Computational vibrational spectroscopy for the detection of molecules in space

CHAPTER SIX

Computational vibrational spectroscopy for the detection of molecules in space Ryan C. Fortenberrya,*, Timothy J. Leeb a

Department of Chemistry & Biochemistry, University of Mississippi, University, MS, United States MS 245-3, NASA Ames Research Center, Moffet Field, CA, United States *Corresponding author: e-mail address: [email protected] b

Contents 1. 2. 3. 4.

Introduction Mathematical framework Early applications of QFFs Modern utilization of QFFs for vibrational spectra 4.1 Composite energy QFFs 4.2 Explicit correlation within QFFs 4.3 Computation of IR intensities 5. The successes of QFFs in astrochemistry 6. Conclusions Acknowledgments References

174 174 177 183 183 187 189 191 193 194 194

Abstract Quartic force fields have been defining the potential portion of the internuclear Hamiltonian for decades. This review discusses the history of their development as a tool for analyzing and producing vibrational and rovibrational spectra for molecules of interest to astrophysical observation. Coupled cluster theory has long been a necessary partner in this development, and correlation consistent basis sets, second-order vibrational perturbation theory, and now explicitly correlated electronic wavefunctions have demonstrated consistent usefulness in the determination of these properties. The current status of this approach has produced vibrational frequencies within 1.0 cm
1 of experiment in many cases, preceded the laboratory observation of specific molecules, and is now becoming a helpful tool for the detection of new molecules in space.

Annual Reports in Computational Chemistry, Volume 15 ISSN 1574-1400 https://doi.org/10.1016/bs.arcc.2019.08.006

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2019 Elsevier B.V. All rights reserved.

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1. Introduction Space is different from Earth. The interstellar medium (ISM) is a cold near-vacuum. The corona of a star is a relatively dense near-hell. The rest of the cosmos typically lies somewhere in between. This diverse set of temperatures and pressures gives rise to molecules both common and curious, durable and dainty, firm and floppy. The detection of these molecules (and what they can tell us about the Universe) relies upon spectroscopic data from million- and even billion-dollar telescopic instruments typically falling in the infrared (IR) and radio/microwave regions of the electromagnetic spectrum. While many of the desired molecular species cooperate for experimental elucidation, others are more coy and combative. Quantum chemistry does not struggle to get the molecules to produce their spectroscopic secrets, but it does struggle to get these data accurate enough for comparison to laboratory experiments and astronomical observation. However, as modern quantum chemical theories have developed, so has their accuracy and reliability. Consequently, the application of quantum chemistry to vibrational, rotational, and rovibrational, spectral characterization of nonterrestrial molecules is a natural fit (1, 2), a veritable “match made in Heaven.”

2. Mathematical framework There are numerous means of solving the vibrational Schr€ odinger equation, but this current text will focus on the rise of quartic force fields (QFFs) as the potential portion within the Watson Hamiltonian. While there have been some cases where such an approach does not work, this review will highlight those where it does and how this has shaped how astronomical observations or laboratory simulations of interstellar environments can detect novel molecules. Hence, a brief discussion of the mathematical framework follows. The fundamental equation of quantum mechanics is Shr€ odinger’s famous equation. At least the cat did not eat that homework assignment. Typically, the Watson Hamiltonian is implemented for solving rovibrational problems and is of the form: H¼

1X 1 X ∂2 1 X ð Jα
π α Þμαβ ð Jβ
π β Þ

μ + V ðQÞ: 2 αβ 2 k ∂Qk2 8 α αα

(1)

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Jα is the total angular momentum of a given cardinal direction (x, y, or z) denoted by α or β in Eq. (1) while π α is the total vibrational angular momentum of the same direction; μαβ is the inverse of the moment of inertia tensor for the given geometric coordinates; Qk is a single normal coordinate; and Q is the set of all normal coordinates (3, 4). Most of these pieces are related to kinetic energy and are solved through derived mathematical equations. However, the potential portion is left for definition on its own. This piece is the most costly in solving vibrational structure computations as it necessarily depends upon the system in question. The potential energy surface (PES) is the most common interpretation of V (Q). These have been computed since the earliest days of quantum chemistry and have provided tremendous insights into the behavior of molecules. While PESs certainly provide numerical data to the human interpreter of the data, they often can be tremendously difficult to compute. They can require tens of thousand of points (or more) to be computed, and if accuracy is desired, each of those points can be a nontrivial computation. However, local PESs can suffice for vibrational frequency computations and even finer localized minima are adequate for rotational constant determination. However, even these minima can require significant computational time. Mostly, the PES and V (Q) term is a function. Under the harmonic approximation and for bond stretches, this potential function is a simple parabola fit to the classical spring constant equation of: 1 V ¼ kx2 : 2

(2)

Fitting a parabola as a function is an easy process. This can be done numerically with ease as only three points are required to fit the function with one point being the minimum. Usage of a mere three points also requires some assumptions about the regularity of the function typically implying that five points are really the best minimum number. These five points are most readily identified as the minimum of the PES and then subsequent  δ and  2δ displacements from there. Quantum chemical programs have been coded to treat these types of harmonic computations for decades, and the equivalent analytic derivatives for certain computational levels are also standard in most modern programs, as well. In order to really solve the nuclear Schr€ odinger equation using the Watson Hamiltonian in Eq. (1), a more complete description of the PES must be formed. This can be done by taking regular steps along the coordinate of interest and fitting the resulting energy to a Morse potential for

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input into Eq. (1). Computations of this nature are incredibly expensive but often are necessary. However, how many points really are enough? While millions of points could be used, the simplest number of points is nine, a very special nine points. Again,  δ displacements from the minimum begin to describe the shape of the function. However, by selecting factors of δ from 1 to 4, regular steps are created. These regular steps actually produce numerical derivatives, and higher-order derivatives can be fit together to define the function in a Taylor series (5). Combinations of various coordinates where no more than four displacements are contained produces an approximation to the potential function. This fourth-order Taylor series expansion of the internuclear potential is the quartic force field or QFF. This is given in the form of: V¼

1X 1X 1X Fij Δi Δj + Fikj Δi Δj Δk + Fikjl Δi Δj Δk Δl , 2 ij 6 ijk 24 ijkl

(3)

where the Fij… terms are the force constants while the ΔiΔj… terms are the displaced distances for coordinates i, j, and so forth (6, 7). This formulation implies that the anharmonic potential can be described for significantly fewer points than a traditionally constructed PES curve. However, the number of points to be computed grows geometrically as the number of atoms increases. Hence, the cost reduction in each coordinate goes down notably, but the computation of the potential is still time consuming even for small systems. All of the above issues related to the number of points on a PES are moot if analytic first, second, third, and fourth derivatives of the energy with respect to nuclear position have been derived for the chosen level of theory. At this point a QFF can be constructed directly from the analytic force constants. However, the complicated, non-Hermitian nature of the most accurate of truncated theories, coupled cluster theory, combined with the relative lack of need for anything beyond second derivatives for most chemical applications has kept analytical derivatives largely confined to lower levels of theory (8–11). Hence, either numerical differentiation, displacements of Hessians (akin to harmonic force fields), or semiglobal PES construction are required for accurate rovibrational quantum chemical predictions. Once the potential has been produced, the Watson Hamiltonian can be solved. The simplest means of solving the vibrational Schr€ odinger equation is with perturbation theory at second-order (VPT2) based on Rayleigh– Schr€ odiner perturbation theory (4, 12–14). Derivatations of the anharmonic

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portion’s contributions to both the vibrational and rotational equations are employed to shift the computed values from their equilibrium and/or harmonic values. This method is quick and reliable for rigid systems. Less rigid systems may require the construction of full, anharmonic vibrational wavefunctions. These are akin to their electronic structure cousins in vibrational self-consistent field and vibrational configuration interaction (VCI) approaches (15, 16). QFFs can be utilized within VCI as long as continuous functions with proper behavior at the asymptote (the so-called proper limiting behavior) like those that fit Morse-cosine coordinates are constructed (17). However, even VCI based on QFFs can fail for floppy systems (18) requiring that semiglobal PESs be constructed. Even so, QFFs with VPT2 are most often reliable and have produced novel insights into molecules of astronomical interest for decades.

3. Early applications of QFFs The advent of coupled cluster theory especially at the singles, doubles, and perturbative triples [CCSD(T)] level ushered in a new era within quantum chemistry where a sweetspot balancing accuracy and computational cost was achieved (19–24). Combining the balance of time and accuracy for vibrational computations from QFFs and for electronic approaches with CCSD(T) was a natural fit. During this time, news of the ozone layer depletion was starting to make its way into policy discussions, and concern over environmental issues was starting to influence scientific research. This confluence of events naturally prompted the first CCSD(T)-based QFF rovibrational analysis to focus on O3. Additionally, previously computed lower levels of theory had gotten the ordering of the vibrational frequencies incorrect with respect to what experiment appeared to indicate. Hence, the need to understand this pertinent molecule better and the advent of new theories led to the application of the former to the latter (25). Contemporary but previous work had shown that then cutting-edge theories of the CCSDT-n series as well as QCISD(T) were performing fairly well for the prediction of harmonic frequencies while CCSD+T(CCSD) was not (26–28). However, CCSD(T) was beginning to show its efficacy. This first application of QFFs based on CCSD(T) with atomic natural orbital (ANO) basis sets for O3 (25) produced a ν1 of 1105 cm
1 while experiment was 1103 cm
1. Similarly, ν2 turned out to be 699 cm
1 with experiment at 701 cm
1. The b2 antisymmetric stretch, however, was 67 cm
1 below the experimental 1042 cm
1 value. Regardless, the fact that the two totally

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symmetric modes performed so well was highly promising for the application of CCSD(T) and QFFs to the prediction of vibrational frequencies for the detection of molecules with interest to the atmospheric and astrophysical sciences. This approach was soon thereafter leveraged into clusters (3 and 4 atoms) of beryllium and their magnesium analogs (29, 30). Herein multireference configuration interaction (MRCI) and CCSD(T) again with ANO basis sets were shown to perform similarly with CCSD(T) not requiring the additional step of defining the active space which can introduce its own errors if not properly selected. In both cases of Be and Mg, CCSD(T) along with QFFs for vibrational frequencies and spectroscopic constants gave highly promising results for even these bizarre inorganic species. Hence, more traditional, p-block inorganic molecules of environmental interest should also be as readily treated. This was validated with the analysis of NO2+ a common member of the NOx (NOX) family of atmospheric pollutants (31). The existing state of the gas-phase experimental data placed the bond length at 1.153 A˚, while the CCSD(T)/ANO computations put this value closer to 1.122 A˚ (31, 32). The resulting vibrational frequencies from this analysis were instrumental in finally allowing for the laboratory detection of this fairly common atmospheric cation within a year’s time (33). This success was following shortly in 1993 by analysis of the greenhouse gases N2O and CO2 (34). In this early CCSD(T)/cc-pVTZ study, the 2350.8 cm
1 theoretical value for the antisymmetric stretch in CO2 was nearly spot on with the 2349.1 cm
1 experimental fundamental frequency. The symmetric stretch came in 7.2 cm
1 below, and the bend was 15.1 cm
1 below experiment. N2O was within 1.0 cm
1 for both stretches and 5.4 cm
1 above experiment for the bend. Similar accuracies were also achieved with ammonia with the QFF computed via CCSD(T)/cc-pVQZ just prior to the success with N2O and CO2 (35). Hence, QFFs based on CCSD(T) with triple-zeta and even quadruple zeta basis sets were beginning to show highly accurate representations for vibrational frequencies even more than 25 years ago. As the successes of accurately computing vibrational frequencies with QFFs, VPT2, and CCSD(T) increased, it was realized that combining such accurate purely ab initio computations with some level of empirical correction for experimentally known observables could further increase the predictive power of computational anharmonic vibrational frequencies. The common terrestrial and known interstellar (36) formaldehyde molecule had been well-classified in its fundamental vibrational frequencies by this

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point in 1993, but the overtones and combination bands as well as data related to the HDCO and D2CO isotopologues had not been as thoroughly analyzed and vetted by the community. CCSD(T)/cc-pVQZ fundamental vibrational frequencies were within 8.0 cm
1 of experiment in every case but one, the ν5 antisymmetric stretch (37). Theory placed this fundamental at 2791.7 cm
1 while experiment was much higher at 2843.3256 cm
1. However, iterative manipulation of the harmonic frequencies allowed for more accurate descriptions of the subsequent anharmonic fundamentals: ½n + 1

ωi

½n

½n

¼ ωi + νi ðexpÞ
νi ðcalcÞ:

(4)

In such a procedure using Eq. (4), the ωi values are modified until convergence is reached. The adjustment that is brought into the force field, in this case for formaldehyde, cleaned up the computed fundamental vibrational frequencies such that they were indistinguishable from experiment (as would be desired) and created overtones and combination bands that often deviated from experiment by less than 2.0 cm
1 and only rarely by more than 10.0 cm
1 (37). Additionally, frequencies of new bands were predicted increasing the known spectral database for this molecule. Even so, this was mostly a proof-of-concept study showing that higher vibrational quanta could be accurately produced theoretically. This further expanded upon the accuracy of computation beyond what plain ab initio computations could do. At this point in the early-to-mid 1990s, theoretical prediction of vibrational frequencies of small molecules was becoming more mainstream. However, molecules of interest for the upper atmosphere, in the ISM, or circumstellar media can be more exotic than formaldehyde and carbon dioxide. One such example is nitroxyl, HNO first observed toward the center of the galaxy in Sagittarius B2 (38). Accuracies for the CCSD(T)/ cc-pVTZ and cc-pVQZ anharmonic vibrational frequency predictions of this molecule were expected to be high. The ν2 and ν3 bending and heavy atom stretch modes, respectively, were 6.1 and 5.4 cm
1 above experiment, but the hydride stretch was below by 16.9 cm
1 (39). Granted this is excellent agreement, but if the expectation is to predict unknown values, better accuracy is needed. The previous work on formaldehyde had employed variational procedures for solving the vibrational structure computations, and such computations had promised accuracy for modes that had some degree of large-amplitude motions. However, VSCF and its “correlated” VCI improvement require continuous functions with proper limiting behavior, i.e., proper asymptotic convergence at the dissociation

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limit. Fitting these points to a Morse potential in the case of bond stretches and cosine functions in the case of bends or torsions can create continuous potential functions with proper limiting behavior. Hence, HNO was the first molecule to which a QFF was fit to a potential with proper limiting behavior comprised of Morse-cosine coordinates (39). The combination of CCSD(T)/cc-pVQZ for the energy points in the QFF and a transformation of the function into Morse-cosine coordinates for use in VCI produced anharmonic vibrational frequencies of 2683.4, 1502.3, and 1568.8 cm
1 each within
0.6, 1.5, and 3.5 of known experiment (39). The era of “spectroscopic accuracy” (differences with experiment on the order of 1.0 cm
1 or less) appeared to be approaching with such accuracies becoming routinely possible. The CCSD(T)/cc-pVQZ QFF of methane produced accurate fundamentals, as well, and further improvements to the QFF through Morse-cosine transformations have made this 1995 paper an established benchmark of methane IR spectral benchmarking (7). While the use of variational anharmonic methods like VCI are highly useful, they can become costly computations in and of themselves. Hence, VPT2 and making improvements to its use is still preferred. However, the accuracies of the pure QFF, VPT2 approach were hampered by the presence of resonances within the fundamental frequency range. The variational results overcome this issue due to the nature of the fundamental mode contributions in the individual fundamental’s wavefunction, but VPT2 approaches are much less computationally intensive. Hence, treatment of resonances within VPT2 can maintain accuracy but still keep vibrational structure computational costs at a minimum. One of the most straightforward sets of resonances are Fermi resonances. Type-1 Fermi resonances have an overtone of a lower-frequency fundamental close to a higher-frequency fundamental. This perturbs and displaces the observable, higher-frequency fundamental changing its energy. Similarly, a type-2 Fermi resonance is a combination of two (or more) lower-energy frequencies combining to perturb a higher-frequency fundamental in the same way. Treatment of such Fermi resonances had been undertaken for some time, and these were included in this analysis of methane (7). However, this methane study showed that two sets of mulitple Fermi resonances of either type perturbing the same fundamental (or multiple fundamentals are perturbed by the same Fermi resonances) produces larger shifts in the physical frequencies than simple resonance inclusion can correct. Hence, this “polyad” of resonances was necessary to further improve accuracy in QFF/VPT2 anharmonic frequency predictions.

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The first insertion of polyads of Fermi resonances into VPT2 computations was analyzed with a CCSD(T)/cc-pVTZ QFF of ethylene (40). While the raw agreement between the QFF values and experiment is quite good, inclusion of the resonant polyads reduces the error for each of the 12 modes to less than 10.0 cm
1 in every case. The example ν2 fundamental frequency was computed to be 1630.4 cm
1 while experiment reported 1625.4 cm
1. However, treating the 2ν10 ¼ ν2 type-1 Fermi reduced the computed frequency to 1621.3 cm
1 reducing the error from 5.0 to
4.1 cm
1. More interestingly, the computed ν11 b1u hydride stretch dropped from 3000.7 to 2979.1 cm
1 when including the ν2 + ν12 ¼ ν3 + ν12 ¼ ν11 resonance polyad (40). While both the raw and polyad-treated ν11 frequencies actually bookend the 2988.6 cm
1 experimental value, the shift brought about by the introduction of the polyad demonstrates that further inclusion of higher terms in the resonance structure is necessary to get the right answer for the right reason. This idea of resonance polyads was further refined for the analysis of trans-HNNH, a molecule of high importance in the search for the spectroscopically dark but essential nitrogen molecule. In Ref. 41, the full set of polyad treatment is laid out and inclusion of further Coriolis resonances are also introduced into the VPT2 treatment. The ν1 symmetric hydride stretch is affected by three resonant overtones and combination bands creating a tetrad of ν1 ¼ 2ν2 ¼ ν2 + ν3 ¼ 2ν3. Additionally, the ν5 antisymmetric hydride stretch has a triad of resonant frequencies creating the ν5 ¼ ν2 + ν6 ¼ ν3 + ν6 resonance equivalence. In the case of the ν1 tetrad, the possible frequencies for this inclusion range from 3004.7 to 3165.9 cm
1 with the best description predicting the frequency 3033.3 cm
1 nearly 100 cm
1 below the experimental 3128 cm
1 frequency. However, this analysis led to the realization that the previously assigned ν1 band actually corresponds to the resonant ν2 + ν3 combination band. The ν5 stretch, on the other hand, computed at 3125.0 cm
1 confirmed the experimental 3120 cm
1 value (41). For the latter, the polyad ranged from 3125.0 to 2807.0 cm
1 with the best description of the function creating the final eigensolution. Additionally, the various deuterated forms were also analyzed giving excellent accuracies compared to experiment (41). These studies showed that vibrational frequencies must be treated with not only Fermi resonances but polyads of them in order to produce accurate results. This has recently become essential in the computation of PAHs where the density of modes in a given frequency range is very high, and small resonances compound having significant effects (42).

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With this inclusion of Fermi resonance polyads as well as the Coriolis and later Darling and Denison resonances, the level of accuracy that could be produced with the vibrational structure computations was maximized for VPT2 at this point. In order to improve the computation of anharmonic vibrational frequencies, the QFF itself would have to be improved. The first step of this came in increasing the basis set space in order to describe fully the Hartree–Fock limit. The Dunning and coworkers’ cc-pVnZ basis sets (43–46) had already shown their value in computing the fundamental frequencies, but the additional property of complete basis set (CBS) limit extrapolation for these basis sets promised an even more accurate definition of the QFF than what even CCSD(T)/cc-pVQZ had delivered previously. The first analysis of this was applied to the ammonium cation (47, 48), but not for the full QFF. The CBS extrapolation was applied to the harmonic frequencies since the CCSD(T)/cc-pVQZ computations on the full QFF were too costly at the time. However, the anharmonic corrections were included from the CCSD(T)/cc-pVTZ QFF. Even for this incomplete CBS-including QFF, VPT2 produced ν3 and ν4 to be 3345.1 and 1447.2 cm
1 as compared with the experimental values of 3343.1399 and 1447.2158 cm
1. The ν4 mode is exceptionally accurate. The CCSD(T)/cc-pVTZ ν3 and ν4 values were also good at 3339.1 and 1446.2 cm
1, but the CBS extrapolation was better. While this study was ongoing, the CBS extrapolation was fully developed producing highly accurate atomization energies and proton affinities (48). The form utilized here and subsequently employed even to now for the computation of vibrational frequencies is     1
4 1
6 EðlÞ ¼ A + B l + +C l+ 2 2

(5)

where l is the highest angular momentum and the cardinal letters represent the energies for the various basis set levels descending from A. Other extrapolation schema exist, but the above has been benchmarked effectively for this application. As the 20th century came to a close, the accuracies of both the electronic and vibrational structure computations were converging to produce highly accurate anharmonic fundamental vibrational frequencies. Hence, these types of computations started to be applied to molecules for which there was either little or conflicting experimental data. However, one last class of molecules had yet to be benchmarked. The first of these was an analysis of the hydroxyl anion, OH
, (49). Diffuse functions were required in the

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computations, but aug-cc-pV5Z (in one of its first implementations) was actually shown to overestimate the vibrational frequency by 7 cm
1. Even so, the rotational constant was shown to be identical, for all intents and purposes, with experiment (49). Consequently, even anions submitted to accurate vibrational analysis utilizing CCSD(T)/cc-pVnZ QFFs with VPT2. At this time, the presence of the cyano anion had long been known in the laboratory but had yet to be observed in high-resolution rovibrational spectroscopic experiments. The radical was well known and was even one of the first molecules observed in the ISM (50, 51). Initial benchmarking of the CCSD(T)/aug-cc-pV5Z QFF and its VPT2 results produced a CN radical fundamental of 2041.8 cm
1 and a rotational constant of 1.88435 cm
1 in excellent agreement with experiment at 2042.42 and 1.89107 cm
1, respectively, (52). As such, the predicted CN
fundamental of 2039.9 and 1.86392 cm
1 rotational constant was highly insightful for the field. In fact, this showed that in any experiment where both the CN neutral radical and the CN
closed-shell anion may be present that the band centers in any vibrational or rovibrational analysis would be directly on top of one another explaining why they had not been resolved previously in the laboratory (52). This insight allowed for the spectra of the radical and the anion to be separated such that CN
was ultimately observed rotationally in the atmosphere of the carbon-rich star IRC +10 216 (53). As a result, vibrational and rovibrational quantum chemical computations were becoming essential tools in the analysis of new molecules in space.

4. Modern utilization of QFFs for vibrational spectra The modern era of QFFs utilizing VPT2 is characterized by two trends: composite energy schema and the incorporation of explicit correlation. The first of these two had been utilized previously to some extent in the QFFs for NH4+ and CN
(47, 52). While CCSD(T)/cc-pVQZ and CCSD(T)/ANO had produced some shockingly accurate vibrationally frequencies, these were not consistent and often were highly accurate for some modes and not for others. Hence, a more consistent accuracy was needed.

4.1 Composite energy QFFs Several approximations are present in most quantum chemical computations, most notably including the Born–Oppenheimer approximation. However, another common one in correlated methods is the frozen core approximation.

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Ryan C. Fortenberry and Timothy J. Lee

Several basis sets have been developed over the past decades to treat coreelectron inclusion in full correlation especially the commonly used ones from Dunning and Peterson (54, 55), but Martin and Taylor developed what is referred to as the MT basis set for the treatment of core-electron correlation in these types of quantum chemical applications (56). Additionally, while CCSD(T) offers a sweetspot of great accuracy for reasonable computational cost, if absolute accuracy is the goal, CCSD(T) is not completely adequate. Furthermore, the electrons have some relativistic character even in hydrocarbons that could be corrected. Finally, full quantum electrodynamics will have some effect, as well, on the final energies of the systems. However, the last of these is incredibly difficult to compute and likely would have less than a 0.1 cm
1 corrective effect for light atoms (57). The issue with all of these necessary inclusions is that a CCSDTQPH/ aug-cc-pwCV9Z-DK QFF further corrected for non-Born–Oppenheimer behavior is impossible to compute in a timely fashion. However, breaking down the individual components and adding them together has been a time-tested approach within quantum chemistry. While there are dozens of such composite schema for various applications, methodologies have been developed for anharmonic vibrational frequency computations by Lee and coworkers (discussed below) as well as by Peterson and coworkers (58–63). While the two are very similar to one another, the following will review that developed by the former. The CBS extrapolation was already the first step in utilizing more than one computation for determining the energy of a single point in a QFF (47, 48) setting the stage for the composite approaches that would require multiple energies to be computed at each point. Lee and Huang (64, 65) implemented what has become a highly successful composite energy scheme that is done in two parts. The first is optimization of the reference geometry, and the second is in computing the energies at each point. Regarding the former, ultimately, the reference geometry is refit into the equilibrium geometry in the creation of the potential itself from the QFF. Hence, the reference geometry does not necessarily have to include all of the desired corrections for the QFF as long as the reference geometry is close to the ultimate equilibrium structure for the full, composite QFF, typically within one displacement value. CCSD(T) was shown to be sufficient for the method and higher-order electron correlation was not necessary. The quintuple-ζ basis level is close enough to the CBS limit for the geometry. Relativity had little effect on the geometry for these light atoms, but core-electron correlation

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185

does. Hence, after exploring several different method/basis set composite combinations, the most efficient reference geometry is computed via: R ¼ R5Z + ½RMT ðcoreÞ
RMT ðvalenceÞ:

(6)

At this stage, the grid of points defining the QFF could be produced, and the energies at each point could be computed. Displacements of 0.005 A˚ for bond lengths and 0.005 radians for bond angles and torsions performed the best for typical molecular systems. Then, at each point, the 3-point basis set extrapolation from Eq. (5) (48) involving aug-cc-pVTZ, aug-cc-pVQZ, and aug-cc-pV5Z (TQ5) was shown to be the most effective CBS definition. The 2-point (66) Q5 extrapolation was nearly as good, but the computational cost of the triple-ζ computations is so small compared to the essential larger basis sets that computation of the aug-cc-pVTZ energies is recommended (64, 65). For instance, the CCSD(T)/CBS(TQ5) ν1 symmetric stretch in water was computed to be 3662.5 cm
1. Experiment was notably lower at 3657.05 cm
1 (64). While this is still a solid result, further enhancements could be produced. Inclusion of simple scalar relativity from the Douglas–Kroll electronic Hamiltonian (67, 68) is incorporated by computing a single-point energy via CCSD(T)/cc-pVTZ-DK twice, once with the relativity turned on and a second time with it excluded. Larger DK basis sets actually led to a loss of precision making the cc-pVTZ-DK set optimal. The difference in these energies is added to the CCSD(T)/CBS value for each energy point. For water (64), the ν1 fundamental vibrational frequency lowered to 3659.7 cm
1, much closer to experiment. The other vibrational frequencies were also lowered by 2 cm
1 each encroaching upon their respective experimental values. In this initial treatment, higher-order electron correlation was included through the use of the averaged coupled-pair functional (ACPF) method (69). The ACPF computation is an internally contracted, multireference procedure that is designed to behave like full-CI due to the inclusion of approximate size-extensivity in the CI expansion. This correction was computed as the energy of ACPF/aug-cc-pVQZ minus CCSD(T)/aug-cc-pVTZ. Addition of this corrective factor further reduced the ν1 frequency of water to 3656.8 cm
1 within 0.3 cm
1 of experiment (64). The other modes differed from experiment by 4.0 and
2.1 cm
1. Hence, all of the modes for a single molecule were now producing accurate vibrational frequencies. This was further borne out for the HO2+ molecule, as well. Later work, discussed below,

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utilized CCSDT as a means of incorporating higher-order electron correlation, and there are numerous ways of doing this even going up to the CCSDT(Q) level (70). This procedure was also confirmed for the CCH
and NH
2 closed-shell anions (65). Additionally, corrections for core-electron correlation were tested utilizing the aug-cc-pCV5Z basis set showing that an additive correction was just as effective as explicitly including core-electron correlation in the CBS-extrapolated energies and cost significantly less computational time. This was later corroborated (71). The use of the MT basis set for the core-electron correlation portion of the composite energy points was implemented in a 2011 study of protonated cyclopropenylidene (c-C3H3+ ) (3). Again, the MT basis set was designed for this purpose, and it showed more consistently accurate frequencies than the core-including correlation consistent basis sets at the triple- and quadruple-ζ levels. The most widely used (in our groups) composite method for QFFs was codified in 2011 in a study of hydrogenated carbon dioxide (72). At this point, the standard energy scheme for the QFF looked like: E ¼CCSDðTÞ=CBSðTQ5Þ + ðCCSDðTÞ=MTðcoreÞ
CCSDðTÞ=MTðvalenceÞÞ + ðCCSDðTÞ=cc-pVTZ-DKðRel:Þ
CCSDðTÞ=cc-pVTZ-DKðNoRel:ÞÞ + ðCCSDT=aug-cc-pVTZ
CCSDðTÞ=aug-cc-pVTZÞ:

(7) (8) (9) (10)

This was called the CcCRE QFF and has been used (along with its derivatives) in our groups since 2011. The “C” term is for CBS, the “cC” is for core correlation, “R” is for relativity (scalar here), and “E” is for higher-order electron correlation. This study (72) found that the ACPF/aug-cc-pVQZ computation was much too expensive for general use. However, the CCSDT/ aug-cc-pVTZ computations were incredibly costly, and the additional gain from the “E” term was negligible for an increase of over 50% in total computational time for the entire set of points in the QFF (72). This implementation of the CcCR QFF within VPT2 produced a ν1 O
H stretching and a ν2 C¼O stretching fundamental of 3640.7 and 1862.2 cm
1 within 5.0 and 10.0 cm
1, respectively, of experiment at 3635.702 and 1852.567 cm
1 (72–74). The CcCR family of QFFs from Eq. (10) was benchmarked with a few modes for these two HOCO and c-C3H3+ studies, but they also provided what are likely highly accurate descriptions of the other vibrational frequencies that

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were not known at the time and largely have not been analyzed in the time since with a few exceptions discussed later. Subsequently, the same CcCR QFF has been employed to study other molecules where few if any vibrational modes are currently available. A few frequencies are known for some of the systems, and the CcCR QFF with VPT2 has produced accuracies of 1.0 cm
1 for HOCO+, NNOH+, protonated acetylene (c-C2H3+ ), SiC2, and HOSi+ among others (75–80). Additionally, it has predicted vibrational frequencies for numerous other molecules for which there are no experimental data providing the first, and likely a highly accurate, insight into the vibrational nature of various molecules. These include CH2CN
believed to be a possible carrier of the diffuse interstellar bands (81, 82) as well as numerous other anions, various noble gas species which are now known to be possible interstellar molecules (83), and even truly inorganic species like dimers of magnesium oxide and fluoride (84–93).

4.2 Explicit correlation within QFFs While the accuracies and applications of the CcCR QFF within VPT2 in the prediction of fundamental vibrational frequencies for the interstellar detection of molecules are quite exciting for increasing the chemical inventory of space, the computational costs of these methods are still rather high. In fact, the largest CcCR QFF thus far analyzed has been on molecules of only six atoms, l/c-C3H3+ , CH2NH2+ , and c-(C)C3H2 each requiring more than 3000 points on the QFF surface (94, 95). The number of points on a QFF, while still sparse, grows geometrically with the increase in the number of atoms regardless of whether these are hydrogen, uranium, or anywhere in between. Hence, similar accuracies as CcCR for lower computational costs are required for more “realistic” (i.e., larger) molecules to be studied via QFFs in order to produce physically meaningful vibrational spectra. The first exploration of explicit correlation for defining QFFs compared CCSD(T)-R12 and -F12 methods (96) to standard CCSD(T) approaches where both utilized basis sets from double-ζ quality all the way to quintuple-ζ and CBS extrapolations for canonical CCSD(T) (97). The tested systems including H2O, N2H+, NO2+ , and C2H2 yielded excellent agreement between explicitly correlated wavefunction-based QFFs and those from CCSD(T)/CBS(TQ5). Additionally, both were excellent in their comparisons to experiment. For example, CCSD(T)-F12/cc-pVQZF12 produced anharmonic vibrational frequencies for water of 3658.7, 1597.4, and 3756.1 cm
1, all within 3.0 cm
1 of experiment with the

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stretches within 1.5 cm
1 (97). It was stated at the time (2010) that explicit correlation would one day replace the need for CBS extrapolations, but, then, the R12 and F12 methods still needed more development at the time specifically in the area of core-electron treatments in order for them to become generically useful in QFF formulations (97). However, more recent work (98) has shown that such considerations for core-electron treatment may not be as necessary as previously thought at least for closed-shell molecules. QFFs for 41 different molecules were computed via CCSD(T)-F12/aug-cc-pVTZ and compared to published CcCR values and experiment where possible. While the overall mean absolute error (MAE) in comparing individual anharmonic vibrational frequencies was not ideal, the largest cause of this error lay with the radicals. Removing them from the set produced a MAE of less than 6.0 cm
1 compared with CcCR. For the few available experimental values, the error was actually smaller at less than 5.0 cm
1 (98). The rotational constants did not compare nearly as well from CCSD(T)-F12 to either CcCR or experiment, but this work showed that CCSD(T)-F12 is viable for producing an accurate QFF used to generate fundamental vibrational frequencies. Additionally, a single set of CCSD(T)-F12 computations is markedly faster than any set of CcCR values. At the triple-ζ level, HOCO+ required a mere 8 h to compute a CCSD(T)-F12 QFF. However, the full CcCR took more than 20,000 h on a comparable computer system. As such, accurate QFFs can be produced in relatively short amounts of time for exciting molecular species as happened with the determination of the vibrational frequencies of H2SS+ believed to be an intermediate in the formation of the known HSS interstellar molecule (99–101) discussed in more detail below. Alternatively, QFFs for larger molecules can also now be explored. For instance, the 3789 QFF points required for the six-atom system, c-(C) C3H2, took less than 72 h to compute with CCSD(T)-F12/cc-pVTZ-F12 while the CcCR version (which only did a TQ, 2-point CBS extrapolation) took months to complete. Even so, ongoing work on the protonated form of c-(C)C3H2 and the allyl cation require more than 10,000 and 20,000 QFF points, respectively. Such numbers of computations are doable with CCSD(T)-F12/cc-pVTZ-F12 utilizing modern computing hardware since each energy point takes less than an hour. Even so, these relatively huge numbers of points in the QFF for molecules with “only” seven and eight atoms are showing that QFFs themselves still have substantial costs. The inclusion of analytic derivatives of any level for CCSD(T)-F12 would reduce the cost of computing the QFF. Gy€ orffy and Werner (102)

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have implemented analytic gradients for CCSD(T)-F12, and such could greatly open the number of atoms in a molecule that can be treated for anharmonic vibrational treatment. Additionally, treatment of core-electron correlation should also be explored with these modern implementations of F12 methods in order to reduce the MAE compared to experiment or higher-level canonical CCSD(T). Hence, there is still more work in the development of QFFs and their use with VPT2 or even VCI for application to the detection of new molecules in space. However, CCSD(T)-based QFFs in general have already shown their use in assisting with novel molecular detections both in the laboratory and even in space.

4.3 Computation of IR intensities QFFs are defined strictly from nuclear displacements and the associated electronic energies resulting from integrals involving the wavefunction and the Hamiltonian. IR intensities, however, require the construction of a dipole moment surface (DMS), and thus result from nuclear perturbations as well as the response to an external electric field. In the simplest form, the DMS is represented by just the dipole derivatives and, when combined with harmonic frequencies, the resulting IR intensities are termed the “doubleharmonic approximation.” The analytic evaluation of dipole derivatives are a second derivative with respect to the electronic energy. Consequently, double harmonic IR intensity computations have been common for more than 30 years. For example, see Ref. (103) and references therein. The calculation of anharmonic IR intensities has received considerably less attention than the calculation of anharmonic vibrational frequencies. Quantum chemical procedures for anharmonic IR intensities using second-order vibrational perturbation theory, including spreading the intensity over polyad resonances, has been developed on at least four occasions (104–107), while variational calculation of IR intensities has been more prevalent (108–110). VPT2 anharmonic IR intensities only require up to the third derivatives of the dipole moment, but, in general, variational calculations require at least a semiglobal DMS. As a result, development in the computation of anharmonic intensities has not paralleled that for the corresponding frequencies. Highly accurate IR intensities, rivaling experiment, can be computed variationally, again, via a semiglobal DMS provided the DMS is computed with a high-level electronic structure method, such as CCSD(T), and with inclusion of sufficient diffuse functions in the one-particle basis set (18, 108, 110, 111).

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Unlike a QFF, however, a simple local or limited DMS probably cannot be defined and still obtain highly accurate IR intensities. In one recent example, semiglobal DMSs were computed for H2O, CO2, and SO2 using the CCSD(T)/aug-cc-pVTZ method, and the resulting rovibrational IR intensities are nearly indistinguishable from experiment (112). The accuracy of this approach means that QFFs combined with these types of DMSs can accurately fill in the gaps in spectral databases where experiment falls short. Further empirical refinements to the QFF can produce global line lists that are more than sufficient to augment high-resolution experimental data in many cases. Further discussion of this could fill an entire review and is reserved for a later narrative. While CCSD(T)/aug-cc-pVTZ produces intensities indistinguishable from experiment, lower-level computations using second-order Møller– Plesset perturbation theory (MP2) (113) with the same basis set have been shown to agree with CCSD(T) to within 10% but cost a fraction of the computational time (112). For instance, the ν3 antisymmetric stretch of SO2 peaks at 3.9  10
20 cm/molecule for CCSD(T) and 4.2  10
20 cm/molecule for MP2. The DMS for CCSD(T) took over 5000 CPU hours to compute while the MP2 DMS took less than 500 CPU hours. Additionally, simple double-harmonic intensities using MP2 have been shown to be sufficient for providing a semiquantitative (and often quantitative) prediction for fundamentals that are not highly influenced by resonances (63). The antisymmetric NN stretch in the NNHNNH+ proton-bound complex (111), for example, has a MP2/aug-cc-pVTZ double-harmonic intensity of 241 km/mol, and a CCSD(T)-F12/cc-pVTZ-F12 semiglobal DMS produces an intensity for this same mode at 214 km/mol. The former took less than 10 min to compute while the latter took more than 1000 CPU hours. Other modes of this molecule produce errors on the order of 20%, but they absolutely provide trends in relative intensities for the various fundamentals for exceptionally low computational cost. More work is needed in method development and protocol benchmarking for ab initio anharmonic intensities. Currently, low-level methods generate semiquantitative data about intensities that are helpful for providing general trends and relative peak heights for vibrational absorption. However, higher-level methods provide fully quantitative predictions for these same observables. Unfortunately, these are costly computations and cannot benefit from similar sparsities as QFFs do. Further work may lead to betterdefined DMSs for efficient and accurate computations of IR intensities, but this is currently an open area of research.

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5. The successes of QFFs in astrochemistry A previous review describes the successes of quantum chemistry as an assisting tool in the discovery of new molecules in space (80). Hence, the rest of this narrative will focus on QFFs in particular. The aromatic cyclopropenylidene (c-C3H2) hydrocarbon molecule was discovered in the ISM in 1985 (114) based on previous computational work providing data regarding the vibrational frequencies and, most importantly, the rotational constants along with the dipole moments (115). In fact, c-C3H2 is so common in space (116, 117) that it has been detected outside of our galaxy (118), in the diffuse ISM (119), and in various isotopic abundances (120). While the initial quantum chemical work that aided in the detection was not a QFF, subsequent QFF work on this molecule has produced the full set of vibrational frequencies for this system. These data should enhance IR characterization of astrophysical regions where this molecule is known to exist (121–124). More importantly for this discussion is that c-C3H2 is believed to form by way of its protonated counterpart, c-C3H3+ , through dissociative attachment with an electron. The QFF for this D3h, aromatic system was produced in 2011 along with the linear isomer (3) and was one of the two papers that helped to codify the CcCR QFF itself. Regardless, the CcCR QFF predicted the ν4 hydride stretching frequency to be 3131.7 cm
1. Subsequent experimental work followed three years later utilizing this frequency as a starting point to search the experimental data. Ultimately, this frequency and the vibrationally excited rotational constants also computed were able to assist in determining that the gas-phase ν4 fundamental is 3131.1447 cm
1 (125). Hence, theory guided experiment in this case. Having such a benchmark for the behavior of the CcCR QFF VPT2 results has led to searches for c-C3H3+ in space based on the theoretical data. While these remain inconclusive at the moment, comparison of theoretical lines for acetylene to those observed toward the Orion nebula has been inspiring for the promise of such interactions between theory and astronomical observation in the future (126, 127). The discovery of the argonium cation (ArH+) in the ISM (83) and subsequent follow-up work (128–130) has shown that truly unexpected molecules are present in space. This first, naturally occurring noble gas molecule is actually found to be in higher abundances than CO in certain environments (129). Additionally, other noble gas molecules have been observed in the

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laboratory even if the helonium ion (HeH+) (131, 132) was notoriously absent from interstellar surveys and searches (133, 134). until the past year (134a,b). ArN2+ and ArHAr+, for example, have been produced experimentally (135, 136), and dozens of noble gas species have been alayzed computationally (137–142). However, the bonding, or the strongest bonding at least, in the lower atomic number noble gas atoms in these ions is almost always dative in nature. Hence, the noble gas atom donates electrons to an electron-deficient ligand. This is true in argonium and in the experimentally produced HArF structure (143). Hence, argon bonding to the hydroxy cation would produce a shared bond. This was first explored computationally in the context of the larger [Ar,O,H]+ PES (144), and quantum chemists proposed the existence of ArOH+ in the ISM. This CcCR QFF VPT2 study provided the first vibrational frequencies and spectroscopic constants for ArOH+ in order to aid in its detection (88). This molecule was subsequently observed in the laboratory via its ν1 O
H stretch (145) showing that the flexibility of theory can lead experiments into potentially difficult but rewarding avenues of exploration. No interstellar searches have reported the presence of ArOH+, but the relative energetics also provided theoretically (88) should assist in abundance modeling necessary for possible observation. Similarly, the HOCO+ is one of the most abundant molecules in space and was observed in 1981 in the Orion Nebula and confirmed in 1984 in Sagittarius B2 (146, 147). As discussed previously, the CcCR QFF VPT2 frequencies were within 1.0 of the experimental 3375.37413 cm
1 ν1 O
H stretch (75, 148, 149). Similarly, the B and C rotational constants in this near-prolate rotor were both within 15 MHz of experiment (150). Consequently, when Bizzocchi and coworkers (151) were classifying HOCO+ for its rest frequencies in order to explore the use of this form as a probe of carbon dioxide with radio telescopes, the CcCR QFF VPT2 data were further benchmarked with the experiment. Then, the CcCR QFF VPT2 predicted rotational constants for the ν5 ¼ 1 vibrationally excited state were utilized to determine accurately the rest frequencies for this state of HOCO+. Differences in theory and experiment were assumed to be 15 MHz once more, and the lines predicted for this vibrational state were observed. A similar analysis followed for DOCO+ using data from the same CcCR QFF VPT2 analysis and showed that interstellar radio lines from the pre-stellar core L1544 thought to belong to DOCO+, in fact, could not be (151). As such, these quantum chemical data are giving necessary insights into astrochemical observations beyond even the laboratory.

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One of the most exciting uses of quantum chemical anharmonic data in the context of astrochemistry comes in the aforementioned story of HSS. Work in our group had explored the possible presence of this radical in the ISM. Previous work had returned a nondetection of HSS in a warm stellar core (152) even though this simple S
S bonding molecule had been abundantly observed in comets (153). The CcCR QFF VPT2 data (100) showed that the vibrational spectrum of this molecule was likely very weak but that the rotational spectrum would be observable for the upper-limits on the HSS abundance previously imposed (152). Then, within 12 months, the detection of HSS (or equivalently S2H) was reported in the Horsehead Nebula (101). The type of lines the CcCR QFF VPT2 results indicated would be best for searches were among those utilized in this molecule’s detection. Subsequent work in our group showed that the gas-phase portion of the molecular synthesis likely would not proceed through HSSH+ but should go through the H2S
S+ isomer instead (99, 154). This has led to questions about oxygen’s ability to engage in this similar “oxywater” bonding isomer (155–158) which may be a possible explanation for the high abundance of O2 in the coma of comet 67P/Churyumov–Gerasimenko (159). Regardless, this work is showing how QFF VPT2-based spectroscopic data and quantum chemistry as a whole are assisting observation in significant ways in order to grow our understanding of the chemical universe.

6. Conclusions Clearly, quantum chemistry is moving into an era where it can interact directly with astrochemical modelers and observational astronomers. Additionally, the interplay between QFF VPT2 data and that from laboratory experiments has never been stronger, and new developments in the field will only strengthen these interactions. Other developments beside these mentioned are ongoing including the use of empirically parameterized and highly accurate limited line lists (112, 160, 161), the continued development of DMSs for the prediction of intensities/absorptivities of anharmonic frequencies (63, 109), and expanding QFFs to even larger systems through exploration of various other electronic structure methods (42). The past development of QFFs in conjunction with VPT2 has shown notable steps in accuracy, and this is continuing. While VPT2 will never be able to properly treat large-amplitude motions, more well-behaved modes within those same molecules have been well-represented (18, 111, 162, 163). Additionally, VCI and other variational vibrational methods are likely

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much too costly to analyze molecules with more than half-a-dozen or so atoms. Hence, VPT2 still will be a necessary tool for predicting anharmonic vibrational frequencies well into the future, and the new ways of defining the QFF itself as developed over the past three decades and continuing into the future promise to provide a trove of data necessary to characterize molecules of astrochemical interest. These quantum chemical data will no doubt be an asset for support of NASA’s Stratospheric Observatory for Infrared Astronomy (SOFIA) and the James Webb Space Telescope (JWST) missions. In the coming years, it may be that the only limit on what type of comparable spectroscopic data that can be generated will be the creativity within the minds that set the computations to work.

Acknowledgments R.C.F. acknowledges NASA grant NNX17AH15G, NSF grant OIA-1757220, and start-up funds provided by the University of Mississippi for support of this work. T.J.L. is supported by the National Aeronautics and Space Administration through the NASA Astrobiology Institute under Cooperative Agreement Notice NNH13ZDA017C issued through the Science Mission Directorate as well as support from the NASA 16-PDART16 2-0080 and NASA 17-APRA17-0051 grants.

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52. Lee, T. J.; Dateo, C. E. Accurate Spectroscopic Characterization of 12C14N
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