Collision-induced Raman scattering and the peculiar case of neon: Anisotropic spectrum, anisotropy, and the inverse scattering problem
Sophie Dixneuf, Florent Rachet, and Michael Chrysos
Citation: The Journal of Chemical Physics 142, 084302 (2015); doi: 10.1063/1.4913212 View online: http://dx.doi.org/10.1063/1.4913212
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/8?ver=pdfcov Published by the AIP Publishing
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Collision-induced Raman scattering and the peculiar case of neon:
Anisotropic spectrum, anisotropy, and the inverse scattering problem
Sophie Dixneuf,1,a)Florent Rachet,2and Michael Chrysos2,b)
1Forschungszentrum Jülich GmbH IEK-8: Troposphere, 52425 Jülich, Germany
2LUNAM Université, Université d’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 Bd Lavoisier, 49045 Angers, France
(Received 2 December 2014; accepted 5 February 2015; published online 25 February 2015) Owing in part to theporbitals of its filledLshell, neon has repeatedly come on stage for its peculiar properties. In the context of collision-induced Raman spectroscopy, in particular, we have shown, in a brief report published a few years ago [M. Chrysoset al., Phys. Rev. A80, 054701 (2009)], that the room-temperature anisotropic Raman lineshape of Ne–Ne exhibits, in the far wing of the spectrum, a peculiar structure with an aspect other than a smooth wing (on a logarithmic plot) which contrasts with any of the existing studies, and whose explanation lies in the distinct way in which overlap and exchange interactions interfere with the classical electrostatic ones in making the polarizability anisotropy,α∥−α⊥. Here, we delve deeper into that study by reporting data for that spectrum up to 450 cm−1 and for even- and odd-order spectral moments up toM6, as well as quantum lineshapes, generated from SCF, CCSD, and CCSD(T) models forα∥−α⊥, which are critically compared with the experiment. On account of the knowledge of the spectrum over the augmented frequency domain, we show how the inverse scattering problem can be tackled both effectively and economically, and we report an analytic function for the anisotropy whose quantum lineshape faithfully reproduces our observations. C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4913212]
I. INTRODUCTION
Collision-induced light scattering by atomic or molecular ensembles is a collective effect rather than an effect occurring exclusively within the realm of individual units.1–3 It takes place when shining a laser onto a gas of nonreactive atoms or molecules and manifests itself with light-scattering spectra.
The intensity of these spectra depends on the frequencies of the incident and scattered radiation (and on temperature) and can be measured and calculated as an appropriate scattering differential cross-section.1,2,4That process has a supramolec- ular origin, since it is a direct manifestation of polarizabilities or hyperpolarizabilities in excess of those of the interacting constituents and, as such, it branches out into two interest- ing areas of scientific inquiry: collision-induced Raman and collision-induced hyper-Raman spectroscopy, respectively.4
The past decade has witnessed a rapid growth in the theo- retical understanding of the collision-induced hyper-Raman process in atomic gases.5–7A number of substantial advances, such as the uncovering of how collision-induced hyper-Raman spectra should be normalized and formally calculated,5–7 or improvements (with the advent of new, flexible and sophisti- cated numerical techniques) in the knowledge of incremental hyperpolarizabilities of rare gas mixtures7–10tend to open up a new avenue for research. This is all the more the case with collision-induced Raman scattering, which is a linear optical
a)Present address: Centre CEA de Grenoble, Laboratoire CEA-bioMérieux, Bât 40.20, 17 rue des Martyrs, 38054 Grenoble, France. Also at BIOASTER, 321, avenue Jean Jaurès, 69007 Lyon, France.
b)Electronic mail: michel.chrysos@univ-angers.fr
process. In that context, impressive progress has been made over the past two decades in the theoretical understanding11–16 and modeling8–10,17–21of incremental polarizability invariants of simple rare-gas systems and also experimentally with the advent of highly reliable collision-induced Raman spectra over large frequency ranges.22–28
The way in which neon gas scatters radiation was inten- sively studied from the late 70s to the late 80s.29–33 Works on rare-gas atoms by our group only started appearing in the late 80s34but they quickly became essential to the community mainly on the basis of a unique Raman equipment and a working protocol intended for high sensitivity spectra.22,25,26,35
In this respect, the issue of Raman scattering by the ever intriguing helium gas was revisited and fully addressed23,25 while the untrodden land of Raman scattering by atomic mix- tures was explored experimentally and was interpreted.27,28 In parallel, a novel approach aiming at effectively solving the inverse scattering problem was developed36,37and it was successfully applied to argon, which is another gas we have lengthily dealt with in our group.12,13,22,35 In the meantime, neon once again came to the attention of the public,16,20,21,26
including some never before revealed facets of its gas calling on scientists to make a rethink over the situation.16,26 Neon continues to surprise today; among its peculiarities, the low stability of its compounds (as compared with helium ones) is still to be further investigated.38Lying in part in the specificity of its electronic structure (occupied p orbitals), the intrigu- ing reduced stability of the neon compounds is in line with recent suggestions to shift He to group 2 of the periodic table, leaving Ne to occupy the top-right position in the noble gas group.38
0021-9606/2015/142(8)/084302/8/$30.00 142, 084302-1 © 2015 AIP Publishing LLC
084302-2 Dixneuf, Rachet, and Chrysos J. Chem. Phys.142, 084302 (2015)
The purpose of this article is to investigate further the Raman process in supramolecular neon. To this end, (i) we report properly calibrated data of the Ne–Ne anisotropic Ra- man spectrum over a very broad frequency domain, (ii) we show how to properly implement our method and consistently solve with it the inverse scattering problem for that system, and (iii) we propose a model for the(Ne)2anisotropy, which reads
β(R)=βDID(R)+ γC6
3α0R6+24α20Cq
R8 + X3
R10
−X1e−X2R+X4R3−E R5, (1)
withX1=1.406 72×104,X2=2.503 03,X3=7.528 38×105, X4=1.015 33×10−2, and E =1.5×10−5(all values are in a.u.;βDIDis the all-orders dipole-induced dipole (DID) contri- bution). A number of conclusions are also drawn, based upon comparisons between our measurements and quantum line shapes (generated by the proposed model or by existing modern ab initioones20,21) as well as between the models themselves.
The measurements that we are reporting here cover the range [5:450] cm−1, that is, a range three to five times greater than the range formerly covered in the experiment by Bérard and Lallemand29 ([6:80] cm−1) and Frommhold et al.30,31 ([10:170] cm−1). That spectral extension reveals a marked change of slope above 300 cm−1 as to the way in which intensity varies with frequency. No similar behavior has been observed in helium gas (in spite of the renowned quantum character of that light atom), even though an equally wide spectral domain has also been explored there.24 A similar feature can, nevertheless, be seen in the theoretical line shape of a simple empirical model for neon, long ago reported by Meinanderet al.,33but at that time, no attention had been paid to that feature, which for lack of experimental evidence had probably been thought of as being spurious. We show below that its occurrence on the spectrum is to some extent predicted by the most recentab initiodata forβ(R), all of which have a local maximum at R≃4.5 bohrs.20,21By shifting artificially that barrier of the anisotropy from its original location to either slightly shorter or slightly larger R, and comparing the generated quantum line shapes with the experiment, we identified the existence of that short-range barrier (and its exact position on theR-axis) as responsible for the occurrence of the spectral feature. As a consequence, the conclusion is drawn that the unconventional line shape in the wing lies with electron correlation at close separations. On account of the general truths that “the higher the level of an ab initio calculation of β(R), the more electronic correlation it contains at short R,” and that “the more the electronic correlation at short distances, the better those distances will reproduce the spectrum in the wing,” the superiority of the model anisotropy that we propose for neon is consistent with its structure at short distances, which becomes more marked than in any of the existing ab initiodata. How to take advantage of the good response of such data to tailor-make an optimal polar- izability model is another interesting point addressed in this article.
II. EXPERIMENTAL
The detailed description of the equipment used for the experiment, and of the protocol we followed for signal detec- tion, calibration, and processing, has been given in Ref.25.
Below, we give only a short summary along with some points that are specific to neon.
The green line (λ0=514.5 nm) of a 2W Ar+laser was used for excitation and a four-window high-pressure cell was used for the sample (high purity neon gas, guaranteed by the “Air Liquide” company for total residual impurities<1 ppm). Gas densities,ρ(in amagat), were deduced from pressure measure- ments, P (in atm), on account of the virial equation of state (T =294.5 K).39 The reception angle was kept fixed at 90◦ with respect to the incident radiation. A double monochromator with two holographic gratings (1800 grooves/mm) was used to disperse the scattered radiation. A−20◦C-cooled bialkali pho- tomultiplier along with an amplifier-discriminator and a photon counter were employed for signal detection in the frequency range [5:250] cm−1, whereas for the much weaker signals in the range [230:450] cm−1, far higher sensitivity was required which was ensured by mapping the signal onto the array of a CCD detector (cooled to 140 K and associated with a Super-Notch holographic filter). The consistency of the detection by the two detectors was checked by using the 20 cm−1-wide common zone in their spectral ranges. On moving from 5 to 250 cm−1 in the Raman frequency range, the resolution was gradually increased from 1 to 10 cm−1in the case of the spectrometer, whereas for the CCD, the resolution of the spectrograph was fixed (1.2 cm−1). The anisotropic signal (denoted below asS∥) was measured for a polarization of the incident beam paral- lel to the direction of the observation40 and for six different values of gas density per Raman frequency,ν. In the frequency ranges [5:25], [25:150], [150:250] and [300:450] cm−1,S∥(ρ, ν) was found to scale purely quadratically as a function of ρup to gas densities as large as 100, 150, 240, and 250 amagat, respectively. Given that optical signals exhibiting a quadratic ρ-dependence are characteristic of binary interactions, density- independent binary quantities were derived for each frequency by analyzing the slope of the linear part of S∥(ρ, ν)/ρ as a function ofρ; these quantities were then calibrated (see below) to yield absolute intensities,I∥(ν), for Ne–Ne. A total absence of ternary interactions and an insignificant contribution from monomers due to impurities in the gas were guaranteed by the density analysis mentioned above.
Once corrected for the aperture of the scattered beam,31 the anisotropic intensities were calibrated on an absolute scale (cm6), according to a formula described in Ref.25. The inte- grated intensity of the S0(0)rotational line of H2served as a reference for the purpose. A satisfactory agreement was found between our I∥(ν)measurements and those taken by Bérard and Lallemand and Frommhold’s group29–31in the respective common frequency intervals. The lowest frequency collision- induced signal that we detected was 5 cm−1 away from the Rayleigh peak. The hitherto unknown wing of the spectrum was explored up to 450 cm−1—thus almost tripling the Raman domain formerly probed.30,31
The values of the measured intensity in absolute unit are given in Table I. The spectrum is shown graphically, as a
TABLE I. Absolute-unit anisotropic intensities (cm6) as a function of frequency shift (cm−1) at 294.5 K. The maximum experimental error is±10%,±15%,±20%,±30%,±45%,±65%, and±80%, atν=5, 12.5, 100, 150, 290, 380, and 440 cm−1, respectively. The interval between 320 and 372 cm−1is empty because of the occurrence of residual H2in this region of the spectrum; this gas was used in our experiment for signal calibration purposes (see Sec.II).
ν I∥ ν I∥ ν I∥
5 1.12×10−55 200 1.08×10−59 380 4.48×10−61
7.5 1.02×10−55 220 5.10×10−60 385 4.31×10−61
10 9.17×10−56 225 4.54×10−60 390 4.08×10−61
12.5 8.00×10−56 230 3.19×10−60 395 3.87×10−61
15 7.04×10−56 240 2.33×10−60 400 3.66×10−61
17.5 6.46×10−56 250 2.00×10−60 405 3.44×10−61
20 5.56×10−56 255 1.70×10−60 410 3.23×10−61
22.5 4.81×10−56 263 1.39×10−60 415 3.01×10−61
25 4.14×10−56 270 1.31×10−60 420 2.81×10−61
50 1.095×10−56 280 1.22×10−60 425 2.59×10−61
75 3.15×10−57 290 1.05×10−60 430 2.42×10−61
100 8.99×10−58 300 7.72×10−61 435 2.17×10−61
125 3.58×10−58 310 6.95×10−61 440 2.01×10−61
150 7.93×10−59 320 6.09×10−61 445 1.77×10−61
175 2.35×10−59 372 4.88×10−61 450 1.56×10−61
function ofν, along with quantum line shapes (plotted for the sake of comparison on the same graph) at the end of the article.
III. THEORETICAL
A. Ab initiocomputed anisotropy and quantum computations of line shapes
Let us now focus on the most recentab initiodata of β, referred to in what follows as SCF (Ref.20), CCSD (Ref.20), CCSD (Ref.21), CCSD(ω) (Ref.21), and CCSD(T) (Ref.20).
These acronyms make a reference to the computation method that was employed, and their meaning is self-consistent field (SCF), coupled-cluster with single and double excitations (CCSD for the static and CCSD(ω) for the dynamic variant), and coupled-cluster with single and double excitations and perturbative triple excitations (CCSD(T)). All these compu- tations are distinguished for their high optimization level and the reliable orbital basis sets they used. In particular, the validity of the combination of a coupled-cluster method with an augmented well-balanced Dunning’s correlation-consistent polarized valence orbital basis set (similar to the one used in Ref. 21) has already been demonstrated for modeling the anisotropy of Ar–Ar18,37 and it is now to be tested in the modeling of the Ne–Ne anisotropy. The frequency-dependent CCSD(ω) data set was obtained through a second-order ap- proximation in the frequency arguments using expansions in Cauchy moments.18,19
For the calculations of quantum line shapes, the compu- tation procedures detailed in Refs.12 and13were used, in which a Fox-Goodwin integrator and discrete variable repre- sentations (DVR) have been implemented to ensure reliable line shapes; the intensity of a typical line shape so computed remains accurate over wide frequency ranges albeit the varia- tions in the intensity amplitude can be several orders of magni- tude. A multitude of properly weighted energy-integrated squared matrix-elements computed up to R=300 bohrs had
to be accounted for in the calculations, and they were added up to span a large interval of rotational quantum numbers for the initial and final supramolecular states. The computation of matrix-elements, especially those corresponding to the most energetic transitions (and affecting primarily the far wing of the spectrum), is known to be extremely sensitive to the choice of the interpolation function of the short-separation anisotropy data (this point has been raised in Ref.21); this interpolation step is critical in the region [4:5] bohrs where some of Maroulis’s models show slight instabilities.20Choice was made to use a unique analytical form to be adjusted to fit each ab initio anisotropy described in this article, on the basis of thePfitfunction proposed by Hättiget al.21The input anisotropy models were prepared as follows. For Hättig’s data,Pfitparameters from Ref.21were used to interpolate the anisotropy function over the range [3:300] bohrs (leading to the models CCSD and CCSD(ω) of Ref.21shown plotted at the end of the article). As for Maroulis’s data, these were first completed by the asymptotic classical DID model,1 beyond the reported values ofR, using the consistentab initioatomic polarizabilities [α0=2.3719, 2.6721, and 2.7149a03, for SCF (Ref.8), CCSD (Ref.8), and CCSD(T) (Ref.8), respectively], and then, a Pfit form was adjusted to the ensemble over the range [3:300] bohrs. The resulting interpolated models for the CCSD (Ref. 21), CCSD(ω) (Ref. 21), SCF (Ref. 20), CCSD (Ref.20), and CCDS(T) (Ref.20) anisotropy are shown plotted below, and the corresponding values for the sevenPfit
parameters are listed in TableII.
B. Solving the inverse scattering problem 1. A classicized spectrum used as input
The way in which we will proceed below to solve the inverse scattering problem is a compromise between efficiency and economy, i.e., an application of a simply implemented method to reach solutions that are reliable enough for our
084302-4 Dixneuf, Rachet, and Chrysos J. Chem. Phys.142, 084302 (2015)
TABLE II. Values of fit parameters of the anisotropy models tested herein. The function readsPfit=A3/R3 +A6/R6+A8/R8+A10/R10−exp(−(b R3+R−s)/ρ)as reported by Hättiget al.for the purpose of fitting their ownab initiodata.21The units ofb,s,ρ,A3, A6,A8, and A10are 10−3×a−20 ,a0,a0,a60,a90,a110, anda013, respectively.
b s ρ A3 A6 A8 A10
SCFa 6.894 04 3.562 92 0.926 214 33.8453 −48.3194 6847.61 −31 711.4
CCSDa 7.977 33 3.873 94 0.981 886 42.9404 71.7086 8232.64 −41 572.1
CCSD(T)a 9.816 14 3.939 49 1.058 72 44.4377 71.4437 6384.34 −31 286.2
CCSDb 8.009 00 3.809 20 0.986 526 40.9336 105.888 6125.93 −28 987.1
aReference20.
bReference21.
purposes. The framework of our approach has been described in Refs.36and37.
Figure1helps the reader grasp better the course of thought and the steps followed in our methodology on going from the
FIG. 1. (a): Pathways①and ②describe how classical and quantum line shapes can be generated from a given input function β(R), respectively.
To come full circle, an appropriate (de-)symmetrization procedure must be used ensuring the passage from one line shape to the other. If that procedure is reliable enough, the two pathways are almost equivalent. It is pathway
②which was used in this article. (b): This cartoon shows howβ(R)can be obtained from a given line shape. In pathway①, the input is the measured (or a quantum-computed) line shape, i.e., an asymmetric profile. This implies that in the inversion procedure to be used to generateβ(R), one must imple- ment the exact sum-rule expressions. Conversely, with pathway②, we keep working with the far simpler classical sum-rule expressions but, in order to preserve internal consistency, the input line shape must be the classical one.
As before, the two pathways become linked provided that the symmetrization procedure is reliable. It is pathway②, which was used in this article.
measured line shape I∥(ν) to a “classical” line shape, I∥c(ν) (“c” stands for “classical”), and then to an induced-anisotropy model, β(R), that closely mimics the physical anisotropy of (Ne)2. In fact, a spectrumI∥(ν)is a transcription, in the fre- quency domain, of an induced anisotropy, β(R), and as such, it can be calculated from that β(R) in either two fashions (Fig.1(a)):①by classical calculations (along with a Fourier transformation of the so produced auto-correlation function) or②straightforwardly, with quantum mechanics. In the first case, the calculated profile,I∥c(ν), is symmetric and cannot be a reliable representation of the physical spectrum. In the second case, the computed spectrum,I∥(ν), is a close representation of the true one, provided thatβ(R)has reliably modeled the actual induced anisotropy. The spectral shape, in that case, is asym- metric and faithfully satisfies the principle of detailed balance.
Note that classical and quantum (or measured) line shapes can, to some extent, be deduced from one another by properly (de-)symmetrizing the input profile. The degree of reliability of that procedure depends on the (de-)symmetrization model used to introduce or cancel the asymmetry.
Figure 1(b) presents the same “picture” the other way around, that is, it offers the explanation of how to rewind back to β(R) when starting with a given line shape. As before, there are two possible pathways. In pathway ①, one has to treat the measured (or the quantum) line shape,I∥(ν), which is asymmetric. The latter will then be used to feed a hypothetical inverse scattering procedure, which implements exact (and particularly cumbersome) sum-rule expressions of both even and odd orders in order to consistently converge to β(R). But pathway②describes another, much smarter way to reach β(R): Now, a symmetrization model is used to build an (as reliable as possible) classical profile,Ic∥(ν)from the measured data. That profile will then be used to feed an all-classical in- verse scattering procedure, i.e., a procedure employing clas- sical even-order sum-rule expressions only. As before, the degree of reliability of that procedure depends on how well Ic∥(ν)mimics the truly classical profile.
There are several ways to de-symmetrize classical line shapes in order for the quantum-corrected line shapes to satisfy the principle of detailed balance. As has been argued “on the basis of an entirely theoretical analysis of the quantum and classical versions of the fluctuation-dissipation theorem, among the simple correction procedures, the harmonic correc- tion affects the low- and high-frequency regimes in a more balanced way than for instance the Schofield or the standard approximation.”41Note that a “harmonic correction” is what
has been referred by Frommhold to as procedure P2 (see Ref. 42). In the framework of this article, the truly classical profile was approximated by a symmetrical function, Ic∥(ν), which was derived from the measured data,I∥(ν), by apply- ing procedure P2. Since the aim here is to symmetrize the measured spectrum (rather than to quantum-correct a classical line shape), the procedure P2 forν >0 should read
Ic∥(ν)= kBT hcν(1−e−
h cν
k BT)I∥(ν). (2)
2. Functional form ofβ(R)
For the optimization of the tricky anisotropy of Ne–Ne, we drew our inspiration from the analytic function, Pfit(R), previously used by Hättig et al. to fit their CCSD data for β(R),21 and we cast the anisotropy of neon in the following five-parameter form:
β= βDID+ γC6
3α0R6 +24α20Cq
R8 + X3
R10 −X1e−X2R+X4R3−E R5. (3) The quantities α0, γ, and Cq designate the static dipole- dipole polarizability, second hyperpolarizability, and quad- rupole polarizability of atomic neon, respectively; C6 is the first long-range dispersion coefficient of Ne–Ne interaction potential energy. The following (experimental or semiempir- ical or ab initio) values were used: α0=2.669 a.u.,32 and γ=109.4 a.u.,43C6=6.447 a.u.,44Cq=3.76 a.u.45The non- expanded form of the classical DID interaction, βDID, along with expressions given by Buckingham for the R−6and R−8
terms was used.46QuantitiesXa(a=1,2,3,4)andEdenote the parameters we sought to optimize, all defined as positive.
The orbital overlap/electronic exchange term, conventionally modeled by a short-range simple decaying exponential func- tion, involving the parameters X1 and X2, had to be sup- plemented here with a positive cubic term in the exponent to allow for generating a barrier in the anisotropy, β(R), at close separations. Similarly, a fixed E-parameter quintic decaying term had to be set in the exponent to prevent the exponential function from non-physical unstoppable growth at large separations (R>12 bohrs). The value of that param- eter was adjusted independently to increase the flexibility of function β(R) and to allow, during the optimization, for efficient relaxation towards the desired spectral-moment input;
proceeding by dichotomy, the valueE=1.5×10−5 a−50 was found to serve all purposes, making it possible to fine-tune the theoretical spectrum to provide best agreement with the measured one. Four input sum-rule relationships were im- plemented in our solver for the optimization of Xa. More details about the way to solve the three-parameter inverse scattering problem can be found in Ref.37. Below, we give a brief description of the approach for the four-parameter problem.
3. Sum rules and spectral moments
The expressions of the three low-order classical sum rules, M0, M2, and M4, in anisotropic Raman scattering are well known and can be found in Ref.37. A fourth sum-rule expres- sion was needed here, which reads47
M6=4π (kBT
µ )3 ∞
0
15β′′′2+ (234
R2 −36
RU′+9U′2 )
β′′2 + (2448
R4 −216 R3U′+24
R2U′′+54
R2U′2+U′′2 )
β′2 +
(7560 R6 −432
R5U′+24 R4U′2
) β2+
(36
R −18U′−3U′′
)
β′′′β′′−144 R2 β′′′β′ +216
R3 β′′′β+ (
−792 R3 +36
R2U′− 6
RU′′+6U′U′′
) β′′β′+
(648 R4 −144
R3U′ )
β′′β +
(
−8208 R5 +792
R4U′−36
R3U′′−72 R3U′2
) β′β
g(R)R2dR.
Spectrally,Mnis related to the classical line shape accord- ing to the expression
Mn= 15 2
(λ0 2π
)4
(2πc)n
+∞
−∞
νnI∥c(ν)dν. (4)
In the above two expressions, kB, µ, and T denote the Boltzmann’s constant, the reduced mass of (Ne)2 and the temperature, respectively; λ0=514.5 nm is the laser wave- length,νis the Raman frequency (in wave number units, cm−1), cis the speed of light; β′, β′′, andβ′′′designate first, second, and third derivatives with respect to R.g(R)is the classical
radial distribution function exp(−U)(U=V(R)/kBT being a dimensionless interaction potential and V(R) is the phys- ical potential of Ne–Ne). The reliable HFD-B (Hartree-Fock- Dispersion-B) model of Aziz and Slaman44was employed for the Ne–Ne potential both in the inverse scattering procedure and in the computation of quantum line shapes.
The classical quantities Ic∥(ν)νn are shown in Figure2 (n=0) and 3 (n=2, 4, 6). In Fig. 2, the non-classicized experimental values given in TableIare also shown for com- parison. Obviously, I∥c(ν) and I∥c(ν)ν2 have decreased to near vanishing values before reaching the upper bound ν
=450 cm−1, thus ensuring the perfect convergence of the
084302-6 Dixneuf, Rachet, and Chrysos J. Chem. Phys.142, 084302 (2015)
FIG. 2. Classicized spectrumI∥c(ν) (cm6), shown at both spectral sides, as a function ofν(cm−1)(solid line curve). Our measurements are shown for comparison (symbols); the gap seen between 320 and 372 cm−1is explained by the occurrence of residual H2 in this region of the spectrum; this gas was used in our experiment for signal calibration purposes (see Sec. II).
The asymmetric spectrum shown in dashed atν >0 is again that of the experiment; atν <0, the dashed-line curve was obtained from the experiment by applying the principle of detailed balance.
momentsM0andM2; at 450 cm−1, the convergence ofM4has been completed to within 1%. Conversely, the sudden slow- down in the way in which the spectrum intensity was found to decrease above 300 cm−1(see Fig.2and TableI) results in a revival of Ic∥(ν)ν6beyond that frequency (see Fig.3,n=6) dramatically compromising any prospect for a converged M6
value; in order to override this obstacle, the value of that moment was taken in such a way as to ensure that the quantum line shape that was generated from the optimized function β(R) matches as closely as possible the experimental line shape in the far wing. The following values were used to feed the inverse scattering algorithm: M0=1.890×10−1 Å9, M2
=4.604 Å9ps−2,M4=7.850×102Å9ps−4, andM6=1.000
×106Å9ps−6.
FIG. 3. Classicized quantitiesI∥c(ν)νn(cm6−n)as a function ofν(cm−1)for n=2(×2000),n=4, andn=6(×10−4).
4. Newton-Raphson in a matrix form
Our Newton-Raphson matrix-equation solver37 Xi+1=Xi−D−1
XiF(Xi) (5) was used and was adapted to the four-parameter optimization problem. The 16 elements of the Jacobian matrixDread37
Dab=4π
∞ 0
fab(X)g(R)R2dR (6) wherea andb (=1, 2, 3, 4)designate the line and the col- umn number ofD, respectively.XandF(X)designate column vectors (X1, X2, X3, X4)T and (∆M0, ∆M2, ∆M4, ∆M6)T, respectively. The latter, defined at each step of the procedure as the difference between trial (running) and measured moment values, corresponds to the quantity sought to be vanished upon completion of the program. The meaning of fab has been adequately explained in Ref.37 (these quantities have been denoted asaab, therein); their formal expressions are given as supplementary material.48
On account of Eq.(3), tedious and lengthy expressions are obtained for the various derivatives ofβwith respect toRand Xa(a=1,2,3,4). These are provided in the second part of supplementary material.48
A few iterations sufficed for convergence. Well defined plateaus of stability were obtained and the following values of optimized parameters were determined: X1=1.406 72
×104a03,X2=2.503 03a−10 ,X3=7.528 38×105a130 , andX4
=1.015 33×10−2a−30 . The value of the independently ob- tained parameterEisE=1.5×10−5a−50 .
IV. RESULTS
Figure 4 shows the (Ne)2 anisotropy as a function of R. Our model (this work) is compared with the ab initio- computed SCF, CCSD, and CCSD(T) data from Ref.20and the CCSD and CCSD(ω) data from Ref. 21. The vertical dashed line indicates the shortest interatomic distance (R=4 bohrs) “seen” by our experiment at 294.5 K.
Focusing on the bump aroundR≃4.5 bohrs, an enhance- ment of the curvature in the structure of theβ(R)is observed, with the top of the barrier getting more pronounced as the sophistication of theab initiocomputation is increased. Fur- thermore, when the same level of optimization is used (CCSD) but with two different orbital bases, the basis-set of Ref.20is seen to generate a more pronounced bump. The highest bump is observed on our model, and this observation (in conjunction with the spectral response of the models, see below) prompted us to suggest that the bump height is a decisive factor in the quality ofβ. Interestingly, in the case of argon, our optimized anisotropy model had once again been seen to exhibit the high- est maximum,37a result corroborating the idea that the solution of the inverse scattering problem can be a reference model for induced-polarizability calculations whatever the system.
The dynamic approximation CCSD(ω)of Ref.21is shown to provide an anisotropy (and also a spectrum, see below) which is almost indistinguishable from that produced by the static CCSD data of Ref.21.
FIG. 4. The Ne–Ne anisotropy,β(in a.u.) as a function ofR(Bohr). The interval [3:6] bohrs has been enhanced in the inset to highlight the effect of the electron correlation on the anisotropy bump (located around 4.5 bohrs). Our model (this work) is compared with theab initiodata of Maroulis [CCSD(T), CCSD, SCF] (Ref.20) and Hättiget al.[CCSD, CCSD(ω)] (Ref.21). The vertical dashed line in the inset indicates the shortest interatomic distance probed by our experiment atT=294.5 K.
Figure5allows for a critical comparison between quan- tum line shapes (lines) generated from the anisotropy models of Fig. 4 and our experiment (•) in the range [0:450] cm−1. All the spectra are seen to exhibit a similar trend, produc- ing in particular a change of slope around the Raman fre- quency where the same feature was observed experimentally.
Beyond that value, the theoretical line shapes (lying system- atically below the measured one) are seen to get better as the level of optimization used for the calculation is getting higher. The progression order of those spectra toward our measurements follows the same (optimization) order as the one that we observed in the corresponding anisotropy models (see Fig.4). The spectrum computed from the model of Eq.(3)is unquestionably the best. It is the only spectrum to be consistent with the uncertainties of the measurements, over the entire frequency range, and also the only to faithfully describe the far wing; at lower frequencies (see inset), the CCSD/CCSD(ω) data set of Ref.21is seen to match exactly the measured data in the interval [5:30] cm−1.
FIG. 5. Absolute-unit anisotropic intensities (cm6) as a function ofν(cm−1).
Our experiment is depicted by symbols. Quantum line shapes generated by the anisotropy models of Fig.4are also shown (curves) over the broad interval [5:450] cm−1. The gap seen between 320 and 372 cm−1is explained by the occurrence of residual H2in this region of the spectrum; this gas was used in our experiment for signal calibration purposes (see Sec.II). The line shape generated by the inverse scattering solution is shown by a solid line curve (this work). The interval [0:30] cm−1has been enhanced in the inset to showcase the contribution of bound and metastable dimers.
TableIIIgathers the values of the spectral moments (both of even and odd orders) up to M6. These values were care- fully obtained by integrating the measured and the quantum line shapes. The striking agreement that is found between the moment values of the spectrum generated from our inversion anisotropy model and those obtained from the measurements lends full credence to our way to proceed throughout this article and to the conclusion drawn from our analysis.
V. SYNOPSIS
We reported the room-temperature anisotropic Raman spectrum of neon in a frequency range as wide as never before.
The procedure and the steps we followed in the experiment, starting with the preparation of the gas and ending up with the detection of the weak signals in the wing of the spec-
TABLE III. Spectral moments calculated from measured (“experiment,” this work) and quantum line shapes. The quantum line shapes have been calculated with the inversion anisotropy model (“inversion,” this work) as well as with theab initioanisotropy data of Maroulis and Hättiget al.On going fromM0toM6, the units for the entry values are Å9, 1011Å9s−1, 1025Å9s−2, 1037Å9s−3, 1051Å9s−4, 1064Å9s−5, and 1078Å9s−6, respectively. The uncertainty in the experimental values ofM0,M1, . . . ,M5is±10%,±15%,±20%,±20%,±20%, and±60%, respectively.
M0 M1 M2 M3 M4 M5 M6
Experiment 0.183 0.590 0.490 1.030 0.886 0.834 (0.88)
Inversion 0.187 0.575 0.478 0.997 0.871 1.190 1.370
CCSD(T)a 0.210 0.578 0.479 0.840 0.720 0.574 0.606
CCSDa 0.196 0.541 0.449 0.788 0.674 0.526 0.554
CCSDb 0.178 0.493 0.409 0.717 0.614 0.473 0.497
CCSD(ω)b 0.195 0.528 0.437 0.704 0.602 0.476 0.502
SCFa 0.122 0.339 0.281 0.500 0.428 0.324 0.339
aReference20.
bReference21.
084302-8 Dixneuf, Rachet, and Chrysos J. Chem. Phys.142, 084302 (2015)
trum and with their calibration in absolute cm6 units, were instrumental in the success of the project. The extension of the anisotropic spectrum much farther into the wing revealed a very distinct feature, which (to the best of our knowledge) had never been observed before in any experiment with atomic gases. This feature was reproduced qualitatively by using modern ab initio-calculated models, and its occurrence was found to be in direct connection with a bump in the Ne–Ne anisotropy at close distances. This bump was shown to become steadily more pronounced with increasing optimization level in theab initiocomputations, an observation helping to better understand the growing role of electron correlation close to the unified atom limit. The behavior of the spectrum in the far wing served as a useful benchmark in the implementation and modeling of the induced anisotropy, and helped us provide a new model for the induced anisotropy of(Ne)2. That model, being fully consistent with the source-experiment, is the best representation ever suggested for that system, forR≥4 bohrs.
We believe that the procedure we have followed in solving the inverse scattering problem is the best device to date for inverting spectra and a powerful alternative to expensive ab initiocalculations.
Aside from being impactful for polarizability science and engineering purposes, our study is expected to be of value to investigations related to collision-induced absorption42(CIA).
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