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A modified Cashion–Herschbach potential for the H3 potential energy surface
6 October 2000
Chemical Physics Letters 328 (2000) 469±472
www.elsevier.nl/locate/cplett
A modi®ed Cashion±Herschbach potential for the H3 potential energy surface T.I. Sachse *, K.T. Tang 1, J.P. Toennies Max-Planck-Institut f ur Str omungsforschung, Bunsenstr. 10, D-37073 G ottingen, Germany Received 12 July 2000; received in ®nal form 4 August 2000
Abstract The H3 potential energy surface has been calculated using the Cashion±Hershbach (CH) potential which contains all orders of two-body Coulomb, exchange, and overlap terms. For collinear con®gurations, the dominant three-body contribution comes from the Axilrod±Teller±Muto triple±dipole dispersion energy. By adding only this term, properly damped, the saddle point region potential agrees with recent ab initio calculations to within 9 10ÿ5 a.u. (0.06 kcal/ mole). Ó 2000 Elsevier Science B.V. All rights reserved.
Starting with the famous London equation [1,2] there has been a long continuing search for a simple expression with which to describe the H3 potential energy hypersurface. One of the early semi-empirical potential models, reported by Cashion and Herschbach (CH) in 1964 [3], was derived from the ®rst order valence bond theory by neglecting the three-body terms and overlap integrals. This surface is known to be equivalent to the diatomics-in-molecules surface also published in 1964 [4]. A recent analysis based on the generalized Heitler±London (GHL) theory [5,6] has shown that the CH surface encompasses far more information than previously recognized. The complete GHL potential energy hypersurface of the triatomic H3 -system is given by [5,6]
*
Corresponding author. Fax: +49-551-517-6607. E-mail address: [email protected] (T.I. Sachse). 1 Permanent address: Department of Physics, Paci®c Lutheran University, Tacoma, WA 98447, USA.
V C RAB C RBC C RAC CABC XABC h 1 X RAB ÿ X RBC 2 X RBC ÿ 2 i1=2 ; ÿ X RAC 2 X RAC ÿ X RAB 2 1 where RAB ; RBC ; RAC are the interatomic distances and C RAB and X RAB are, respectively, the twobody Coulomb and exchange energies between atom A and B in the presence of atom C. The nonadditive three-body Coulomb and the three-body cyclic exchange energies are denoted by CABC and XABC , respectively. For an isolated H2 molecule, the Coulomb and exchange energies are de®ned in terms of the singlet Es and triplet Et energies [5] 1 C RAB Es RAB Et RAB ; 2
2
1 X RAB Es RAB ÿ Et RAB : 2
3
0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 0 9 1 8 - 0
470
T.I. Sachse et al. / Chemical Physics Letters 328 (2000) 469±472
The GHL energy expressions for the H2 molecule are P Jn Kn ; 4 Es n1 P 1 n0 Sn P Jn ÿ Kn ; 5 Et n1 P 1 ÿ n0 Sn where Jn , Kn and Sn are, respectively, the nth order Coulomb, exchange and overlap integrals [5]. For comparison, in the usual Heitler±London approximation, the singlet and triplet energies are given by EsHL
J1 K1 ; 1 S0
6
EtHL
J1 ÿ K1 ; 1 ÿ S0
7
where, in the common usage, J1 , K1 and S0 are referred to as the Coulomb, exchange and overlap integrals. If in Eq. (1) all three-body eects, all orders of Sn , and, ®nally, all orders of Jn and Kn except J1 and K1 are neglected then Eq. (1) becomes equivalent to the London formula: VLondon J1 RAB J1 RBC J1 RAC h 1 2 K1 RAB ÿ K1 RBC ÿ 2 2
K1 RBC ÿ K1 RAC
2
K1 RAC ÿ K1 RAB
i1=2
:
8
In their derivation, Cashion and Herschbach started with this equation but used the Heitler± London energies of Eqs. (6) and (7) with S0 neglected to approximate J1 and K1 by ®tting to the available ab initio singlet and triplet energies 1 J1 Es Et ; 2 1 K1 Es ÿ Et : 2
9
But since J1 and K1 in this approximation are identical to C and X as de®ned in Eqs. (2) and (3) [5,6], Cashion and Herschbach have, in fact, introduced the full Coulomb and exchange energies
as de®ned in Eqs. (2) and (3). Therefore, their surface is given by VCH C RAB C RBC C RAC h 1 2 X RAB ÿ X RBC ÿ 2 2
X RBC ÿ X RAC X RBC i1=2 2 : ÿ X RAB
10
A comparison with Eq. (1) with consideration of Eqs. (2)±(5) reveals that this surface includes not only the overlap integrals, but also all higher orders of two-body Coulomb and exchange integrals, which had been thought to have been neglected. In view of this new insight, it is interesting to reexamine the accuracy of the CH potential. Using the `exact' singlet and triplet potential energies for the H2 molecule [7,8], the CH potential has been compared with the recently calculated accurate ab initio H3 potential energy surface [9±11]. For the important linear saddle point region, at which the hydrogen exchange reaction takes place the classical barrier height of the CH potential is, however, about 3 kcal/mole (0.005 a.u.) greater than the accepted value of 9.6 kcal/mole. Apparently, the neglected three-body eects are still necessary. The GHL theory, Eq. (1), shows that the main three-body eects are: (1) the cyclic exchange energy XABC [12±14]; (2) the implicit three-body eect on the two-body exchange energies due to the presence of the third atom [15,16]; and (3) the three-body Coulomb energy CABC [17±19]. The cyclic and the implicit three-body eects on the exchange energies in Eq. (1) can be calculated using a generalization of the surface integral method [12±14], which is based on the concept that the electrons are continuously trading places. This has been shown previously to give excellent results for the two-body exchange energy of the H2 -molecule [20,21]. In H3 , the cyclic exchange of the three electrons is responsible for the three-body exchange energy XABC [12±14]. For the cyclic exchange, one would expect that XABC is negligibly small in the linear con®guration, because one atom is in the middle. The few explicit results available
T.I. Sachse et al. / Chemical Physics Letters 328 (2000) 469±472
[13,14] indicate, indeed, that XABC rapidly decreases in magnitude as the three atoms approach the linear con®guration. The same is true for the three-body eects on the two-body exchange energies [15,16]. Therefore, it is not unreasonable to assume that the dominant three-body eect in the linear con®guration comes from the leading threebody Coulomb term, which is also known as the three-body dispersion energy. The main contribution to CABC is from the nonadditive triple±dipole interaction. Axilrod and Teller [17] and Muto [18] were the ®rst to give an analytic formula for the long-range behavior of this interaction CABC !
C9 R3AB R3BC R3AC 1 3 cos hA cos hB cos hC ;
11
1=3 b 3! ÿ C6He±H 2C6H±H =C9 :
471
14
With the well-known numerical values for the dispersion coecients in atomic units C6H±H 6:499 [23], C6He±H 2:82 [24] and C9 21:643 [25], the value of the damping parameter for H3 is b 1:4130:
15
In Fig. 1, the semi-empirical CH surface for the linear con®guration for symmetric stretching of both coordinates RAB RBC R passing through the reaction saddle point is plotted as a function of R (dotted line). The CH energies were evaluated using the accurate Coulomb and exchange energies extracted from the accurate ab initio H2 singlet and triplet potentials [7,8]. The CH surface for H3 plus Eq. (12) with Eq. (15) (solid line) is compared
where hA ; hB and hC are the interior angles at the respective atoms. Recently, we have introduced a new damping function [19] to extend the validity of this equation to smaller interatomic distances. This new damping function for the three-body dispersion term was found to give results in very good agreement with recent third order perturbation calculations for H3 [22]. The advantage of the new method is that it is applicable to all three-body systems if the respective two-body coecients in the united atom limits involved are available. For the H3 -system, the damped third order dispersion energy has the form [19] CABC f2 RAB f2 RBC f2 RAC
C9 R3AB R3BC R3AC
1 f0 RAB f0 RBC f0 RAC 3 cos hA cos hB cos hC ;
12
where the f2n are the Tang±Toennies damping functions [23] given by f2n R 1 ÿ eÿbR
k 2n X bR : k! k0
13
The f2n depend only on a single-range parameter b which is determined by taking into account the united atom limit of H±H which is the helium atom. As derived in Ref. [19], the result for the damping parameter for the H3 -system is given by
Fig. 1. Potential energy of H3 in the symmetrial linear con®guration RAB RBC passing through the saddle point (SP). The dotted line is calculated using the CH [3] method based on the accurate ab initio H2 potential energy of Refs. [7,8]. The open circles are the recent accurate ab initio results [9]. The solid line is the present result which is the sum of CH and the damped three-body polarization energy. It agrees to within 10ÿ4 a.u. with the ab initio result.
472
T.I. Sachse et al. / Chemical Physics Letters 328 (2000) 469±472
with the accurate ab initio results [9] (open circles). As expected the pure CH potential surface agrees with the ab initio results at large distances RAB RBC P 3:5 a.u. The agreement is to within 10ÿ4 a.u. since there the three-body eects are negligibly small. But at the minimum RAB RBC 1:757 a.u., which is also the saddle point for the reaction, the CH energy is ÿ0:15454 a.u., compared to the most accurate ab initio value [9±11] of ÿ0:1591760 a.u. If now CABC calculated from Eq. (12), which is equal to ÿ0:00453 a.u. is added to the CH energy, the saddle point energy becomes ÿ0:15908 a.u., which is only 0.00009 a.u. (0.06 kcal/mole) greater than the ab initio result. Thus the CH potential provides a simple yet very accurate description of the H3 potential energy surface. Moreover, for the important linear saddle point region the inclusion of the damped three-body polarization energy accounts for the remaining dierence between the CH potential and the best ab initio potentials. For non-linear con®gurations, the CH potential surface should also be a good starting approximation, since it contains all orders of the two-body interactions. However, for an accurate description, one must also include the three-body eects in the exchange energy, which in the linear con®guration are negligibly small. Although they can be calculated by the surface integral method, at present only numerical results at some isolated points are available [13±16]. Obviously, for non-linear con®gurations more work is needed to explore the analytic ®t to the three-body corrections to the exchange energy. In the future, it will be interesting to explore the validity of the CH procedure for other reactive systems. Acknowledgements We thank D. Herschbach and A.J.C. Varandas for helpful comments. K.T.T. wishes to thank the
Alexander von Humboldt±Stiftung for ®nancial support. References [1] F. London, Probleme der Modernen Physik, Sommerfeld± Festschrift, S. Hirzel, Leipzig, 1928, 104. [2] F. London, Z. Elektrochem. 35 (1929) 552. [3] J.K. Cashion, D.R. Herschbach, J. Chem. Phys. 40 (1964) 2358. [4] F.O. Ellison, J. Chem. Phys. 41 (1964) 2198. [5] K.T. Tang, J.P. Toennies, C.L. Yiu, Int. Rev. Phys. Chem. 17 (1998) 370. [6] U. Kleinekath ofer, K.T. Tang, J.P. Toennies, C.L. Yiu, J. Chem. Phys. 111 (1999) 3377. [7] W. Kolos, L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429. [8] W. Kolos, L. Wolniewicz, Chem. Phys. Lett. 24 (1974) 457. [9] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 104 (1996) 7139. [10] H. Partridge, C.W. Bauschlicher, J.R. Stallcop, E. Levin, J. Chem. Phys. 99 (1993) 5951. [11] D.L. Diedrich, J.B. Anderson, Science 258 (1992) 786. [12] E.J. Shipsey, Int. J. Quant. Chem. 25 (1984) 891. [13] U. Kleinekath ofer, T.I. Sachse, K.T. Tang, J.P. Toennies, C.L. Yiu, J. Chem. Phys. 113 (2000) 948. [14] U. Kleinekath ofer, Ph.D. Thesis, University of G ottingen, 1996. [15] T.I. Sachse, U. Kleinekath ofer, K.T. Tang, J.P. Toennies, in preparation. [16] T.I. Sachse, Ph.D. Thesis, University of G ottingen, 2000. [17] B.M. Axilrod, E. Teller, J. Chem. Phys. 11 (1943) 299. [18] Y. Muto, Proc. Phys. Soc. Jpn. 17 (1953) 629. [19] T.I. Sachse, K.T. Tang, J.P. Toennies, Chem. Phys. Lett. 317 (2000) 347. [20] K.T. Tang, J.P. Toennies, C.L. Yiu, J. Chem. Phys. 99 (1993) 377. [21] C. Herring, M. Flicker, Phys. Rev. 134 (1964) A362. [22] T. Korona, R. Moszynski, B. Jeziorski, J. Chem. Phys. 105 (1996) 8178. [23] K.T. Tang, J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [24] K.T. Tang, J.P. Toennies, J. Chem. Phys. 66 (1977) 1496. [25] Z.C. Zhang, A.R. Allnatt, J.D. Talman, W.J. Meath, Mol. Phys. 81 (1994) 1425.
Chemical Physics Letters 328 (2000) 469±472
www.elsevier.nl/locate/cplett
A modi®ed Cashion±Herschbach potential for the H3 potential energy surface T.I. Sachse *, K.T. Tang 1, J.P. Toennies Max-Planck-Institut f ur Str omungsforschung, Bunsenstr. 10, D-37073 G ottingen, Germany Received 12 July 2000; received in ®nal form 4 August 2000
Abstract The H3 potential energy surface has been calculated using the Cashion±Hershbach (CH) potential which contains all orders of two-body Coulomb, exchange, and overlap terms. For collinear con®gurations, the dominant three-body contribution comes from the Axilrod±Teller±Muto triple±dipole dispersion energy. By adding only this term, properly damped, the saddle point region potential agrees with recent ab initio calculations to within 9 10ÿ5 a.u. (0.06 kcal/ mole). Ó 2000 Elsevier Science B.V. All rights reserved.
Starting with the famous London equation [1,2] there has been a long continuing search for a simple expression with which to describe the H3 potential energy hypersurface. One of the early semi-empirical potential models, reported by Cashion and Herschbach (CH) in 1964 [3], was derived from the ®rst order valence bond theory by neglecting the three-body terms and overlap integrals. This surface is known to be equivalent to the diatomics-in-molecules surface also published in 1964 [4]. A recent analysis based on the generalized Heitler±London (GHL) theory [5,6] has shown that the CH surface encompasses far more information than previously recognized. The complete GHL potential energy hypersurface of the triatomic H3 -system is given by [5,6]
*
Corresponding author. Fax: +49-551-517-6607. E-mail address: [email protected] (T.I. Sachse). 1 Permanent address: Department of Physics, Paci®c Lutheran University, Tacoma, WA 98447, USA.
V C RAB C RBC C RAC CABC XABC h 1 X RAB ÿ X RBC 2 X RBC ÿ 2 i1=2 ; ÿ X RAC 2 X RAC ÿ X RAB 2 1 where RAB ; RBC ; RAC are the interatomic distances and C RAB and X RAB are, respectively, the twobody Coulomb and exchange energies between atom A and B in the presence of atom C. The nonadditive three-body Coulomb and the three-body cyclic exchange energies are denoted by CABC and XABC , respectively. For an isolated H2 molecule, the Coulomb and exchange energies are de®ned in terms of the singlet Es and triplet Et energies [5] 1 C RAB Es RAB Et RAB ; 2
2
1 X RAB Es RAB ÿ Et RAB : 2
3
0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 0 9 1 8 - 0
470
T.I. Sachse et al. / Chemical Physics Letters 328 (2000) 469±472
The GHL energy expressions for the H2 molecule are P Jn Kn ; 4 Es n1 P 1 n0 Sn P Jn ÿ Kn ; 5 Et n1 P 1 ÿ n0 Sn where Jn , Kn and Sn are, respectively, the nth order Coulomb, exchange and overlap integrals [5]. For comparison, in the usual Heitler±London approximation, the singlet and triplet energies are given by EsHL
J1 K1 ; 1 S0
6
EtHL
J1 ÿ K1 ; 1 ÿ S0
7
where, in the common usage, J1 , K1 and S0 are referred to as the Coulomb, exchange and overlap integrals. If in Eq. (1) all three-body eects, all orders of Sn , and, ®nally, all orders of Jn and Kn except J1 and K1 are neglected then Eq. (1) becomes equivalent to the London formula: VLondon J1 RAB J1 RBC J1 RAC h 1 2 K1 RAB ÿ K1 RBC ÿ 2 2
K1 RBC ÿ K1 RAC
2
K1 RAC ÿ K1 RAB
i1=2
:
8
In their derivation, Cashion and Herschbach started with this equation but used the Heitler± London energies of Eqs. (6) and (7) with S0 neglected to approximate J1 and K1 by ®tting to the available ab initio singlet and triplet energies 1 J1 Es Et ; 2 1 K1 Es ÿ Et : 2
9
But since J1 and K1 in this approximation are identical to C and X as de®ned in Eqs. (2) and (3) [5,6], Cashion and Herschbach have, in fact, introduced the full Coulomb and exchange energies
as de®ned in Eqs. (2) and (3). Therefore, their surface is given by VCH C RAB C RBC C RAC h 1 2 X RAB ÿ X RBC ÿ 2 2
X RBC ÿ X RAC X RBC i1=2 2 : ÿ X RAB
10
A comparison with Eq. (1) with consideration of Eqs. (2)±(5) reveals that this surface includes not only the overlap integrals, but also all higher orders of two-body Coulomb and exchange integrals, which had been thought to have been neglected. In view of this new insight, it is interesting to reexamine the accuracy of the CH potential. Using the `exact' singlet and triplet potential energies for the H2 molecule [7,8], the CH potential has been compared with the recently calculated accurate ab initio H3 potential energy surface [9±11]. For the important linear saddle point region, at which the hydrogen exchange reaction takes place the classical barrier height of the CH potential is, however, about 3 kcal/mole (0.005 a.u.) greater than the accepted value of 9.6 kcal/mole. Apparently, the neglected three-body eects are still necessary. The GHL theory, Eq. (1), shows that the main three-body eects are: (1) the cyclic exchange energy XABC [12±14]; (2) the implicit three-body eect on the two-body exchange energies due to the presence of the third atom [15,16]; and (3) the three-body Coulomb energy CABC [17±19]. The cyclic and the implicit three-body eects on the exchange energies in Eq. (1) can be calculated using a generalization of the surface integral method [12±14], which is based on the concept that the electrons are continuously trading places. This has been shown previously to give excellent results for the two-body exchange energy of the H2 -molecule [20,21]. In H3 , the cyclic exchange of the three electrons is responsible for the three-body exchange energy XABC [12±14]. For the cyclic exchange, one would expect that XABC is negligibly small in the linear con®guration, because one atom is in the middle. The few explicit results available
T.I. Sachse et al. / Chemical Physics Letters 328 (2000) 469±472
[13,14] indicate, indeed, that XABC rapidly decreases in magnitude as the three atoms approach the linear con®guration. The same is true for the three-body eects on the two-body exchange energies [15,16]. Therefore, it is not unreasonable to assume that the dominant three-body eect in the linear con®guration comes from the leading threebody Coulomb term, which is also known as the three-body dispersion energy. The main contribution to CABC is from the nonadditive triple±dipole interaction. Axilrod and Teller [17] and Muto [18] were the ®rst to give an analytic formula for the long-range behavior of this interaction CABC !
C9 R3AB R3BC R3AC 1 3 cos hA cos hB cos hC ;
11
1=3 b 3! ÿ C6He±H 2C6H±H =C9 :
471
14
With the well-known numerical values for the dispersion coecients in atomic units C6H±H 6:499 [23], C6He±H 2:82 [24] and C9 21:643 [25], the value of the damping parameter for H3 is b 1:4130:
15
In Fig. 1, the semi-empirical CH surface for the linear con®guration for symmetric stretching of both coordinates RAB RBC R passing through the reaction saddle point is plotted as a function of R (dotted line). The CH energies were evaluated using the accurate Coulomb and exchange energies extracted from the accurate ab initio H2 singlet and triplet potentials [7,8]. The CH surface for H3 plus Eq. (12) with Eq. (15) (solid line) is compared
where hA ; hB and hC are the interior angles at the respective atoms. Recently, we have introduced a new damping function [19] to extend the validity of this equation to smaller interatomic distances. This new damping function for the three-body dispersion term was found to give results in very good agreement with recent third order perturbation calculations for H3 [22]. The advantage of the new method is that it is applicable to all three-body systems if the respective two-body coecients in the united atom limits involved are available. For the H3 -system, the damped third order dispersion energy has the form [19] CABC f2 RAB f2 RBC f2 RAC
C9 R3AB R3BC R3AC
1 f0 RAB f0 RBC f0 RAC 3 cos hA cos hB cos hC ;
12
where the f2n are the Tang±Toennies damping functions [23] given by f2n R 1 ÿ eÿbR
k 2n X bR : k! k0
13
The f2n depend only on a single-range parameter b which is determined by taking into account the united atom limit of H±H which is the helium atom. As derived in Ref. [19], the result for the damping parameter for the H3 -system is given by
Fig. 1. Potential energy of H3 in the symmetrial linear con®guration RAB RBC passing through the saddle point (SP). The dotted line is calculated using the CH [3] method based on the accurate ab initio H2 potential energy of Refs. [7,8]. The open circles are the recent accurate ab initio results [9]. The solid line is the present result which is the sum of CH and the damped three-body polarization energy. It agrees to within 10ÿ4 a.u. with the ab initio result.
472
T.I. Sachse et al. / Chemical Physics Letters 328 (2000) 469±472
with the accurate ab initio results [9] (open circles). As expected the pure CH potential surface agrees with the ab initio results at large distances RAB RBC P 3:5 a.u. The agreement is to within 10ÿ4 a.u. since there the three-body eects are negligibly small. But at the minimum RAB RBC 1:757 a.u., which is also the saddle point for the reaction, the CH energy is ÿ0:15454 a.u., compared to the most accurate ab initio value [9±11] of ÿ0:1591760 a.u. If now CABC calculated from Eq. (12), which is equal to ÿ0:00453 a.u. is added to the CH energy, the saddle point energy becomes ÿ0:15908 a.u., which is only 0.00009 a.u. (0.06 kcal/mole) greater than the ab initio result. Thus the CH potential provides a simple yet very accurate description of the H3 potential energy surface. Moreover, for the important linear saddle point region the inclusion of the damped three-body polarization energy accounts for the remaining dierence between the CH potential and the best ab initio potentials. For non-linear con®gurations, the CH potential surface should also be a good starting approximation, since it contains all orders of the two-body interactions. However, for an accurate description, one must also include the three-body eects in the exchange energy, which in the linear con®guration are negligibly small. Although they can be calculated by the surface integral method, at present only numerical results at some isolated points are available [13±16]. Obviously, for non-linear con®gurations more work is needed to explore the analytic ®t to the three-body corrections to the exchange energy. In the future, it will be interesting to explore the validity of the CH procedure for other reactive systems. Acknowledgements We thank D. Herschbach and A.J.C. Varandas for helpful comments. K.T.T. wishes to thank the
Alexander von Humboldt±Stiftung for ®nancial support. References [1] F. London, Probleme der Modernen Physik, Sommerfeld± Festschrift, S. Hirzel, Leipzig, 1928, 104. [2] F. London, Z. Elektrochem. 35 (1929) 552. [3] J.K. Cashion, D.R. Herschbach, J. Chem. Phys. 40 (1964) 2358. [4] F.O. Ellison, J. Chem. Phys. 41 (1964) 2198. [5] K.T. Tang, J.P. Toennies, C.L. Yiu, Int. Rev. Phys. Chem. 17 (1998) 370. [6] U. Kleinekath ofer, K.T. Tang, J.P. Toennies, C.L. Yiu, J. Chem. Phys. 111 (1999) 3377. [7] W. Kolos, L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429. [8] W. Kolos, L. Wolniewicz, Chem. Phys. Lett. 24 (1974) 457. [9] A.I. Boothroyd, W.J. Keogh, P.G. Martin, M.R. Peterson, J. Chem. Phys. 104 (1996) 7139. [10] H. Partridge, C.W. Bauschlicher, J.R. Stallcop, E. Levin, J. Chem. Phys. 99 (1993) 5951. [11] D.L. Diedrich, J.B. Anderson, Science 258 (1992) 786. [12] E.J. Shipsey, Int. J. Quant. Chem. 25 (1984) 891. [13] U. Kleinekath ofer, T.I. Sachse, K.T. Tang, J.P. Toennies, C.L. Yiu, J. Chem. Phys. 113 (2000) 948. [14] U. Kleinekath ofer, Ph.D. Thesis, University of G ottingen, 1996. [15] T.I. Sachse, U. Kleinekath ofer, K.T. Tang, J.P. Toennies, in preparation. [16] T.I. Sachse, Ph.D. Thesis, University of G ottingen, 2000. [17] B.M. Axilrod, E. Teller, J. Chem. Phys. 11 (1943) 299. [18] Y. Muto, Proc. Phys. Soc. Jpn. 17 (1953) 629. [19] T.I. Sachse, K.T. Tang, J.P. Toennies, Chem. Phys. Lett. 317 (2000) 347. [20] K.T. Tang, J.P. Toennies, C.L. Yiu, J. Chem. Phys. 99 (1993) 377. [21] C. Herring, M. Flicker, Phys. Rev. 134 (1964) A362. [22] T. Korona, R. Moszynski, B. Jeziorski, J. Chem. Phys. 105 (1996) 8178. [23] K.T. Tang, J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [24] K.T. Tang, J.P. Toennies, J. Chem. Phys. 66 (1977) 1496. [25] Z.C. Zhang, A.R. Allnatt, J.D. Talman, W.J. Meath, Mol. Phys. 81 (1994) 1425.