THE JOURNAL OF CHEMICAL PHYSICS 138, 084301 (2013) Potential energy curves for the interaction of Ag(5s) and Ag(5 p) with noble gas atoms J. Loreau, H. R. Sadeghpour, and A. Dalgarno ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA (Received 27 August 2012; accepted 22 January 2013; published online 22 February 2013) We investigate the interaction of ground and excited states of a silver atom with noble gases (NG), including helium. Born-Oppenheimer potential energy curves are calculated with quantum chemistry methods and spin-orbit effects in the excited states are included by assuming a spin-orbit splitting independent of the internuclear distance. We compare our results with experimentally available spectroscopic data, as well as with previous calculations. Because of strong spin-orbit interactions, excited Ag-NG potential energy curves cannot be fitted to Morse-like potentials. We find that the labeling of the observed vibrational levels has to be shifted by one unit. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4790586] I. INTRODUCTION There is a great deal of interest in trapping and cooling of atomic and molecular species, ostensibly for a detailed manipulation of interatomic interactions, and precision spectroscopy.1,2 Cold and ultracold atomic and molecular ensembles are also employed as prototypes to simulate many-body quantum condensed-phase matter,3 to study processes far from equilibrium,4 and to create qubits for quantum logic operations.5 Nearly all of the current focus has been on cooling and trapping of alkali metal atoms and associated molecular species, because of the availability of accessible cycling transitions for laser cooling. Alkali atoms can be treated as one-electron atoms, which makes them amenable to accurate numerical calculations of their properties. The paradigm shift to other atoms in the periodic table occurred with the advent of general-purpose magnetic and off-resonant optical trapping schemes in recent years. In magnetic trapping of atoms or molecules, a first necessary ingredient is a species with a spin projection. One such atom is silver, which has been confined in a magneto-optical trap,6 and in a buffer-gas cooled magnetic trap.7 It was found that in a high density He buffer gas cooled trap, Ag has a sizeable propensity to undergo three-body recombination (Ag-He-He → AgHe (v = 0, J )∗ - He → AgHe(0, 0) - He) and form van der Waals (vdW) complexes.8 This process of formation of weakly-bound molecules shows up as a loss of Ag atoms from the trap. Collisions of optically-pumped spin-polarized atoms with 3He have been shown to be highly efficient in transfer of spin polarization to 3He nuclei. Silver was shown in a recent work to be even more efficient than commonly used alkali-metal atoms for polarization transfer.9 The cold and ultracold molecules come in two main flavors: they are either weakly-bound highly vibrationallyexcited Feshbach molecules, created by pairing ultracold atoms, or are deeply-bound molecules, which can be paramagnetic, for trapping in a buffer-gas trap, or polar, for slowing in an electric field decelerator and eventually trapped. A third class of trappable molecules is the vdW molecules, which are bound solely by long-range dispersion interaction and are weakly bound. Among these, the interaction of Ag and other coinage metals with noble gases has been the subject of numerous experimental studies.10–17 Spectroscopic studies on these complexes have focused on the molecular absorption corresponding to the strong atomic 2P ← 2S transition. The understanding of the bonding of such vdW complexes can be used to improve models of atom-surface interaction and study of chemical reaction dynamics,18–20 while their decay by chemical exchange, pre-dissociation, and dissociation, can be controlled by external fields. Silver complexes with noble gas atoms can also be used for application to magnetometry,21 and the pressure broadening and shift of the D1 line of Ag in collisions with Ar and He were recently measured.22 In this work, we describe the molecular states resulting from the interaction of Ag(5s) and Ag(5p) with all the noble gases and focus on the Ag-Ar system in order to establish a comparison with experimental data. A schematic diagram of the potential energy curves (PECs) of the low-lying spectrum of the Ag-NG complexes, including the spin-orbit coupling, is shown in Fig. 1. The ground state of these vdW complexes is the X 2 + state that correlates to Ag(5s 2S) + NG(1S) and is attractive for all noble gases. The first excitation of the silver atom, Ag(5p 2P), gives rise to the A 2 and the B 2 + molecular states. The B 2 + state is expected to be less strongly bound than the ground state as it corresponds to the interaction between the NG(1S) and the p orbital of Ag oriented parallel to the intermolecular axis. On the other hand, in the case of the A 2 , the p orbital is perpendicular to the intermolecular axis, leading to a more attractive state. Ag−He is an exception to this rule since He does not have a p shell, and the absence of p − p repulsion leads to an even more deeply bound A 2 state. The spin-orbit (SO) interaction cannot be neglected in systems involving silver. Its effect is to split the 2P state of Ag into the doublet states 2P1/2 and 2P3/2 at 29552.1 cm−1 and 30472.7 cm−1, respectively, separated by 920.6 cm−1. In the Ag−NG complexes, the effect of the SO interaction is to mix the A 2 and B 2 + states into a 2 1/2 state 0021-9606/2013/138(8)/084301/8/$30.00 138, 084301-1 © 2013 American Institute of Physics 084301-2 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) 31500 31000 30500 30000 29500 B 2Σ1/2+ A 2Π3/2 A 2Π1/2 Ag(2P3/2)+NG Ag(2D5/2)+NG Ag(2P1/2)+NG Energy (cm-1) 600 400 X 2Σ+ 200 0 Ag(2S)+NG 4 6 8 10 12 Internuclear distance (a0) FIG. 1. Schematic representation of the PECs of the low-lying states of Ag−NG Van der Waals complexes. (dissociating to 2P1/2) and 2 3/2 and 2 + 1/2 states (dissociat- ing to 2P3/2). However, the major complication in the theo- retical treatment of the Ag−NG systems is the fact that the second excited state of silver corresponds to the configura- tion 4d95s2 (2D5/2). This 2D5/2 component overlaps with the 4d105p (2P3/2) state, rendering an accurate calculation a considerable task. Therefore, we treat the SO interaction analyti- cally by approximating the coupling by its atomic value, and discuss for Ag−Ar the validity of this approximation. Previous theoretical ab initio calculations of the poten- tial energy curves of the ground state of Ag-noble gases systems include studies of Ag−He complexes,23,24 and Ag with He, Ne and Ar.25 Similar calculations were performed by Gardner et al.,26 in addition to the PEC of the Ag−Kr, Ag−Xe, and Ag−Rn complexes. The interaction potential of Ag with N2 has also been recently reported,27 completing the study of the interaction of silver in its ground state with buffer gases. However, except for the case of Ag−He,23 the excited states of these vdW complexes have never been investigated theoretically. On the experimental front, two spectroscopic studies of silver-noble gases complexes ex- ist: Jouvet et al. investigated the Ag−Ar complex using laser-induced fluorescence,10 while Brock and Duncan used resonance-enhanced multiphoton ionization (REMPI) tech- nique to study Ag−Ar, Ag−Kr, and Ag−Xe complexes.11 Bands were observed for the transitions A 2 1/2 ← X 2 + and A 2 3/2 ← X 2 + and the spectroscopic parameters of these excited states were extracted. No B + 1/2 ← X2 + bands were observed. As will be shown in Sec. IV, the excited B + state, which also correlates asymptotically to Ag(5p), is weakly bound and has an equilibrium geometry at a much larger internuclear distance, such that the transition to the ground state is not favored. In this work, we compute PECs for the Ag−NG com- plexes dissociating into Ag(5s) + NG(1S) and Ag(5p) + NG(1S). We describe the computational method in Sec. II and present the potential energy curves without spin-orbit coupling in Sec. III. The inclusion of the spin-orbit interaction as a perturbation and the resulting PECs are discussed in Sec. IV, and we make a detailed comparison with experimental results in Sec. V. II. COMPUTATIONAL METHOD We described the silver atom using the aug-cc-pwCVnZPP basis set,28 with n = Q,5. This basis set is based on a small core relativistic effective core potential (ECP) that replaces the 1s − 3d core,29 and was constructed to describe accurately the remaining 19 electrons, including core-valence correlation. For the noble gases He, Ne, and Ar, we used the aug-cc-pVnZ basis sets30–32 with n = Q,5. Calculations involving the heavier atoms Kr, Xe, and Rn were realized with a small core relativistic ECP (describing, respectively, the 10, 28, or 60 inner electrons of these noble gases) while the augcc-pVnZ-PP (n = Q,5) basis set33 was used in order to explicitly describe the outer-core (n − 1)spd shells and the nsp valence shells. The characterization of the Ag−NG interaction is improved by the inclusion of a set of (3s3p2d2f1g) bond functions located at midway between the two atoms. This set of functions is well suited for interactions involving noble gases.34 The use of bond functions removes the need for the complete basis set (CBS) extrapolation while producing results that are in good agreement with the CBS limit.35 The X 2 + and A 2 PECs were calculated using the spin-unrestricted coupled cluster method with single, double, and perturbative triple excitations (UCCSD(T)),36,37 as implemented in the MOLPRO 2009.1 package.38 The reference wave functions employed in the coupled cluster calculations were generated with the spin restricted Hartree-Fock (ROHF) method. In these calculations, we correlated not only the valence, but also the outer-core electrons. This means that for silver, the effect of the 4s24p64d105s (5p) electrons was included. For He, Ne, and Ar, all electrons were correlated while for Kr, Xe, and Rn, the (n − 1)spd and nsp electrons were kept active. The PEC of the B 2 + state was calculated with the multireference configuration interaction (MRCISD) method,39,40 including the Davidson correction. The reference wave function was generated using the complete active space selfconsistent field (CASSCF) method. For these calculations, we correlated the valence and 4d10 electrons of Ag and the nsp electrons of the noble gas. The PECs were calculated on a grid of internuclear distances R between 3.5 a0 and 20 a0. At each point, we corrected the energy using the counterpoise method in order to account for the basis set superposition error.41 The energies and wave functions of the rovibrational levels were obtained by solving the radial Schrödinger equation using a B-spline method.42 The spectroscopic constants were determined by fitting the vibrational energies to the standard form E(v) = ωe(v + 1/2) − ωexe(v + 1/2)2, using a nonlinear least-squares Marquardt-Levenberg algorithm. 084301-3 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) TABLE I. Spectroscopic parameters of the ground state X 2 + of the Ag−NG molecules. Ag−He does not have enough vibrational levels to extract ωe and ωexe. Complex Re(a0) De (cm−1) D0 (cm−1) ωe (cm−1) ωexe (cm−1) Ag−He Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn 8.69 8.67a 8.78b 8.69c 8.80d 7.80 7.80a 7.87b 7.57 7.53a 7.63b 7.46 7.48e 7.38 7.43e 7.01 7.13e 7.31 7.5a 6.81b 7.42c 6.80d 27.54 28.1a 26.4b 112.93 113.9a 107.2b 173.67 169.3e 264.81 253.9e 384.80 355.8e 2.00 2.2a 1.4d 21.04 21.7a 102.62 104.2a 163.61 160.3e 254.25 244.6e 372.64 345.8e 16.6 13.5a 13.2b 20.0 19.8a 19.0b 18.9 18.3e 19.6 18.7e 21.7 20.0e aRCCSD(T) calculations of Ref. 26 including core-valence correlation. bCCSD(T) calculations from Ref. 25. cCCSDT calculations from Ref. 24. dRCCSD(T) calculations from Ref. 18. eRCCSD(T) calculations of Ref. 26 without core-valence correlation. 1.72 1.68a 1.73b 0.90 0.88a 0.83b 0.52 0.48e 0.36 0.31e 0.31 0.26e If rovibrational levels close to the dissociation limit are of interest, it is necessary to know the behavior of the PECs for internuclear distances larger than 20 a0 as calculated in this work. The PECs can be obtained for all internuclear dis- tances by fitting the ab initio points to the asymptotic poten- tial C8, Vas = − and C10, n Cn/Rn which were using the dispersion coefficients previously calculated for the X 2 C6, +, A 2 , and B 2 + states by Zhang et al.43 III. POTENTIAL ENERGY CURVES WITHOUT SPIN-ORBIT INTERACTION A. Ground state The spectroscopic parameters of the ground state X 2 + are presented in Table I and compared with previous theoretical works. The results presented in Table I were obtained with 5Z basis sets for both Ag and the noble gases. Because bond functions were employed in this calculation, the dependence of the spectroscopic constants on the basis set is expected to be small. The use of a 5Z basis set instead of a QZ basis set modifies De by less than 1%, and the effect is even smaller on Re. Interestingly, for the noble gases He, Ne, and Ar, we obtain a larger value for De with the 5Z basis set than with the QZ basis set, while the effect is reversed for the heavier gases Kr, Xe, and Rn. This phenomenon can be explained by the fact that the bond functions break the hierarchy of the AVnZ basis sets, and it is therefore not recommended to extrapolate the results to the complete basis set limit. The PEC of the ground state of Ag−He was already studied by various groups18,24–26 using the CCSD method. As can be seen from Table I, our results agree quite well with previous calculations, despite using different basis sets. For TABLE II. Permanent dipole moment of the Ag−NG species in debye. Complex Ag−He Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn Dipole moment (D) 0.010 0.046 0.137 0.201 0.297 0.409 Ag−Ne and Ag−Ar, the agreement between the present calculations and the results of Refs. 25 and 26 is excellent, although our values for De and Re are slightly closer to those of Ref. 26. For Ag−Kr, -Xe, and -Rn, we find again good agreement with the values reported by Gardner et al.26 However, we obtain larger values for De and smaller values for Re, and the discrepancy increases with the noble gas mass. This can be explained by the fact that Gardner et al. did not correlate the inner-valence electrons of the atoms in their calculations. While this has no effect for the complexes involving He, Ne, and Ar (as mentioned in Ref. 26), this is not the case for the heavier noble gases. For Ag−Rn, the effect of corevalence interactions can be as much as 10% of the value of De. Therefore, while freezing the core or inner-valence electrons significantly reduces computational cost, it can also lead to a dramatic underestimate of De and an overestimate of Re for heavy complexes. In Table I, we did not include the values of D0 determined experimentally.10,11 These values were extrapolated from transitions between the ground state of the complexes and various excited states, and are strongly isotopic- and statedependent. For example, for Ag−Kr, Ref. 11 provides values for D0 between 68 and 138 cm−1, and we believe that these values are not precise enough to allow for detailed comparison. The trends in the spectroscopic parameters De, Re, and ωe along the noble gas series have been discussed by Gardner et al.26 In particular, De increases with NG atomic number, while Re decreases. The trend in De is expected as the polarizability is larger for heavier noble gases, which enhances the vdW interaction. The dissociation energies are plotted in Fig. 2(a) as a function of NG polarizability, exhibiting a nearly linear dependence, as expected. The trend in Re is more surprising as increasing the NG mass results in an increase of the vdW radius, which in turn would be expected to lead to larger equilibrium distances. However, as extensively discussed in Ref. 26, a combination of other factors, such as sp hybridization, results in a decrease in Re along the noble gas series. B. Permanent electric dipole moments The vdW molecules can become polar and possess permanent electric dipole moments. In Table II, we present the vibrationally averaged dipole moments in the X 2 + PEC for Ag−NG species. There is a monotonic increase of the dipole moment with the NG atom mass. Overall, the dipole moments are relatively small in the ground electronic and vibrational 084301-4 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) Dissociation Energy (cm-1) 500 (a) 400 300 200 100 0 HeNe 4000 (b) X 2Σ+ state B 2Σ+ state Ar Kr Xe Rn 3000 2000 1000 0 0 A 2Π state 10 20 Polarizability (a03) 30 FIG. 2. Dissociation energy of the Ag−NG complexes as a function of the polarizability of the noble gases. (a) For the X 2 + and B + states; (b) for the A 2 state. The polarizability of the noble gases increases with their mass. states, while we expect the dipole moments to be larger in the excited states. C. Excited state PECs dissociating to Ag(5p 2P) The equilibrium distance and dissociation energy of the two excited states A 2 and B 2 +, which correlate asymptotically to Ag(5p 2P) + NG, are presented in Table III, while their dissociation energy is plotted as a function of the noble gas polarizability in Fig. 2. The dissociation energy of the B 2 + state is found to increase linearly with the noble gas polarizability, as was observed for the ground state. However, for a given complex, the value of De is always smaller than that of the ground state. This was expected since the B 2 + state corresponds to the interaction between the noble gas and the p orbital of Ag oriented parallel to the intermolecular axis, which enhances repulsion. This also leads to a much larger equilibrium geometry than for the ground state. Finally, the value of Re decreases with increasing NG atomic number. The A 2 state corresponds to the interaction between the noble gas and the p orbital of Ag oriented perpendicularly to the intermolecular axis. Therefore, this state is much more deeply bound than the X 2 + or B 2 + states. The dissociation energy increases linearly from Ne to Rn, but the interaction with He results in a larger De than with Ne (cf. Fig. 2(b)). This occurs because He does not have a p shell, and the absence of p − p repulsion leads to an even more deeply bound A 2 state. This behavior was already observed in other systems, e.g., involving alkali metals–noble gas complexes.44,45 The PEC of the A 2 state was previously investigated for Ag−He.18,23,46 Our results agree well with the calculations of Brahms et al.,18 which were also performed using the CCSD(T) method. On the other hand, we find large discrepancies with the two other sets of calculations, performed using MP223 and CISDT methods,46 with values for De differing by as much as 50%. IV. POTENTIAL ENERGY CURVES WITH SPIN-ORBIT INTERACTION As previously mentioned, the spin-orbit interaction cannot be neglected in the 5p shell of the silver atom as the splitting between the 2P1/2 and the 2P3/2 states is = 920.6 cm−1. The most accurate description of the molecular states arising from Ag(5p 2P1/2) and Ag(5p 2P3/2) would be realized by performing a CASSCF + MRCI calculation including the 2 1+/2, 2 3/2, and 2 5/2 states dissociating into Ag(4d95s2 2D5/2) + NG(1S) as well as the spin-orbit interaction. In particular, when the SO coupling is taken into account, a non-adiabatic interaction between the A 2 3/2 state and the 2 3/2 state arising from Ag(2D5/2) can be expected, as illustrated schematically in Fig. 1. Unfortunately, this approach is computationally demanding and poses considerable difficulties even for the simplest molecule studied here, Ag−He.46 Therefore, we instead assume that the spin-orbit Hamiltonian is given by the atomic interaction, Hso = ξ l · s, where the spin-orbit parameter ξ is Rindependent. This approximation has been previously used with success to describe systems for which the spin-orbit interaction makes a full MRCI calculation intractable.17,23,46,47 While this approximation will clearly fail at small distances, where the excited state PECs are mostly repulsive, we will test the merits of this approximation by comparing molecular parameters and vibrational energies with other calculations, when available, and observations. In order to obtain TABLE III. Spectroscopic parameters of the excited A 2 and B 2 + states of the Ag−NG molecules. State A2 Complex Ag−He B2 + Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn Ag−He Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn aRCCSD(T) calculation from Ref. 18. bCISDT calculations from Ref. 46. cMP2 calculations from Ref. 23. Re(a0) 4.40 4.42a 4.76b 5.16c 5.54 5.18 5.19 5.23 5.34 13.80 14.91a 12.35 11.86 11.64 11.77 11.34 De (cm−1) 477.8 463.6a 272.1b 349.9c 187.4 1302.6 2138.6 3796.8 4328.3 2.30 0.95a 7.89 35.40 56.61 78.72 113.04 084301-5 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) Energy (cm-1) Energy (cm-1) 2000 1000 0 -1000 -2000 -3000 -4000 A 2Π3/2 Ag(2P3/2)+NG Ag-He Ag-Ne Ag-Ar Ag-Kr Ag-Xe Ag-Rn 2000 1000 0 A 2Π1/2 Ag(2P1/2)+NG -1000 100 -2000 0 -3000 -4000 4 -100 -200 468 6 8 10 Internuclear distance (a0) 10 12 12 FIG. 3. Potential energy curves of the A 2 3/2 (top panel) and A 2 1/2 (bottom panel) states of the Ag(5p)-NG complexes. The inset shows the double well structure that appear for He and Ne as a consequence of the spin-orbit interaction. the spin-orbit coupled PECs, it is necessary to evaluate the matrix elements of the spin-orbit Hamiltonian. Hso is diago- nal in the quantum number = + and its matrix ele- ments can therefore be easily computed in the |LSJ ⟩ repre- sentation. However, the ab initio calculations are performed in the |LS ⟩ (spin-uncoupled) representation, so that it is necessary to transform the matrix elements of Hso in this representation. From the 2 + and the 2 states dissociating into Ag(5p), we can form the 2 3/2 state that correlates asymptot- ically to 2P3/2 and has projection | | = 3/2 onto the internu- clear axis, and the 2 1/2 and 2 + 1/2 states, which dissociate, respectively, into 2P1/2 and 2P3/2 and correspond to the | | = 1/2 projection. The total Hamiltonian in the |LS ⟩ representation is block-diagonal in and has the following form: | | = 1/2 : U (R√) − 1 2 ξ √2 2 ξ 2 2 ξ U (R) , (1a) | | = 3/2 : U (R) + 1 2 ξ , (1b) where U (R) and U (R) denote the PECs of the 2 and 2 + states, respectively. The spin-orbit parameter is equal to two thirds of the atomic splitting, ξ = 2 /3 = 613.7 cm−1. The 250 Ag-He 200 B 2Σ1/2+ Ag-Ne Ag-Ar Ag-Kr 150 Ag-Xe Ag-Rn 100 50 Ag(2P3/2)+NG 0 -50 -100 8 10 12 14 16 18 20 Internuclear distance (a0) FIG. 4. Potential energy curves of the B 2 complexes. + 1/2 state of the Ag(5p)−NG diagonalization of (1) yields the SO-coupled potentials. The resulting PECs are presented in Figs. 3 and 4, and the spec- troscopic parameters are given in Table IV. The 2 3/2 state, due to symmetry, is not affected by the SO interaction. The values of Re and De are therefore identical to those discussed in Sec. III. We do not observe a general trend in the behavior of ωe for this state, while ωexe decreases with increasing NG atomic number. The 2 1/2 PEC arises from the mixing of pπ and pσ orbitals by the SO interaction. Since in the 2 molecular sym- metry, the pπ orbital is perpendicular to the internuclear axis, the mixing in of the pσ orbital results in repulsion when the Ag and NG atoms interact at short internuclear distances. For Ag−He and Ag−Ne, orbital repulsion is large enough to overcome the attractive character, leading to a short-range barrier and a double well structure illustrated in the inset of Fig. 3. This behavior was already observed in alkali-noble gas PECs.45 The 2 1/2 state of Ag−He presents two minima separated by a barrier peaking at R = 6.95a0 with a maximum energy of 91.2 cm−1. The first minimum is located at R = 4.41a0, which is almost the same value as in the 2 3/2 state. However, the dissociation energy is less than half that of the 2 3/2 state (200.6 cm−1 compared to 477.8 cm−1). Moreover, the 2 1/2 state only supports one bound vibrational level, whereas the 2 3/2 state supports 6 bound levels. The second minimum is located at large internuclear distance, Re = 13.02a0, and is due to the mixing with the B 2 + state. This well has a dissociation energy of De = 1.63 cm−1 but does not support any vibrational levels. The PEC of the 2 1/2 state of Ag−Ne also presents a barrier, located at R = 6.78a0 with a height of 67 cm−1. The first minimum of the potential is situ- ated at R = 5.70a0 but has positive energy. The second minimum is located at Re = 11.41a0 with a dissociation energy De = 6.14 cm−1, which is due to the 2 + state. It supports 2 vibrational levels. For the heavier Ag−NG species, the attractive character is strong enough so that the PEC of the 2 1/2 state is purely attractive. We find that the equilibrium distance is identical to that of the 2 3/2 state, but that the dissociation energy is systematically smaller. The main effect of the spin- orbit interaction is to dramatically modify the intermediate- and long-range part of the PEC, as shown in the inset of 084301-6 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) TABLE IV. Spectroscopic parameters of the excited 2 1+/2, 2 1/2, and 2 3/2 states of the Ag−NG molecules for natural abundances. For Ag−He in the 2 1/2 state, the values of Re and De for the two potential wells are given (see text). The 2 1/2 and 2 + 1/2 states of Ag−He and Ag−Ne do not support enough vibrational states to extract ωe and ωexe. State Complex Re(a0) 2 1/2 2+ 1/2 2 3/2 Ag−He Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn Ag−He Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn Ag−He Ag−Ne Ag−Ar Ag−Kr Ag−Xe Ag−Rn 4.41, 13.02 5.22a 11.41 5.18 5.19 5.23 5.34 13.62 12.10 11.49 11.20 11.25 10.69 4.40 5.54 5.18 5.19 5.23 5.34 aMP2 calculations of Ref. 23. De (cm−1) 200.6, 1.63 89.2a 6.14 1028.3 1861.8 3514.5 4050.8 1.61 6.70 31.1 50.8 72.8 109.1 477.8 187.4 1302.6 2138.6 3796.8 4328.3 D0 (cm−1) 119.2 3.89 977.0 1810.1 3455.1 3993.7 4.20 26.19 45.13 65.82 100.93 392.6 163.4 1250.9 2087.1 3738.0 4272.0 ωe (cm−1) 106.2 103.4 118.1 111.0 8.05 7.89 8.27 8.83 174.3 47.0 100.6 100.6 115.5 109.3 ωexe (cm−1) 2.76 1.44 1.00 0.77 0.51 0.31 0.23 0.18 16.1 3.07 2.00 1.21 0.89 0.70 Fig. 3. Because of this, the PEC of the 2 1/2 state cannot be fitted to a Morse-like potential and the spectroscopic param- eters ωe and ωexe presented in Table IV do not provide an accurate representation of the potential. This is true in par- ticular for the high vibrational levels, which lie close to the dissociation limit. Finally, the effect of the spin-orbit interaction on the PEC of the 2 + 1/2 state is to reduce the value of the equilibrium dis- tance and the dissociation energy by a few percent compared to the PEC of the B 2 + state without spin-orbit. For Ag−He and Ag−Ne, the potential does not support enough vibra- tional levels (0 and 2, respectively) to extract spectroscopic parameters. V. COMPARISON WITH THE EXPERIMENT The experimentally determined spectroscopic parameters10, 11 of the 2 1/2 and 2 3/2 states for specific isotopes of the Ag−Ar, Ag−Kr, and Ag−Xe complexes are presented in Tables V–VII and compared with our calculations. These parameters are the Morse vibrational terms, ωe and ωexe (the energy of the levels is given by E(v) = ωe(v + 1/2) − ωexe(v + 1/2)2), the dissociation energy D0 (i.e., the binding energy of the v = 0 state), obtained by a Birge-Sponer extrapolation, and the transition frequency ν00 (between the v = 0 level of either the 2 1/2 or the 2 3/2 state and the v = 0 level of the ground state), also extrapolated using Birge-Sponer analysis. The dissociation energy of the ground state can also be extrapolated using the same method. We observe that the agreement between theory and experiment for the 2 3/2 PEC is good for all species, even though we neglected the interaction of this state with the 2 3/2 state dissociating into Ag(2D5/2) + NG(1S). This interaction probably occurs at large internuclear distances so that it does not affect the spectroscopic parameters. We note that the largest discrepancy between theory and experiment occurs for the Ag−Kr system, for which the experimental data are subject to large errors.11 For the 2 1/2 state, which is expected to be TABLE V. Comparison of the spectroscopic parameters calculated in this work with the experimental values for 107Ag−40Ar. All parameters are in units of cm−1. Exp. 1 and Exp. 2 corresponds to the experimental values determined in Refs. 10, 11, respectively. Param. 2 1/2 This work Exp. 1 Exp. 2 2 3/2 This work Exp. 1 Exp. 2 ωe 106.3 109.2 112.9 100.6 100.3 100.2 ωexe 2.77 2.83 3.33 2.00 2.04 2.01 D0 977 999 903 1251 1184 1199 ν00 28 677 28 714 29 324 29 325 TABLE VI. Comparison of the spectroscopic parameters calculated in this work with the experimental values from Ref. 11 for 107Ag−83Kr. All parameters are in units of cm−1. Param. 2 1/2 This work Exp. 2 3/2 This work Exp. ωe 103.8 121.6 101.1 108.1 ωexe 1.45 1.58 1.23 1.26 D0 1809.8 2286 2086.8 2267 ν 00 27 905 27 404 28 549 28 274 084301-7 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) TABLE VII. Comparison of the spectroscopic parameters calculated in this work with the experimental values from Ref. 11 for 107Ag−129Xe. All parameters are in units of cm−1. Param. 2 1/2 This work Exp. 2 3/2 This work Exp. ωe 118.9 123.8 116.2 115.8 ωexe 1.01 1.01 0.90 0.91 D0 3455 3728 3738 3630 ν 00 26 352 26 100 26 989 27 021 the most strongly affected by the spin-orbit interaction, the pa- rameters do not show the same level of agreement. The source of the discrepancy can be traced back to the different behavior of SO-coupled 2 1/2 and 2 3/2 PECs, as discussed in Sec. IV. While the 2 3/2 PEC can be represented by a Morse-like potential, this is not the case for the 2 1/2 state, as was already mentioned in Ref. 11. Therefore, the values of the spec- troscopic parameters ωe and ωexe cannot be expected to reflect correctly the properties of the potential, especially close to the dissociation limit. Furthermore, the parameters D0 and ν00 are extracted using a Birge-Sponer extrapolation, which is not expected to be particularly accurate as these parameters depend more on the low vibrational levels, while the experi- mental data terminate on the low end at v = 7. In order to establish a more comprehensive comparison between theory and experiment, we focus for the remainder of this work on the Ag−Ar complex. Rather than comparing with the experimentally-determined spectroscopic param- eters, we find it more instructive to study directly the vibra- tional energies of the observed transitions, for which accurate values have been reported.11 Table VIII contains the energy between successive vibrational levels in the 2 1/2 state. These energies are determined from the reported transition frequencies between vibrational levels (v and v′) in the 2 1/2 PEC, and the ground electronic and vibrational state. At first glance, the agreement is not satisfactory. However, Brock and Duncan11 state that there might be an error of ±1 in their assignment of the vibrational levels, which is based on the isotopic shift. If we assume that the experimental levels are shifted by one unit (i.e., the level v = 7 is now the level v = 8, TABLE VIII. Comparison of the vibrational spacing in the 2 1/2 state of 107Ag−40Ar with the experimental values of Ref. 11. The spacings are ob- tained from (v − v′′) − (v′ − v′′) = v − v′, where v, v′ are the vibrational levels in the 2 1/2 PEC, and v′′ is a label for vibrational levels in the X 2 + 1/2 PEC. v → v′ Exp.11 Theory 8→7 9→8 10 → 9 11 → 10 12 → 11 13 → 12 14 → 13 15 → 14 57.9 53.0 47.3 41.3 33.5 27.3 18.4 11.0 63.5 58.5 53.4 48.1 42.6 36.6 29.6 19.7 TABLE IX. Vibrational dependence of the spin-orbit splitting δso(v) in the 2 state of 107Ag−40Ar, δso(v) = Ev(2 3/2) − Ev(2 1/2). The second column contains the results of our calculations, while the results of the third column are obtained by shifting the numbering of the vibrational levels in the 2 1/2 state by one unit of v. The last column contains the experimental results of Ref. 11. v Theory Theory, shifted Exp. 0 647.2 1 648.5 2 650.0 3 651.7 4 653.6 5 655.9 6 658.5 7 661.5 8 665.1 9 669.3 10 674.4 11 680.6 12 688.3 13 698.0 14 711.0 15 731.1 16 763.1 17 789.2 18 811.8 19 831.5 550.1 556.1 562.3 568.8 575.6 582.6 590.0 597.9 606.4 615.8 626.2 638.0 651.9 668.9 692.9 728.7 758.4 784.5 807.5 827.4 595.2 605.2 616.5 628.4 641.6 657.9 676.7 708.9 730.5 and so forth), the agreement between theory and experiment is excellent: the average discrepancy is about 1 cm−1. We believe that this is not a coincidence and that the assignment of the experimental vibrational levels should be shifted by unity. We also compared the vibrational dependence of the spin-orbit splitting in the 2 state, i.e., δso(v) = Ev(2 3/2) − Ev(2 1/2). These results are presented in Table IX. We find once again that our results agree well with the observed level separation only if the vibrational levels are shifted by unity in the 2 1/2 state, but not in the 2 3/2 state. The agreement could probably be improved if the coupling with the 2 3/2 state arising from the Ag(2D5/2) asymptote was taken into account. We note that δso(v) increases monotonically with v. This conclusion contradicts that presented in Ref. 11, where it is stated that δso(v) presents a minimum for v = 7, but should increase for lower v. However, this conclusion was based on the assumption that the potential can be fitted to a Morse potential, which is not the case. We obtain a value of δso(0) = 647 cm−1 for the 0 − 0 band, larger than the expected value of 2 /3 = 614 cm−1. Despite the large number of electrons in these systems and the approximation of an R-independent spin-orbit splitting, we obtain an excellent agreement with the experiment for both the 2 1/2 and the 2 3/2 states, provided that the assignment of the vibrational levels of the 2 1/2 state is shifted by one unit. VI. CONCLUSIONS We calculated the Born-Oppenheimer potential energy curves for the interaction of ground and excited state 084301-8 Loreau, Sadeghpour, and Dalgarno J. Chem. Phys. 138, 084301 (2013) silver atoms with noble gas atoms, and compared the spectroscopic parameters with available theoretical and observed data. We discussed the effect of the spin-orbit interaction using a simple model. While the molecular states dissociating into Ag(2D5/2) + NG were neglected, we obtained good agreement with the experimental data for the 2 3/2 state. 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