PCCP. A theoretical spectroscopic study of HeI and HeBr. 1. Introduction. 2. Electronic calculations.

Transcription

1 RESEARCH PAPER A theoretical spectroscopic study of and C. Le onard,* a F. Le Que ré a and K. A. Peterson b a Laboratoire de Chimie The orique, Universite de Marne la Valle e, 5 boulevard Descartes, Champs sur Marne, F Marne la Vallee Cedex 2, France. celine.leonard@univ-mlv.fr; Fax: þ ; Tel: þ b Department of Chemistry, Washington State University, Pullman, Washington, , USA PCCP Received 25th January 2005, Accepted 21st February 2005 First published as an Advance Article on the web 9th March 2005 Highly accurate potential energy functions for the and molecules have been calculated using an ab initio treatment that included basis set extrapolation to the complete basis set, as well as spin orbit coupling in the ground 2 S 1 and first 2 P excited doublet states. The rovibronic bound state energies and resonance lifetimes were also evaluated by a Prony analysis of the autocorrelation function of the evolving wave packet. DOI: /b501253h 1. Introduction In recent experiments, 1 processes such as the photodissociation of CH 3 I and CF 3 I inside helium droplets were studied. It was shown that small clusters of IHe n (n ¼ 1, 12) were ejected out of the droplet after the breaking of the carbon iodine bond. In these clusters, as well as in other halogen rare gas van der Waals molecules, the interaction of the halogen with helium is often described by a Morse potential, which can be used in pairwise additive potentials. For instance, in 2 4 or 5,6 2 the well depth in the Morse expansion is between 16.5 cm 14 and 18 cm 1, 3 and the equilibrium distance varies between 7.56 a 3,4 0 and 8.69 a 0. 6 In this paper, we performed an ab initio study of the helium iodine and helium bromine interaction, including ab initio spin orbit coupling. Since no previous ab initio and experimental data was available for, the helium bromine system, which had already been treated by de Lara-Castells et al. 7 using a model for the spin orbit coupling, was used as a benchmark in order to validate our ab initio approach. To study the stability of the systems, we determined the rovibrational bound state energies and resonance lifetimes in the Hund s case (c) framework using a wave-packet time propagation followed by a Prony analysis of the autocorrelation function, as suggested by Gray Electronic calculations 2.1. Determination of the electronic potential curves and both exhibit two electronic states, 2 S 1 and 2 P, that dissociate to the first dissociation asymptote, X ( 2 P) þ He ( 1 S), X being Br or I. In the present work we use the notation V S and V P, respectively, for these states (see Fig. 1a). For both and, potential energy curves were computed for both electronic states at the ROHF-RCCSD(T) level of theory 9 with a series of correlation consistent basis sets. A total of 24 points were sampled on each potential curve ranging from 24.0 to 6.0 a 0. The zero of energy was defined by calculations at a separation of 100 a 0. For Br and I the newly developed small-core relativistic pseudopotentials and their corresponding aug-cc-pvnz-pp (n ¼ T, Q, 5) basis sets 10 were used. These choices of pseudopotentials result in 25 active electrons for these atoms (3s3p3d4s4p for Br and 4s4p4d5s5p for I). In the case of the He atom, the standard aug-cc-pvnz basis sets 11 were used. These basis sets will be denoted below simply as AVTZ, AVQZ, and AV5Z for n ¼ T, Q, 5, respectively. In order to increase the convergence rate with respect to the basis set limit for these weakly-bound systems, the AVnZ basis sets were extended by an additional diffuse function in each angular momentum symmetry using an even tempered extrapolation of the last 2 exponents. These sets, generally denoted d-aug-ccpvnz, will be abbreviated by DAVnZ (n ¼ T, Q, 5) in the remainder of this work. The systematic convergence with basis set exhibited by these correlation consistent sets has been exploited in the present work by extrapolation of the correlation component of the CCSD(T) binding energies at each geometry by a 1/n 3 formula. 12 The AVQZ/AV5Z and DAVQZ/ DAV5Z pairs were used in each case. The Hartree Fock (HF) component of the binding energy was observed to be converged to better than 0.1 cm 1 at the AV5Z level of theory, hence the 5Z HF values (AV5Z for the AVnZ series and DAV5Z for the DAVnZ series) were used together with the extrapolated CCSD(T) correlation contributions to yield complete basis set (CBS) limits for the binding energies. Furthermore, in order to test the accuracy of the pseudopotential approximation in this work, relativistic all-electron calculations were carried out for the near-equilibrium potential curves using the Douglas Kroll Hess Hamiltonian 13 and aug-ccpv5z-dk (AV5Z-DK) basis sets for He, 14 Br, 14 and I. 10 Since the binding energies are very small for both and, the basis set superposition error was corrected at each distance by using the function counterpoise (FCP) scheme of Boys and Bernardi. 15 V(r) ¼ E HeX E {HeX} {HeX} He E X (1) where r is the internuclear distance, and E HeX, E {HeX} He and E {HeX} X are the electronic energies of HeX, He, and X (X ¼ Br, I), respectively, obtained with the dimer basis set of HeX. In all cases, the MOLPRO 16 suite of ab initio programs was used throughout this work. Results for the equilibrium geometries and dissociation energies are given in Table 1 for each basis set used in the present work. These results were obtained by fitting the nearequilibrium portions of the full potentials to 5th-order polynomials in simple displacement coordinates. Within the AVnZ series of basis sets, regular convergence towards the apparent CBS limit is observed. Nearly identical CBS limits are obtained 1694 Phys. Chem. Chem. Phys., 2005, 7, This journal is & The Owner Societies 2005

2 Fig. 1 (a) Ab initio electronic energy potential energy curves (see text) associated to the first dissociation asymptote of. (b) Ab initio spin-electronic energy potential energy curves (see text) associated to the two first dissociation asymptotes of. (c) Ab initio rotational (J ¼ 11/2) spin-electronic energy potential energy curves (see text) associated to the two first dissociation asymptotes of. from either the CP-corrected or uncorrected results. The 2 S 1 state of is calculated to be more strongly bound than the same state in by about 3.5 cm 1. The 2 P states differ by only about 1 cm 1. Also shown in Table 1 are the AV5Z-DK results for and. The differences of these dissociation energies and bond lengths with the AV5Z ones are very small, about 0.1 cm 1 in D e and a maximum of A for r e.it should also be mentioned that the use of a spin unrestricted CCSD(T) approach, ROHF-UCCSD(T), was also tested with the AV5Z basis sets and differences on the order of only 0.1 cm 1 and A were calculated. The most accurate potential energy curves for these systems were obtained using the doubly augmented basis set sequences, DAVnZ. As expected, the convergence rate is observed to be improved and the CBS limits using the DAVQZ and DAV5Z basis sets are 0.4 to 0.8 cm 1 more strongly bound that those derived from the AVnZ series. Also shown in Table 1 for are the CCSD(T) results of de Lara-Castells et al., 7 who also employed basis set extrapolation methods with aug-cc-pvnz basis sets, as well as the CCSD(T) results of Partridge et al. 17 who utilized the aug-ccpvqz basis sets but with an additional set of bond functions. Not surprisingly, the CBS extrapolated results of de Lara- Castells are in very good agreement with our CBS-AVnZ values since very similar methods were used. They utilized a slightly different extrapolation formula, which resulted in slightly different D e and r e values. It should be noted, however, that the use of the AVnZ series of basis sets results in an underestimation of the well depth by nearly 1 cm 1 regardless of the choice of extrapolation formula. The results of Partridge et al. are nearly identical to our DAV5Z values, which are consistent with their expectations. Further basis set extensions, however, are predicted to further increase the dissociation energies to the values shown at the CBS-DAVnZ level in Table 1. Hence our results shown in Table 1 are expected to be the most accurate to date for these states of and. For use in the dynamics calculations, full potential energy functions were determined from the counterpoised-corrected CBS-DAVnZ data using polynomial expansions up to the power of 8 VðrÞ ¼ X8 n¼0 r 6:6 n C n ð2þ r where 6.0 r r r 24.0 a 0. The reference for the fitted functions was always set to 6.6 a 0 for convenience. Table 2 shows the fitted coefficients of the computed electronic energy curves of the 2 P and 2 S 1 states for both molecules. Table 1 RCCSD(T) equilibrium geometries and dissociation energies for and in the uncoupled doublet states. The results uncorrected for BSSE are given in parentheses 2 S 1 2 P D e /cm 1 r e /A D e /cm 1 r e /Å AVQZ (27.52) (2.530) (15.00) (4.007) AV5Z (28.11) (3.515) (15.21) (3.993) AV5Z-DK (28.20) (3.516) (15.41) (3.993) CBS-AVnZ (29.04) (3.499) (15.53) (3.974) DAVQZ (33.01) (3.498) (20.05) (3.956) DAV5Z (31.67) (3.496) (18.29) (3.947) CBS-DAVnZ (30.81) (3.492) (16.78) (3.936) Ref Ref AVQZ (23.84) (3.848) (13.65) (4.320) AV5Z (24.52) (3.833) (13.98) (4.303) AV5Z-DK (24.84) (3.834) (14.23) (4.302) CBS-AVnZ (25.48) (3.815) (14.38) (4.284) DAVQZ (29.13) (3.805) (18.44) (4.262) DAV5Z (27.53) (3.809) (16.47) (4.260) CBS-DAVnZ (26.32) (3.809) (14.75) (4.258) This journal is & The Owner Societies 2005 Phys. Chem. Chem. Phys., 2005, 7,

3 Table 2 Expansion coefficients of the fitted ab initio potential energy functions (hartree). See eqn. (2) for the functional form 2.2. Determination of the spin orbit coupling In the case of I and Br atoms, the spin orbit (SO) interaction is large and cannot be neglected. To evaluate spin orbit matrix elements, we used the Breit Pauli operator. 18 The SO matrix elements between the X 2 S 1 and 2 P states 19 were calculated at the MCSCF level 20 using the relativistic pseudopotential and uncontracted functions extracted from the AV5Z basis sets described in the previous section, however only spd functions were used for the halogens and sp functions for He. This choice was motivated by the implementation of spin orbit matrix elements within MOLPRO. 16 The spin orbit effect in systems such as rare gas halogen atoms can also be treated by a model that considers only the atomic component of the spin orbit interaction. 21 The V S and V P states defined above will mix and split into 3 components (see Fig 1b). According to the Hund s case (c) notations, 22 the resulting spin-electronic eigenvalues for a given internuclear distance r are: U X1=2 ¼ðV S þ V P þ D dþ=2; U I3=2 ¼ V P ; Coefficients/au X 2 S 1 2 P X 2 S 1 2 P C C C C C C C C C Coefficients/au X1/2 I3/2 II1/2 C C C C C C C C C Coefficients/au X1/2 I3/2 II1/2 C C C C C C C C C U II1=2 ¼ðV S þ V P þ D þ dþ=2; where D 4 0 is the spin orbit splitting in the halogen atom and d ¼ [(V P V S ) 2 þ 3/2(V P V S )D þ D 2 ] 1/2 (4) ð3þ Table 3 Atomic spin orbit splitting D ¼ E( 2 P 1/2 ) E( 2 P 3/2 ) Atoms ( 2 P) D exp. a /cm 1 D calc. ab initio/cm 1 Br I a See ref. 23. Table 3 compares the spin orbit splitting in the electronic 2 P ground state of Br and I obtained by the ab initio calculations described above to the experimental data. 23 The ab initio splitting is too low by 160 cm 1 for Br and 347 cm 1 for I but these differences do not have a significant impact on the values obtained for the equilibrium geometries and dissociation energies when the splitting is used in the model defined by eqn. (3). Moreover, in Table 4 the full spin orbit ab initio treatment for and is compared to the model of eqn. (3) showing very similar values for the molecular constants presented. The dissociation energies differ by less than 1 cm 1 and the equilibrium geometries by less than 0.07 a 0. In conclusion, the full ab initio treatment has been retained for the dynamical calculations on. Rotational constants and harmonic wavenumbers are presented in Table Dynamical calculations For the and molecules, the potential well is very shallow and only a few vibrational states can be expected. Therefore, the role of rotation cannot be neglected since it may easily lead to the dissociation of the molecule. Hund s case (c) gives us a general framework for the treatment of rotation while retaining only one degree of freedom for the description of the nuclear wavefunction The calculation of rovibronic bound states The framework given by Veseth 22 for the Hund s case (c) systems was used in order to determine the rovibronic states of and. In Hund s case (c), the total angular momentum J ˆ and the sum of electronic angular momentum and spin, Jˆ a, are related by Rˆ ¼ J ˆ J ˆ a (5) where Rˆ is the angular momentum for the molecular rotation. The rotational (angular) part of the molecular hamiltonian is then Hˆ rot ¼ hcb Rˆ2 ¼ hcb( J ˆ2 þ J ˆ a 2J ˆ J ˆ a ) (6) where hcb ¼ 1 2mr 2 ð7þ with m the reduced mass and r the internuclear separation. Table 4 Equilibrium geometries and dissociation energies for and taking into account spin orbit coupling X1/2 I3/2 II1/2 D e /cm 1 r e /a 0 D e /cm 1 r e /a 0 D e /cm 1 r e /a 0 þ SO a þ SO mod b þ SO a þ SO mod b a Complete ab initio treatment of the SO coupling. b Spin orbit model of eqn. (3) Phys. Chem. Chem. Phys., 2005, 7, This journal is & The Owner Societies 2005

4 Table 5 Harmonic wavenumbers and rotational constants using the CBS extrapolated potentials and ab initio SO treatment B e /cm 1 o e /cm 1 þ SO B e /cm 1 o e /cm 1 2S X1/ P I3/ II1/ B e /cm 1 o e /cm 1 þ SO B e /cm 1 o e /cm 1 2 S X1/ P I3/ II1/ Eigenvalues of Jˆ 2 and J ˆ 2 a are the usual J(J þ 1) and J a (J a þ 1) (in atomic units) and the eigenvalue of the last term in eqn. (6) Jˆ J ˆ a ¼ Jˆ x Jˆ ax þ Jˆ y Jˆ ay þ Jˆ z Jˆ az (8) is O 2, O being the projection of both J ˆ a and J ˆ on the internuclear axis. The first two terms are responsible for a rotational coupling with DO ¼1. In the basis of the eigenfunctions of the parity operator jj a JOi ¼ p 1 ffiffi ðjj a JOijJ a J OiÞ 2 the matrix elements of the operator Hˆ rot are given by hj a JO j ^H rot jj a JOi ¼U Ja;OðrÞ þ 1 JðJ þ 1Þ 2mr2 ½ þj a ðj a þ 1Þ 2O 2 ¼ V Ja;OðrÞðr; JÞ ð9þ ð10þ hj a JO j ^H rot jj a JðO 1Þi ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mr 2 ðj a þ OÞðJ a O þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðj þ OÞðJ O þ 1Þ ð11þ The functions U Ja;O (r) correspond to the spin-electronic potentials determined in the previous section with U X1/2 ¼ U 3/2;1/2 ; U I3/2 ¼ U 3/2;3/2 ; U II1/2 ¼ U 1/2;1/2 The diagonalisation of this matrix gives rise to effective potentials labeled P (J, J a, )(r). The general formula for these potentials for J a ¼ 3/2 are P ðj; 3=2; þþ ¼ 1 2 V 3=2;1=2ðJÞþV 3=2;3=2 ðjþþa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4b 2 þð V 3=2;1=2 ðjþþv 3=2;3=2 ðjþ aþ 2 P ðj; 3=2; Þ ¼ 1 2 V 3=2;1=2ðJÞþV 3=2;3=2 ðjþ a ð12þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4b 2 þð V 3=2;1=2 ðjþþv 3=2;3=2 ðjþþaþ 2 a ¼ 1 ðj þ 1=2Þ mr2 ð14þ ð13þ pffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ 2mr 2 ðj þ 3=2ÞðJ 3=2 þ 1Þ ð15þ and for J a ¼ 1/2 P(J, 1/2, ) ¼ V 1/2;1/2 (J ) a 0 (16) a 0 ¼ 1 ðj þ 1=2Þ 2mr2 ð17þ One can notice that J a is not a good quantum number and is undefined in the energy range where the spin-electronic potentials associated to the same O are close. However, due to the large spin orbit coupling this problem only concerns high energies in the repulsive part of the potentials and has been neglected. Fig. 1c shows the effect of rotational coupling on spin electronic states of for J ¼ 11/2. We then obtained the spin-rovibronic eigenstates by diagonalizing the radial part of the molecular 2 ^H ¼ 1 2 þ P ðj; J a ; ÞðrÞ ð18þ with the Lanczos algorithm 24 using a grid of 256 points for the representation of the wavefunction and potential. The grid limits are set to 6 and 40 a 0 in order to have enough points to represent the rotational barrier and the time propagation of resonances. Since the development of V i (r) in eqn. (2) is only valid up to 24 a 0, it is manually set to the asymptotic 24 a 0 value of V i (r) between 24 and 40 a 0 where the rotational functions are the only non-negligible terms in eqns. (12), (13) and (15) Energies and lifetimes of the spin-rovibronic resonances States below the rotational barrier and above the dissociation asymptote can tunnel through the barrier and form metastable resonant states. A simple diagonalization as described above is not able to precisely find those resonant states because the grid used is a closed box and all eigenvalues obtained from diagonalization have zero amplitude on the grid boundaries, thus limiting the possibilities for the dissociative part of the resonance wave function. However, one can observe eigenfunctions with non-zero amplitude both inside and outside the potential well, which is an indication of resonant behavior. We used these states, computed from a shorter grid between 6 and 24 a 0, as starting functions for wave packet propagations on the extended grid from 6 to 40 a 0 with an absorbing potential in order to avoid reflection on the edge of the grid. The propagation scheme we used is the short iterative Lanczos method 25 with timesteps of Dt ¼ 1.00 fs. Since our initial state is supposed to have a large overlap with existing resonances, the method proposed by Gray 8 in his study of resonances in HCO radical gave good results. In this method, the autocorrelation complex function C(t) ¼ hc(0) c(t)i is expressed via a Prony analysis 26 in the form of a sum CðtÞ ¼ X s b s expð ie s tþ expð G s t=2þ ð19þ where b s ¼ hj s c(0)i 2, and E s is the energy of the eigenstate j s. If j s is a bound state, then the decay width G s will be zero, and if j s is a resonance, its lifetime will be equal to t s ¼ 1/G s.a great advantage of the Prony analysis is its very fast convergence with respect to time: a short propagation period will be sufficient to extract lifetimes that are much larger than the propagation total time. In our case, we propagated the wave packet up to 10 ps and we were able to obtain converged lifetimes of the order of 500 ps. The Prony analysis is particularly well suited for analysis of theoretical results since it is This journal is & The Owner Societies 2005 Phys. Chem. Chem. Phys., 2005, 7,

5 Table 6 and rotational energies (cm 1 ) and lifetimes (ps) (in brackets) of bound states and resonances (v ¼ 0 in all cases). For positive energies, the values are obtained from Prony analysis, otherwise the values are direct eigenstates obtained by Lanczos diagonalization J P (J, 3/2, þ) a P (J, 3/2, ) a P 1 (J, 3/2, þ) a P 1 (J, 3/2, ) a P (J, 1/2, þ) b P(J, 1/2, þ) b 1/ c 7.59 c / / / (4500) / (4500) 2.42 (125) (4500) 11/ (33) 4.87 (10) 0.18 (4500) 2.45 d (272 d ) 13/ (53) 4.42 (8) 2.45 d (272 d ) 4.95 (10) 15/ (110) 4.94 (10) 4.95 (10) J P (J, 3/2, þ) a P (J, 3/2, þ) a P 1 (J, 3/2, þ) a P 1 (J, 3/2, ) a P(J, 1/2, þ) b P(J, 1/2, ) b 1/ c 7.53 c / / / / (4500) 1.56 (261) / (41) 3.51 (18) (81) 13/ (451) 3.64 (10) 1.71 (81) 15/ (4500) 4.02 (19) 17/ (27) a The energy levels are given with respect to the dissociation limit: He (X 1 S) þ X(X 2 P 3/2 )(cf. Table 3). b The energy levels are given with respect to the dissociation limit: He (X 1 S) þ X(X 2 P 1/2 )(cf. Table 3). c For these levels, the components O ¼ 3/2 are not defined and the potentials P are given by setting V 3/2;3/2 ¼ 0 and b ¼ 0 in eqns. (12) and (13). d Presence of another resonance of energy 2.81 cm 1 and lifetime 43 ps. known to be numerically sensitive to noises that might be present in the experimental data Results and discussion Table 6 summaries the rovibronic energies and lifetimes found using the method described above for 1/2 r J r 17/2. We always found only one energy level (v ¼ 0) on each of the P (J, J a, ) potentials for and (with a single exception Fig. 2 Rovibronic levels of and using the values of Table 6 for J a ¼ 3/2. in as noted in Table 6). The rotational energy was such that this level becomes purely dissociative for J values greater than 17/2 for and 15/2 for. For a given effective potential, the two highest values of J before dissociation lead to a resonance in most of the cases. The first resonance presents a lifetime of the order of a few hundreds of picoseconds and the second one has a shorter lifetime. Several very long lifetimes were not computed because the energies were very close to the dissociation limit and far from the top of the rotational barrier. In these cases, the potential should have been extended to larger values of the configuration space and the propagation time increased in order to obtain results, but the numerical errors would have interfered with the Prony analysis. There is a degeneracy of states P(J,1/2, ) and P(J þ 1,1/2, þ) that can be found using eqns. (10), (16) and (17). The values of Table 6 for J a ¼ 3/2 are plotted in Fig. 2 and one can see that for the smallest values of J (J r 5/2), we can assign the two lowest levels to O ¼ 1/2 and the two highest to O ¼ 3/2, showing that the O-doubling is strong for both molecules. For intermediate J values, the O-doubling is weaker for O ¼ 3/2 states than for O ¼ 1/2 states, but cannot be neglected as in Hund s case (a). Finally, for larger values of J, the O mixing (due to b in eqns. (12) and (13)) increases strongly with J and O is no more a good quantum number. We also checked our potential against Morse potentials used in other studies of van der Waals molecules. We fitted all of our ab initio data for the 2 S 1 state and the spin electronic X1/2 state by a Morse function hcm(r) ¼ hcd e ({1 exp[ a(r r e )]} 2 1) (20) for 6.0 r r r 24.0 a 0 and obtained a rms error of about 0.4 cm 1. This rms value could be considered too important if very accurate results are required. For comparison, the fitted functions used for our dynamics calculations (eqn. (2)) have rms errors of about 0.03 cm 1. Table 7 shows a comparison between our Morse fits and different Morse parameters found in the literature and used in dynamical calculations. For, the agreement with the parameters of de Lara-Castells et al. 7 is good as expected Phys. Chem. Chem. Phys., 2005, 7, This journal is & The Owner Societies 2005

6 Table 7 The parameters of the potential used by Rohrbacher et al. 27 are close to those of the X1/2 potential, but the equilibrium distance seems slightly too large. In the case of the cited references do not explicitly refer to the SO coupled or uncoupled nature of the state. However, the match is closer with our parameters associated to the X1/2 state. Acknowledgements We wish to thank Dr S. K. Gray for letting us use his code for the Prony analysis of autocorrelation functions, and for stimulating discussions on this theme. We also acknowledge Dr A. Mitrushenkov for fruitful discussions. KAP was supported by the US National Science Foundation (CHE ). References Morse parameters a for dynamics D e /cm 1 r e /a 0 a/a 0 1 Ref. 7 2 S Ref. 7 X1/ Ref. 27 X This work 2 S This work X1/ D e /cm 1 r e /a 0 a/a 0 1 Ref Ref Ref This work 2 S This work X1/ a M(r) ¼ D e ({1 exp [ a(r r e )]} 2 1). 1 A. Braun, PhD Thesis N2967, Ecole Polytechnique Fédérale de Lausanne, A. B. McCoy, J. P. Darr, D. S. Boucher, R. R. Winter, M. D. Bradke and R. A. Loomis, J. Chem. Phys., 2004, 120, S. K. Gray and C. E. Wozny, J. Chem. Phys., 1991, 94, R. L. Waterland, M. I. Lester and N. Halberstadt, J. Chem. Phys., 1990, 92, R. H. Bisseling, R. Kossloff, R. B. Gerber, M. A. Ratner, L. Gibson and C. Cerjan, J. Chem. Phys., 1987, 87, C. Cerjan and S. A. Rice, J. Chem. Phys., 1983, 78, (a) M. P. de Lara-Castells, R. V. Krems, A. A. Buchachenko, G. Delgado-Barrio and P. Villarreal, J. Chem. Phys., 2001, 115, ; (b) M. P. de Lara-Castells, A. A. Buchachenko, G. Delgado-Barrio and P. Villarreal, J. Chem. Phys., 2004, 120, S. K. Gray, J. Chem. Phys., 1992, 96, (a) P. J. Knowles, C. Hampel and H.-J. Werner, J. Chem. Phys., 1993, 99, 5219; (b) P. J. Knowles, C. Hampel and H.-J. Werner, J. Chem. Phys., 2000, 112, 3106, Erratum. 10 K. A. Peterson, D. Figgen, E. Goll, H. Stoll and M. Dolg, J. Chem. Phys., 2003, 119, (a) T. H. Dunning Jr, J. Chem. Phys., 1989, 90, 1007; (b) R.A. Kendall, T. H. Dunning Jr and R. J. Harrison, J. Chem. Phys., 1992, 96, A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, H. Koch, J. Olsen and A. K. Wilson, Chem. Phys. Lett., 1998, 286, (a) M. Douglas and N, M. Kroll, Ann. Phys. (New York), 1974, 82, 89; (b) G. Jansen and B. A. Hess, Phys. Rev. A, 1989, 39, W. A. de Jong, R. J. Harrison and D. A. Dixon, J. Chem. Phys., 2001, 114, S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, MOLPRO, a package of ab initio programs designed by H.-J. Werner and P. J. Knowles, version , R. D. Amos, A. Bernhardsson, A. Berning, P. Celani, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, P. J. Knowles, T. Korona, R. Lindh, A. W. Lloyd, S. J. McNicholas, F. R. Manby, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, R. Pitzer, G. Rauhut, M. Schu tz, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni and T. Thorsteinsson, H.-J. Werner, H. Partridge, J. R. Stallcop and E. Levin, J. Chem. Phys., 2001, 115, (a) S. R. Langhoff and C. W. Kern, ModernTheoretical Chemistry, Plenum Press, New York, 1977, vol. 3; (b) H. A. Bethe and E. E. Salpeter, in Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag, Berlin, A. Berning, M. Schweizer, H.-J. Werner, P. J. Knowles and P. Palmieri, Mol. Phys., 2000, 98, (a) H.-J. Werner and P. J. Knowles, J. Chem. Phys., 1985, 82, 5053; (b) P. J. Knowles and H.-J. Werner, Chem. Phys. Lett., 1985, 115, R. Burcl, R. V. Krems, A. A. Buchachenko, M. M. Szcze s niak, G. Chalasin ski and S. M. Cybulski, J. Chem. Phys., 1998, 109, L. Veseth, J. Phys. B, 1973, 6, (a) Ch. E. Moore, Atomic Energy Levels, Circular of the National Bureau of Standards 467, US Government Printing Office, Washington, DC, 1952, Vol. II; (b) Ch. E. Moore, Atomic Energy Levels, Circular of the National Bureau of Standards 467, US Government Printing Office, Washington, DC, 1958, Vol. III. 24 C. Lanczos, J. Res. Natl. Bur. Stand., 1950, 45, T. J. Park and J. C. Light, J. Chem. Phys., 1986, 85, S. Marple, Jr, Digital Spectral Analysis with Applications, Prentice- Hall, Englewood Cliffs, A. Rohrbacher, T. Ruchti, K. C. Janda, A. A. Buchachenko, M. I. Hernàndez, T. Gonzàlez-Lezana, P. Villarreal and G. Delgado- Barrio, J. Chem. Phys., 1999, 110, This journal is & The Owner Societies 2005 Phys. Chem. Chem. Phys., 2005, 7,