Decomposing anharmonicity and mode-coupling from matrix eﬀects in the IR spectra of matrix-isolated carbon dioxide and methane †‡

Gas-phase IR spectra of carbon dioxide and methane are nowadays well understood, as a consequence of their pivotal roles in atmospheric-and astrochemistry. However, once those molecules are trapped in noble gas matrices, their spectroscopic properties become diﬃcult to conceptualize. Still, such spectra provide valuable insights into the vibrational structure. In this study, we combine new matrix-isolation infrared (MI-IR) spectra at 6 K in argon and neon with in vacuo anharmonic spectra computed by vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI). The aim is to separate anharmonicity from matrix eﬀects in the mid-infrared spectra of 12 C 16 O 2 , 12 CH 4 , and 12 CD 4 . The accurate description of anharmonic potential energy surfaces including mode-coupling allows to reproduce gas-phase data with deviations of below 3 cm (cid:2) 1 . Consequently, the remaining diﬀerence between MI-IR and VSCF/VCI can be attributed to matrix eﬀects. Frequency shifts and splitting patterns turn out to be unsystematic and dependent on the particular combination of analyte and noble gas. While in the case of neon matrices these eﬀects are small, they are pronounced in xenon, krypton, and argon matrices. Our strategy allows us to suggest that methane rotates in neon matrices – in contrast to previous reports.


Introduction
Matrix-isolation infrared (MI-IR) spectroscopy, where a guest molecule (or analyte) is trapped in an IR inactive host, can be considered as an ideal technique to study small polyatomic molecules.Trapping at cryo-conditions allows for investigation of single entities by quenching all rotational and translational degrees of freedom.In an ideal MI-IR experiment, very narrow bands are recorded and they are characteristic for the vibrational structure of a single molecule.However, compared to gas-phase experiments, the vibrational frequencies are shifted due to weak interactions between the host and the guest.Such matrix frequency shifts have been extensively evaluated, e.g., by Jacox for diatomics 1 and transient species. 2,3][6][7][8][9][10] That means, rotation is not always perfectly quenched.Additionally, most MI-IR spectra exhibit band splitting of several wavenumbers that cannot be explained by rotation.Those matrix effects are usually interpreted by a distortion of the analyte from its equilibrium structure as a consequence of the specific geometry of the trapping sites within the matrix. 11Further complications of the spectrum are due to the formation of larger clusters of the analyte itself (oligomerization) or with impurities.
Although matrix effects are fairly well studied, their manifestation in spectra of different molecules is not easily conceptualized.9][40][41] However, these models are not readily transferable to other systems without further ado.On the other side, approaches for an efficient and accurate calculation of various host-guest combinations rely on approximations that limit the incorporation of different matrix effects and anharmonicity.Some approaches based on molecular dynamics, 42 vibrational self-consistent field 43 and density functional theory 44 have been investigated.
In order to separate matrix effects from anharmonicity, an essential starting point in the spectral assignment is a flexible ab initio computational approach that correctly predicts the anharmonic vibrational structure of the analyte, without explicit consideration of matrix effects.As a matter of fact, the last three decades of algorithm development in computational chemistry have brought up a variety of tools to predict anharmonic vibrational spectra of polyatomic molecules.This field of quantum chemistry is well-reviewed [45][46][47][48][49][50][51] and the current state of quantum chemistry itself is even dedicated to molecules in motion. 48n these grounds, we see the possibility for thorough studies that combine both high-quality IR experiments and ab initio calculations of anharmonic spectra.From the experimental side, MI-IR spectroscopy is suitable for single-molecule investigations, of course only if the before mentioned matrix effects can be handled.For the calculation, we rely on the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) approaches that are based on multimode representations of the potential energy surface (PES), [52][53][54] as implemented by Rauhut et al. 47,[55][56][57] This approach allows for accurate calculation of anharmonic spectra and is flexibly applicable for very different polyatomic systems without relying on empirically fitted analytical PESs.
The major breakthroughs in the spectroscopic investigation of carbon dioxide (CO 2 ) are shown on a timeline in the upper part of Fig. 1.Today, CO 2 is described as a triatomic molecule, collinear and highly symmetric (D Nh point group), with a vibrational structure dissected into asymmetric stretch n 3 , bending n 2 and symmetric stretch n 1 .The first elaborate investigations in the 1920s considered a triangular molecular structure as possible. 58,59In 1929, while the discussion on collinear vs. triangular structure was still ongoing, 60 the first Raman experiments 61 have revealed IR inactive vibrational transitions.A rigorous theoretical model of a collinear, symmetric CO 2 was presented by Dennison 62 in 1931 and only one year later, Fermi 63 interpreted the ''Fermi doublet'' at 1286 cm À1 und 1389 cm À1 in the Raman spectrum, based on the collinear model, as a resonance between symmetric stretch n 1 and the bending overtone 2n 2 .From 1931 to 1933, Dennison and Adel enhanced the theoretical model on the collinear CO 2 [64][65][66] to perfectly interpret gas-phase experiments, which steadily increased in accuracy and revealed fine-splitting that could only be interpreted as rotational-vibrational transitions. 67In the advent of quantum mechanical models for polyatomic molecular systems during the 1930s, the CO 2 surely was the most prominently studied linear rotator, as also highlighted in Herzberg's spectroscopy reference in 1945. 68The early theoretical proceedings were extensively reviewed in 1951 by Nielsen 69  This journal is © the Owner Societies 2020 One particular cornerstone of this theory is the representation of the molecular force field as a power series around the equilibrium structure.With the success of this theoretical model, the parameters of molecular force fields for the CO 2 molecule have been derived from experimental observation, [71][72][73][74] but also from ab initio calculations. 75This allows for the prediction of further rotational-vibrational transitions to provide all characteristic lines of gaseous CO 2 .Today, a vast amounts of experimental gas-phase data are evaluated in combination with computational predictions. 76,77n MI-IR spectroscopy, CO 2 is a good example for the need of understanding matrix effects.In an ideal MI-IR experiment, CO 2 should exhibit the two fundamental vibrational transitions n 2 and n 3 and some overtones and combination bands.In reality, multiple band splitting has been observed.During the last 50 years, different interpretations of this band splitting based on trapping sites, dimerization and complexation have been provided by various authors.The first MI-IR study of CO 2 by Fredin et al. 12 in 1974 was accompanied by a discussion on the occurrence of the CO 2 dimer in Ar matrix to reason the band splitting.They favored a T-shaped dimer (C 2v point group) previously identified in phase IR experiments, 78 and ruled out a staggered parallel dimer (C 2h point group).Furthermore, they found only one trapping site to occur in N 2 matrices, yet, at least two trapping sites in Ar matrices.The latter assumption is also made in the Kr MI-IR study by Loewenschuss et al. 13 Here, additional MI-Raman experiments also show the Fermi resonance (n 1 and 2n 2 ).In 1978, the hypothesis of a T-shaped CO 2 dimer was challenged by Guasti et al. 15 With reference to calculated dimer structures, 79 they favored the staggered parallel dimer and verified this by extensive Ar, Kr and Xe MI-IR experiments.For Ar matrices, they confirmed two trapping sites, assumed that also the dimer bands are split due to two trapping sites, assumed that even larger aggregates can occur at higher temperatures, and they even assigned a band to the CO 2 -H 2 O cluster.In Kr and Xe matrices, the spectra are reported to be less complicated.Most findings up to the year 1982 were summarized by Irvine et al., 16 who provided further Ar MI-IR 17 and D 2 MI-IR 16 experiments that basically consolidated the picture of CO 2 monomers trapped in two substitutional sites and the occurrence of staggered parallel CO 2 dimers.While in 1994, Nxumalo et al. 18 came to similar conclusions, only one year later in 1995 Kno ¨zinger et al. 19 came up with a reassignment of the Ar MI-IR spectrum.Based on mass spectroscopy studies of the aggregation of carbon dioxide (CO 2 ) n in gas-phase mixtures with Ar and Kr and MI-IR investigation of the n 3 region of 12 CO 2 and 13 CO 2 in those mixtures, they concluded that the four bands in Ar matrices cannot be explained by dimerization but by four trapping sites of the CO 2 monomer.In 2000, Schriver et al. 21proposed, aware of the controversies, three trapping sites, a CO 2 -N 2 complex and a tentative assignment of CO 2 staggered parallel dimers in Ar matrices.In 2008, however, a reinvestigation of the Ar and N 2 MI-IR spectrum of CO 2 by Castano et al. 22 14 He nanodroplets 80 and para-H 2 matrix 20,81 have been presented.In He nanodroplets the CO 2 molecule rotates, whereas in para-H 2 matrices it does not, yet, dimerization and up to three trapping sites are discussed.
Striking parallels can be found in the history of the spectroscopy of the methane (CH 4 ) molecule (cf.lower part of Fig. 1).The molecular structure of CH 4 was identified as a tetrahedron based on its stereochemistry, 82 way before vibrational spectroscopy became available.The vibrational structure of the highly symmetric (T d point group) tetrahedral CH 4 molecule is described by the asymmetric deformation n 4 , asymmetric stretch n 3 , symmetric deformation n 2 , and the symmetric stretch n 1 , since the first gas-phase IR experiments of good resolution by Cooley 83 and a first theoretical model by Dennison 84 in 1925 were reported.However, in 1927 a pyramidal model of CH 4 was considered, 85,86 as it is capable of explaining the rotational fine structure in Cooley's spectrum, which implied at least two different moments of inertia in the molecule.At about the same time it was discussed that both models actually could occur as ''dynamical isomers''. 879][90][91][92][93] Even Herzberg dedicated this puzzling discussion a rather lengthy part in his seminal compendium. 68This should be warning enough that spectra can be misinterpreted even for very simple systems and especially when adequate theoretical models have not been developed.][109] Considering MI-IR spectroscopy, the first Xe, Kr, and Ar MI-IR experiments of CH 4 and CD 4 by Cabana et al. 23 in 1963 have revealed the protruding complexity of the spectra.The authors considered dimerization, various trapping sites, symmetry breakdown, and rotation of the molecule within the matrix.Based on thorough theoretical grounds, 38 they concluded that methane exhibits a hindered rotation in their MI-IR experiments, in contrast to its free rotation in gas-phase.They confirmed this conclusion by Kr MI-Raman experiments. 24ater, Frayer and Ewing 29,30 studied the rotational-vibrational transitions of the methane molecule in Ar MI-IR with a focus on its nuclear spin, as it was previously described by Wilson. 96 Chamberland et al. 31 have further consolidated the hypothesis of hindered rotation in Ar, Kr and Xe matrices.In the 1980s, two pure rotational transitions of CH 4 in Ar, Kr, and Xe matrices were observed by Nanba et al. 32 in the far-IR region, proving the assumption of methane's rotation in the matrix environment.Kr and Xe experiments by Sagara et al. 33 resolved the fine structure also in the Raman spectrum of CH 4 and CD 4 , which was previously not observed.Nelander 110 confirmed the hindered rotation of CH 4 and CD 4 in N 2 MI-IR experiments.Further Ar, Kr, Xe and Ne experiments of high-resolution by Jones et al. 34,35 led to the conclusion that methane is trapped in two substitutional trapping sites, where it can undergo a hindered rotation in only one of those sites, in all matrix materials except neon.In the late 1990s, Momose 36 and Hoshina 27 have provided the first para-H 2 MI-IR experiments, where their interpretation and assignment rests on an almost free rotation of methane.At the same time, Govender et al. 111 explained the observed fine structure by dimerization of methane rather than its rotation.Their assignment based on in vacuo harmonic calculations is somewhat controversial, because the lack of anharmonicity and mode-coupling leaves significant discrepancies between theory and experiment while the matrix environment introduce further discrepancies on top of that.Howsoever, the rotation of CH 4 remained commonly accepted.Almost simultaneously with the dimerization studies by Govender et al., Nauta et al. 112 have provided the first IR experiments in He nanodroplets.They have observed a rotational fine structure similar to the gas-phase and argue that methane is essentially a free rotor in He nanodroplets, too.On the other hand, Ogilvie et al. 28 provided Ar and Ne MI-IR spectra in 2011, making the case that methane is not rotating in Ne matrices.
Despite this extensive work, controversies remain.In MI-IR spectroscopy, deviations of theory from experiment are due to matrix effects as well as anharmonicity.While some theoretical investigations have fostered the conceptualization of specific matrix effects, the inability to accurately account for anharmonicity and mode-coupling limited the applicability.By minimizing the error due to anharmonicity, calculated in vacuo spectra provide a reliable starting point for the subsequent identification of matrix effects.Based on this combined experimental and theoretical methodology, we revisit central concepts in MI-IR spectroscopy related to the impact of the matrix on the vibrational spectra of carbon dioxide (CO 2 ) and methane (CH 4 , CD 4 ) trapped in various noble gas matrices.

Matrix isolation infrared spectroscopy (MI-IR)
The matrix-isolation (MI) setup used in this study has been previously described by Galvez et al., 113 and a detailed depiction of the apparatus is found in the supplementary information of ref. 114.The highlights of the setup are the 1801 rotatable high-vacuum cryostat, which can be cooled to 5.8 K to allow for stable solid Ne matrices, and the Bruker spectrometer Vertex 80v, which is operating under 2 mbar with a liquid nitrogen cooled MCT detector to eliminate all atmospheric noise in the IR region.This is especially necessary when studying the IR spectrum of the omnipresent carbon dioxide molecule.The spectrometer itself is vibration-cushioned and uncoupled from the MI apparatus to avoid vibration-induction.The arising gap between the spectrometer and the MI apparatus is constantly flushed with pure N 2 gas.In all experiments, the atmospheric carbon dioxide is not observed with this setup.Analyte-noble gas mixtures are prepared by barometric monitoring in a volume of about 2 liters.A mixture of 2 mbar of the analyte in dilution with about 990-1010 mbar of noble gas, is denoted as 1 : 500.It is injected with a constant flow of 4 mbar min À1 from a volume of about 200 ml and a pressure of 900-980 mbar.The gas is deposited as a matrix onto a gold plate in the highvacuum cryostat chamber at 10 À7 mbar and a temperature of 5.8 K.The deposition time is about 45 min per layer.Background spectra of the gold plate are taken before each sample measurement.The spectra shown in the figures are corrected for this background.To record spectra 512 scans are accumulated at a resolution of 0.3 cm À1 .We measure a sample of 12 C 16 O 2 gas (99.9995%,Messer Austria) at dilutions of 1 : 200, 1 : 1000 and 1 : 5000, both in Ar and Ne.The sample of 12 CH 4 (99.995%,Messer Austria) is measured at dilutions of 1 : 500, 1 : 1000, 1 : 2000 in Ar and at dilutions of 1 : 500, 1 : 1000, 1 : 2000, 1 : 4000, 1 : 8000 and 1 : 32 000 in Ne.Spectra of the 1 : 500 mixtures, both in Ar and Ne, are repeatedly recorded over a time of 2 h.The sample of 12 CD 4 (99.99%,Euriso-top France) is measured at dilutions of 1 : 1000 and 1 : 2000 in Ar and at 1 : 2000 and 1 : 32 000 in Ne.In Ar matrix, temperature change experiments (5.8 K, 7.8 K, 10.4 K, 5.8 K) are carried out for the 1 : 1000 and 1 : 2000 mixtures of both the 12 CH 4 and the 12 CD 4 species.The CO 2 IR spectra are recorded in the MIR region from 7500-500 cm À1 , whereas the CH 4 and CD 4 spectra are recorded from 4000-500 cm À1 .

Vibrational self consistent field (VSCF) and vibrational configuration interaction (VCI)
All calculations are performed with the MOLPRO software package. 115,116The Born-Oppenheimer equilibrium structures of the CO 2 and CH 4 monomers are optimized at coupled cluster level with singles, doubles and perturbative triples excitations and the explicit correlation of interelectronic distances, i.e., the CCSD(T)-F12 ansatz. 117,118This ansatz allows for faster basisset convergence than conventional coupled-cluster approaches with a little additional computational cost.The accuracy of this method using triple-z (VTZ-F12) basis sets 119,120 corresponds roughly to a conventional calculation using quintuple-z basis sets, which is sufficiently close to the complete basis set limit.Harmonic vibrational energies are calculated at the same level of theory by numerical second derivatives and diagonalization of the Hessian matrix, which also yields normal coordinates.Although the term normal mode actually describes the change along a normal coordinate in time, we shall use it interchangeably for the term normal coordinate.The normal modes q i are used as a coordinate system for the subsequent construction of a multi-mode representation of the potential energy surface (PES), a sum of one-mode (1D) potentials, two-mode (2D) potentials, etc., as depicted in Fig. 2. The MOLPRO SURF algorithm constructs the PES on a grid, where each grid point is generated by elongation along the normal modes, with the 1D sub-potentials along one normal mode V(q i ), the 2D sub-potentials along two normal modes V(q i ,q j ), etc. Symmetry considerations, iterative interpolation, and prescreening techniques significantly lower the amount of ab initio electronic structure single point energy calculations needed, with negligible loss of accuracy in the final PES, as pioneered by Rauhut. 55The grid form of the multi-mode PES representation is transformed into an analytical form by a polynomial fitting procedure. 121Based on the multi-mode representation of the PES, the nuclear Schro ¨dinger equation using the Watson operator is solved variationally within the VSCF, [52][53][54] where the wave function is approximated as a Hartree product of one-mode functions.This approximated solution is subsequently correlation-corrected by the configuration-selective state-specific VCI. 47,56,122It yields the VCI wave function as a linear combination of various excited Hartree products, with the anharmonic vibrational energies corresponding to specific VCI states.An analysis of the dominant configurations that contribute to a specific energy allows identification of resonances.The nuclear wave function also allows for the calculation of IR intensities by employing a multi-mode representation of the dipole moment surface.The latter is calculated here at HF/VTZ-F12 level of theory.Furthermore, the VSCF/VCI approach based on a multi-mode PES inherits the normal mode notation in the VCI states, which in further consequence can be directly compared to the spectroscopist's notation of vibrational transitions that has evolved from normal mode analyses.The whole approach is ab initio and is analogously employed for both the CO 2 and the CH 4 molecules.The CO 2 normal modes q i are derived within the Abelian D 2h subgroup as q 4 asymmetric stretch, q 3 symmetric stretch and q 2 /q 1 bend.Considering the actual D Nh point group and reordering by the irreducible representations, those normal modes give rise to the spectroscopic labels for the fundamental vibrations as n 3 asym.str., n 1 sym.str., and n 2 bend.The CH 4 normal modes q i are derived within the Abelian C 2v subgroup, as q 9 /q 8 /q 7 asymmetric stretch, q 6 symmetric stretch, q 5 /q 4 symmetric deformation and q 3 /q 2 /q 1 asymmetric deformation.In the actual T d point group, and after reordering by the irreducible representations, those normal modes give rise to the spectroscopic labels as n 3 asym.str., n 1 sym.str., n 2 sym.def., n 4 asym.def.For both CO 2 and CH 4 the PES is calculated as a four-mode representation within the respective q i coordinate system, where 1D to 4D sub-potentials are calculated using CCSD(T)-F12 single points with frozen-core (fc) approximation and the cc-pVTZ-F12 basis 119,120,123,124 in the case of CO 2 and with all-electron (ae) correlation and the cc-pCVTZ-F12 [123][124][125][126] basis in the case of CH 4 .In Fig. 2, the grid form of the one-mode (1D) and two-mode (2D) sub-potentials are depicted for the normal modes q 4 , q 3 , and q 1 in the case of CO 2 (Fig. 2A) and for the normal modes q 9 , q 6 , q 5 and q 1 in the case of CH 4 (Fig. 2B).As one may comprehend from the graphs in the CO 2 case, the 1D sub-potentials V(q 4 ) and V(q 1 ) constitute a rather harmonic shape (quadratic and quartic potentials), while the V(q 3 ) is definitely of anharmonic shape.The latter propagates into the 2D sub-potentials V(q 3 ,q 1 ) and V(q 4 ,q 3 ), which also illustrates the occurrence of mode-coupling.Depending on the magnitude of anharmonicity and modecoupling, the calculated VSCF/VCI vibrational frequencies can be expected to deviate from the results of the harmonic approximation.Similar statements can be made in the case of CH 4 .

Results
Carbon dioxide ( 12 C 16 O 2 ) All observed features of carbon dioxide 12 C 16 O 2 in the Ar and Ne MI-IR spectra are shown in Fig. 3.The Ar MI-IR spectrum reveals four distinct regions, each comprising a regular splitting pattern of two bands.In the Ne MI-IR spectrum, the same four regions can be identified.The energies are slightly blueshifted compared to the Ar MI-IR spectrum and the occurrence of band splitting is less regular.The spectra were recorded up to 7500 cm À1 , yet, no features were observed above 4000 cm À1 .Changes in the sample dilution have a more pronounced influence on the Ne MI-IR spectrum than on the Ar MI-IR spectrum.
Both spectra can be uniquely assigned, purely by the comparison with VCI calculation.The assignment is summarized in Table 1 and discussed in Section 4. In general, the VCI calculated vibrational transitions are in very good agreement with the observation in gas-phase IR (MAD: 1.5 cm À1 ).Similarly, the transitions observed in Ne MI-IR agree very well with gas-phase IR (MAD: 1.1 cm À1 ).For other host systems, the frequencies are systematically shifted towards lower energies (MAD: 6.7 cm À1 for Ar, 6.3 cm À1 for Kr, 12.1 cm À1 for Xe).
In both Ar and Ne MI-IR, some transitions with rather low intensity are observed.Some arise from aggregation and vanish with increasing dilution.Others are due to the isotopomers 13 S1).
Methane ( 12 CH 4 and 12 CD 4 ) In Fig. 4 and 5, the MI-IR spectra of both the 12 CH 4 and the 12 CD 4 isotopologue are depicted.We observe temperaturedependent band shapes in the experiments performed with Ar matrices (Fig. 4).In both, Ar and Ne matrices, dilutiondependent band shapes can be observed (Fig. 5).
Temperature-change experiments in Ar matrix (Fig. 4) show that already a slight increase in temperature strongly influences the shape of the observed bands.For the 12 CH 4 isotopoloque (cf.Fig. 4, left), the width of the bands significantly increases upon heating from 5.8 to 10.4 K.In the region of 3050-3000 cm À1 , the intensities of the observed bands equalize and the weakest sideband even disappears at 10.4 K.In the region of 1310-1290 cm À1 , the originally most intense band weakens, while the intensity of its nearest neighbor increases.At 10.4 K, there are only two out of six bands left.However, when cooling back to 5.8 K, the initial band shapes in both mentioned regions perfectly recover.In temperature-change experiments of 12 CD 4 in Ar matrices (cf.Fig. 4, right), heating to 10.4 K leads to dissolution of multiple bands into one band, each in the region of 2280-2250 cm À1 and 1010-980 cm À1 .By cooling back to 5.8 K, again, the original band shapes perfectly recover.
For the 12 CH 4 isotopoloque in Ar at 5.8 K and a dilution of 1 : 500 (cf.Fig. 5, top left), we observe four strongly overlapping bands with similar intensity in the region of 3050-3000 cm À1 , and seven bands with very different intensities in the region of 1310-1290 cm À1 .Those two patterns do not significantly change with increasing dilution.They are basically the same for 1 : 500, 1 : 1000, and 1 : 2000.In contrast to that, the Ne MI-IR experiments tend to be more sensitive toward increasing dilution (cf.Fig. 4, bottom left).In each of the two mentioned spectral regions, one band is observed at a dilution of 1 : 500.With increasing dilution to 1 : 1000 and 1 : 2000, the band at 3020 cm À1 loses intensity, while the band at 1307 cm À1 slightly blue-shifts.At a dilution of 1 : 4000 the bands begin to split, while this splitting is best observed at a dilution of 1 : 8000.At a dilution of 1 : 32 000, the splitting of the band at 3020 cm À1 becomes difficult to be distinguished from spectral noise.In contrast to that, the intensity of the bands around 1307 cm À1 is high enough so that the splitting pattern may also be analyzed in the 1 : 32 000 experiment.We here rely on the 1 : 8000 experiment for subsequent assignment.In this experiment, the shape of the band at 3020 cm À1 is somewhat mirrored to the corresponding band in Ar matrix.The band shape at 1307 cm À1 is similar to the corresponding band in the Ar MI-IR spectrum.For both regions, the bands in the Ne spectrum are much closer to each other than the corresponding bands in the Ar spectrum.
Considering the 12 CD 4 isotopologue, similar observations are made.The Ar MI-IR spectra (cf.Fig. 5, top right) are rather insensitive to dilution, exhibiting two or three strongly overlapping bands in the region of 2280-2250 cm À1 and four partly overlapping bands in the region of 1010-980 cm À1 .The Ne MI-IR spectra (cf.Fig. 5, bottom right) drastically change with dilution.Here, one broad band is observed each in the regions of 2270-2250 cm À1 and 1010-980 cm À1 .Upon dilution of 1 : 32 000, those bands transform to very narrow bands   In between two panels, the notation of the vibrational states are given in grey boxes.Unlabeled spectral features are due to dimers or 13 C and 18 O isotopomers (cf.ESI ‡).
This journal is © the Owner Societies 2020 Phys.Chem.Chem.Phys., 2020, 22, 17932--17947 | 17939 comprising also some band splitting pattern.The intensities in the 12 CD 4 experiments are stronger than in the 12 CH 4 experiments.Thus, for the 12 CD 4 experiments, we can consider the high dilution experiment of 1 : 32 000 in our assignment, as this spectrum is not complicated by spectral noise.The band assignment of methane is summarized in Table 2 and discussed in Section 4. Considering the CH 4 isotopologue, the VCI vibrational transitions are in good agreement with the observation in gas-phase (MAD: 3.7 cm À1 ), and the Ne MI-IR experiments are very close to the gas-phase IR data (MAD: 0.4 cm À1 ).For the other host systems, the frequencies are shifted towards lower energies (MAD: 6.7 cm À1 for Ar, 3.4 cm À1 for Kr, 11.8 cm À1 for Xe).It should be seen that these shifts are not systematically decreasing from xenon to neon.Similar results are obtained for CD 4 .
An important observation is made for the line spacing of the rotational-vibrational transitions.Those are in general smaller in the MI-IR experiments than their counterparts in the gasphase spectra, as can be seen in Table 3.For example, in the gas-phase spectra of the CH 4 isotopologue, the line spacing between P(1) and Q(1) for the n 3 fundamental is 19.7 cm À1 .In matrix-isolation spectra, this particular spacing is 13.0 cm À1 in Xe, 11.7 cm À1 in Kr, 11.7 cm À1 in Ar and 8.5 cm À1 in Ne.Considering the n 4 fundamental vibration, the same line spacing between P(1) and Q(1) is narrower, with 11.0 cm À1 in gas-phase, 9.1 cm À1 in Xe, 7.8 cm À1 in Kr, 7.1 cm À1 in Ar and 3.2 cm À1 in Ne.Similar results are observed for other line spacings (cf.Table 3).For the CD 4 isotopologue, the line spacings are in general smaller than for the CH 4 isotopologue.Also here, the spacings show a systematical decrease when going from Xe to Ne, whereas they are generally lower than observed in the gas-phase.For example, the line spacing between P(1) and Q(1) considering the n 4 fundamental vibration are 9.7 cm À1 in gas-phase, 5.1 cm À1 in Xe, 5.4 cm À1 in Kr, 4.6 cm À1 in Ar and 3.5 cm À1 in Ne.
In the here observed spectral range, two non-fundamental transitions occur with detectable intensity.According to the VCI calculations, there should be the 2n 4 overtone at 2615.0 cm À1 for CH 4 (resp.at 1992.8 cm À1 for CD 4 ) and the n 2 + n 4 combination band at 2834.2 cm À1 (resp.at 2092.0 cm À1 for CD 4 ).With very weak intensities, we observe in the Ar MI-IR spectra the 2n 4 transition at 2616.2 cm À1 for CH 4 (resp.at 1981.9 cm À1 for CD 4 ) and the n 2 + n 4 transition at 2834.3 cm À1 for CH 4 (not observed for CD 4 ).In Ne matrix, no overtones and combination bands are observed.This journal is © the Owner Societies 2020

Discussion
In the introduction we presented a historical retrospect of CO 2 and CH 4 infrared spectroscopy.This retrospect shows that wellresolved gas-phase IR spectra need the help of a generalizable theoretical model for conclusive interpretation.It also highlights that the assignment of gas-phase IR spectra was inherently linked to controversial discussions of molecular structure of CO 2 and CH 4 .In the 1960s, roughly 30 years after the gas-phase IR spectra have been successfully interpreted by quantum theoretical models, the novel matrix-isolation technique led to new insights but also new difficulties in describing the vibrational structure of the trapped molecules.The unambiguous interpretation of MI-IR spectra requires not only knowledge of the molecular structure of the analyte but also of the structure of a host-guest system.Therefore, the assignment of MI-IR spectra is inherently linked to a controversial discussion of matrix effects that arise from the peculiarities of the host-guest structure.
First of all, the laboratory preparation of a matrix is an intricate process depending on various experimental parameters.The phase transition of the analyte-noble gas mixture to a solid matrix depends on, inter alia, the deposition speed, the pressure  a Rotational transitions ( J 0 ' J 00 ) are labeled as R( J 00 ) for DJ = J' À J 00 = +1, as Q( J 00 ) for DJ = 0 and as P( Table 3 Line spacings between the rotational-vibrational transitions R(1), R(0), Q and P(1) in the IR spectra of methane isolated in various noble gas matrices and in gas-phase.Data derived from this work in bold letters.Data derived from literature in italics (cf.references in Table 2) and temperature during deposition, the deposition technique (pulsed or continuous) and the mixing behaviour of the analyte and the noble gas, which may not be the same for different mixing ratios.Changing these parameters leads to different properties of the solid matrix.For example, it is often not known whether the matrix is in an amorphous or crystalline phase.In most studies, however, the local environment around the analyte is assumed to resemble the crystal-like ordering of the noble gas that serves as host.
Although assumptions on the local host-guest structure may be valid, it is actually not possible to directly derive the hostguest structure from an MI-IR spectrum.However, the matrix effects impacting the spectra can be very well investigated and rationalized by some basic concepts of the local host-guest structure.Such matrix effects are, among others, band-splitting due to differences in trapping sites, frequency shifts due to the interaction of the host with the guest, and analyte rotation within the matrix trapping site.In the following, we discuss these matrix effects and their conceptualization in a systematic series of host-guest combinations and evaluate the transferability of successful concepts from one host-guest system to another.

Matrix trapping sites
CO 2 in Ar matrices exhibits a band-splitting pattern that has been extensively and controversially discussed in the literature. 12,17,19,21,22Two equally intense bands are observed in each spectral window (cf.Fig. 3).This doublet can be reasonably explained by assuming that CO 2 monomers are trapped in two different trapping sites.As the band-splitting pattern is not changing with increasing dilution, dimerization cannot be the reason for this doublet.The shoulders observed for the bands at 2345.6 cm À1 and 2339.5 cm À1 may be reasoned by further trapping sites, as claimed by Kno ¨zinger et al. 19 or as dimers, as claimed originally by Fredin et al. 12 As these shoulders are barely existent in our experiments, it is sufficient to explain our spectra by the hypothesis of two trapping sites.In literature, similar band-splitting has been observed in Kr matrices, yet, not in Xe matrices. 15However, as experimental literature data are rather sparse for Xe MI-IR, it can be assumed that the band-splitting was simply not reported.
In a previous study on CO 2 in Ne matrices, no discussion on band splitting and matrix trapping sites has been provided, 14 perhaps because the band splitting in Ne matrices is much simpler than in corresponding Ar matrices.Only for the region around 668 cm À1 , a variety of bands occur, yet, their intensity is heavily dependent on the dilution.It is reasonable to assume that a great part of those bands is due to dimerization as most of them disappear upon dilution.However, the doublet remains in this region even with the highest dilution (1 : 5000).It is possible to explain this by two different trapping sites for Ne matrices.However, this is problematic to justify because these two trapping sites should lead to a similar splitting pattern in the other spectral regions, too, which is not the case (cf.Fig. 3).The difference in the Ar and Ne MI-IR spectra of carbon dioxide is a vivid example on how band-splitting patterns are dictated by the chosen matrix material.
In our CH 4 and CD 4 experiments, we observe characteristic band splitting patterns, both for CH 4 and CD 4 in Ar and Ne matrices (cf.Fig. 5).However, these patterns are not as systematic as one may expect from the occurrence of distinct trapping sites, e.g., as seen for the doublet pattern in CO 2 :Ar (cf.Fig. 3).For both CH 4 and CD 4 , the splitting in the v 4 region is very different to the splitting in the v 3 region.In the pioneering studies of CH 4 and CD 4 trapped in noble gas matrices, no direct evidence for distinct trapping sites was given. 23,31Based on high-resolution experiments in Xe, Kr, and Ar matrices, Jones et al. 34 demonstrated later for the v 4 region that CH 4 may enter two different trapping sites.However, they showed that only one of these trapping sites is predominant and that this predominant trapping site is linked to the characteristic band splitting pattern observed in most other studies.We will discuss the assignment of this pattern as rotational-vibrational transitions in the following.

Rotation in the matrix
The n 3 and n 2 region of the CO 2 spectrum have been previously studied by MI-IR with all types of noble gas matrices. 13,15,17,18,21ith reference to those studies, our observations agree with the interpretation that CO 2 does not rotate in solid noble gas matrices.When the main band splitting pattern is assigned by the occurrence of two trapping sites, there remain basically no bands that one could assign as rotational-vibrational transitions.
In various MI-IR studies of CH 4 , the characteristic band splitting patterns (cf.[27][28][29][30][31]34,35,110 As the temperature is very low in MI-IR experiments, only a handful of rotational-vibrational transitions can be observed in the spectra.Considering the isolation of CH 4 in Ar matrices, our observations in various dilution and temperature change experiments directly support this interpretation.High dilution experiments help to rule out oligomerization and complexation as the origin of these band splitting patterns.At high dilutions, the methane molecules are well separated and diffusion within the Ar solid at 5.8 K is hardly possible, thus, oligomerization or complexation is considered as inhibited in our matrices.Oligomerization and complexation may be induced by so-called annealing experiments (e.g. in ref. 23), where diffusion is increased by heating and partial melting of the matrix.However, the chance of complexation and oligomerization in our temperature-change experiments is negligible, as we performed those experiments at a high dilution of 1 : 1000, where the amount of methane molecules within the solid is very low.Consequently, the here observed temperature-change induced alternation of the spectral pattern in Ar matrices is due to nuclear-spin interconversion, as shown by Frayer et al., 30 and is an evidence for rotation within the matrix.Most authors agree on the assignment of rotational-vibrational transitions, particularly R(1), R(0), Q(1) and P(1), for the characteristic band splitting pattern.We adopt this notation in our Ar and Ne MI-IR experiments and update previous literature assignments for the sake of consistency (cf.Table 2).
While it is established in literature to assign rotationalvibrational transitions in the spectra of CH 4 and CD 4 in Xe, Kr, and Ar matrices, this is not the case for Ne matrices.However, our new data strongly suggest an assignment of rotationalvibrational transitions for CH 4 and CD 4 in the Ne MI-IR spectra (cf.Fig. 5, bottom).Our assignment is rationalized based on following reasoning: (1) Assuming rotation of CH 4 in noble gas matrices, the rotational barrier should increase from Xe to Ne, resulting in decreasing rotational-vibrational line spacings.This trend is confirmed in Table 3, individually for the n 3 and n 4 fundamentals of both CH 4 and CD 4 .(2) When going from CH 4 to CD 4 , the rotational constants should decrease by a factor of 2, and with that the rotational-vibrational line spacings should halve approximately.This is observed for all noble gases, also for Ne (cf.Table 3).

Matrix frequency shifts
Combining our results for CO 2 in Ar and Ne matrices with the results in other noble gases from previous studies, 13,15,17,18,21 we observe an anticipated trend when going from Xe to Ne: The frequencies are successively blue-shifted toward the values obtained from gas-phase experiments (cf.Table 1).This holds, at least, for the n 3 and n 2 fundamental vibrations.For CO 2 in noble gas matrices, the following widely-accepted concept holds: The number of electrons decreases from Xe to Ne, which leads to a systematic decrease in the non-bonding interactions of the host with the analyte.Thus, the host's influence on the vibrational structure of the analyte is successively diminished, i.e., the matrix shift is minimized.In the Ne matrix, CO 2 behaves like single molecules in highly diluted gas-phase.Consequently, a very good agreement between Ne MI-IR spectra, gas-phase IR spectra and the VCI calculations based on the model of collinear CO 2 in vacuo was found.
From Raman experiments of CO 2 , the overtone 2n 2 is expected to be in resonance with the n 1 fundamental. 63This has been observed in Kr MI-Raman experiments, 13 where the frequencies are red-shifted by approximately 6 cm À1 compared to gas-phase.Another such resonance is present for the IR active n 1 + n 3 and 2n 2 + n 3 combination bands.In accordance with previous studies, 17,18,21 we observe this resonance in Ar MI-IR.Here, each of the two resonant transitions exhibits a onefold band-splitting (cf.Fig. 3), which can be reasoned with two trapping sites.The centres of those singly split bands are each red-shifted by approximately 11 cm À1 compared to gas-phase.In Ne MI-IR, we do not observe an additional bandsplitting of the resonance, which is in accordance with previous experiments. 14Here, the frequencies of the resonant transitions are only red-shifted by approximately 1 cm À1 compared to gas-phase.To the best of our knowledge, there are no Kr and Xe MI-IR experiments published considering the n 1 + n 3 and 2n 2 + n 3 combination bands for spectral interpretation.Yet, recent He nanodroplet experiments 80 found those resonating states at 3612.42 cm À1 and 3714.44 cm À1 , very close to our observation in Ne MI-IR (3612.4cm À1 and 3714.7 cm À1 ).From these observations, one may deduce that the matrix frequency shifts of the Fermi resonance of n 1 + n 3 and 2n 2 + n 3 show a similar systematic trend as for the fundamental transitions.However, further experiments are needed to prove this.
Comparing the non-rotating Q(1) transition of CH 4 in matrix-isolation with the fundamental vibrational transition in gas-phase, the absolute matrix shift is about 12 cm À1 for Xe and 6 cm À1 for Ar, while it is only about 3 cm À1 for Kr and 1 cm À1 for Ne.In other words, for CH 4 and CD 4 the matrix frequency shift is not systematically decreasing when going from Xe to Ne.This is in central contradiction to the observation made for CO 2 and undermines the concept that the matrix frequency shifts systematically change with the electronic structure of the host system.The concept may be extended by incorporation of other factors that have an impact on the matrix frequency shifts in order to interpret the case of CH 4 and CD 4 , e.g., by considering the crystal structure of the host system.However, such ideas cannot be transferred from one host-guest system to the other, because they are intrinsic to the specific size distribution and electronic interaction between analyte and noble gas.This will be discussed in the up-following paragraph.

Matrix effects in various host-guest combinations
At first glance, the MI-IR spectra of CO 2 and CH 4 (resp.CD 4 ) are very simple compared to the vast number of bands in the gas-phase IR spectra.At a closer look, matrix frequency shifts and splitting patterns complicate the spectra and its interpretation is not straight-forward.While oligomerization can be identified by probing various dilutions of the analyte, further spectral features are rather unsystematic for both species in different matrix materials.The linear molecule CO 2 is best described as a non-rotating monomer in all noble gas matrices.In Ar matrices, it has at least two different matrix trapping sites, whereas (potentially) only one in Ne.The spectra of the spherical rotator CH 4 (resp.CD 4 ) may best be explained by a hindered rotation in all matrices.Different matrix trapping sites can mostly be ruled out in both Ar and Ne.Although matrix frequency shifts systematically decrease from Xe to Ne for CO 2 , this systematic trend is not observed for CH 4 (resp.CD 4 ).In other words, the impact of the matrix on the vibrational structure depends on the molecular symmetry of the analyte, the properties of the host material and the intricate interplay between the two.Table 4 shows that this statement holds true also in a bigger context when we consider our previous Ne and Ar MI-IR studies on water 10 and fluoroethane. 131The matrix effects depend on the specific host-guest combination.
These findings demonstrate the challenges for a theoretical model to accurately calculate MI-IR spectra.The fundamental question is, whether matrix effects can be rationalized by a hostguest structure localized around the molecule under study.If so, one may ask how this host-guest structure can be obtained and how it is most accurately described in a theoretical model.
Matrix frequency shifts can be computed as the difference between the vibrational spectrum of the host-guest structure and the vibrational spectrum of the molecule in vacuo.Thus, several distinct host-guest structures may have to be tested to identify the one that best matches to the experimental spectrum.Several approaches can be invoked.For example, the host can be modelled with an implicit force field or by placement of host atoms.With a VSCF/VCI approach based on multi-mode PES representations, an extension of the molecular framework by the addition of noble gas atoms was successfully demonstrated for iodine in argon. 43he computation of matrix splitting patterns is more tedious, because observation of band splitting in experimental spectra typically implies the occurrence of different trapping sites.Consequently, a variety of host-guest structures have to be evaluated to reproduce these experimental findings.This approach has recently been applied in a theoretical study of CO 2 in argon within the harmonic approximation, showing promising results. 44The agreement with experimental spectra would further profit from incorporation of anharmonicity and mode-coupling in the calculations.At this level, most matrix effects should be predictable from theory with high accuracy.

Conclusion
The correct account of anharmonicity and mode-coupling in computations of molecular vibration, opens a promising route towards the identification of matrix effects in a MI-IR spectrum.In this study, we demonstrate that in vacuo VSCF/VCI calculations based on one-the-fly calculated multi-mode PES representations, i.e., a theoretical model derived for the description of gas-phase spectra, provide a suitable starting point for the interpretation of MI-IR spectra.Especially when Ne matrices are used, the discrepancy between theory and experiment is within 1-3 cm À1 , considering the here presented results of methane and carbon dioxide.With the access to accurate anharmonic molecular spectra, it remains to address matrix effects.Those are observed in all kind of noble gas matrices.Incorporating also observations from our previous MI-IR and VCI studies on water and fluoroethane, we see that matrix effects cannot systematically be accounted for but intricately depend on the host-guest combinations.Historically, the conclusive interpretation of gas-phase IR spectra was ultimately linked to the development of a general and transferable theoretical description of molecular vibration.Similarly, we suppose that the unambiguous interpretation of matrix effects in MI-IR spectra requires general and transferable concepts to model the hostguest structure.Such a universally applicable model deems necessary to ultimately solve controversies, such as the occurrence of trapping sites or rotation within the matrix.

70 Fig. 1
Fig. 1 Schematic timeline of spectroscopic progress of gas-phase IR/Raman and matrix-isolation IR of CH 4 and CO 2 , together with the major theoretical foundations to their interpretation.The textboxes are colored by the type of experiment (grey = gas-phase, blue = Kr, Ar, Xe and N 2 matrices, red = Ne matrix, green = para-H 2 matrix, yellow = He nanodroplets).
C 16 O 2 and 12 C 18 O 16 O.VCI calculations for those species show the same good agreement with the experiment as mentioned for 12 C 16 O 2 .Those results are provided in the ESI ‡ (cf.Table

a
Fermi resonant transitions are marked with FR. b Data from this work are in bold letters (CO 2 : Ar 1 : 5000 at 5.8 K, CO 2 : Ne 1 : 5000 at 5.8 K), data from the literature in italics.c VSCF/VCI calculation with up to quintuple excitations on a 4-mode PES at CCSD(T)-F12A/cc-pVTZ-F12. Harmonic approximation (Harm.) at CCSD(T)-F12A/cc-pVTZ-F12. d Taken from ref. 17 (Xe/CO 2 ratio between 100 and 23 000, at 6-8 K). e Taken from ref. 13 (Kr/CO 2 ratio between 500 and 30 000, at 20 K). f Taken from ref. 77.g Taken from ref. 127.h Mean absolute deviation (MAD) and maximum deviation (MAX) with reference to gas-phase IR.For Ar and Kr the higher-lying energies of the split bands have been considered.

Fig. 3
Fig.3Mid-infrared spectrum of carbon dioxide ( 12 C 16 O 2 ) isolated in argon (top, blue) and neon (bottom, red) matrices at 5.8 K.The spectra depicted as colored solid lines are considered in the assignment.Frequencies assigned to the monomer are labeled with their relative intensity in brackets.In between two panels, the notation of the vibrational states are given in grey boxes.Unlabeled spectral features are due to dimers or13 C and18 O isotopomers (cf.ESI ‡).

Fig. 4
Fig. 4 Impact of temperature changes in the mid-infrared spectrum of a methane-argon matrix (left: 12 CH 4 , right: 12 CD 4 ), initially deposited at 5.8 K.

Fig. 5
Fig.5Mid-infrared spectra of methane (left:12 CH 4 , right: 12 CD 4 ) isolated in argon (top, blue) and neon (bottom, red) matrices at 5.8 K and various dilutions.The spectra depicted as colored solid lines are considered in the assignment.Frequencies assigned to the monomer are labeled with their relative intensity in brackets.In between two panels, the notation of the vibrational states are given in grey boxes.
again revived the ''two trapping sites'' hypothesis in Ar matrices.Based on computed harmonic frequencies, the authors corroborated the assumption of Schriver et al. on the CO 2 -N 2 complex and reintroduced the CO 2 dimer in a similar assignment as Fredin et al. from 1974.They claimed to resolve all controversies, without explicitly disproving the arguments by Kno ¨zinger et al. from 1995.Since 2000, further MI-IR experiments in Ne matrix, They observed time-dependent changes in the IR absorption intensity, which served as evidence for nuclear spin interconversion from ortho-to meta-methane.In 1970, as a continuation of the work by Cabana et al., the MI-IR temperature change studies of

Table 1
Directly observed fundamentals, overtones, combination bands of the carbon dioxide ( 12 C 16 O 2 ) monomer in various noble gas matrices, in gasphase and from in vacuo calculation

Table 2
Rotational-vibrational transitions of methane ( 12 CH 4 and 12 CD 4 ) in various noble gas matrices, in gas-phase and from ab initio VCI calculation l MAD (MAX) for Q-branch

Table 4
Matrix effects in neon and argon, as observed for polyatomic molecules of different point groupsH 2 O [C 2v ] a CH 4 [T d ] C O 2 [D Nh ] CH 3 CH 2 F b [C s ]