Perspective: Accurate ro-vibrational calculations on small molecules
aa r X i v : . [ phy s i c s . c h e m - ph ] S e p Perspective: Accurate ro-vibrational calculations on smallmolecules
Jonathan TennysonDepartment of Physics and Astronomy, UniversityCollege London, Gower Street, WC1E 6BT London, UK (Dated: October 16, 2018)
Abstract
In what has been described as the fourth age of Quantum Chemistry, variational nuclear motionprograms are now routinely being used to obtain the vibration-rotation levels and correspondingwavefunctions of small molecules to the sort of high accuracy demanded by comparison with spec-troscopy. In this perspective I will discuss the current state-of-the-art which, for example, showsthat these calculations are increasingly competitive with measurements or, indeed, replacing themand thus becoming the primary source of data on key processes. To achieve this accuracy ab initio requires consideration small effects, routinely ignored in standard calculations, such those due toquantum electrodynamics (QED). Variational calculations are being used to generate huge list oftransitions which provide the input for models of radiative transport through hot atmospheresand to fill in or even replace measured transition intensities. Future prospects such as study ofmolecular states near dissociation, which can provide a link with low-energy chemical reactions,are discussed. . INTRODUCTION Quantum chemistry has for the last fifty years placed tremendous emphasis on solvingthe molecular electronic structure problem but the nuclei in molecules also move. Observingthese moving nuclei is at the heart of high resolution spectroscopy and their energy levels alsoprovide the means of quantifying a wealth of thermodynamic properties via the partitionfunction.[1] Traditionally nuclear motion was treated using perturbation theory based onharmonic vibrational motion and rigid rotational motions. This model provides much ofthe language of spectroscopy and approaches based on it are continuing to be developed,for example through use of quartic force fields [2] and vibrational second-order perturbationtheory (VPT2) [3]. However, the harmonic-oscillator rigid-rotor model is firmly rooted in thenotion of small amplitude motion about an equilibrium geometry so must be limited in itsregion of applicability: it must break for all systems as they are excited towards dissociation.Nuclear motion methods based on direct solution of the time-independent Schr¨odingerequation are increasingly being used to compute rotation-vibration energy levels for a rangeof states up to and even above dissociation for important small molecules. While such com-putations used to require a national supercomputer,[4, 5] they can now be performed on agood workstation.[6–8] Increasingly nuclear motion calculations are becoming the primarysource of information on small molecules as the results of these calculations are competitivewith or, in some cases, more reliable than measurements. In this context I note that thevalue for “spectroscopic accuracy” of 1 cm − oft-quoted by theoreticians appears to havebeen chosen more for quantum chemical convenience than because it is true, or indeed useful,value. Rotation-vibration spectra can only be considered to be high resolution at accuraciesapproaching 0.01 cm − , a value I would suggest should be used for “spectroscopic accu-racy”. This perspective will discuss situations where theory is competitive with or replacingobservation as the primary source of data. In other words will address situations wherefirst principles calculations are replacing experiment for key data because either they canbe computed more accurately or are too difficult to measure reliably. In this context I notethat if one is providing computed data for use in models or other applications then it shouldalso be incumbent on the provider to also supply some estimated associated uncertaintyof these data [9, 10]. The perspective will also mention some key areas in small moleculespectroscopy where high accuracy remains a distant goal.2he nuclear motion methods used to solve spectroscopic problems are generically knownas variational methods. This is because, at least in their early implementations, they in-volved obtaining direct solutions of the nuclear motion Schr¨odinger equation using suit-able basis functions to represent the wavefunction. Within the limitations of the Born-Oppenheimer (BO) approximation, these solutions are variationally exact for a given po-tential energy surface (PES).[11] The name variational has stuck even though many codesnow adopt the grid-based discrete variable representations (DVRs) to represent the vibra-tional wavefunctions.[12] DVR methods are not strictly variational and can show convergencefrom below,[13] but have proved robust and reliable in practical calculations. Developmentand improvement of these methods continues apace[14] and the whole area of high accu-racy treatment of nuclear motion calculations has been dubbed the fourth age of quantumchemistry.[15]For the purposes of this perspective I will take small molecules to mean ones containingup to five atoms. For these systems, use of variational nuclear motion methods almost alwaysmeans that the errors arising from these nuclear motion calculation reflect the underlyinginaccuracy of the PES employed and, possibly, issues with the BO approximation. I willconsider in turn the small but growing number of cases were full ab initio treatments areproviding benchmark accuracy; the more standard case which makes use of experimentaldata to help provide accurate results; finally I consider future prospects and, in particular,states around the dissociation limit and the link with chemical reactions. Before doing thisI will outline the various motivations for performing such calculations. II. USES OF BOUND STATE NUCLEAR MOTION CALCULATIONS
Like others, I originally started performing nuclear motion calculation to test poten-tial energy surfaces. High resolution spectroscopy can obtain transition frequencies withexquisite precision[16] and therefore provides a stringent test of potentials. For some timenow, this process has routinely been treated as an inverse problems with nuclear motioncalculations used to determine spectroscopically-accurate PESs using observed data.[17–22]This procedure is now the main source of high-accuracy PES and has got to the point thatthe most accurately determined geometry for the water molecules comes from a refinedPES;[23] other structural determinations are also increasingly relying on high accuracy the-3retical calculations.[24, 25]Even without the need for refining the PES, variational nuclear motion calculations havebeen used to predict[26] and assign[27, 28] spectra. They have also been used to probe thefundamental behaviour of molecules revealing the clustering of energy levels at high rota-tional angular momentum[29, 30], the rearrangement of the levels around the monodromypoint which occurs when a bent molecule becomes linear,[31] and the quantal behavior ofclassically chaotic systems.[32] More recently such calculations are playing a role in guidingobservations of processes important for fundamental physics such as a possible change inthe proton-to-electron mass ratio.[33, 34]Explicit summation of energy levels can be used to give temperature-dependent partitionfunctions and other thermodynamic properties such as the specific heat.[1, 35] These datacan be combined to give equilibrium constants as a function of temperature.[36] For hightemperatures, typically
T >> et al. [48] in 1985; they computed an extensive, if not accurate by modern stan-dards, list of spectral lines for hot (2000 K) HCN. They showed that use of this line list ina model atmosphere of a ‘cool’ carbon star made a huge difference: extending the model ofthe atmosphere by a factor of 5, and lowering the gas pressure in the surface layers by oneor two orders of magnitude. Subsequent calculations on water showed it has a similar line4
Wavelength, m m F ( W mm ) l - - - m VSTAR STDS CH VSTAR 10to10 CH FIG. 1: Infrared spectrum of a T-dwarf 2MASS J055591915–1404489 as observed asobserved using the Infrared Telescope Facility (IRTF) [40] as modelled using the codeVSTAR [41] and the empirical spherical top data system (STDS) [42] for methane or usingthe 10to10 methane variational line list [43].blanketing effect in oxygen-rich cool stars.[49] This has led to the computation of extensiveline list of transitions for hot molecules by a number of groups.[19, 50–55] Recent work hasparticularly focussed on providing line lists for hot methane,[51, 56–58] the use of which havealso been shown to have a dramatic effect on models of astronomical objects, [43] see Fig. 1Although the driver for computing hot line lists has largely been astronomical applications,there are actually many terrestrial applications in areas such as combustion, enviromentalmonitoring and plasma discharge studies for which they are also routinely being used. Theseline list can also be used to give other properties such as cooling functions [59] and radiativelife times of individual states.[60] They also can form the input to models of electric-fieldinteractions with polar molecules such as strong-field induced ro-vibrational dynamics andoptoelectrical Sisyphus cooling.[61]While much attention is focused on the calculation of energy levels and hence transtitionfrequencies, most practical applications also require transition intensities. As discussedbelow, the provision of transition intensities is becoming an increasingly important reason5or performing nuclear motion calculations.
III. HYDROGENIC SYSTEMS AS BENCHMARKS
Since the pioneering work of Kolos and Wolniewicz [62], H has always provided the ab initio benchmark for high-accuracy spectroscopic studies. Recent theoretical calcula-tions on the frequency of the fundamental vibration of H agrees with observation withintheir mutual uncertainty of 2 × − cm − .[63] This work demonstrates what is needed forthe precise ab initio determination of ro-vibrational energy levels. It transpires that thenon-relativistic problem can be solved equally accurately using a direct fully-nonadiabaticapproach [64] or using the more traditional BO separation approach of solving the frozengeometry electronic structure problem[65] augmented by diagonal (adiabatic)[66] and off-diagonal (non-adiabatic)[67] corrections to the BO approximation. Rather remarkably, thecurrent largest source of uncertainty is the treatment of quantum electrodynamic (QED)effects;[68] in this it echoes high precision calculations on the isoelectronic helium atom.[69]The spectrum of H provides another probe of possible electron-to-proton mass variation[70],a phenomenon whose strength is sensitive to terms which arise from BO breakdown.For diatomic systems high-accuracy studies are increasingly becoming based on theuse explicitly-correlated Gaussians to treat both the electronic and nuclear motionsimultaneously.[71] However, this methodology has yet to make significant impact in poly-atomic systems, even ones with containing few electrons such as H +3 . Here a more pragmaticmodel based on high-accuracy electronic structure calculations using explicitly-correlatedGaussians[72] and a simplified treatment of non-adiabatic effects using effective vibrationaland rotational masses[73] has been found to provide excellent predictions of ro-vibrationaltransition frequencies[74] and intensities[75]. This work demonstrated the importance forhigh accuracy of both using an extensive grid of points in the electronic structure calcula-tion and being careful in how they are fitted to the functional form used to represent thePES.[76] The subsequent focus in these studies has been on including the effects of quantumelectrodynamics[77] and improving the treatment of non-adiabatic effects.[78–80] The recenthigh-precision spectra recorded for H +3 and its isotopologues[81, 82] will in due course serveas benchmarks against which improved ab initio procedures can be tested.While improving the accuracy with which ab initio calculations can predict measured6IG. 2: A small portion of the photodissociation spectrum of H +3 ; as reported by Kemp etal. [83]. Reproduced with permission from Phil. Trans. R. Soc. Lond. A (2000) 358,2407. Copyright 2000 The Royal Society.high resolution spectra of H +3 and its isotopologues has made steady progress, the problemof treating the spectrum of these species near dissociation remains unsolved. The denseand complicated near-dissociation spectrum of H +3 and its isotopologues was systematicallyrecorded by Carrington and co-workers for a decade starting in 1983.[84–86] Figure 2 showsa small (0.12 cm − ) region of the near-dissociation spectrum of H +3 illustrating the densityof lines and their variable widths, which reflects the different decay lifetimes of the variousstates. This spectrum was recorded by monitoring by the dissociation of the molecularion into H + H + and thus records transitions to temporarily bound states sitting abovedissociation. Attempts to model these spectra quantum-mechanically have so far givenlittle insight into the underlying physical processes involved and certainly provide nothingapproaching any line assignments.[87–89]Before moving to larger systems it is worth mentioning the intriguing H +5 system. Thisfour-electron ion does not provide a benchmark for accuracy but instead provides a funda-mental fluxional system in which the atoms freely interchange even at energies easily probedby spectroscopy. The stable CH +5 ion provides a similar system.[90, 91]. Accurate rep-7esentation of the rotation-vibration states of H +5 and its multiple isotopically-substitutedforms has proved very challenging. In particular, new methods of treating the ro-vibrationalsymmetry of these fluxional systems have had to be developed.[92, 93]. Obtaining reliableanalytic fits to a PES which shows ten[94–96] energetically-accessible stationary points haveproved difficult.[97] These systems display unusual spectroscopic properties [98] and newmethods for treating the nuclear motion problem have been developed.[99–105] Work onthese fluxional system is far from complete. IV. HEAVIER SYSTEMS
The stand form for storing spectroscopic data is usually referred to as a line list. A linelist comprises two components: (a) a list of energy levels which can be used to give transitionfrequencies and (b) a list of transition probabilities. For very large lists it is recommendedthat these are stored separately to minimize disk usage.[106, 107] For both energies andintensities, calculations start from electronic structure calculations giving an initial ab initio
PES and dipole moment surfaces (DMS). There is increasing evidence that best resultsrequire the consideration of effects, such as QED,[108] which are usually considered to betoo small to be important for chemically important molecules. Furthermore, even if thePES only provides a starting point for a fit to spectroscopic data, these fit improve fairlysystematically as the ab initio model used as the starting point is improved.The basic procedure for systematically improving ab initio is the use of the focal-pointanalysis (FPA) [109] or closely-related variants. In this procedure a base (focal point)calculation is performed at some high but affordable level of theory. The magnitude of theeffects neglected or approximated in focal point calculation are corrected for individually byperforming additional calculations. These calculations consider effects such extrapolation tothe complete basis set limit, core correlation, high-order correlation, scalar relativistic effects,QED correction, spin-orbit effects and corrections to the Born-Oppenheimer approximation.The top-up calculations are performed using a mixture of larger (variational) calculations andperturbation theory as is appropriate for each effect. Examples are given in the papers citedelsewhere in this perspective.[109–114] Of course, for simpler systems with fewer electrons,more of these effects can be included in the base calculation; for example starting from anall-electron calculation eliminates the need to consider separately correlation of the core8lectrons.Nuclear motion calculations on ab initio
PES’s for systems containing more than H atomsrarely reproduce observed transition frequencies to much better than 1 cm − .[22, 110, 112,113] However there are well-worked procedures for improving the PES by fitting to spec-troscopic data.[21, 115–117] Features of these procedures include the use of the Hellmann-Feynman theorem to avoid having to compute derives of the energy with respect to param-eters of the PES by numerical finite differences, the inclusion of rotationally excited statesas fits to vibrational energy levels alone are prone to give false minima in the fit,[118] andthe use of the initial ab initio points to constrain the fit [117] which helps to stop the PESfrom becoming unphysical in regions where it is not determined by the available experimen-tal data. Spectroscopically determined PES’s are capable of reproducing the observed datawith accuracies approaching 0.01 cm − . There are residual issues with how to deal withissues arising from failure of the BO approximations with some fits simply using a singlePES to represent all isotopologues[21] and others explicitly considering non-BO terms aspart of the fit.[116]Finally, for larger molecules, where it can be difficult to employ large enough basis setsto fully converge the nuclear motion calculation, some of this convergence error is eitherknowingly or unknowingly absorbed into the spectroscopically determined PES. Such effec-tive PES’s have been found to be very useful,[119] but, of course, cannot straightforwardlybe used with other basis set parameters or indeed other nuclear motion programs.For the DMS the strategy is somewhat different and it is usual to simply use a high-quality ab initio surface. Indeed the evidence is that this produces better results thanempirical fits,[120] although care must be taken to base the surface on a sufficiently highquality electronic structure calculation.[121] and an appropriate, dense grid of points.[122]Dipole moments can usually be computed as expectation values of a given wavefunction butcan also be calculated using finite differences between energies perturbed by a small electricfield. The Hellmann-Feynman theorem shows that these two methods are equivalent forexact wavefunctions. High accuracy tends to favor use of finite differences despite the extracomputational cost: the finite difference method is more accurate but, perhaps more impor-tantly, as an energy-based method it allows small corrections to the DMS to be introducedalong the lines of the FPA procedure used for PES.[123, 124]9 . LINE LISTS FOR HOT MOLECULES As mentioned above there is a major demand from astrophysics and elsewhere of com-prehensive lists of transitions for hot species important in the atmospheres of cool stars andextrasolar planets. Many of these species are closed shell polyatomic molecules composedof elements such H, C, O, N and S. My own ExoMol project [125] has already producedmore than 20 such line lists, see Tennyson et al. [106] for a review of the current status.Other groups notably from Reims[55] and NASA Ames,[126, 127] are also computing linelists for hot species, with more experimentally driven line list being provided by Bernathand co-workers [128, 129] Amongst the theoreticians is a certain consensus on how best toperform such calculations. There are reviews available detailing how to compute accuraterotation-vibration line lists,[130, 131] so here I will just give brief examples which illustratestrategic issues.Figure 3 illustrates the results achievable by comparing the computed spectrum of H O with measured spectra currently available in the HITRAN database;[132] the main sourceof spectroscopic data for atmospheric models. The APTY H O variational line list [133]is based an empirically adjusted version of a high quality ab initio PES [134, 135] and acompletely ab initio
DMS [136]. It contains around 20 billion transitions and is designed tobe complete for wavenumbers up to 6000 cm − and temperatures up to 1250 K.Thie perspective gives little about the nuclear motion programs employed in the calcu-lations. This is because these programs generally give very precise solutions to the nuclearmotion problem and, when inter-comparisons have been performed, the codes have beenshown to give the same results for a given PES [21, 54, 110] or DMS.[54, 137] However, hotmolecules probe many vibrationally and rotationally excited levels and the resulting lists oftransitions between these levels can be huge. Programs have therefore had to be adaptedto cope with both the numerical and computational demands of these calculations.[138] Inparticular my group has developed methods of using graphical processing units (GPUs) toaccelerate the computation of the many billions of transitions needed.[139] The speed-upsfrom this approach are large and should aid studies on larger molecules, such as hydrocar-bons beyond methane, which are thought to be important in the atmospheres of hot Jupiterexoplanets.The line lists generated to model the spectroscopic behavior of small molecules are so large10IG. 3: Comparison of the 296 K absorption spectrum of hydrogen peroxide generatedusing the APTY variational line list [133] and taken from the 2012 release of HITRANdatabase [132]. Note that HITRAN currently contains no lines for H O at wavenumbershigher than 1500 cm − .because as the temperature rises the number of states involved in transitions grows rapidly.Tests on methane [43] showed that good results illustrated in Fig. 1 could only be obtained byretaining 3.2 billion out just under 10 billion transitions provided by the full 10to10 line list.This meant the explicit consideration of transitions four orders-of-magnitude weaker than thestandard intensity cut-off used by the HITRAN database. Figure 4 shows the temperaturedependence of the absorption spectrum of H S as modelled by the AYT2 line list [140] whichcontains 114 million vibration-rotation transitions computed using an empirically-adjustedPES [140] and an ab initio
DMS.[141] The shape of the spectrum changes significantly withtemperature as the transitions involving highly excited rotational states and vibrational hotbands act to smooth out the sharp peaks and troughs observed at low temperatures.To avoid leaving the impression that accurate results can be obtained in all cases, itshould be noted that inclusion of a transition metal atom, even in a diatomic molecule,11 -30 -28 -26 -24 -22 -20 -18 T=1000 KT=3000 KT=2000 K C r o ss - s e c t i on s / c m m o l e c u l e - wavenumber, cm -1 T=298 K
FIG. 4: Temperature-dependent spectra of H S generated using the ATY2 line list [140].Note how the depth of the minima (windows) decreases monotonically with temperature.makes solution of the electronic structure problem very much harder. As discussed at lengthelsewhere,[142–144] limitations on the accuracy of the ab initio potential functions in thiscase makes it difficult to use such calculations in a truly predictive fashion.
VI. TRANSITION INTENSITIES
Extensive line lists, which contain tens of billions of transitions may be useful for astro-physics where observations are rarely made at very high resolving power. However, remotesensing studies of our own atmosphere are often made at resolutions approaching that ob-tainable in the laboratory. Frequencies from variational nuclear motion calculations arerarely able to match these accuracy requirements but the same is not true for computedtransition intensities. It is much more challenging experimentally to measure precise, abso-lute transition intensities; the success of remote sensing missions such OCO2[145] demandsmore precise transition intensities than are available in standard data compilations such as12ITRAN 2012.[132]Theory has long provided transition intensities for species, such H +3 , for which measure-ments were not available.[146] Recently, however, it has become apparent that it is possibleto predict transition intensities to within 1 % or better based on the use of high accuracy abinitio DMS [124, 147] and the judicious use of wavefunctions from variation nuclear motioncalculations.[148, 149] One major advantage of this approach is that transition intensities forisotopically substituted species can be computed with some confidence at a similar level ofaccuracy. Thus, for example, recent calculations have provided transition intensives for theimportant, radio-active trace species O CO with what can be assumed to be same accuracyas those computed for the main, O CO, isotopologue.[150]The ab initio computation of precise transition intensities requires some adaption tostandard procedures used for both electronic stucture and nuclear motion calculations. Inparticular it is becoming apparant that the calculation of accurate dipole moments requiresmore extending the treatement of correlation by, for example, using larger reference spacesin multi-reference configuration interaction (MRCI) treatments.[124]. For reliable results itis important dipole moment surfaces are smooth [122, 151] and that reliable surfaces canonly be obtained by using very extensive grids of points.[22]. An uncertainty quantificationprocedures[10] has been developed for transition intensity calculations based on calculationswith multiple PES and DMS.[148] I would expect use of this procedure to become morewidespread. Finally, I note that the use of computed transition dipoles for modeling electric-field effects requires retention of the phase information in the dipoles.[61] This informationis lost when the dipoles are used to compute Einstein A coefficients or transition intensities,which are the standard quantities stored in data compilations. Changes to compilations oftheoretical transition information are therefore needed to accommodate for this use.
VII. FUTURE DIRECTIONS
The discussions above have essentially concentrated on spectra at infrared (or possiblyvisible) wavelengths for molecules with thermally occupied levels. However, there are circum-stances where it is desirable to move beyond this region, not least because not all observedspectra are thermal in origin.[152] Laboratory rotation-vibration spectra of water have beenobserved in the near ultra-violet[153] and there is increased interest in the atmospheric conse-13uences of absorption by ro-vibrational excitation of water at even shorter wavelengths thanthis.[154, 155] There are variational line lists which cover these wavelengths[156, 157] buttests show that while it is possible to obtain reasonable predictions for transition frequencieswith a good PES, it is hard to get reliable predictions for the transition intensity.[155] Itwould appear that this problem is associated with difficulties in obtaining a suitable DMSfunction. In particular other studies have already shown that computed transition inten-sities are sensitive the fit of the DMS to the ab initio data even at visible wavelengths[122, 123, 158], and that calculations of these high overtone transitions require care withthe numerics.[151] So far, despite use of extended grids of ab initio dipoles, a satisfactoryfit has not been found the water dipole moment which allows the stable computation oftransition intensities for the nine or ten quanta overtone transitions that occur at ultravioletwavelengths.Moving further up the energy levels, Boyarkin, Rizzo and co-workers performed a seriesof multi-photon experiments which probed rotation-vibration levels of water below [159–163], above [114, 164] and, indeed, at dissociation [114, 165]. These experiments have themajor advantage over the near-dissociation experiments on H +3 that their multiphoton natureboth greatly simplifies the resulting spectrum and makes the assignment of the rotationalquantum numbers to the final state relatively straightforward. Variational calculations onthe bound levels gives good general agreement [166] with the observations although thedifferences increase markedly just below dissociation suggesting that the PES in this regionis less accurate than at lower energies. Thus far global water potentials have not includedthe effect of spin-orbit coupling [167], which is known to be unimportant at low energies[168] but almost certainly becomes significant near dissociation. Theoretical studies of thespectra above dissociation remain more preliminary.[6, 7]The ability to compute levels above dissociation raises a direct link not only with pho-todissociation but also with reactive scattering at low energies. The idea of using variationalrotation-vibration calculations extending above the dissociation limit as a basis for theoreti-cal studies of cold and ultra-cold reactions is currently being explored within my group.[169]14 cknowledgements I thank various members of the ExoMol group for letting me use their work in this article;in particular, I thank Ahmed Al-Refaie and Sergey Yurchenko for help with the illustrationsand Laura McKemmish for helpful comments on my manuscript. Much of my work discussedhere was supported by the ERC under the Advanced Investigator Project 267219 and the UKresearch councils NERC (under grants NE/J010316 and NE/N001508) and STFC (undergrants ST/I001050, ST/M001334 and ACLP15).15
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