- Email: [email protected]
Ab initio calculations and modeling of diabatic potential energy surfaces for the Van der Waals complex Cl(2P)⋯CH4(X1A1)
20 June 2002
Chemical Physics Letters 359 (2002) 309–313 www.elsevier.com/locate/cplett
Ab initio calculations and modeling of diabatic potential energy surfaces for the Van der Waals complex Clð2PÞ CH4ðX 1A1Þ Jacek Kłos Institute of Theoretical Chemistry, NSRIM, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands Received 10 April 2002; in final form 29 April 2002
Abstract The three lowest potential energy surfaces, R, Px and Py of the Van der Waals complex Clð2 PÞ þ CH4 ðX 1 A1 Þ are derived from accurate ab initio calculations for the vertex, edge and face geometries. The restricted coupled cluster singles, doubles and non-iterative triples excitations [RCCSD(T)] level of theory is applied with a large basis set. The on the R surface and is located in the global Van der Waals minimum, which is 348 cm1 deep, occurs at R ¼ 3:3 A vicinity of the face arrangement. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The study of the interaction between atomic chlorine and the methane molecule is of great importance in the chemistry of the stratosphere. The reaction Clð2 PÞ þ CH4 ! CH3 þ HCl is the main pathway of chlorine removal in the ozone depletion cycle [1–3]. This reaction is the subject of many experimental investigations (cf. [4–6] and references therein) in which various quantities have been measured, mainly rate constants. Most of the theoretical investigations of the potential energy surface of Cl–CH4 system were
E-mail address: [email protected] (J. Kłos).
focused on the transition state and reactive region [7,8]. Reactive scattering calculations and thermal and vibrationally selected rate constants have been calculated and compared to experimental results by several groups. Some potentials [9] give an unphysical description of the Van der Waals region, as was pointed out in [10]. Therefore our goal in this Letter is to describe the Van der Waals region of the entrance channel to the reaction between Cl and methane. This reaction is an example of a heavy-light-heavy system where a hydrogen atom is abstracted. Recently, we published [11] a study of the entrance channel region to the reaction Clð2 PÞ þ HCl, which is of similar type. We will use state-of-theart methods of electronic structure calculations. Atomic chlorine is an open-shell system with a 2 P ground state. The threefold spatial degeneracy is removed when a methane molecule starts to
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 7 1 8 - 2
310
J. Kłos / Chemical Physics Letters 359 (2002) 309–313
interact with the chlorine atom. There are three orientations of the singly occupied 3p chlorine orbital with respect to the vector ~ R. One orientation, which defines one of the three diabatic states, has the singly occupied 3p orbital pointing toward the carbon atom along ~ R. We will refer to this potential surface as the R diabat. The two orientations of the singly occupied chlorine orbital perpendicular to ~ R define two diabatic surfaces referred to as Px and Py .
shown to be both effective and economical for a quite a number of Van der Waals complexes [17– 20,11]. 2.2. Ab initio calculations of interaction energies
2. Computational methods
All calculations reported in this Letter were performed by the aid of the MO L P R O package [21]. The supermolecular method was used in calculations of three diabatic potential energy surfaces. This method derives the interaction energy as the difference between the energies of the dimer AB and the monomers A and B
2.1. Geometries and basis sets
DEðnÞ ¼ EAB EA EB :
ðnÞ
The position of the chlorine atom is described by the usual polar coordinates ðR; h; UÞ with respect to a frame fixed to methane [12], as is shown in Fig. 1. This frame is such that the xz-plane is a mirror plane, i.e., the two protons above the xyplane are in the xz plane. The CH4 monomer was kept rigid and the CH bond length was fixed at the close to the experimental equivalue r ¼ 1:085 A librium value. Calculations employed the augmented correlation-consistent polarized basis sets of triple zeta quality of Dunning and his coworkers [13–15], referred to as AVTZ basis. Calculations included also bond functions of Tao and Pan [16], with the exponents: sp 0.9, 0.3, 0.1; d 0.6, 0.2 denoted as (3 3 2). Bond functions that are centered in the middle of the vector ~ R have been
ðnÞ
ðnÞ
ð1Þ
The superscript (n) denotes the level of ab initio theory. In CCSD(T) calculations the use of the above equation is straightforward, and free from arbitrary choices, as long as the dimer and monomer energies are calculated with the same dimer centered basis set (DCBS). The basis set superposition error was removed according to counterpoise procedure of Boys and Bernardi [22]. For a given diabatic state, the orientation of the singly occupied orbital for the Cl monomer in the DCBS was kept the same as in dimer. The CCSD(T) method is well-known to be very effective in recovering electron correlation effects in Van der Waals complexes calculations [23,24], and is the preferred method of choice as long as a single-reference approach is valid and efficient. 2.3. One-dimensional fitting Interaction energies for the vertex, edge and face orientations were fit to the following expression due Degli-Esposti and Werner [25] " # 7 X C 2i V ðRÞ ¼ GðRÞ eaðRRe Þ T ðRÞ ð2Þ R2i i¼3 with GðRÞ ¼
8 X
g j Rj :
ð3Þ
j¼0
The damping function T is Fig. 1. Molecule fixed frame for the Cl–CH4 complex.
T ðRÞ ¼ 12ð1 þ tanhð1 þ tRÞÞ:
ð4Þ
J. Kłos / Chemical Physics Letters 359 (2002) 309–313
311
The parameters a, gj , t and Ci are optimized. The largest root mean square error of the fit was 0:004 cm1 .
3. Modeling of the anisotropy To fit the anisotropy we expanded the potential for each state in a series of tetrahedral harmonics Tl ðh; UÞ transforming according to the A1 representation of the group Td : X V ðR; h; UÞ ¼ Vl ðRÞTl ðh; UÞ: ð5Þ l
The radial Vl coefficients were determined by solving systems of algebraic equations using the fact that we have already radial fits for the vertex, edge and face geometries. The expansion was truncated after the first three tetrahedral harmonics, T0 , T3 and T4 [26] adapted to the current frame: T0 ðh; UÞ ¼ C0;0 ðh; UÞ;
ð6Þ
T3 ðh; UÞ ¼ C3;2 ðh; UÞ;
ð7Þ
pffiffiffiffiffi pffiffiffiffiffi 15 21 C4;4 ðh; UÞ; C4;0 ðh; UÞ T4 ðh; UÞ ¼ 6 6
ð8Þ
Fig. 3. Contour plot of the Px diabatic potential energy surface for U ¼ 0°. Energies in cm1 .
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl mÞ! m P ðhÞ cosðmUÞ: Cl;m ðh; UÞ ¼ 2 ðl þ mÞ! l
ð9Þ
Fig. 4. Contour plot of the Py diabatic potential energy surface for U ¼ 0°. Energies in cm1 .
The Plm are associated Legendre functions [27]. Harmonics with l ¼ 1; 2 do not appear in the description of tetrahedral anisotropy. Thus, the tetrahedral symmetry of the potential is assured. Figs. 2–4 show contour plots of the R, Px and Py diabat for the angle U ¼ 0°. Potentials are available upon request from Kłos [28].
4. Features of the potential energy surfaces Fig. 2. Contour plot of the R diabatic potential energy surface for U ¼ 0°. Energies in cm1 .
The global Van der Waals minimum occurs for the R diabat in the proximity of a face of the tet-
312
J. Kłos / Chemical Physics Letters 359 (2002) 309–313
rahedron, which means that the complex has C3v symmetry. The well depth is 348 cm1 and chlo away from the carbon atom. rine is located 3.3 A There is also a local minimum for h ¼ 0°, which corresponds to a geometry with chlorine on the edge of the tetrahedron. This minimum is 276 cm1 deep. The vertex geometry is a saddle point on the R surface of 110 cm1 high. This saddle point corresponds to a hydrogen bonded . complex with radial minimum located at 3.9 A The diabats of P character are roughly three times shallower than the R diabat. In the case of the vertex arrangement the Px and Py , diabats are practically identical in the bonding region. Even in the repulsive region they are practically the same. To a good approximation the P diabats are degenerate for the vertex geometry. This is not the case for the edge and face orientations, where the two different perpendicular orientations of the 3p chlorine orbital result in different interaction energies. The differences are not large, though, on the order of 10 cm1 or less, so that we can conclude that they are the same in very good approximation. The Px diabat reveals one minimum for the and is 120 cm1 face geometry around R ¼ 3:75 A deep. The vertex and edge orientations are connected by a valley of 100 cm1 deep. The Py diabat has two Van der Waals minima, for the face and vertex orientations, respectively. The minimum for the face arrangement is approximately 10 cm1 deeper than for the vertex geometry and is 110 cm1 deep. It is interesting to compare the R diabat shown in Fig. 2 with the Ar–CH4 surface reported in [12] and shown there in Fig. 2. The anisotropy is very similar. The face well is approximately 2.5 deeper for Cl–CH4 than for the Ar–CH4 system.
5. Summary and conclusions The three lowest diabatic potential energy surfaces of the Van der Waals complex Clð2 PÞ–CH4 have been presented, calculated at the RCCSD(T) level of theory with the AVTZ + 332 basis set. The global Van der Waals minimum occurs on the R diabat and the well depth is 348 cm1 . Diabats of P character are three times shallower and less
anisotropic, and are very similar to each other. Diabatic potentials developed in this work model interaction of a 2 P chlorine atom with the ground state methane. The model is approximate, as offdiagonal nonadiabatic coupling matrix elements needed for transformation to adiabatic surfaces have been disregarded. Yet, for the first time in the literature, presented surfaces provide description of weak interactions in the Cl–CH4 complex, based on highly accurate ab initio calculations.
Acknowledgements The author thanks Prof. A. van der Avoird, Dr. P.E.S. Wormer and G. Chałasi nski for discussions and their critical reading of the manuscript. The author acknowledges support from the European Research Training Network (THEONET II). This work is supported also by the National Science Foundation (Grant No. CHE-0078533) and by the Polish Committee for Scientific Research (KBN) (Grant No. 3 T09A 112 18).
References [1] R.S. Stolarski, R.J. Cicerone, Can. J. Chem. 52 (1974) 1610. [2] F.S. Rowland, M.J. Molina, Rev. Geophys. Space Phys. 13 (1975) 1. [3] R.D. Hudson, Chlorofluoromethanes and the Stratosphere, NASA Ref. Publ. 1010, 1977. [4] A.R. Ravishankara, P.H. Wine, J. Chem. Phys. 72 (1980) 25. [5] Z.H. Kim, A.J. Alexander, H.A. Bechtel, R.N. Zare, J. Chem. Phys. 115 (2001) 179. [6] Y.-J. Chen, L.-K. Chu, S.-R. Lin, Y.-P. Lee, J. Chem. Phys. 115 (2001) 6513. [7] T.N. Truong, D.G. Truhlar, K.K. Baldridge, M.S. Gordon, R. Steckler, J. Chem. Phys. 90 (1989) 7137. [8] K.D. Dobbs, D.A. Dixon, J. Phys. Chem. 98 (1994) 12584. [9] J. Espinosa-Garcıa, J.C. Corchado, J. Chem. Phys. 105 (1996) 3517. [10] H.-G. Yu, G. Nyman, Phys. Chem. Chem. Phys. 1 (1999) 1181. [11] J. Kłos, G. Chałasi nski, M.M. Szczezsniak, H.-J. Werner, J. Chem. Phys. 115 (2001) 3085. [12] T.G.A. Heijmen, T. Korona, R. Moszy nski, P.E.S. Wormer, A. van der Avoird, J. Chem. Phys. 107 (1997) 902. [13] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.
J. Kłos / Chemical Physics Letters 359 (2002) 309–313 [14] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [15] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. [16] F.-M. Tao, Y.-K. Pan, J. Chem. Phys. 97 (1992) 4989. [17] S.M. Cybulski, R.A. Kendall, G. Chałasi nski, M.W. Severson, M.M. Szczezsniak, J. Chem. Phys. 106 (1997) 7731. [18] R.A. Kendall, G. Chałasi nski, J. Kłos, R. Bukowski, M.W. Severson, M.M. Szczezsniak, S.M. Cybulski, J. Chem. Phys. 108 (1998) 3235. [19] S.M. Cybulski, R. Burcl, G. Chałasi nski, M.M. Szczezsniak, J. Chem. Phys. 103 (1995) 10116. [20] S.M. Cybulski, R. Toczyłowski, J. Chem. Phys. 111 (1999) 10520.
313
[21] R.D. Amos et al., H.-J. Werner, P.J. Knowles (Eds.), MOLPRO is a Package of ab initio Programs. [22] S.F. Boys, F. Bernardi, Mol. Phys. 19 (1970) 553. [23] G. Chałasi nski, M.M. Szczezsniak, Chem. Rev. 94 (1994) 1723. [24] G. Chałasi nski, M.M. Szczezsniak, Chem. Rev. 100 (2000) 4253. [25] A.D. Esposti, H.-J. Werner, J. Chem. Phys. 93 (1990) 3351. [26] P.E.S. Wormer, Mol. Phys. 99 (2001) 1973. [27] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [28] E-mail address: [email protected].
Chemical Physics Letters 359 (2002) 309–313 www.elsevier.com/locate/cplett
Ab initio calculations and modeling of diabatic potential energy surfaces for the Van der Waals complex Clð2PÞ CH4ðX 1A1Þ Jacek Kłos Institute of Theoretical Chemistry, NSRIM, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands Received 10 April 2002; in final form 29 April 2002
Abstract The three lowest potential energy surfaces, R, Px and Py of the Van der Waals complex Clð2 PÞ þ CH4 ðX 1 A1 Þ are derived from accurate ab initio calculations for the vertex, edge and face geometries. The restricted coupled cluster singles, doubles and non-iterative triples excitations [RCCSD(T)] level of theory is applied with a large basis set. The on the R surface and is located in the global Van der Waals minimum, which is 348 cm1 deep, occurs at R ¼ 3:3 A vicinity of the face arrangement. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The study of the interaction between atomic chlorine and the methane molecule is of great importance in the chemistry of the stratosphere. The reaction Clð2 PÞ þ CH4 ! CH3 þ HCl is the main pathway of chlorine removal in the ozone depletion cycle [1–3]. This reaction is the subject of many experimental investigations (cf. [4–6] and references therein) in which various quantities have been measured, mainly rate constants. Most of the theoretical investigations of the potential energy surface of Cl–CH4 system were
E-mail address: [email protected] (J. Kłos).
focused on the transition state and reactive region [7,8]. Reactive scattering calculations and thermal and vibrationally selected rate constants have been calculated and compared to experimental results by several groups. Some potentials [9] give an unphysical description of the Van der Waals region, as was pointed out in [10]. Therefore our goal in this Letter is to describe the Van der Waals region of the entrance channel to the reaction between Cl and methane. This reaction is an example of a heavy-light-heavy system where a hydrogen atom is abstracted. Recently, we published [11] a study of the entrance channel region to the reaction Clð2 PÞ þ HCl, which is of similar type. We will use state-of-theart methods of electronic structure calculations. Atomic chlorine is an open-shell system with a 2 P ground state. The threefold spatial degeneracy is removed when a methane molecule starts to
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 7 1 8 - 2
310
J. Kłos / Chemical Physics Letters 359 (2002) 309–313
interact with the chlorine atom. There are three orientations of the singly occupied 3p chlorine orbital with respect to the vector ~ R. One orientation, which defines one of the three diabatic states, has the singly occupied 3p orbital pointing toward the carbon atom along ~ R. We will refer to this potential surface as the R diabat. The two orientations of the singly occupied chlorine orbital perpendicular to ~ R define two diabatic surfaces referred to as Px and Py .
shown to be both effective and economical for a quite a number of Van der Waals complexes [17– 20,11]. 2.2. Ab initio calculations of interaction energies
2. Computational methods
All calculations reported in this Letter were performed by the aid of the MO L P R O package [21]. The supermolecular method was used in calculations of three diabatic potential energy surfaces. This method derives the interaction energy as the difference between the energies of the dimer AB and the monomers A and B
2.1. Geometries and basis sets
DEðnÞ ¼ EAB EA EB :
ðnÞ
The position of the chlorine atom is described by the usual polar coordinates ðR; h; UÞ with respect to a frame fixed to methane [12], as is shown in Fig. 1. This frame is such that the xz-plane is a mirror plane, i.e., the two protons above the xyplane are in the xz plane. The CH4 monomer was kept rigid and the CH bond length was fixed at the close to the experimental equivalue r ¼ 1:085 A librium value. Calculations employed the augmented correlation-consistent polarized basis sets of triple zeta quality of Dunning and his coworkers [13–15], referred to as AVTZ basis. Calculations included also bond functions of Tao and Pan [16], with the exponents: sp 0.9, 0.3, 0.1; d 0.6, 0.2 denoted as (3 3 2). Bond functions that are centered in the middle of the vector ~ R have been
ðnÞ
ðnÞ
ð1Þ
The superscript (n) denotes the level of ab initio theory. In CCSD(T) calculations the use of the above equation is straightforward, and free from arbitrary choices, as long as the dimer and monomer energies are calculated with the same dimer centered basis set (DCBS). The basis set superposition error was removed according to counterpoise procedure of Boys and Bernardi [22]. For a given diabatic state, the orientation of the singly occupied orbital for the Cl monomer in the DCBS was kept the same as in dimer. The CCSD(T) method is well-known to be very effective in recovering electron correlation effects in Van der Waals complexes calculations [23,24], and is the preferred method of choice as long as a single-reference approach is valid and efficient. 2.3. One-dimensional fitting Interaction energies for the vertex, edge and face orientations were fit to the following expression due Degli-Esposti and Werner [25] " # 7 X C 2i V ðRÞ ¼ GðRÞ eaðRRe Þ T ðRÞ ð2Þ R2i i¼3 with GðRÞ ¼
8 X
g j Rj :
ð3Þ
j¼0
The damping function T is Fig. 1. Molecule fixed frame for the Cl–CH4 complex.
T ðRÞ ¼ 12ð1 þ tanhð1 þ tRÞÞ:
ð4Þ
J. Kłos / Chemical Physics Letters 359 (2002) 309–313
311
The parameters a, gj , t and Ci are optimized. The largest root mean square error of the fit was 0:004 cm1 .
3. Modeling of the anisotropy To fit the anisotropy we expanded the potential for each state in a series of tetrahedral harmonics Tl ðh; UÞ transforming according to the A1 representation of the group Td : X V ðR; h; UÞ ¼ Vl ðRÞTl ðh; UÞ: ð5Þ l
The radial Vl coefficients were determined by solving systems of algebraic equations using the fact that we have already radial fits for the vertex, edge and face geometries. The expansion was truncated after the first three tetrahedral harmonics, T0 , T3 and T4 [26] adapted to the current frame: T0 ðh; UÞ ¼ C0;0 ðh; UÞ;
ð6Þ
T3 ðh; UÞ ¼ C3;2 ðh; UÞ;
ð7Þ
pffiffiffiffiffi pffiffiffiffiffi 15 21 C4;4 ðh; UÞ; C4;0 ðh; UÞ T4 ðh; UÞ ¼ 6 6
ð8Þ
Fig. 3. Contour plot of the Px diabatic potential energy surface for U ¼ 0°. Energies in cm1 .
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl mÞ! m P ðhÞ cosðmUÞ: Cl;m ðh; UÞ ¼ 2 ðl þ mÞ! l
ð9Þ
Fig. 4. Contour plot of the Py diabatic potential energy surface for U ¼ 0°. Energies in cm1 .
The Plm are associated Legendre functions [27]. Harmonics with l ¼ 1; 2 do not appear in the description of tetrahedral anisotropy. Thus, the tetrahedral symmetry of the potential is assured. Figs. 2–4 show contour plots of the R, Px and Py diabat for the angle U ¼ 0°. Potentials are available upon request from Kłos [28].
4. Features of the potential energy surfaces Fig. 2. Contour plot of the R diabatic potential energy surface for U ¼ 0°. Energies in cm1 .
The global Van der Waals minimum occurs for the R diabat in the proximity of a face of the tet-
312
J. Kłos / Chemical Physics Letters 359 (2002) 309–313
rahedron, which means that the complex has C3v symmetry. The well depth is 348 cm1 and chlo away from the carbon atom. rine is located 3.3 A There is also a local minimum for h ¼ 0°, which corresponds to a geometry with chlorine on the edge of the tetrahedron. This minimum is 276 cm1 deep. The vertex geometry is a saddle point on the R surface of 110 cm1 high. This saddle point corresponds to a hydrogen bonded . complex with radial minimum located at 3.9 A The diabats of P character are roughly three times shallower than the R diabat. In the case of the vertex arrangement the Px and Py , diabats are practically identical in the bonding region. Even in the repulsive region they are practically the same. To a good approximation the P diabats are degenerate for the vertex geometry. This is not the case for the edge and face orientations, where the two different perpendicular orientations of the 3p chlorine orbital result in different interaction energies. The differences are not large, though, on the order of 10 cm1 or less, so that we can conclude that they are the same in very good approximation. The Px diabat reveals one minimum for the and is 120 cm1 face geometry around R ¼ 3:75 A deep. The vertex and edge orientations are connected by a valley of 100 cm1 deep. The Py diabat has two Van der Waals minima, for the face and vertex orientations, respectively. The minimum for the face arrangement is approximately 10 cm1 deeper than for the vertex geometry and is 110 cm1 deep. It is interesting to compare the R diabat shown in Fig. 2 with the Ar–CH4 surface reported in [12] and shown there in Fig. 2. The anisotropy is very similar. The face well is approximately 2.5 deeper for Cl–CH4 than for the Ar–CH4 system.
5. Summary and conclusions The three lowest diabatic potential energy surfaces of the Van der Waals complex Clð2 PÞ–CH4 have been presented, calculated at the RCCSD(T) level of theory with the AVTZ + 332 basis set. The global Van der Waals minimum occurs on the R diabat and the well depth is 348 cm1 . Diabats of P character are three times shallower and less
anisotropic, and are very similar to each other. Diabatic potentials developed in this work model interaction of a 2 P chlorine atom with the ground state methane. The model is approximate, as offdiagonal nonadiabatic coupling matrix elements needed for transformation to adiabatic surfaces have been disregarded. Yet, for the first time in the literature, presented surfaces provide description of weak interactions in the Cl–CH4 complex, based on highly accurate ab initio calculations.
Acknowledgements The author thanks Prof. A. van der Avoird, Dr. P.E.S. Wormer and G. Chałasi nski for discussions and their critical reading of the manuscript. The author acknowledges support from the European Research Training Network (THEONET II). This work is supported also by the National Science Foundation (Grant No. CHE-0078533) and by the Polish Committee for Scientific Research (KBN) (Grant No. 3 T09A 112 18).
References [1] R.S. Stolarski, R.J. Cicerone, Can. J. Chem. 52 (1974) 1610. [2] F.S. Rowland, M.J. Molina, Rev. Geophys. Space Phys. 13 (1975) 1. [3] R.D. Hudson, Chlorofluoromethanes and the Stratosphere, NASA Ref. Publ. 1010, 1977. [4] A.R. Ravishankara, P.H. Wine, J. Chem. Phys. 72 (1980) 25. [5] Z.H. Kim, A.J. Alexander, H.A. Bechtel, R.N. Zare, J. Chem. Phys. 115 (2001) 179. [6] Y.-J. Chen, L.-K. Chu, S.-R. Lin, Y.-P. Lee, J. Chem. Phys. 115 (2001) 6513. [7] T.N. Truong, D.G. Truhlar, K.K. Baldridge, M.S. Gordon, R. Steckler, J. Chem. Phys. 90 (1989) 7137. [8] K.D. Dobbs, D.A. Dixon, J. Phys. Chem. 98 (1994) 12584. [9] J. Espinosa-Garcıa, J.C. Corchado, J. Chem. Phys. 105 (1996) 3517. [10] H.-G. Yu, G. Nyman, Phys. Chem. Chem. Phys. 1 (1999) 1181. [11] J. Kłos, G. Chałasi nski, M.M. Szczezsniak, H.-J. Werner, J. Chem. Phys. 115 (2001) 3085. [12] T.G.A. Heijmen, T. Korona, R. Moszy nski, P.E.S. Wormer, A. van der Avoird, J. Chem. Phys. 107 (1997) 902. [13] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.
J. Kłos / Chemical Physics Letters 359 (2002) 309–313 [14] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [15] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. [16] F.-M. Tao, Y.-K. Pan, J. Chem. Phys. 97 (1992) 4989. [17] S.M. Cybulski, R.A. Kendall, G. Chałasi nski, M.W. Severson, M.M. Szczezsniak, J. Chem. Phys. 106 (1997) 7731. [18] R.A. Kendall, G. Chałasi nski, J. Kłos, R. Bukowski, M.W. Severson, M.M. Szczezsniak, S.M. Cybulski, J. Chem. Phys. 108 (1998) 3235. [19] S.M. Cybulski, R. Burcl, G. Chałasi nski, M.M. Szczezsniak, J. Chem. Phys. 103 (1995) 10116. [20] S.M. Cybulski, R. Toczyłowski, J. Chem. Phys. 111 (1999) 10520.
313
[21] R.D. Amos et al., H.-J. Werner, P.J. Knowles (Eds.), MOLPRO is a Package of ab initio Programs. [22] S.F. Boys, F. Bernardi, Mol. Phys. 19 (1970) 553. [23] G. Chałasi nski, M.M. Szczezsniak, Chem. Rev. 94 (1994) 1723. [24] G. Chałasi nski, M.M. Szczezsniak, Chem. Rev. 100 (2000) 4253. [25] A.D. Esposti, H.-J. Werner, J. Chem. Phys. 93 (1990) 3351. [26] P.E.S. Wormer, Mol. Phys. 99 (2001) 1973. [27] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [28] E-mail address: [email protected].