number 3
CHEMICAL
[ 121 J.C. Tully and CM. Truesdale, J. Chem. Phys. 65 (1976) 1002. [13] B-T. Pickup, Proc. Roy. Sot. A333 (1973) 69. [14] K.K. Docken and J. Hinze, J. Chem. Phys. 57 (1972) 4928. [15] RK. Preston and J.C. TulJy, J. Chem. Phys. 54 (1971) 4297.
PHYSICS LETTERS
15 June 1981
[ 161 L-D. Landau and EM. L&hi& Quantum mechanics, non-relativistic theory, 2nd Ed. (Addison-Wesley, Reading, 1965). [17] E E. Nikitin and L. Ztilicke, Theory of chen&aJ elemcntary processes {Springer, Berlin, 1978). [18] E.E. Nikitin, Advan. Quantum Chem. 5 (1970) 135. [ 191 J.C. Tully, in: Dynamics of molecular cohisrons, ed. W.H. Miller (Plenum Press, New York, 1976) p. 217.
573
Volume 80, number 3
SURFACE CROSSING IN INTERACTION
15 June 1981
OF ATOMIC HYDROGEN
WITH A LITHIUM METAL CLUSTER J. VOJT&, J. FIgER * and R POLk J. Heyrovskj, Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, 22138 Prague, Czechoslovakia Received 3 March 1981
Interaction of atomic hydrogen with a (4,1,4) lithium cluster, simulating the (100) metal surface, is studied using the diatom&in-molecules method.Ground-and excited&ate potential curves for normal approach of H to some attack positions on the surface intersect or pseudo-intersect. The results reveal possiile nonadiabatic character of the absorption process.
1. Introduction
2.
Model and method
Recently much interest has been directed toward the theoretical study of chemisorption on metal surfaces modelled by atomic clusters. These systems are rather complicated, so the quantum-chemical methods employed have generally been less sophisticated than those applied to treating simple chemical reactions in the gas phase. For example, in theoretical studies of the interaction of hydrogen with Li clusters, the calculations have been performed by the ab initio SCF method [l-3], semiempirical crystal orbital techniques [4] and the CNDO scheme ES]. Due to the inherent features of these methods, the studies have been confined to the ground-state energy hypersurface and the problem of whether excited states play a role in the adsorption process has largely been ignored. We have carried out calculations of the interaction of atomic hydrogen with the (4,1,4) lithium metal cluster, including excited states. The method used is the diatomics-in-molecules (DIM) approach [6---81 which has turned out to be quite successful in the theoretical interpretation of gas-phase reactions, including those where excited states and their coupling with the ground-state energy hypersurfaces play an important role.
The (100) face of lithium metal, assumed to have a perfect bee lattice, is modelled by the (4,1,4) cluster (fig. 1). Because of the Lig cluster size, we were forced to limit ourselves to the minimum basis DIM calculation, including only 2S states on each atom. A singlet wavefunction of the LigH system is expanded in terms of 252 polyatomic basis functions (PBFs) with &fs = 0. With this basis, each diatomic fragment can occur in a singlet or triplet state, so that the spin parts of the spin-adapted singlet PBF can be taken as those of the Serber type 191. Specifically, the basis is constructed in two steps [ 10,l l]. First, the space formed by the primitive (unantisymrnetrized) basis functions is split into the eigens aces of the diatomic f fragment spin operators Sfz: S34, .... SG 10 and of time reversal. The 42 singlet (see branching diagram)
* Present address: Department of Physical Chemistry, Charles University,128 40 Prague,Czechoslovakia.
Fig. 1. The (4,1,4)
0 009-2614/81/0000-OOOO/%
lithium clusterand adsorption sites: X, cross bond; S, supraatomic; H, hole; N, “near hole”,
02.50 0 North-Holland Publishing Company
569
Volume 80, number 3
primitive basis functions, denoted by lQln, are then obtained by the direct diagonalization of the total S* operator in these subspaces. The S-l H matrix of the non-hermitean DIM formulation f13-] for the Ll,H system is 9 &‘-lH
-7
10
= KG1 LqKTcKL
15 June 1981
CXEMICAL PHYSICS LETTERS
$z(~~)T;;KL
-8
)-- 8 5 hcK) .(I) K=l
As our basis includes only S-type states, the TcKt> matrices do not depend on the geometry of the system. Furthermore, due to the method of construction of the 1’4, Ibasis, we have TtKL) = 1 (identity matrix) for KL E { 12,34, . .. . 9 lo} . Each of the other T[KL) matrices relates the basis { 1 Qpn} to a new one which (after proper antisymmetrhation) is symmetry adapted with respect to the constants of motion of the corresponding fragment diatomic and is obtained as a representative in {I@, } of an operator permuting appropriately the fragment atoms. The diatomic fragment matrices /lcKL) are found to be diagonal so that with our minimum basis, the non-hermitean DIM formulation, which exactly excIudes overIap from the scheme, happens to lead to a hermitean matrix and is equivalent to the hermitean version of the DIM method with the neglect of overlap [ 13]_ For the evaluation of the diagonal elements of the fragment matrices, 1 q and 3 Zz potential energy curves (PECs) of Li, are taken from the work by Pickup [13], while z Zlf and 3 Cc PECs of LiH are based on the MC SCF results of Docken and Hinze [ 143 _ In evaluating the atomic parts of the total hamiltonian, the primitive functions ‘9, are assumed to be eigenfunctions of the atomic hamiltonians [6] _
3 z -9
-10 I
I
I
I
1
2
3
4
r(A)
Fig. 2. Interaction energy curves for the supra&omicapproach of the atomic hydrogen to the lithmm metal cluster.
system has C, symmetry. According to fig. 2, the system 1~11 follow the lowest (A’) PEC, leading to an equilibrium distance (I-61 A) and interaction energy (-1.89 eV) in reasonable accord with other calcula-
3. Results The cube edge in the (4,1,4) cluster is taken to be 3.491 a. Interaction energy curves obtained for normal approach of atomic hydrogen to the metal surface for four adsorption sites - supra-atomic, cross bond, hole, and “near hole” (fig_ l), together with their symmetry notation, are shown in figs. 24. In the asymptotic region, the differences between the PECs correspond to the relevant excitation energies of the Lig cluster. For the supra-atomic approach (fig. Z), the L.ig H 570
-11
I
-2
1
-1
I
1
I
I
0
1
2
3
L
4
AA)
Fig. 3. PECs for the cross-bond site approach of the atomic hydrogen to the lithium metal cIuster.
CHEMICAL PHYSICS LElTERS
Volume 80, number 3 -7
-
A¶ a)
15 June 1981
5
b,
E
.4
A, -8
2
2
-
2
Bl
1
-9 -
-10
-
bE:;;
2
81
1
A, -11
’ -1
I 0
1 1
I
2
1
3
t
4 6)
1 -1
I 0
20
L :: II 11C,, :: :! :: ‘1’I_
,OG u’ I
2
1
3
’
4
0
Fig. 4. PECs for the normal approach of the atomic hydrogen to (a) the hole site and (b) the poution lying at a distance of 0.1 A from the hole site. The latter case is supplemented by the non-adiabatic coupling element Cl2 calculated according to ref. [ 151.
tions for normal supra-atomic approach [2,4,5]. Unlike the previous case, in the hole and cross-bond site attacks (figs_ 4a and 3) the PECs that are the most energetically favourable in the asymptotic region are intersected at distances of x0.9 and 0.2 A, respectively. According to the general rules about the possible arrangement of two PECs [ 161, such an intersection can occur only between the PECs of different symmetry. The PECs in fig. 4b are obtained for the normal approach to the attack possition which lies 0.1 a from the hole site and is chosen so that the L&-H system has no spatial symmetry. According to the abovementioned rules, the PECs avoid crossing, indicating that even a very simple interpretation of the (elementary) adsorption process should include corrections to the Born-Oppenheimer approximation.
4. Discussion and conclusions First we shall briefly mention the most obvious effects on the course of the normal approach of H to the surface, caused by the potential curve crossing and pseudo-crossing. According to the semiclassical theory of non-adiabatic behaviour 1171, the transition probability be-
tween two adiabatic electronic
states \k, (R) and \k2(R) (R stands for a set of nuclear coordinates) is COMeCted with the scalar product of the coupling matrix element
and the classical nuclear velocity u of the system. If the scalar product Cl2 -u vanishes as the system passes through a certain region of configuration space, no transition is induced [17]. This situation is encountered in the normal approach of H to the hole site of the cluster surface, governed in the asymptotic region by the PEC of BI symmetry (cf. fig. 4a), provided we take \kl(R) as corresponding to this B, PEC ’ and denote by e2(R) the state closest in energy. Indeed, as the C,, symmetry of the system is conserved during the normal approach, the component of Cl2 which is relevant to the motion considered will, on symmetry grounds, vanish regardless of whether the lowest PECs get close or even cross_ We can therefore conclude that during this idealized normal approach the system will not reach the over-all minimum on the Al PEC. On the contrary, the system wilI follow the Bi PEC leading to the equilibrium distance of 0.4 A and interaction energy -1.77 eV. For a norma! 571
Volume 80, number 3
CHEMICAL PHYSICS LETTERS
approach of H to the cross-bond site (fig. 3), anab gous reasoning leads to the values re = 0.25 A and Eint = -2.42 eV. Now let us turn to the attack position which lies in the vicinity of the hole site. According to fig. 4b, the Li9H system ~111 follow the lowest curve until it reaches the non-adiabatic region characterized by a large magnitude of the coupling matrix element C12_ The motion of the system through this narrow region will cause a transition from the ground to the excited state. In order to get a rough estimate of the probability PI, of transition from the adiabatic state *I(R) to the state e,(R), resulting from a single passing of the coupling region, we use the Landau-Zener model [ 1 S] _ Aswming the normal component of the hydrogen atom velocity at the pseudo-crossing point to be 7 X 10d4 au (corresponding to the mean-square speed at room temperature), and taking the other parameters required by the model from the curves in fig. 4b, we arrive at PI2 = 0.6. If we ignore other possible transitions, we find that the application of this simple model to the normal approach of H to the attack posirion near the hole site leads to two pairs of adsorption characteristics. The values re = -0-2 & E lnt = -2.83 eV apply with probability ~0.4, while the other pair, r, = 0.4 A, Eint = -1.77 eV, applies with probability m-6. Thus, analogous to the hole and cross-bond attack, we arrive at the result which is in deep contrast to that which can be deduced when only one PEC is available. It indicates that - besides the ground state the excited state should be taken into consideration when interpreting experimental adsorption data. Of course, a more sophisticated treatment of the non-adiabatic behaviour of the system during the adsorption process would require a thorough investigation of its kinematics, including an appropriate averaging of the calculated data for specific initial conditions, e.g. adsorbing particle velocities, sites and angles of attack (cf. surface hopping trajectory method [ 191). A necessary prerequisite for such a treatment of the problem would be knowledge of the interaction energy hypersurfaces and non-adiabatic coupling between them in the configuration space relevant to the model. Concerning the interpretation of non-adiabatic behaviour of the system, it should be noted that the DIM calculation was carried out using a minimum 572
15 June 1981
basis set and consequently no charge transfer could occur. Also, other common causes of non-adiabatic behaviour (quenching, radiationless transitions, etc.) cannot count with our model. In an attempt to shed some light on the cause of non-adiabatic behaviour, we have evaluated those parts of the total energy in the lowest states which correspond to the singlet and triplet contributions to the interaction of atomic hydrogen with individual lithium atoms. While the subsequent analysis has not revealed any apparent relation between the changes in these quantities and the non-adiabatic behaviour, the results seem to indicate that the crossing might be connected With changes in that part of the energy and/or wavefunction corresponding to the (interacting) metal cluster. At present we are not able to specify whether these changes are confined ro a certain bond of the cluster or whether they are of a more collective nature. These calculations represent an application of the DIM method to a system with the largest number of atoms. However, it should be noted that we should have preferred to base our discussion on results obtained for a larger Li cluster and with a more flexible basis than the minimum one. So our results should be viewed with caution. Clearly, caloulations at a relatively very high level of sophistication are necessary to draw definite conclusions about the possible role of non-adiabatic effects on the course of adsorption. Though calculations of such a kind on clusters of sufficient size would be very expensive, our results indicate that they are justifiable.
References [ 11 A-B. Kum, D-J. Mickishand P.W. Deutsch, SoIid State Commun. 13 (1973) 35. 121 H. StoII and H. Preuss, Surface Sci. 65 (1977) 229. 131 J-B. Moffat, Surface Sci. 84 (1979) 65. [4] R. Lavery and 1-H. HiIIier, J_ Mol. CataI. 4 (1978) 9. [S] A-L. Companion, Chem. Phys. 14 (1976) I. [61 F-0. EIIison, J- Am. Chem. Sot. 85 (1963) 3540. 171 J-C- Tully, J. Chem. Phys. 58 (1973) 1396. [81 E. Steiner, P-R Ceitain and P-J_ Kuntz, J. Chem. Phys. 59 (1973) 47_ [9] R. Serber, Phys. Rev. 45 (1934) 461.
[IO1 W-I. Salmon, K. Ruedenbergand L.M. Cheung, J. Chem.
Phys. 57 (1972) 2787. 1111 J. Vojtik and J. Fser, Theoret. Chim. Acta 45 (1977) 301.
Volume 80, number 3
CHEMICAL
[ 121 J.C. Tully and CM. Truesdale, J. Chem. Phys. 65 (1976) 1002. [13] B-T. Pickup, Proc. Roy. Sot. A333 (1973) 69. [14] K.K. Docken and J. Hinze, J. Chem. Phys. 57 (1972) 4928. [15] RK. Preston and J.C. TulJy, J. Chem. Phys. 54 (1971) 4297.
PHYSICS LETTERS
15 June 1981
[ 161 L-D. Landau and EM. L&hi& Quantum mechanics, non-relativistic theory, 2nd Ed. (Addison-Wesley, Reading, 1965). [17] E E. Nikitin and L. Ztilicke, Theory of chen&aJ elemcntary processes {Springer, Berlin, 1978). [18] E.E. Nikitin, Advan. Quantum Chem. 5 (1970) 135. [ 191 J.C. Tully, in: Dynamics of molecular cohisrons, ed. W.H. Miller (Plenum Press, New York, 1976) p. 217.
573