On the isotropic Raman spectrum of Ar2 and how to benchmark ab initio calculations of small atomic clusters: Paradox lost Michael Chrysos, Sophie Dixneuf, and Florent Rachet Citation: The Journal of Chemical Physics 143, 024304 (2015); doi: 10.1063/1.4923370 View online: http://dx.doi.org/10.1063/1.4923370 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Finite temperature path integral Monte Carlo simulations of structural and dynamical properties of Ar N −CO2 clusters J. Chem. Phys. 137, 074308 (2012); 10.1063/1.4746941 Fluxional and aromatic behavior in small magic silicon clusters: A full ab initio study of Si n , Si n 1 − , Si n 2 − , and Si n 1 + , n = 6 , 10 clusters J. Chem. Phys. 127, 014314 (2007); 10.1063/1.2746030 Frequency-dependent hyperpolarizabilities of the Ne, Ar, and Kr atoms using the approximate coupled cluster triples model CC3 J. Chem. Phys. 123, 094303 (2005); 10.1063/1.2008211 Polarizabilities and hyperpolarizabilities for the atoms Al, Si, P, S, Cl, and Ar: Coupled cluster calculations J. Chem. Phys. 122, 044301 (2005); 10.1063/1.1834512 Structures and stabilities of small silicon clusters: Ab initio molecular-orbital calculations of Si 7 – Si 11 J. Chem. Phys. 118, 3558 (2003); 10.1063/1.1535906

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THE JOURNAL OF CHEMICAL PHYSICS 143, 024304 (2015)

On the isotropic Raman spectrum of Ar2 and how to benchmark ab initio calculations of small atomic clusters: Paradox lost Michael Chrysos,1,a) Sophie Dixneuf,2,b) and Florent Rachet1

1

LUNAM Université, Université d’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 Bd Lavoisier, 49045 Angers, France 2 Centre du Commissariat à l’Énergie Atomique de Grenoble, Laboratoire CEA-bioMérieux, Bât 40.20, 17 rue des Martyrs, 38054 Grenoble, France

(Received 15 May 2015; accepted 22 June 2015; published online 8 July 2015) This is the long-overdue answer to the discrepancies observed between theory and experiment in Ar2 regarding both the isotropic Raman spectrum and the second refractivity virial coefficient, BR [Gaye et al., Phys. Rev. A 55, 3484 (1997)]. At the origin of this progress is the advent (posterior to 1997) of advanced computational methods for weakly interconnected neutral species at close separations. Here, we report agreement between the previously taken Raman measurements and quantum lineshapes now computed with the employ of large-scale CCSD or smartly constructed MP2 induced-polarizability data. By using these measurements as a benchmark tool, we assess the degree of performance of various other ab initio computed data for the mean polarizability α, and we show that an excellent agreement with the most recently measured value of BR is reached. We propose an even more refined model for α, which is solution of the inverse-scattering problem and whose lineshape matches exactly the measured spectrum over the entire frequency-shift range probed. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4923370]

I. INTRODUCTION

Collision-induced Raman scattering is a commonly used term to define the inelastic scattering of a photon by pairs or ensembles of weakly interacting atoms or molecules in a gas or a liquid.1 In the early 1980s, a series of groundbreaking experimental and numerical developments opened new frontiers in this field of scientific investigations.2–4 Although less investigated than in the past, this field continues nowadays to attract attention, with a special interest in isotropic scattering of small atomic and molecular species and mixtures.5–11 Currently, new and prolific areas of experimental investigation are being developed, intended, among other things, to measure the polarizability anisotropy of rare-gas diatomic molecules;12,13 in this promising research topic (which belongs to the wider topic of multiphoton processes and strong laser fields), short and intense laser pulses are utilized to cause alignment, orientation, and deflection, as well as ionization and Coulomb explosion of the dimers. Although recent developments with circularly polarized laser pulses suggest that some of these techniques are also applicable to the mean polarizability of the dimer,14 none of them seems suitable for probing unbound rare-gas pairs and, by extension, for determining the way in which the induced polarizability of the pair depends on interatomic separation R. Clearly, collision-induced Raman scattering remains one of the only truly effective methods for the latter systems and properties. However, owing to the difficulty of detecting and processing weak Raman signals, the measurement of isotropic Raman spectra by pairs of weakly interacting a)Electronic address: [email protected] b)Also at BIOASTER, 321 Avenue Jean Jaurès, 69007 Lyon, France.

0021-9606/2015/143(2)/024304/7/$30.00

neutral atoms is a high-risk task and all the more if far-wing data are needed.2,6,7 Quite analogous is the situation with the components of the induced polarizability tensor responsible for these spectra for two closely approaching atoms, because (in contradistinction to the anisotropy) incremental polarizability traces, α(R), are higher-order corrections to the total polarizability (in terms of their R−1-series expansion).9,15–23 Thirty years after the first faltering steps of interactioninduced spectroscopy, a wealth of ab initio post Hartree-Fock methods and data is now accessible for the induced polarizability average and anisotropy of small systems. Good knowledge of these quantities for close interatomic separations can be of great utility to physicists because short R are instrumental in representing the collision-induced lineshapes in the far wings of the spectra. Although these ab initio data are undoubtedly a considerable improvement over earlier calculations,3,4 their translation to spectra for certain of these systems has only poorly been addressed so far. Among rare gas atoms, Ar2 pair has long interested the academic literature and in particular our group;5,24–28 however, some of the characteristics of its spectra and polarizabilities are still not entirely understood. Specifically, the reliable computation of Ar2 α as a function of R is a task needing carefully chosen orbital basis sets and advanced quantumchemistry computational tools to be applied. For this reason, its precise, full ab initio description had for a long time been a numerical challenge.3–5 In this context, the disagreement of Dacre’s original predictions for α (Refs. 3 and 4) with the measured isotropic Raman spectrum of Ar2 and (at the same time) with results from independently made experiments for the second refractivity or dielectric virial coefficient (Ref. 29) remains a sobering thought. The latest word in this issue goes

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back to 1997, to a work tackling the isotropic component of Ar2 in room-temperature gas.5 A strikingly regular, absolutely calibrated isotropic spectrum with an intensity Iiso(ν) shown to fall off purely exponentially for ν > 60 cm−1 had been reported therein within a frequency range ν as wide as never before for that system; this feature being in accord with later experiments in He2 (Refs. 6 and 7), NeAr (Ref. 10), and Ne2 (Ref. 30) offers the validation of a seemingly universal property of binary isotropic rare-gas atomic spectra to decrease like e−λν (λ > 0) with increasing ν asymptotically.31 Equally interesting is another finding reported therein, namely, that the measured Ar2 spectrum (Ref. 5) and the quantum lineshape due to Dacre’s self-consistent field (SCF) data for α(R) (Refs. 3 and 4) are identical up to a factor a = 0.7 (Ref. 5); in view of this finding, it had at that time appeared tempting to us to postulate a correction √ to Dacre’s SCF α data by introducing the scaling factor a (Ref. 5). However, proceeding that way had the side effect of further damaging the already poor agreement between theoretical and experimental29 values for the second refractivity virial coefficient BR , destroying any prospect for consistency between Iiso(ν) and BR . The “rescaled model” (i.e., the one which arises √ by multiplying Dacre’s SCF data3 by the empirical factor a = 0.837 (Ref. 5)) will be referred to hereafter as SCF-expt. Although it generates a quantum lineshape in excellent agreement with the experimental spectrum of Ref. 5, its meaning is questionable mainly because of its serious shortcoming of providing a value 30 times smaller than the most recent data measurements of the second virial coefficient.29 Similarly, unsatisfactory is the response of the two original Dacre’s models (Refs. 3 and 4), both of which provide values ∼20-25 times smaller than expected. What is the exact origin of these contradictions? Is there any hope for calculated and measured virial coefficients and spectra to reconcile? It is the purpose of this article to answer these questions. The advent, in the 1990s, of advanced computational tools15,17–19 intended for the purpose of accounting for exchange and orbital-overlap interactions and electron correlation effects in systems made up of weakly interconnected neutral species at close separations is what permitted this progress. How to benchmark the different levels of ab initio computations for the induced polarizability of Ar2 is one more purpose of this article.

II. REFRACTIVITY AND RAMAN SPECTRA: THEORY VS OBSERVATION A. The second refractivity virial coefficient

The second dielectric or refractivity virial coefficient was calculated from data sets for α that are posterior to 1990, at the conditions of the experiment of Ref. 5 (T = 294.5 K, λ = 514.5 nm). Expressions BR (ω) ≈ Bε + ω2 BR(2) + · · ·,  8π 2 N A2 ∞ Bε = α(R)g(R)R2dR, 3 0 and

(1) (2)

TABLE I. Theoretical values for B ε of Ar2 at the temperature of the Raman experiment (294.5 K) calculated with the employ of ab initio data for α(R) reported by various authors for various levels of optimization. Values generated by Dacre’s SCF (Ref. 3) and by its scaled variants SCF-COR (Ref. 4) and SCF-expt (Ref. 5) are also shown to make the comparisons easier; let the reader be reminded that the latter two models are defined as: SCF-COR = 1.216 × SCF and SCF-expt = 0.837 × SCF. CCSD(ω)

CCSD

1.418a

1.387a 1.355b

MP2

SCF

0.973c 0.079d

−0.523c

HF

−0.907d 0.064e 0.076f 0.053g

a Reference

18 (CCSD/d-aug-cc-pVQZ-33211). 17 (CCSD/d-aug-cc-pVTZ-33211). c Reference 19. d Reference 15. e Reference 3. f Reference 4. g Reference 5. b Reference

BR(2)

8π 2 N A2 = 3

 0

∞

∆Sα (−4)g(R)R2dR,

(3)

with ω = 2πc λ (ω is in atomic units; 1 a.u. of ω is 4.134 14 × 1016 s−1), were used for the purpose; Bε , BR(2), g(R), N A, and ∆Sα (−4) designate the second dielectric virial coefficient, the dispersion contribution, the classical radial distribution function, the Avogadro constant, and the fourth-order Cauchy moment, respectively. The Bε values are shown in Table I. Three of them make a reference to Dacre’s SCF model (Ref. 3) and to its semi-empirically corrected (SCF-COR) TABLE II. Values for the B R (λ, T ) coefficient of Ar2 from refractivity (R) and dielectric-constant (ε) experiments, for various wavelengths at near room temperature in the reverse chronological order. The italicized entries correspond to conditions of T and λ that are close to those of our experiment (294.5 K, 514.5 nm). Experimental errors are shown in parentheses. T , λ, (2) B R , B ε , and B R are expressed in units of K, nm, cm6 mol−2, cm6 mol−2, −2 6 and cm mol /(a.u.)2, respectively. Method Ra Ra Rb εc εc εe Rf Rf Rf Rg

T

λ

296.83 296.83 323 242.95 303.15 303.15 298.2 298.2 298.2 298.2

543.5 325.5 633.0 514.5 514.5 514.5 632.8 514.5 457.9 632.8

Bε

1.84(7) 1.22(9) 1.23(5)

(2)

B R (T )

5.49 5.17 5.17

B R (λ, T ) 1.73(34) 1.81(34) 1.76(5) 1.88(11)d 1.26(13)d 1.27(9)d 1.57(58) 1.55(74) 1.53(32) 1.49(15)

a Reference

29. 33. c Reference 34. d The values of B were obtained from B measurements once corrected for dispersion, R ε (2) B R (T ), by means of the Cauchy moment ∆S α (−4) values for the mean polarizability reported in Ref. 18. e Reference 35. f Reference 36. g Reference 37. b Reference

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FIG. 1. Absolute-unit isotropic Raman scattering intensities (cm6) by freefree Ar pairs as a function of ν (cm−1) at room temperature. The measurements of Ref. 5 are depicted by symbols (•, expt). Quantum lineshapes are illustrated by lines. In order to best bring out the spectral features of unbound Ar pairs close to resonance, the contributions from bound and predissociating dimers are not shown. The SCF-expt lineshape (suggested in Ref. 5 as the best fit to the experiment) is shown (solid line curve) along with Dacre’s3,4 SCF and SCF-COR lineshapes and HF/SCF (Refs. 15 and 19) ones (dashed-line curves). The wing of the spectrum is shown in inset. Although the response of SCF-expt looks consistent with expt over the entire ν range, this agreement does not imply adequacy of the model (see Table I).

FIG. 3. The mean polarizability of Ar–Ar, α (in a.u.), as a function of R (Bohr). The Dacre’s scaled SCF model (SCF-expt) introduced in Ref. 5 as the best fit to our experiment (expt, see Fig. 1) is shown (solid-line curve) along with other ab initio computed SCF and HF data (dashed-line curves). The range [7:9] bohrs is shown in inset to highlight the bump and its possible relation to electron correlation effects. Note that the pronounced bump of SCF-COR is only because of the scaling factor by which Dacre’s SCF data have been multiplied to provide SCF-COR (see footnote Ref. 32) and not because of any explicit consideration of additional correlation effects in the latter model.

(Ref. 4) and rescaled (SCF-expt) (Ref. 5) variants, respectively (shown for the sole purpose of making the comparisons easier).32 What is most interesting with Table I is that the data follow a clearly monotonous trend, going from algebraically small to algebraically large values as the optimization level and the quality of the orbital basis set used in the ab initio computations are increased: Hartree-Fock (HF),15 SCF,19 secondorder Møller-Plesset perturbation theory (MP2),15,19 coupled cluster singles and doubles response theory (CCSD) with triple (CCSD/d-aug-cc-pVTZ-33211)17 or quadruple (CCSD/d-

aug-cc-pVQZ-33211)18 zeta basis sets, dynamic CCSD(ω) coupled cluster approximation.18 The largest value is obtained with CCSD(ω)/d-aug-cc-pVQZ-33211, which is the highest available optimization level.18 This value matches remarkably well the measured value29 BR = 1.73 ± 0.34 cm6 mol−2 from a refractivity experiment in conditions of wavelength (543.5 nm) and temperature (296.83 K) close to those of our Raman experiment.5 The latter as well as other values of Bε and BR , measured by various authors29,33–37 in conditions of temperature and wavelength close to those of our experiment, are gathered in Table II.

FIG. 2. Same as Fig. 1 but for post Hartree-Fock computations for α (dashedline curves). The quantum lineshape generated by the inverse-scattering solution of Eq. (4) (see text, below) is shown with a solid line. The experimental uncertainty in the region [150:300] cm−1 is of the order of the size of data points (•).

FIG. 4. Same as Fig. 3 but for CCSD(ω), CCSD, and MP2 calculations (dashed-line curves). The inverse-scattering solution of Eq. (4) is shown with a solid line. Distances < 5 bohrs are not detectable experimentally at 294.5 K. Bumps in this figure are much more pronounced than they are in Fig. 3, a property in direct relation with the effect of electron correlation.

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B. The Raman spectrum

In Figures 1 and 2, the isotropic Ar–Ar spectrum is illustrated as a function of ν in the interval [0:400] cm−1. Only free-free contributions are shown. The spectral wing [300:400] cm−1 is emphasized in inset. Measurements taken in our group5 are depicted by symbols. Quantum lineshapes are represented by lines. Data for α(R) computed ab initio posterior to 1990 were used for the purpose. To ensure reliably computed lineshapes, smart numerical procedures were utilized for quantum spectral intensity calculations25,26 with a spatial grid for polarizability matrix-elements extended over separations as wide as 300 bohrs. Regarding measurements, we remind the reader that (i) high-sensitivity equipment entirely designed in our laboratory has been used to conduct the experiment (the way to proceed is outlined in Ref. 5; the interested reader is addressed to Refs. 7 and 38 for details), (ii) independently measured calibrated components I⊥(ν) or I∥(ν) (depending on whether the incident-beam polarization is perpendicular, ⊥, or parallel, ∥, to the scattering plane) have been processed, and (iii) the linear combination Iiso(ν) = 1.010I⊥(ν) − 1.176I∥(ν) (by which ideal combination39 Iiso(ν) = I⊥(ν) − 76 I∥(ν) is corrected for the effects of the finite aperture of the scattered beam) has been formed to generate the isotropic spectrum. In this respect, let us stress that I⊥(ν) and 76 I∥(ν) cancel out for small values of ν as lightscattering by rare-gas atoms is a process generating depolarized spectra close to resonance; as a result, the extraction of Iiso(ν) in Ar2 below 60 cm−1 is subject to huge statistical errors and is an unreliable operation. Only measurements taken above 60 cm−1 are numerically meaningful and are shown (•) in Figs. 1 and 2. In Fig. 1, only the low optimization-level computations (SCF, HF) are examined. Lineshapes due to Dacre’s SCF data as well as due to the semi-empirically corrected SCFCOR and SCF-expt are also shown, as these models and spectra are engaged in the discussion below. Fig. 2 is devoted to higher level computations: MP2, CCSD, CCSD(ω); the lineshape due to the inverse scattering solution proposed in this article (see below) is also shown. The curves α(R) are shown in Figs. 3 and 4: Fig. 3 illustrates the data used for the lineshapes of Fig. 1; Fig. 4 shows the data used for the lineshapes of Fig. 2. In Sec. III, an inventory of these polarizability computations and data is made for Ar2, along with a critical analysis of the various functions for α(R), their spectral response, and the BR values they generate.

III. RESULTS AND ANALYSIS THROUGH A SURVEY OF MODERN AB INITIO METHODS AND DATA FOR α

For large R, the primary cause of the changes of α in response to an external electric field is a change in the electric field experienced by one atom as a result of polarization of the second. For shorter R, further changes arise as a consequence of overlapping atomic charge clouds and of significant electronic exchange interactions. To properly account for these effects, the use of large basis sets and of diffuse orbitals seems indispensable. Specifically, for electron correlation effects, sophisticated post Hartree-Fock methods are necessary at the MP2 level or beyond. To which extent MP2 level is

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sufficient to describe the relevant features of α due to electron correlation for two closely separated Ar atoms is one of the issues discussed in the remainder of this article. It is also part of its purposes to benchmark the choice of the basis set and its significance in the success of the Ar2 spectrum modeling. A. From HF/SCF calculations to Møller-Plesset perturbation theory

The first extensive computations at the MP2 level for Ar2 were reported in 1996. For these computations, large basis sets and diffuse functions were utilized, which were shown to contribute significantly to α even if the anisotropy were largely unchanged.15 Furthermore, the uncontracted [15s11p3d] version of Dacre’s basis3 was employed and more importantly also a further set of 3s3p3d diffuse orbitals was included.15 Standard counterpoise techniques intended for the purpose of eliminating basis-set superposition errors were used to treat the extreme sensitivity of α. Correlation effects were included at the MP2 level with frozen core, believed to incorporate the bulk of valence electron correlation.15 The authors of Ref. 15 also reported data at the HF level and found significant differences from Dacre’s estimates, a result reflecting (according to these authors) the extreme sensitivity of α to the quality of the orbital basis set for Ar2.15 However, as shown in Figure 1, poor agreement is now obtained between the lineshape generated by the HF data of Ref. 15 and our experiment (Ref. 5), in spite of the effort put into the choice of the orbital basis and the computation of α. As for the MP2 data set of Ref. 15, its performance is still poor when the comparison is made with our experiment, even though an improved quality lineshape was found (see Fig. 2) as compared to the HF lineshape (see Fig. 1). As shown in Figs. 3 and 4, the HF and MP2 data of Ref. 15 for α take overall small values as a function of R and display a very short bump near or above 8.5 bohrs; this feature is largely responsible for the disappointing response of these data. Interestingly, the more pronounced these bumps, the deeper and steeper are the dives seen around 10 cm−1 in the lineshapes of Figs. 1 and 2. Clearly, neither the HF nor the MP2 data of Ref. 15 were able to generate any such V-shaped profiles. The disappointing response of the HF and MP2 data of Ref. 15 is likewise confirmed toward BR (Table I): As anticipated in Sec. II A (and in a way consistent with expectations), the agreement between the calculated BR value and the one measured by Hohm at λ = 543.5 nm (BR = 1.73(34) cm6 mol−2, Ref. 29) is improving with increasing level of optimization. In the case of MP2 of Ref. 15, the two values differ by a factor of 20; in the case of HF, the calculated value has the wrong sign. A more advanced treatment, also relying on finite-field many-body MP2 calculations for the polarizability of Ar2, appeared in 2000, using experience for the performance of hierarchies of Møller-Plesset perturbation theory methods in single-system calculations.19 These calculations were done in such a way as to answer questions related to the size of the interaction effects and the importance of the basis, as well as to the role of the incompleteness of a basis set and the way in which this incompleteness is related with the level of theory and the quality of the predictions. Aside from MP2, also finite-

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field SCF calculations were reported therein.19 For both these calculations, a flexibly constructed [8s6p5d4f] basis set for Ar was used, starting from an initial TZV (15s9p)[6s4p] basis set. On account of the lineshapes of Figs. 1 and 2, of the αcurves of Figs. 3 and 4, and of the BR data of Table I, the following conclusions are drawn about the performance of the MP2 and SCF data of Ref. 19: A shallow but well-defined trough is now seen in the SCF lineshape around 10 cm−1 (see Fig. 1), which contrasts with the absence of dip in the lineshape of the HF data of Ref. 15. In the case of the MP2 data of Ref. 19, an even more pronounced dip is observed in the lineshape (see Fig. 2). This feature is evidence of the successful control of electron correlation effects in the MP2 level as compared to SCF/HF computations, as well as of the superiority of Maroulis’s carefully constructed orbital basis at equal levels of optimization (MP2). Whereas the SCF lineshape is still too far from the experiment (see Fig. 1), the lineshape due to the MP2 data of Ref. 19 closely follows our measurements over the entire range of ν, with the agreement improving gradually with increasing ν (see Fig. 2). This observation is compatible with the bump displayed by these MP2 data19 in Fig. 4, which is substantially higher than in any of the aforementioned calculations. Furthermore, it is again compatible with the trend found in the BR data of Table I, as now the value generated by the MP2 data of Ref. 19 has the correct sign and a magnitude accounting for 70% of Hohm’s experimental BR value. It is worth pointing out that the experimental uncertainty in the region [150:300] cm−1 of the spectrum (Figs. 1 and 2) is of the order of the size of the data points (•), a precision suggesting this Raman experiment as a powerful benchmark tool for α.

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a particularly slow convergence of α with the one-particle basis sets. A d-aug-cc-pVTZ-33211 basis set was used for the calculations of α in Ref. 17; this basis set is a compromise between computational cost and accuracy since α calculations in the quadruple and quintuple zeta basis sets are particularly demanding. For the calculations reported in the subsequent article (Ref. 18), a CCSD/d-aug-cc-pVQZ-33211 basis set was employed. The behavior of these two basis sets against R and their response regarding refractivity and scattering are shown in Table I and in Figs. 2 and 4. The CCSD data of Ref. 18 (d-aug-cc-pVQZ-33211) are perfectly consistent both with the measured isotropic spectrum (see Fig. 2) and with the measured value of BR (Table I). A very similar response is observed with the CCSD data of Ref. 17 (d-aug-ccpVTZ-33211), even though these data lag slightly behind the CCSD/d-aug-cc-pVQZ-33211 as seen by carefully examining the corresponding values of Table I and the curves of Figs. 2 and 4 for those two series of data.

IV. SOLVING THE INVERSE SCATTERING PROBLEM: ANALYTICAL MODELING VS AB INITIO DATA FOR α

The augmented knowledge of the isotropic spectrum (Ref. 5) allowed us to obtain converged values of even-numbered and odd-numbered spectral moments M0, M1, . . . , M6. These quantities turned out to be instrumental in solving the inverse scattering problem for α. The essentials of this approach have been discussed elsewhere;27,28,38 below, only the specifics about the mean polarizability of Ar2 will be given. The incremental mean polarizability of Ar2 was given the following parametric form: α(R) = α D I D (R) +

B. CCSD theory and the use of triple or quadruple zeta basis sets

While neither MP4 calculations nor their less expensive partial-fourth-order alternative is available for Ar2, there are calculations on Ar2 as part of a more general effort on argon clusters at the CCSD level of theory (coupled cluster with single and double excitations). Coupled-cluster theory is known as an even more reliable way to account for the subtle interplay between overlap/exchange and electrostatic effects in the induced polarizability for short R, but its implementation is cumbersome and expensive. Systematic CCSD/CCSD(ω) calculations for Ar2 were first reported in 1999 by Fernández et al. (Ref. 17) and a few months later by Hättig et al. (Ref. 18). For those calculations, the authors started with a basis-set study of the static limit (CCSD) by using Dunning’s correlation consistent polarized valence basis sets. They ended with the inclusion of the dispersion of the properties through second order in the frequency arguments by means of the Sα (−4) Cauchy moment (CCSD(ω)). The coupled cluster response approach was employed with frozen 1s22s22p6 orbitals of the argon atoms and CCSD wavefunctions for the polarizabilities and Cauchy moments; the induced properties were counterpoise corrected. The selection of the basis set and the estimation of the remaining basis-set error were one of the most crucial steps in the calculation of the induced properties, showing

+

20α02Cq 5γC6 + 9α0 R6 R8

C D + − Ae−B R , R10 R12

(4)

with A, B, C, and D as the parameters we seek to optimize (A and B being defined as positive). The inspiration for this form came from the function Pfit = A6 R−6 + A8 R−8 + C R−10 R−σ + DR−12 − e− ρ , employed in Ref. 18 for the purpose of fitting CCSD data. The values of the optimized A, B, C, and D are gathered in Table III. For the sake of completeness, the values of the corresponding Pfit parameters for CCSD are σ also shown (A = e ρ , B = ρ−1). The symbols α D I D (R), α0, γ, Cq , and C6 in Eq. (4) denote the all-orders dynamic dipoleinduced dipole (DID) mean polarizability (Ref. 2), the atomic polarizability (α0(ω) = 11.337 a03), the second hyperpolarizability, the quadrupole polarizability, and the first long-range TABLE III. Optimized values for parameters A, B, C, and D of Eq. (4); the units are a 03, a 0−1, a 013, and a 015, respectively.

Inversion CCSDa a Fit

A

B

C

D

3738.2046 6395.6610

1.355 75 1.422 88

7.7097 × 107 1.2318 × 108

−1.1910 × 109 −1.7660 × 109

parameters values for the CCSD data reported in Ref. 18 are also given for the sake

of completeness (A = e

σ ρ , B = ρ −1).

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FIG. 5. Integrands ν n Iiso(ν), in units of cm6−n , shown at both spectral sides as a function of ν (cm−1) for even numbered spectral moments (n = 2, 4, 6). Inversion-, CCSD-, and experimental-lineshape integrands are shown plotted with solid-line, dashed-line, and dotted-dahed-line curves, respectively. To treat the lack of reliable experimental data below 60 cm−1, the spectrum portion in the region [0:60] cm−1 was extrapolated by using the CCSD(ω) profile. The principle of detailed balance was used to generate the experimental quantities at ν < 0. The illustrated curves make a reference to free-free pairs only.

dispersion coefficient of the Ar · · · Ar interaction potential, respectively. Classical sum-rules expressions for M0, M2, and M4 along with Eqs. (1)–(3) for BR were used as criteria for convergence for the optimization of A, B, C, and D. In order to ensure reliable input values for the classical sum-rules used to feed the inverse-scattering calculation, it was essential to take proper account of bound and predissociating Ar-Ar dimers. M0 was practically the only input quantity to be affected by those species (M0,inp = 1.047 10−2 Å9). The Aziz and Slaman HFDTCS2 (Hartree-Fock-Dispersion Total Cross Section potential, type 2) model was employed as the interaction potential for Ar · · · Ar (Ref. 40) in the computations of lineshapes and the inversion procedure. As shown below via a posteriori comparison, the close agreement between the values of spectral moments M0, M1, M2, . . . , M5, M6 derived from the experiment with those derived from the quantum lineshape due to the optimized model of Eq. (4) lends full credence to the quality of the proposed model for α. In Figures 5 and 6, even- and odd-numbered moment integrands ν n Iiso (ν) (n = 1, 2, . . . , 6) are shown; free-free lineshapes generated by the inverse-scattering solution (solid line curves) or the CCSD calculation18 (dashed-line curves), along with the lineshape of the experiment5 (dotted-dashed curves) were employed for the purpose. Differences in each of these families of curves are hardly distinguishable. The fact that the curves have reached near-zero levels at ν = ±400 cm−1 shows that the values of the spectrally calculated moments have converged. To compensate for the lack of experimental data below 60 cm−1, choice was made to employ the quantum lineshape of the CCSD(ω) data of Ref. 18; as with the isotropic spectrum of Ne2 (Ref. 30), criteria based upon the level of optimization, the extent of the basis set, and the spectral response of CCSD(ω) motivated this choice.

J. Chem. Phys. 143, 024304 (2015)

FIG. 6. Same as Figure 5 but for the odd-numbered spectral moments n = 1, 3, and 5.

Table IV shows values for even- and odd-numbered moments obtained from measured (expt, Ref. 5) and quantum lineshapes (free-free). The inverse-scattering solution (inversion, this work) of Eq. (4) and ab initio-computed data from Refs. 15 and 17–19, for α, were employed for the purpose. Results for Dacre’s SCF model (Ref. 3) and for its two semiempirical scaled variants (Refs. 4 and 5, see text) are shown listed for a more complete understanding. Clearly, the higher the level of optimization in the ab initio computation of α, the closer is the agreement with the measured (expt, Ref. 5) TABLE IV. Spectral moments obtained from measured (expt, Ref. 5) and quantum lineshapes (free-free), the latter being calculated with the employ of the inverse-scattering solution (inversion, this work) or ab initio -computed data for α. For the sake of completeness, the last three rows make a reference to Dacre’s SCF model (Ref. 3) and to its empirical “scaled” variants (Refs. 4 and 5, see text). The values of M0, M1, M2, M3, M4, M5, and M6, have been expressed in units of 10−3 Å9, 109 Å9 s−1, 1023 Å9 s−2, 1035 Å9 s−3, 1049 Å9 s−4, 1063 Å9 s−5, and 1077 Å9 s−6, respectively. Our calculations have shown that the greatest contribution to each of the six spectral moment values M i (i = 1, . . ., 6) comes from the frequency region around 65, 80, 100, 120, 150, and 200 cm−1, respectively.

Expta Inversion CCSD(ω)b CCSDb CCSDc MP2d MP2e SCFd HFe SCFf SCF-CORg SCF-expta

M0

M1

M2

M3

M4

M5

M6

9.49 9.81 9.56 8.28 9.13 12.76 18.94 21.02 21.96 13.26 19.56 9.28

16.95 17.36 17.61 15.41 17.15 23.14 30.46 27.00 28.67 23.21 34.25 16.25

13.39 13.71 13.92 12.19 13.56 18.26 24.00 21.28 22.57 18.32 27.05 12.83

73.93 76.88 79.68 71.66 78.67 97.62 121.14 105.64 107.42 99.96 147.53 69.97

59.88 62.28 64.53 58.09 63.78 78.93 97.85 86.56 86.74 81.05 119.62 56.73

6.79 7.23 7.33 6.80 7.44 8.75 10.54 10.49 9.27 9.63 14.21 6.74

5.73 6.06 6.19 5.70 6.24 7.32 8.86 9.21 7.87 8.10 11.96 5.67

a Reference

5. 18. c Reference 17. d Reference 19. e Reference 15. f Reference 3. g Reference 4. b Reference

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Chrysos, Dixneuf, and Rachet

moment values. SCF-expt makes an exception to this rule as it provides moment values that are very close to the measured ones in spite of its empirical nature and the low optimization level in which it originates. Nevertheless, this agreement is only accidental as SCF-expt is unable to account for the measured BR value; it is all the more so with Dacre’s SCF and SCF-COR, which heavily contradict both the Raman experiment of Ref. 5 and the refractivity experiment of Ref. 29. The CCSD(ω) results are remarkably close to the experimental values; quite similar is the situation at the static CCSD level (Refs. 17 and 18). At the MP2 level of theory, a good agreement with the experiment is found insofar as the carefully designed basis set of Ref. 19 has been considered. Finally, the close agreement that we observe between the moment values calculated from the “inverted” and the experimental lineshape is once more the most convincing way to prove the validity of the inverse-scattering solution proposed herein and of the methodology outlined in recent articles.30,38 The percent relative deviation between the calculated and the experimental integrated spectra (see M0, Table IV) offers a more quantitative way to globally size up the response of the various ab initio data: CCSD(ω) calculations are shown to agree with the measurements to within