and Jacek K los 1) Chemical Physics Program, University of Maryland, College Park,

Transcription

1 New XDM-corrected potential energy surfaces for ArNO(X 2 Π): A comparison with CCSD(T) calculations and experiments Michael Warehime, 1, a) Erin R. Johnson, 2, b) 3, c) and Jacek K los 1) Chemical Physics Program, University of Maryland, College Park, MD 2742 USA 2) Chemistry and Chemical Biology, University of California, Merced, Merced, CA USA 3) Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 2742 USA (Dated: 18 December 214) We report new potential energy surfaces for the ground state ArNO(X 2 Π) van der Waals system calculated using the unrestricted Hartree-Fock (UHF) method with the addition of the Becke-Roussel correlation functional and exchange-hole dipole moment dispersion correction (XDM). We compare UHFBR-XDM surfaces and those previously reported by Alexander from coupled cluster CCSD(T) calculations [J. Chem. Phys. 111, 7426 (1999)]. The bound states of ArNO have been investigated with these new UHFBR-XDM surfaces, including relative energy-level spacing, adiabatic bender states and wave functions, and spectroscopic data. These results have been found to be in good agreement with calculations based on the CCSD(T) PESs. These new PESs are used to investigate the inelastic scattering of NO(X) by Ar. Full close-coupling integral cross sections at collision energies of 442 cm 1, 1774 cm 1 and differential cross sections at collision energy of 53 cm 1, were determined for transitions out of the lowest NO(X) rotational level (j = ω = 1/2,f). These cross sections are in good agreement with those calculated with CCSD(T) and accordingly in good agreement with the most recent initial and final state resolved experimental data. The UHFBR-XDM scheme yields high-quality potential surfaces with computational cost comparable to the Hartree-Fock method and our results may serve as a benchmark for application of this scheme to collisions between larger molecules. a) Electronic mail: mickwarehime@gmail.com b) Electronic mail: ejohnson29@ucmerced.edu c) Electronic mail: jklos@umd.edu, Corresponding author 1

2 I. INTRODUCTION The computational efficiency of density-functional theory (DFT) makes it an appealing option to calculate the electronic structure of molecules. Using DFT to model intermolecular complexes, one must account for weak dispersion forces for a complete representation of the electronic environment. The standard test of DFT functionals is to benchmark them against a set of molecules with well-known properties, usually closed-shell molecules in their ground states. These benchmarks do not typically include long-range, dispersion interactions. Whether or not DFT methods can systematically describe van-der-waals complexes involving open-shell molecules, or molecules in electronically excited states, is not well known. Recently, Ershova et al. asked if the interaction between the open-shell NO radical and a rare-gas atom could be described by DFT methods. 1 Their study, which tested a set of functionals with different long-range corrections, produced a high quality description of the ground electronic state, X 2 Π. However, none of the tested functionals could accurately describe the NO-Ar system in the first excited electronic state, A 2 Σ. In the present work we rigorously test new DFT calculations with novel long range, dispersion corrections for the ArNO ground state. These results are in very good agreement with results from PESs calculated from the more computationally-intensive CCSD(T) method for this system. Ershova s question about how well DFT can model the electronically-excited state of this moiety remains an open one. The inelastic dynamics of open-shell diatomics with noble gases, such as the collisions of NO and Ar, have been well studied theoretically and experimentally. 214 The ArNO system has been a focus of interference studies in differential cross sections 15 and sophisticated experimental measurements that are capable of resolving the Λ-doublet fine structure. 16 These fine-structure results were confirmed by state-of-the-art scattering calculations based on Alexander s CCSD(T) surface. 17 The ArNO system has also been used to study the angular momentum orientation of the NO molecule after collisions with Ar atoms. 18 Furthermore, there is a long history to the refinement of the two PESs required to describe the ground state of ArNO(X 2 Π). 25,19 This wealth of experimental and theoretical knowledge of the system make it suitable as a benchmark for our new dispersion-corrected DFT calculations. A post-hartree-fock scheme with corrections for dispersion interactions (UHFBR-XDM) developed by Johnson and Becke 2 has produced a new set of PESs for the ground state ArNO(X 2 Π) system. To add dispersion interactions to the UHFBR energies, we applied the exchange-hole dipole moment (XDM) dispersion model 21 using the exact Hartree-Fock exchange-hole. Hartree-Fock with Becke-Roussel dynamical correlation 22 and the XDM dispersion correction, as applied here, has timings similar to standard Hartree-Fock, which are much faster than than the CCSD(T) calculations. These UHFBR-XDM PESs are qualitatively and quantitatively very similar to earlier CCSD(T) calculations 4,5 and the more recent RCCSD(T) PESs by Cybulski et al. 19 To test the new UHFBR-XDM PESs against known experimental cross sections we perform fully-quantum, close-coupling scattering calculations of the integral sections at collision 2

3 energies of 442 cm 1 and 1774 cm 1, and differential cross sections at a collision energy of 53 cm 1. These UHFBR-XDM potentials show the promise of using DFT with dispersion corrections to describe the physical properties of small intermolecular complexes, even for open-shell systems. II. POTENTIAL ENERGY SURFACES In this work we use the standard Jacobi coordinates, (R, r, θ), to describe the triatomic system where we define θ = as the collinear approach Ar-NO and the NO bond length is fixed at r e = Å. Due to the reflection symmetry in the triatomic plane, the approach of a structureless atom to a diatomic in a 2 Π state gives rise to two PESs, A and A. The A and A ground states PESs were generated from self-consistent Hartree-Fock calculations, performed using the basis-set-free NUMOL program. 23 Dynamical correlation and dispersion effects were included in a post-hf manner 2 using the Becke-Roussel correlation functional 22 and the exact-exchange formulation of the XDM dispersion model. 21 The two parameters in the XDM damping function were assigned values of a 1 =.75 and a 2 = 1.39 Å. This approach gives a mean absolute error of 2 cm 1 for the binding energies of the noble-gas pairs consisting of He, Ne, and Ar atoms. 24 Note that XDM-corrected potentialenergy surfaces for larger molecules can easily be generated with Gaussian, or any electronic structure program capable of writing a wavefunction file, using the postg program. 25,26 As can be seen in Figs. 1-2, the PESs calculated using UHFBR-XDM and CCSD(T) are qualitatively very similar. In Figure 2 we show the half-sum and half-difference potentials used in the dynamic calculations, defined as follows: V sum (R, θ) = 1 l max 2 [V A (R, θ) V A (R, θ)] = V l (R)d l (θ), (1) V dif (R, θ) = 1 l max 2 [V A (R, θ) V A (R, θ)] = V l2 (R)d l 2(θ), (2) where d l m(θ) denotes the reduced Wigner rotation matrix elements. In this work we use l max = 1. The wells in the A and A surfaces, as predicted by UHFBR-XDM calculations are all slightly deeper than those from CCSD(T) calculations. The UHFBR-XDM calculations also predict a lower half-sum potential relative to the CCSD(T) calculations. The right panel of Figure 2 shows the half-difference potential as predicted by UHFBR-XDM calculations is in good agreement with the CCSD(T) potentials, though the UHFBR-XDM half-difference potential is slightly broader and higher than the corresponding CCSD(T) PES. Table I compares the minimum geometry, (R e, r e, θ e ), and minimum energy, D e, for both of the ArNO PESs as predicted by our UHFBR-XDM method, Alexander s CCSD(T) PESs 4,5, Alexander s coupled electron pair method (CEPA) calculations 2 and Cybulski s RCCSD(T) PESs. 19 The UHFBR-XDM values are in good agreement with both coupledcluster methods. The A well depth predicted by UHFBR-XDM is about 7% and 4% deeper than the CCSD(T) and RCCSD(T) predictions, respectively. The well depths of the A 3 l= l=2

4 adiabatic PES are quite similar for all listed methods except CEPA, which is much shallower. The CCSD(T) PESs based on Alexander s calculations 4 are reproduced in the present work to ensure consistency in the comparisons with the new UHFBR-XDM calculations. As in the case of the UHFBR-XDM calculations, the CCSD(T) PESs were calculated with the NO bond length fixed at Å. III. BOUND STATES In this section we briefly describe the parameters needed to converge the bound state wave functions and obtain rotational constants for the triatomic complex. The formal expressions for the wave functions of the ArNO complex have been developed previously. 4,6 The bound states of ArNO(X 2 Π) were calculated using the HIBRIDON quantum chemistry package. 27 The radial part of the wave function was represented by an equidistant distribution of Gaussian basis functions. 28 The fine-structure rotational channels of the 2 Π NO molecule were defined by the experimental values 29 of rotational constant of B = cm 1 and the spin-orbit constant of A SO = cm 1, with the reduced mass of the ArNO complex set to a.m.u. calculated from the masses of the most abundant isotopes. Each rotational level splits into two Λ-doublets where the Λ-doubling parameters 29 are p =.117 cm 1 and q = cm 1. In order converge the bound state for values of the total angular momentum in the range J =.5-6.5, the NO rotational basis included channels up to j max = 18. Our predictions of the bound-state energies of the ArNO complex are in good agreement with the CCSD(T) calculations, as can be seen in Table S-1 in the supplemental material 3 and Figure 3. We also calculated the bound-state energies within the coupled states (CS) approximation, which ignores the Coriolis coupling. For these CS calculations we used the same parameters and separately performed calculations for all possible values of the body-fixed frame projection quantum number, P, of the total angular momentum, J. These CS calculations were used to identify the P quantum number for the close-coupling bound-state energies shown in Table S-1 of the supplemental material. 3 The bound state predictions using the UHFBR-XDM PESs are lower in energy relative to those calculated using the CCSD(T) potentials. This is consistent with the deeper wells in both the V A and V A UHFBR-XDM PESs. The predicted dissociation energy, D, of the lowest bound state of the ArNO complex from the our PES is closer in energy to the experimental value of D than from the CCSD(T) calculations, as can be seen in Table II. The dissociation energy predicted by the UHFBR-XDM PES is very close to both the RCCSD(T) results of Cybulski et al. 19 and the experimental value of Tsuji et al. 8 The bound state parities, as predicted by UHFBR-XDM and CCSD(T) calculations, are shown in Figure 4. The lowest bound state of the ArNO complex should have positive parity and the next lowest bound states have the same parities as those predicted by the CCSD(T) calculations. 6 The UHFBR-XDM PESs incorrectly predict that the lowest bound state will have negative parity. Table S-2 shows that the incorrect assignment of the bound-state parities leads to qualitatively incorrect predictions for transition energies out of 4

5 several of the lowest bound states of the ArNO complex, i.e. negative energies. The slight differences between V dif from the UHFBR-XDM and CCSD(T) calculations may give rise to this discrepancy in the parity of the bound states. The rotational constants for the ArNO complex as predicted from both the CCSD(T) and UHFBR-XDM potentials are in good agreement with the CCSD(T) predictions and experimental values of Wen et al. 14 These rotational constants are shown in Table III. The rotational constants were found with the following fit: E ν,j,p = E ν,p B ν,p J(J 1) a(j 1 ), (3) 2 where B ν,p is the rotational constant listed in Table III. We also present predictions of the bound-state calculations using both potentials within Close-Coupling (CC), Centrifugal Decoupled (CD) and Adiabatic Bender (AB) approaches, shown in Table IV. The predictions of the relative energies of the lowest bound states by UHFBR-XDM and CCSD(T) calculations are in good agreement. At higher energies the relative spacing between states is not consistent, which reflects the subtle differences between the UHFBR-XDM and CCSD(T) PESs, including the deeper UHFBR-XDM wells. The CD and AB energies from UHFBR-XDM are in good agreement with CCSD(T) as is shown in Table IV. With the exception of the bound state parities, our results are in very good agreement with those from Alexander s CCSD(T) PESs. 4,5 IV. ADIABATIC BENDER STATES The UHFBR-XDM adiabatic bender potentials, shown in Figure 5, are very similar to those based on CCSD(T) calculations. However, the UHFBR-XDM P = 1/2, n = 1, 2 adiabatic bender potentials do not have the same strongly avoided crossing as do the equivalent CCSD(T) potentials. This is related to the anisotropy of the V A PES from the linear to T-shaped geometry. As can be seen in lower panels of Figure 1, there is an increasing barrier from θ = to θ = 9 in the CCSD(T) PES, whereas the UHFBR-XDM PES has a very small barrier in this region. Accordingly, the lowest two CCSD(T) adiabatic bender potentials have a much stronger avoided crossing. The P = 1/2, 3/2, n = 1, 2 bender potentials are predicted to be about 5 cm 1 lower in the UHFBR-XDM calculations, while the P = 1/2, n = 3, 4 bender potentials are predicted to be at about the same energy with both methods. The CCSD(T) and UHFBR-XDM predictions of the distribution function, ρ P n (R, θ), which shows the probability of finding the Ar atom for given values of R and θ, is shown in Figures S-2 and S-3 in the supplementary material. 3 In general, the distribution functions predicted by UHFBR-XDM and CCSD(T) calculations are in good agreement. However, the UHFBR-XDM calculations tend to predict more localized distribution functions than those predicted by CCSD(T) calculations, which reflects the incorrect anisotropy of the UHFBR-XDM potential energy surfaces. V. SCATTERING CALCULATIONS 5

6 Initial and final state resolved differential cross section calculations with collision energy of E col = 53 cm 1 for the UHFBR-XDM and CCSD(T) potentials are shown in Figure 6 for spin-orbit conserving collisions and Figure S-4 3 for spin-orbit changing collisions. We show the DCS for both parity-conserving and parity-changing transitions to compare with the recent experiment of Eyles et al. 15 The theoretical cross sections were averaged over a Gaussian distribution of angles with a FWHM = 8. We scale the theoretical DCSs such that the integral of the DCS matches experiment for both parity-conserving and paritychanging collisions. The integral cross sections calculated for E col = 442 cm 1 and E col = 1774 cm 1, are shown in Figures 7 and S-6 3, respectively. For scattering calculations at the highest collision energy of 1774 cm 1, we used j max = 33. The integral cross sections predicted by UHFBR- XDM and CCSD(T) at these two energies are in good agreement. The theoretical cross sections at E col = 442 cm 1 were averaged over a Gaussian distribution of collisional energies with a FWHM = 1% of E col. The calculations for E col = 1774 cm 1 were found using a 4:1 relative population of the j = 1/2 and j = 3/2 rotational levels of the NO molecule to simulate thermal distribution of initial rotational states. To maintain a collisional energy of 1774 cm 1 the calculations for the j = 3/2 rotational level were performed at a total energy of 1779 cm 1, as done previously. 4 VI. DISCUSSION In this work we show the performance of the UHFBR-XDM potentials for the open-shell ArNO ground state system derived from the Becke-Johnson dispersion correction formalism in the bound state and scattering calculations. The new potentials are compared to the previous ones based on the CCSD(T) ArNO potentials extrapolated to the basis set limit by Alexander 5 and newer RCCSD(T) PESs by Cybulski 19. Our new UHFBR-XDM potentials agree remarkably well with the CCSD(T) surfaces. The positions of minima are very similar, with the well being slightly more attractive in case of the UHFBR-XDM potentials, especially for the A adiabatic surface. The fact that the well depth is deeper by about 8 cm 1 for the A adiabatic surface shifts the zero-pointcorrected dissociation energy closer to the experimental value of Tsuji et al. 8 (Table II). The agreement with CCSD(T) is promising if we keep in mind that the UHFBR-XDM model is by far the less computationally demanding method of the two. The computational cost of computing the XDM dispersion is negligible relative to the HF calculation. Thus we have obtained accuracy close to CCSD(T) level with essentially the cost of HF. The UHFBR-XDM potentials (Figures 1 and S-1 3 ) exhibit somewhat different anisotropy in the vicinity of collinear arrangements of the atoms. Specifically near the collinear approaches, the UHFBR-XDM PESs are more repulsive and the A adiabat is flatter in comparison to the CCSD(T) PES. The diabatic surfaces, V sum and V dif, for both UHFBR-XDM and CCSD(T) are shown in Fig. 2. The UHFBR-XDM model predicts saddle points for collinear geometries, while the CCSD(T) exhibits small local minima. The general anisotropy is similar with the global minimum being in the T-shape geometry (near 9 degrees) showing the near-homonuclear 6

7 character of the V sum PES. In case of UHFBR-XDM, the difference potential calculations show a slightly wider repulsive region than the CCSD(T) diabat. The dissociation energy predicted from UHFBR-XDM is in better agreement with both Cybulski s RCCSD(T) PESs and experiment as compared to the CCSD(T) predictions. The UHFBR-XDM stretching frequency, ω stretch, shown in Table II, is also slightly closer to experiment. While the theoretical frequencies are off by about 5%, one has to keep in mind that the experimental value can be ambiguous. 31 As shown in Figure 4, the UHFBR-XDM PESs predict the incorrect parity of the bound-state wave functions. The incorrect assignment of the bound-state parities leads to qualitatively incorrect predictions of the transition energies out of the lowest bound states (Table S-2). The source of this discrepancy between the UHFBR-XDM and CCSD(T) is not well understood and may serve as a target for improving future DFT surfaces for open-shell systems. The Close-Coupling bound states were used to estimate the rotational constants of the ArNO complex using both potential models. The UHFBR-XDM PES results, shown in Table III, are in very good agreement with both CCSD(T) and experimental results. In the Adiabatic Bender approximation (Fig. 5), the avoided crossing region in the UHFBR-XDM curves is slightly weaker than for the CCSD(T) curves. We also show ro-vibrational wave functions obtained from the Adiabatic Bender approximation in Figures S-2 and S-3 in the supplementary material 3. The wave functions corresponding to the UHFBR-XDM potential are more localized due to the deeper well, but generally similar to those using CCSD(T). From the scattering calculations we have obtained observables such as integral cross sections (ICSs) and differential cross sections (DCSs) using both potential models. We find very good agreement with the initial and final state resolved experiments of Eyles et al 15 for both UHFBF-XDM and CCSD(T) DCSs for spin-orbit conserving transitions (Figure 6) and spin-orbit changing transitions (Figure S-4) at E col = 53 cm 1. The theoretical cross sections tend to over-estimate the amount of backscattering and the DFT results universally predict more backscattering than the CCSD(T) results. However, our CCSD(T) results are consistent with Eyles 15 and the new DFT potential performs remarkably well when compared to experiment. In Figure 7 the ICSs from the UHFBR-XDM and CCSD(T) calculations are compared with the experimental results of Joswig et al. 1 The total experimental ICS s are scaled to match the corresponding total theoretical cross sections. The total ICS is composed of transitions to both F1 and F2 spin-orbit manifolds, ICS tot = j 2 i=1 σ(j, F i ) As one can see, both UHFBR-XDM and CCSD(T) results reproduce experiment quite well in case of the SO-conserving transitions. The UHFBR-XDM potential gives better agreement with experiment for the lowest j quantum numbers. The propensities in SOchanging transitions are well reproduced by both theoretical potentials, but the magnitude 7

8 of the experimental cross sections is approximately twice as high as than those from calculations for j up to 7.5. For a pure homo-nuclear PES only j = even transitions are allowed. The near homo-nuclear character of the ArNO PESs angular anisotropy allows for all transitions but maintains a propensity for j = even transitions. Both UHFBR-XDM and CCSD(T) ICS results show this j = even propensity. To probe the repulsive part of the UHFBR-XDM PESs we performed scattering calculations of the ICSs at collision energy of 1774 cm 1 with respect to the j = 1/2 and j = 3/2 initial rotational states. The inelastic cross sections for spin-orbit manifold conserving and spin-orbit manifold changing transitions are shown in Figure S-6. The j = 3/2 state was added with a weight of.2 to.8 of the j = 1/2 cross sections to simulate the thermal distribution of the rotational states of NO. The UHFBR-XDM cross sections are slightly smaller especially for the j = 1.5 final rotational state, but for higher j they are similar to CCSD(T). The repulsive wall is reproduced quite well and the agreement with experiment is almost as good as CCSD(T). One could use the low cost UHFBR-XDM method and extend the potential to include vibrational modes of the NO molecule to investigate the NO vibrational de-excitation upon collisions with Ar in future studies. VII. CONCLUSIONS We report a comparison of the bound states and scattering results obtained on our new potential energy surfaces for the ground state ArNO(X 2 Π) system and those previously reported with Alexander s CCSD(T) PESs. The new UHFBR-XDM PESs are qualitatively and quantitatively similar to CCSD(T) calculations. The UHFBR-XDM V sum PES is characterized by moderately deeper van der Waals well and a lower zero-order corrected dissociation energy, which is in better agreement with Cybulski s recent PES and experiment relative to previous Alexander s CCSD(T) results. The anisotropy of the UHFBR-XDM surfaces does not agree with that of the CCSD(T) surfaces, especially in the vicinity of collinear geometries. Similarly, the anisotropies of the half-difference potential, V dif, in the T-shape region are markedly different for these two surfaces. These facts may explain why the UHFBR-XDM potential predicts opposite parity splittings to those obtained with CCSD(T) PESs. One possible source of discrepancy between the XDM results and the reference CCSD(T) potential could be neglect of the three-body contribution to the dispersion energy, which will stabilize the collinear geometries. However, tests using the many-body generalization of the XDM model 32 indicate that this effect is not sufficiently large to account for the error and it is more likely due to the underlying dynamical correlation functional. In the scattering calculations, the UHFBR-XDM PES performs very well compared to CCSD(T) and experiment, in spite of a pronounced preference for back scattering. The integral cross sections presented in this work agree fairly well with experiment, especially for the lowest rotational quantum numbers. The UHFBR-XDM scheme, as applied for the Ar-NO system, is in good agreement with both coupled-cluster methods and with experiments with the added benefit of great savings in computational time. The agreement between DFT, existing theory and experiment presented in this work is promising for the use of DFT with dispersion functionals to accurately model small, open-shell systems and 8

9 serves as a benchmark for application of this method to larger molecular colliders. ACKNOWLEDGMENTS We acknowledge support by the Chemical, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U. S. Department of Energy, under Grant No. DESC2323 to Prof. M. H. Alexander. REFERENCES 1 O. V. Ershova and N. A. Besley, Journal of Chemical Physics, 136, (212). 2 M. H. Alexander, Differential and integral cross sections for the inelastic scattering (X 2 Π) by Ar based on a new a & initio potential energy surface of NO, Journal of Chemical Physics, 99, 7725 (1993). 3 G. C. Nielson, P. G. A., and R. T. Pack, Journal of Chemical Physics, 66, 1396 (1977). 4 M. H. Alexander, A new, fully ab initio investigation of the NO(X 2 Π)Ar System. I. Potential energy surfaces and inelastic scattering, Journal of Chemical Physics, 111, 7426 (1999). 5 M. H. Alexander, A new, fully ab initio investigation of the NO(X2Pi)Ar System. II. Bound states of the Ar-NO complex, Journal of Chemical Physics, 111, 7435 (1999). 6 P. D. A. Mills, C. M. Western, and B. J. Howard, Rotational Spectra of Rare Gas-Nitric Oxide van der Waals Molecules. 1. Theory of the Rotational Energy Levels, J. Phys. Chem., 9, 3331 (1986). 7 P. D. A. Mills, C. M. Western, and B. J. Howard, Rotational Spectra of Rare Gas - Nitric Oxide van der Waals Molecules. 2. The structure and spectrum of Argon-Nitric Oxide, J. Phys. Chem., 9, 4961 (1986). 8 K. Tsuji, K. Shibuya, and K. Obi, Bound-bound transition of NO-Ar van der Waals complexs, Journal of Chemical Physics, 1, 5441 (1994). 9 H. Kohguchi, T. Suzuki, and M. H. Alexander, Molecule by Ar Fully State-Resolved Differential Cross Sections for the Inelastic Scattering of the Open-Shell NO Molecule by Ar, Science, 294, 832 (21). 1 H. Joswig, A. P., and R. Schinke, J. Chem. Phys., 85, 194 (1986). 11 M. S. Elioff and D. W. Chandler, State-to-state differential cross sections for spin multiplet-changing collisions of NO(X 2 Π) with Argon, Journal of Chemical Physics, 117, 6455 (22). 12 T. Schmelz, P. Rosmus, and M. H. Alexander, Theoretical Study of Bound States of Ar-NO, Phys. Chem, 98, 173 (1994). 13 Y. Kim, J. Fleniken, H. Meyer, M. H. Alexander, and P. J. Dagdigian, A joint theoreticalexperimental investigation of the lower bound states of no(x 2 π)-ar complex, Journal of Chemical Physics, 113, 73 (2). 14 B. Wen, Y. Kim, H. Meyer, J. K los, and M. H. Alexander, Ir-rempi double resonance spectroscopy : The near-ir spectrum of no-ar revisited, Phys. Chem. A, 112, 9483 (28). 15 C. J. Eyles, M. Brouard, C.-H. Yang, J. K los, F. J. Aoiz, A. Gijsbertsen, A. E.Wiskerke, and S. Stolte, Nature Chemistry, 3, 597 (211). 16 C. J. Eyles, M. Brouard, H. Chadwick, B. Hornung, B. Nichols, C.-H. Yang, J. K los, F. J. 9

10 Aoiz, A. Gijsbertsen, A. E. Wiskerke, and S. Stolte, Phys. Chem. Chem. Phys., 14, 543 (212). 17 C. J. Eyles, M. Brouard, H. Chadwick, F. J. Aoiz, J. K los, A. Gijsbertsen, X. Zhang, and S. Stolte, Phys. Chem. Chem. Phys., 14, 542 (212). 18 P. G. Jambrina, J. K los, F. J. Aoiz, and M. P. de Miranda, Phys. Chem. Chem. Phys., 14, 9826 (212). 19 H. Cybulski and B. Fernández, Journal of Physical Chemistry A, 66, 7319 (212). 2 E. R. Johnson and A. D. Becke, A post-hartree-fock model of intermolecular interactions: Inclusion of higher-order corrections, Journal of Chemical Physics, 124, (26). 21 A. D. Becke and E. R. Johnson, Exchange-hole dipole moment and the dispersion interaction revisited, J. Chem. Phys., 127, (27). 22 A. D. Becke, Thermochemical tests of a kinetic-energy dependent exchange-correlation approximation, International Journal of Quantum Chemistry Symposium, 52, 625 (1994). 23 A. D. Becke and R. M. Dickson, Numerical solution of Schrödinger s equation in polyatomic molecules, Journal of Chemical Physics, 92, 361 (199). 24 K. T. Tang and J. P. Toennies, The van der Waals potentials between all the rare gas atoms from He to Rn, Journal of Chemical Physics, 118, 4976 (23). 25 A. Otero-de-la Roza and E. R. Johnson, Non-covalent interactions and thermochemistry using xdm-corrected hybrid and range-separated hybrid density functionals, J. Chem. Phys., 138, 2419 (213). 26 See 27 hibridon is a package of programs for the time-independent quantum treatment of inelastic collisions and photodissociation written by M. H. Alexander, D. E. Manolopoulos, H.-J. Werner, and B. Follmeg, with contributions by P. F. Vohralik, D. Lemoine, G. Corey, R. Gordon, B. Johnson, T. Orlikowski, A. Berning, A. Degli-Esposti, C. Rist, P. J. Dagdigian, B. Pouilly, G. van der Sanden, M. Yang, F. de Weerd, S. Gregurick, J. K los and F. Lique. More information and/or a copy of the code can be obtained from the website 28 I. P. Hamilton and J. C. Light, Journal of Chemical Physics, 84, 36 (1986). 29 J. W. C. Johns, J. Reid, and D. W. Lepard, J. Mol. Spectrosc., 65, 155 (1977). 3 See supplemental material at [url will be inserted by aip] for additional tables and figures.. 31 J. Miller and W.-C. Cheng, Multiphoton ionization, J. Phys. Chem., 89, 1647 (1985). 32 A. Otero-de-la Roza and E. R. Johnson, Many-body dispersion interactions from the exchange-hole dipole moment model, J. Chem. Phys., 138, 5413 (213). 33 M. H. Alexander, S. Gregurick, and P. J. Dagdigian, Potential energy surfaces for the interaction of bh(x 1,a 1) with ar and a theoretical investigation of the stretchbend levels of the arbh(a) van der waals molecule, J. Chem. Phys., 11, 2887 (1994). 1

11 TABLE I. Minimum geometry (R e, θ e ) in bohr and degrees and minimum energy, D e, in cm 1 for the ArNO(X 2 Π) A and A PESs calculated with different methods. a This work b Ref. 5 c Ref. 2 d Ref. 19 Method R e (A ) θ e (A ) D e (A ) R e (A ) θ e (A ) D e (A ) UHFBR XDM a CCSD(T) b CEPA c RCCSD(T) d TABLE II. Spectroscopic data, in cm 1, for the ArNO complex. D ω a stretch ω b bend UHFBR XDM CCSD(T) RCCSD(T) 86.7 c Experiment 87.8 d 4 e a Difference in the two lowest J = 1/2, P = 1/2, n = 1 states, see Table IV b Difference between the lowest J = 1/2, P = 1/2, n = 1 and lowest J = 3/2, P = 3/2, n = 1 states, see Table IV c Cybulski et al. Ref. 19 d Ref. 8 e Ref

12 a Band labels consistent with Reference 14 b Reference 14 c this work TABLE III. Rotational constants, B ν,p, in cm 1. Band a Exp. b CCSD(T) c UHFBR-XDM c A B B C D E D F G H TABLE IV. Relative energies in cm 1 of the lowest bound states of the ArNO complex. CC a CC CD b AB c J d P e n f ν g s π h = 1 π = 1 π = 1 π = 1 CCSD(T) XDM CCSD(T) XDM CCSD(T) XDM CCSD(T) XDM a Close-coupled calculations. b Centrifugal-decoupled calculations. c Lowest vibrational energy from the adiabatic bender potentials. d Nominal value of the total angular momentum, J, along ArNO bond axis. e Nominal value of the projection of total angular momentum, J. f Adiabatic bender state index, see Fig. 5 g Nominal stretching quantum number, i.e. the number of nodes in the adiabatic bender wave function. h π = ±1 refers to positive or negative parity. 12

13 CCSD(T) 7 UHFXD R / bohr energy / cm θ / degrees θ / degrees FIG. 1. (upper) Contour plot of the A PES. (lower) The minimum energy profiles of the A PES as a function of θ. The contours are labeled in cm 1 relative to the minimum of the potential (for CCSD(T), D e =116.6 cm 1 and for UHFBR-XDM, D e =124.8 cm 1 ). 13

14 R / bohr R / bohr UHFXDM CCSD(T) θ / degrees UHFXDM CCSD(T) θ / degrees FIG. 2. (left) V sum (R, θ) (right) V dif (R, θ). The thick black curve in the right panels indicates the beginning of the repulsive wall, where V sum (R, θ) =. The blue circles indicate the minimum geometries of V A and V A. The minimum of V sum for UHFBR-XDM and CCSD(T) are -116 cm 1 and cm 1, respectively. 14

15 energy / cm -1 energy / cm P = 3/2 P = 1/2 2 P = 1/ P = 1/ P = 1/ P = 1/2 P = 3/2 P = 3/2 P = 5/2 P = 5/2 CCSD(T) UHFXDM 5 P = 3/2 P = 1/ J tot FIG. 3. Relative positions of the lowest bend-stretch states of the ArNO complex. Only the positive parity states are shown. The states are labeled with the nominal value of P, which corresponds to the projection of the total angular momentum, J, onto the ArNO bond axis, R. The dependence of the energy of the states is shown as a function of total angular momentum. The dashed levels correspond to the first excited state with P = 3/2 to help distinguish these states from the second excited states with P = 1/2. 15

16 energy / cm -1 1 CCSD(T) UHFXDM o.1 e e o o e J = 1/2 3/2 5/2 J = 1/2 3/2 5/2 FIG. 4. Relative positions of the lowest bend-stretch states of the ArNO complex with P = 1/2. The zero of energies are cm 1 and -87. cm 1 for the CCSD(T) and UHFBR-XDM predictions respectively. The and labels indicate the total parity of each state. The e/o labeling is a shorthand for determining allowed transitions and is consistent with Ref energy / cm -1 energy / cm n = 4 n = 3 n = 2 n = 1 n = 4 n = 3 n = 2 n = 1 n = 2 n = 1 95 n = 1 n = R / Bohr FIG. 5. Adiabatic bender potential energy curves for the ArNO complex. The solid and dashed curves correspond to P = 1/2 and P = 3/2, respectively. The states are labeled with by the value of n, which correspond to the n th -eigenvector of the W (R) matrix 33 for a given value of P. The horizontal lines correspond to the lowest vibrational level for each adiabatic bender potential, see Table IV. Note: the P = 1/2, n = 1 and n = 2, as well as, P = 3/2, n = 1 and n = 2 vibrational energies are nearly indistinguishable graphically. 16

17 dσ/dω / Å 2 dσ/dω / Å 2 dσ/dω / Å θ θ j' = 6.5 j' = DFT CCSD(T) Exp θ j' = 1.5 dσ/dω / Å 2 dσ/dω / Å 2 dσ/dω / Å 2.6 j' = θ θ θ j' = 9.5 j' = FIG. 6. DCS for spin-orbit conserving transitions with collision energy E col = 53 cm 1 from DFT (red) CCSD(T) (blue) and experimental results of Eyleset al 1517 (black) for the ArNO( 2 Π 1/2, v =, j = 1/2, p = 1) ArNO( 2 Π 1/2, v =, j = j, p = p ). The overall parity of the state p = ϵ( 1) j 1/2 and ϵ = 1 for e and ϵ = 1 for f. Final states with positive parity (parity conserving) are shown with solid lines band those with overall negative parity (parity changing) are shown with dashed lines. 17

18 2 ICS / Å Joswig UHFXDM CCSD(T) F F 2 ICS / Å j FIG. 7. Integral cross sections with E col = 442 cm 1. (upper) SO-conserving, F 1, ArNO( 2 Π 1/2, v =, j = 1/2, p = 1) ArNO( 2 Π 1/2, v =, j = j, p = p ). (lower) SOchanging, F 2, ArNO( 2 Π 1/2, v =, j = 1/2, p = 1) ArNO( 2 Π 3/2, v =, j = j, p = p ). The black triangles correspond to the Joswig experiments 1. These cross sections are normalized such that the total cross section, F 1 F 2, for both theoretical calculations and the experimental values are equal. 18

19 CCSD(T) 7 UHFXD R / bohr energy / cm θ / degrees θ / degrees

20 R / bohr R / bohr UHFXDM CCSD(T) θ / degrees UHFXDM CCSD(T) θ / degrees

21 3 CCSD(T) 25 energy / cm -1 energy / cm P = 1/2 P = 1/2 P = 1/2 P = 1/2 P = 1/2 P = 3/2 P = 5/2 P = 3/2 P = 3/2 P = 5/2 UHFXDM 5 P = 3/2 P = 1/ J tot

22 energy / cm -1 1 CCSD(T) UHFXDM o e e o o e J = 1/2 3/2 5/2 J = 1/2 3/2 5/2

23 5 6 energy / cm -1 energy / cm n = 3 n = 1 n = 3 n = 4 n = 2 n = 4 n = 2 n = 1 n = 2 n = 1 95 n = 1 n = R / Bohr

24 dσ/dω / Å 2 dσ/dω / Å 2 dσ/dω / Å θ θ j' = 6.5 j' = DFT CCSD(T) Exp θ j' = 1.5 dσ/dω / Å 2 dσ/dω / Å 2 dσ/dω / Å 2.6 j' = θ θ θ j' = 9.5 j' =

25 ICS / Å Joswig UHFXDM CCSD(T) F F 2 ICS / Å j