Spectroscopy of YO from first principles
Alexander N. Smirnov, Victor G. Solomonik, Sergei N. Yurchenko, Jonathan Tennyson
SSpectroscopy of YO from first principles † Alexander N. Smirnov a , Victor G. Solomonik a , Sergei N. Yurchenko ∗ b and Jonathan Tennyson b February 25, 2020
Abstract
We report an ab initio study on the spectroscopy of the open-shelldiatomic molecule yttrium oxide, YO. The study considers the sixlowest doublet states, X Σ + , A (cid:48) ∆ , A Π , B Σ + , C Π , D Σ + , and afew higher-lying quartet states using high levels of electronic struc-ture theory and accurate nuclear motion calculations. The coupledcluster singles, doubles, and perturbative triples, CCSD(T), andmultireference configuration interaction (MRCI) methods are em-ployed in conjunction with a relativistic pseudopotential on the yt-trium atom and a series of correlation-consistent basis sets rangingin size from triple- ζ to quintuple- ζ quality. Core–valence corre-lation effects are taken into account and complete basis set limitextrapolation is performed for CCSD(T). Spin-orbit coupling is in-cluded through the use of both MRCI state-interaction with spin–orbit (SI-SO) approach and four-component relativistic equation-of-motion CCSD calculations. Using the ab initio data for bondlengths ranging from 1.0 to 2.5 Å, we compute 6 potential energy,12 spin–orbit, 8 electronic angular momentum, 6 electric dipolemoment and 12 transition dipole moment (4 parallel and 8 per-pendicular) curves which provide a complete description of thespectroscopy of the system of six lowest doublet states. The D UO nuclear motion program is used to solve the coupled nuclear mo-tion Schrödinger equation for these six electronic states. The spec-tra of Y O simulated for different temperatures are comparedwith several available high resolution experimental studies; goodagreement is found once minor adjustments are made to the elec-tronic excitation energies.
Oxides of transition metals and lanthanides have rich and com-plex spectra due to the presence of many low-lying excited elec-tronic states. This complexity poses particular challenges for ex-perimental and theoretical studies. The yttrium oxide, YO, is anexample of a rare-earth oxide whose electronic structure is verydifficult to explore. Yttrium is a relatively abundant rare-earth el-ement both on Earth (the 28th most abundant element ) and inspace (the second most abundant rare-earth metal ). As a result,the spectrum of YO has been the subject of many astrophysicalobservations. In particular, YO has been observed in a variety ofspectra of cool stars including R-Cygni , Pi-Gruis , V838 Mon ,and V4332 Sgr . The spectrum of YO has been extensively usedas a probe to study high temperature materials at the focus of asolar furnace . The A Π / electronic state YO has a relativelyshort life time of 33 ns with large diagonal Franck-Condon fac-tors, which makes this molecule well suited for cooling experi-ments with the potential in quantum information applications. Yttrium oxide is one of the very few molecules that have been laser a Department of Physics, Ivanovo State University of Chemistry and Technology,Ivanovo 153000, Russia b Department of Physics & Astronomy, University College London, LondonWC1E 6BT, UK ∗ Corresponding author; E-mail: [email protected] † Electronic Supplementary Information (ESI) available. See DOI:10.1039/b000000x/ cooled and trapped in a magneto-optical trap .A considerable number of experimental studies have beenperformed probing the A Π – X Σ + , ? –34 B Σ + – X Σ + , A (cid:48) ∆ – X Σ + , and D Σ + – X Σ + bands of YO, as well as its microwave rotational spectrum and its hyperfine structure. Chemiluminescence spectraof YO have also been investigated.
Many of these spectrawere recorded using YO samples which were not in thermody-namic equilibrium, thus, at best, only providing information on therelative intensities. For YO, relative intensity measurements werecarried out for the A Π – X Σ + system by Bagare and Murthy .However, the permanent dipole moments of YO in both the X Σ + and A Π states were measured using the Stark technique. In case of the absence of direct intensity measurements, mea-sured lifetimes can provide important information on Einstein Acoefficients and hence transition dipole moments. The lifetimesof some lower lying vibrational states of YO in its A Π , B Σ + ,and D Σ + states were measured by Liu and Parson and Zhang et al. .YO is a strongly bound system. The compilation by Gaydon reports its dissociation energy to be 7.0 ± recommended a D value of 7.290(87) eV based onmass spectrometric determinations.A few theoretical investigations of YO are available in the liter-ature. The most comprehensive one was carried out by Langhoffand Bauschlicher who reported the spectroscopic constants forthe lowest five doublet, X Σ + , A (cid:48) ∆ , A Π , B Σ + , C Π , and four-teen quartet electronic states of YO. The doublets were studied atthe multireference single and double excitation configuration in-teraction (MRCI) level of theory and, in the case of the X Σ + , A (cid:48) ∆ , and A Π states, also using the modified coupled-pair func-tional (MCPF) method. All the quartet states were consideredat the CASSCF level, and that with the lowest energy, reportedly Φ , at the MCPF level as well. Zhang et al. have recently re-ported the CASPT2 spectroscopic constants and excitation ener-gies for a set of lowest doublet states of YO including the D Σ + state in addition to the doublets studied previously by Langhoffand Bauschlicher . In all of the previous theoretical studies, onlymodest double- ζ or triple- ζ basis sets were employed. RKRcurves and some Franck-Condon factors of YO were computed bySriramachandran and Shanmugavel .The main objective of the present study is to characterize boththe electronic ground state and the plethora of low-lying excitedstates of YO with high-level ab initio methods, and to accuratelydescribe from first principles the spectroscopy of YO via producingthe potential energy curves (PECs) and other data needed to calcu-late the rovibronic energies and transition probabilities comprisinga so-called line list for this molecule. The generation of such linelists is a major object of the ExoMol project. Thus far, ExoMol studies of open-shell transition metal (TM)diatomics have struggled due to difficulties in providing reliable ab initio starting points.
The intrinsic challenge to theoryposed by open-shell systems is associated with several types of1 a r X i v : . [ phy s i c s . c h e m - ph ] F e b roblems including spin contamination, symmetry breaking in thereference function, strong nondynamical electron correlation ef-fects, avoided crossings between adiabatic potential energy sur-faces, etc. (for the discussion, see, e.g., Refs. 59,60). In the open-shell TM-containing species, these problems are exacerbated bystronger relativistic effects than those in the molecules made up ofrelatively light main group elements, and greater number of elec-tronic excited states governing the spectroscopic behaviour of amolecule and hence deserving to be taken into account in a studyaimed at accurate description of its spectroscopy. Moreover, thelow-lying electronic states of TM species are commonly degener-ate or near-degenerate, which complicates their theoretical treat-ment even more. Multireference methods of quantum chemistrybest suited for describing closely spaced electronic states mightseem to be the natural choice for studying these systems. However,most routine multireference methods, such as MRCI, are incapableof properly handling dynamical electron correlation and thereforedo not provide high accuracy description of TM-containing speciescommonly featuring strong dynamical correlation effects. Such ef-fects are best treated with single reference coupled cluster (CC)theory known for its capability to predict highly accurate proper-ties even for molecules with mild to moderate MR character. Un-fortunately, the higher likelihood of severe multireference char-acter in the ground and/or low-lying electronic excited states ofopen-shell TM-containing species makes their treatment by singlereference methods very problematic, if possible at all. Particularlythis is true for the studies aimed at a description of the molecularpotential energy surfaces over a wide range of geometries. It istherefore not surprising that the high-level coupled cluster stud-ies on the open-shell TM-containing species, where a few excitedstates are treated on an equal footing with the ground state, arevery uncommon and only deal with near-equilibrium regions ofthese states (see, e.g., Refs. 61–64). Such a study on a manifoldof electronic excited states of a TM-containing diatomic moleculeover a wider bond length range has not been reported so far.It is thus clear that none of routine methods of modern quan-tum chemistry are entirely satisfactory in all respects for accuratelydescribing from first principles the spectroscopy of open-shell TM-containing species. Nevertheless, one can try to solve this challeng-ing task via the so-called composite approach by which the desiredset of molecular properties is obtained using multiple methods ofdifferent nature and sophistication rather than a single method.In this paper, we have examined efficiency of such an ap-proach taking the example of YO. The PECs for the six low-est doublet electronic states of this molecule, X Σ + , A (cid:48) ∆ , A Π , B Σ + , C Π , D Σ + , were obtained from the extensive high-levelcoupled cluster calculations addressing core–valence correlationand basis set convergence issues, whereas the spin–orbit curves(SOCs), electronic angular momentum curves (EAMCs), elec-tric dipole moment curves (DMCs), and transition dipole mo-ment curves (TDMCs) were obtained at the MRCI level of theory.These curves, with some simple adjustment of the minimum en-ergies of the PECs, are used to solve the coupled nuclear-motionSchrödinger equation with the program D UO . The spectroscopicmodel and ab initio curves are provided as part of the supplemen-tary material. Our open source code D UO can be accessed via http://exomol.com/software/ . Multireference single and double excitation configuration inter-action, MRCI, and underlying complete active space self- consistent field, CASSCF, calculations were carried out usinga relativistic energy-consistent 28-electron core pseudopotential(PP) accompanied with the aug-cc-pVTZ-PP basis set on the Yatom, and the aug-cc-pVTZ all-electron basis set on the O atom(this combination of sets is hereafter referred to as aVTZ). To ob-tain a consistent MRCI data set in the widest possible range of bondlengths, the state-averaged CASSCF procedure was employed withdensity matrix averaging over 22 doublet (six Σ + , seven Π , five ∆ ,two Φ , and two Σ − ) and 9 quartet (two Σ + , three Π , two ∆ , one Φ , and one Σ − ) states, with equal weights for each of the roots.The active space included 7 electrons distributed in 13 orbitals (6 a , 3 b , 3 b , 1 a ) that had predominantly O 2p and Y 4d, 5s, 5p,and 6s character; all lower energy orbitals were constrained to bedoubly occupied. All valence electrons (4d, 5s Y; 2s, 2p O) wereincluded in the MRCI correlation treatment.Potential energy, spin–orbit coupling, and dipole momentcurves, as well as electronic angular momentum and transitiondipole matrix elements were obtained at the MRCI level for the sixlowest doublet states. Moreover, the potential curves were calcu-lated using the extended multi-state complete active space second-order perturbation theory, XMS-CASPT2, with the basis sets aug-cc-pwCVTZ-PP on Y and aug-cc-pwCVTZ on O (henceforth ab-breviated as awCVTZ). In the respective SA-CASSCF calculations,the (7e,13o) active space was employed together with averagingover the lowest six doublet states. In order to remedy issues per-taining to intruder states, a level shift of 0.4 and an IPEA (ioni-sation potential, electron affinity) shift of 0.5 were employed forXMS-CASPT2.To calculate the molecular Ω states and respective spin–orbitcurves, we used the spin–orbit – MRCI state-interacting ap-proach : the spin-coupled eigenstates were obtained by diago-nalizing H es + H SO in a basis of MRCI eigenstates of electrostaticHamiltonian H es . The matrix elements of H SO were constructed us-ing the one-electron spin–orbit operator accompanying the yttriumpseudopotential.Spin–orbit effects were also treated more rigorously in rela-tivistic four-component (4c) all-electron calculations employinga Gaussian nuclear model and an accurate approximation to thefull Dirac–Coulomb Hamiltonian. The respective spin-free re-sults were obtained with the spin-free Hamiltonian of Dyall. Inthese calculations, the relativistic TZ-quality basis sets of Dyall were used for the Y and O atoms (hereafter referred to as TZ D ).The basis sets were kept uncontracted to provide sufficient flexibil-ity. Electron correlation was taken into account via the equation-of-motion CCSD (EOM-CCSD) method with the Y outer-core (4sand 4p) electrons correlated together with the valence electrons.The EOM-EA scheme (adding 1 electron to the closed shell) wasapplied with the reference defined by the YO + cation and the ac-tive space comprising 12 spinors (Y 5s and 4d). For the YO elec-tronic states inaccessible via the EOM-EA procedure, we employedthe EOM-IP scheme (removing 1 electron from the closed shell)with the YO − (Y 5s ) anion taken as the reference and an ac-tive space composed of 8 spinors (Y 5s and O 2p). The virtualorbital space was truncated by deleting all virtual spinors with or-bital energies larger than 15 a.u. In the relativistic calculations ofdipole moments, a finite-field perturbation scheme was employedby adding the z-dipole moment operator as a small perturbationto the Hamiltonian. Perturbations with electric field strengths of ± ∆ E SO , utilized in the calcula-tions of the YO atomization energy were obtained from the exper-imental J -averaged zero-field splittings of the ground state atomic2erms : ∆ E SO = − -1 (O) and − -1 (Y).The most sophisticated PEC computations were performed atthe coupled cluster singles, doubles, and perturbative triples,CCSD(T), level of theory with a restricted open-shell Hartree-Fock reference and with an allowance for a small amount of spincontamination in the solution of the CCSD equations, i.e., RHF-UCCSD(T). Symmetry equivalencing of the ROHF orbitalswas performed for the degenerate atomic and molecular electronicstates. Both valence (4d, 5s Y; 2s, 2p O) and outer-core (4s,4p Y; 1s O) electrons were correlated. Scalar relativistic effectswere treated with the yttrium pseudopotential described above.Sequences of aug-cc-pwCV n Z-PP ( n = T, Q, 5) basis sets for Ywere used in conjunction with the corresponding all-electron ba-sis sets aug-cc-pwCV n Z for the O atom. These combinations ofbasis sets are denoted below as awCVTZ, awCVQZ, and awCV5Z,respectively.For each point in a grid of r (Y–O) bond lengths, the CCSD(T)calculated energies were extrapolated to the complete basis set(CBS) limit. Three extrapolation schemes were employed. First,a two-point extrapolation of total energies was performed usingthe formula : E n = E CBS + A ( n + / ) , (1)where n = 4 and 5 for the awCVQZ and awCV5Z basis sets. Thisscheme is denoted as CBS1. Second, we employed alternative two-point (Q5) extrapolations of the Hartree–Fock and correlation en-ergy components. These implied using Eq. 1 for the correlationpart and the Karton and Martin formula for the HF energy: E n = E CBS + A ( n + ) exp ( − . n / ) ; (2)this is denoted as CBS2. Third, the CBS estimates were also ob-tained using the awCVTZ, awCVQZ, and awCV5Z total energies viathe three-parameter, mixed Gaussian/exponential expression : E n = E CBS + A exp ( − ( n − )) + B exp ( − ( n − ) ) , (3)where n = 3, 4 and 5 for the awCVTZ, awCVQZ and awCV5Z basissets, respectively. This is denoted as CBS3.The spectroscopic constants r e , ω e , ω e x e , and α e of YO were ob-tained from a conventional Dunham analysis using polynomialfits of total energies for bond lengths in the vicinity of the mini-mum for a given electronic state.The CCSD(T) calculations of the equilibrium dipole moments, µ e , for a few lowest states were carried out at the correspondingCCSD(T)/CBS1 equilibrium bond lengths. The dipole momentswere computed by numerical differentiation of the total energy inthe presence of a weak electric field. Finite perturbations withelectric field strengths of ± The relativistic 4c-EOM-CCSD calculations were done with the use of the DIRAC pro-gram. We use the program D UO to solve the coupled Schrödingerequation for 6 lowest electronic states of YO. D UO is a varia-tional program capable of solving rovibronic problems for a gen-eral (open-shell) diatomic molecule with an arbitrary number ofcouplings, see, for example, Refs. 58,92–94. All ab initio couplingsbetween these 6 states are taken into account as described below. The goal of this paper is to provide a qualitative simulation of theelectronic spectra of YO based on the ab initio curves. We there-fore do not attempt a systematic refinement of the ab initio curvesby fitting to the experiment, which will be the subject of futurework. In order to facilitate the comparison with the experimentaldata, we, however, perform some shifts of the T e values and simplescaling of the SOCs (see below).In D UO calculations, the coupled Schrödinger equation wassolved on an equidistant grid of 301 bond lengths r i ranging from r = ab initio curves arerepresented by sparser and less extended grids (see below). Forthe bond length values r i overlapping with the ab initio ranges, the ab initio curves were projected onto the denser D UO grid using thecubic spline interpolation. The PECs outside the ab initio rangewere reconstructed using the standard Morse potential form f PEC ( r ) = V e + D e (cid:16) − e − a ( r − r e ) (cid:17) . For other curves the following function forms were used : f shortTDMC ( r ) = Ar + Br , f shortother ( r ) = A + Br , for the short range and f longEAMC ( r ) = A + Br , f longother ( r ) = A / r + B / r . for the long range, where A and B are stitching parameters.The vibrational basis set was taken as eigensolutions of the sixuncoupled 1D problems for each PEC. The corresponding basis setconstructed from 6 ×
301 eigenfunctions was then contracted to in-clude 60 lowest (in terms of the vibrational energy) X functionsand 20 from each other state (160 in total). These vibrational ba-sis functions were then combined with the spherical harmonics forthe rotational and electronic spin basis set functions. All calcula-tions were performed for Y O using atomic masses.This study is the first where a D UO calculation has been per-formed for a system with avoided crossings between curves of thesame symmetry. The current version of D UO does not allow fornon-adiabatic couplings, and therefore these were ignored in thisstudy. However, despite the expectation that the non-adiabaticcoupling effects should be relatively important in the regions nearan avoided curve crossing, as we show below, our model neglect-ing these effects is justified by close agreement with the availableexperimental spectra. An overview of the CASSCF PECs for all doublet and quartet statesincluded in the SA-CASSCF procedure is provided in Fig. 1. Inthe vicinity of the ground state minimum (at ∼ ∼ -1 above the ground state.The lowest six doublet MRCI PECs ( X Σ + , A (cid:48) ∆ , A Π , B Σ + , C Π , D Σ + ) are shown in Fig. 2. For the states X Σ + , A (cid:48) ∆ , A Π ,and B Σ + , the PECs were obtained in the full bond length rangeamenable to the underlying CASSCF procedure, . ≤ r ≤ . Å,while the C Π and D Σ + curves were calculated through r = . Å and . Å, respectively. Extending the MRCI curves for the two3 igure 1
CASSCF/aVTZ spin-free potential energy curves of the low-lyingdoublet (black) and quartet (red) states in YO. upper states beyond those distances would require requesting agreater number of states (while exceeding 4 in a single irreduciblerepresentation with the chosen active space would make the MRCIcomputation unfeasible on the hardware used) or selecting the or-der of the states in the initial internal CI (e.g., 1, 2, 3, 5 rather than1, 2, 3, 4), leaving some of them out in each MRCI point. Thisappeared to affect the smoothness of the resulting PECs. There-fore, we refrained from further pursuing the computations withthe same number of MRCI roots in the entire bond length rangeand simply reduced the number of requested states for longer in-ternuclear distances. For all six doublet states, the XMS-CASPT2PECs could be obtained in the range . ≤ r ≤ . Å, Fig. 3;at larger bond lengths the underlying CASSCF procedure failed toconverge. As can be seen from Figs. 2 and 3, the A Π and C Π curves feature an avoided crossing at bond lengths around 2 Å, asdo the B Σ + and D Σ + curves in approximately the same region,see Fig. 3. The avoided crossings of both pairs of curves are alsoseen in the EOM-IP-CCSD calculated PECs, Fig. 4, albeit at shorterdistances (1.8 – 1.9 Å).In order to provide deeper insight into the electronic structure ofYO, we have analyzed the dominant configurations (configurationstate functions) in the MRCI wave functions for the lowest elec-tronic states, Table 1, and the leading atomic orbital contributionsin the respective molecular orbitals, Table 2. Figure 2
MRCI/aVTZ spin-free potential energy curves of YO.
The analysis indicates that the X Σ + ground state consistsmainly of ...10 σ σ π σ electron configuration. The prin- cipal contribution to the singly occupied 12 σ MO comes from theyttrium 5s atomic orbital, with an admixture of the 5p AO particu-larly noticeable at longer internuclear distances. The three lowestactive MOs, 10 σ , 11 σ , and 5 π +( − ) , primarily consist of the oxy-gen 2s and 2p orbitals whose contributions increase with the bondstretching. Therefore, the bonding in the X Σ + YO ground statecan be roughly described as ionic, Y + O − , however, with signifi-cant covalent character mainly owing to an appreciable participa-tion of the yttrium 4p σ AO in the 10 σ MO. Upon the Y–O bondstretching, there is a rapid decrease in the 4p contribution (seeTable 2) reducing the covalency and resulting in a concomitant in-crease in the magnitude of the electric dipole moment in the YOground state (see below).
Figure 3
XMS-CASPT2/awCVTZ spin-free potential energy curves of YO.
The first excited state, A (cid:48) ∆ , mainly consists of...10 σ σ π δ configuration with the 2 δ MO clearlyassigned to the Y 4d δ orbital. At shorter bond lengths this statecan be reasonably described by the single electron excitation fromthe ground state, 5s → δ . Upon bond elongation, the weightof the main configuration gradually decreases, approaching ∼ ∼ σ σ α σ β π δ α configuration (with a weight of 22% at2.19 Å) that corresponds to the Y + O − bonding (Y 5s , O 2p ). Figure 4
The avoided crossing regions of the EOM-IP-CCSD/TZ D spin-free potential energy curves for the A Π , B Σ + , C Π , and D Σ + electronicstates of YO. At bond lengths up to ∼ A Π and B Σ + excited states, ...10 σ σ π π and...10 σ σ π σ , respectively, include the 6 π and 13 σ singlyoccupied MOs mainly composed of the yttrium 5p π and 5p σ atomicorbitals, respectively, yet with a significant admixture of the 4dAOs. Therefore, these states can be viewed as arising from 5s → π and 5s → σ atomic electron promotions. In the samerange of bond length values, the C Π and D Σ + upper lying statesconsist mainly of ...10 σ σ σ π and ...10 σ σ σ π electron configurations, respectively. The YO bonding in the C Π and D Σ + states is hence well described by the Y + O − scheme con-sistent with the Y 5s , O 2p electron configuration.As the Y–O distance approaches the avoided crossing point frombelow, the principal configurations of the A Π and B Σ + stateschange to those specified above for the C Π and D Σ + states, re-spectively, while, vice versa, the principal configurations of the C Π and D Σ + states turn into those being the main ones forthe A Π and B Σ + states at shorter bond lengths. This alterationof main configurations describes an oxygen-to-metal charge back-transfer, Y + O − → Y + O − , in the A Π and B Σ + states of YO uponthe Y–O bond stretching through the avoided crossing region ofbond lengths.Specifically, the avoided crossing point, r ac , chosen to be thepoint of closest approach of two curves, for the A Π and C Π statesamounts to 2.046 Å (XMS-CASPT2) and 1.994 Å (MRCI), with theenergy gap, ∆ E ac , of 366 cm -1 and 243 cm -1 , respectively. For theXMS-CASPT2 B Σ + and D Σ + curves, r ac = 2.064 Å and ∆ E ac =1484 cm -1 . Notably, the principal configuration interchange be-tween the B Σ + and D Σ + states occurs at a slightly shorter in-ternuclear distance: ∼ ∼ r ac = 1.911 Å, ∆ E ac = 231 cm -1 for A Π and C Π curves, and r ac = 1.832 Å, ∆ E ac = 272 cm -1 for B Σ + and D Σ + curves.The CCSD(T) calculations were carried out for the six lowestdoublet states. The reference configurations for each state were asfollows: X Σ + ...10 σ σ π σ A (cid:48) ∆ ...10 σ σ π δ A Π ...10 σ σ π π B Σ + ...10 σ σ π σ C Π ...10 σ σ σ π D Σ + ...10 σ σ σ π The CCSD(T) energies were obtained in the ranges: . ≤ r ≤ . Å for X Σ + , A (cid:48) ∆ and A Π ; . ≤ r ≤ . Å for B Σ + ; . ≤ r ≤ . Åfor C Π ; . ≤ r ≤ . Å for D Σ + . At longer distances (as well asshorter ones for C Π and D Σ + ), the coupled-cluster calculationsfailed due to severe CCSD convergence problems. For the X Σ + , A (cid:48) ∆ , A Π , and B Σ + states, the distances shorter than 1.0 Å werenot considered because relative energies of excited states alreadyexceed 350000 cm -1 at this point. The respective CCSD(T)/CBS1PECs are shown in Fig. 5. Since single reference methods arenot suitable for describing avoided crossings, e.g., those betweenthe PECs of the A Π and C Π , and B Σ + and D Σ + states, theCCSD(T) calculated PECs can be viewed as corresponding to thediabatic presentation of these states. It is clearly seen that theCCSD(T) A Π and C Π curves cross each other at 2.031 Å, as dothe B Σ + and D Σ + ones at 2.013 Å. Figure 5
CCSD(T)/CBS1 spin-free potential energy curves of YO.
Table 3 summarizes the optimized bond lengths r e , harmonic vi-brational frequencies ω e , equilibrium dipole moments µ e , and adi-abatic excitation energies T e of the low-lying doublet states calcu-lated at the XMS-CASPT2, MRCI and EOM-CCSD levels as well asthe results from earlier theoretical studies. Our data in thecolumn entitled " C Π " were obtained from the second minimumin the adiabatic PEC of the A Π state, i.e., they can be ascribed tothe diabatic representation of the A Π and C Π states.The results of our EOM-CCSD calculations indicate that the an-ionic reference is less suitable for describing YO than the cationicone. In Table 3, more accurate cationic-reference EOM-EA-CCSDspectroscopic constants are listed for all states except for the C Π and D Σ + ones which are not accessible via the electron attach-ment procedure and therefore were described at the EOM-CCSDlevel only via EOM-IP.The results given in Table 3 are obviously inferior to thoseobtained from high-level CCSD(T) calculations including core–valence correlation and extrapolation to the CBS limit. TheCCSD(T) results are collected in Table 4 together with the exper-imental data available to date. The spread in the CCSD(T)/CBSresults from different CBS extrapolation schemes serve as a roughestimate of the uncertainty in extrapolation. The good agreementbetween the CBS estimates and experimentally determined spec-troscopic properties of the X Σ + , A (cid:48) ∆ , A Π , and B Σ + electronicstates demonstrates the high accuracy in the CCSD(T)/CBS PECsof these states for bond lengths in the vicinity of the PECs min-ima, and is indicative of the mild MR character of the respectiveelectronic wave functions.5 able 1 Weights of the leading configurations (configuration state functions) in the low-lying electronic states of YO derived from analyzing theMRCI/aVTZ wave functions at bond lengths of 1.79, 2.04 and 2.19 Å. Contributions of 2 % and higher are shown. Configurations a Weights, % State 10 σ σ σ σ π + π − π + π − δ + δ − X Σ + − − + 2 2 0 0 0 0 2.3 2.82 2 + 0 2 + 0 − − − − A (cid:48) ∆ ( a ) 2 2 0 0 2 2 0 0 + 0 78.9 67.3 53.12 + − − A Π ( b ) 2 2 0 0 2 2 + 0 0 0 79.92 + 0 − B Σ + C Π ( b ) 2 2 2 0 + 2 0 0 0 0 83.22 2 0 0 + 2 2 0 0 0 2.82 2 0 0 + 2 0 2 0 0 2.32 2 0 0 2 2 + 0 0 0 68.72 + − − D Σ + a The orbital names π + , π − , δ + , and δ − indicate π ( b ) , π ( b ) , δ ( a ) , and δ ( a ) orbitals, respectively. The MO occupanciesrepresented by 2, 0 and + or − denote double, zero, and single occupancies with the total spin raised or lowered by 1/2. Table 2
Analysis of the YO molecular orbitals in terms of leading atomic orbital contributions (above 10 % ) at bond lengths of 1.79, 2.04 and 2.19 Å. MO 1.79 Å 2.04 Å 2.19 Å10 σ
52% 2s O + 43% 4p σ Y 71% 2s O + 27% 4p σ Y 84% 2s O + 15% 4p σ Y11 σ
63% 2p σ O + 10% 4d σ Y 72% 2p σ O 77% 2p σ O12 σ
86% 5s Y 82% 5s Y + 12% 5p σ Y 81% 5s Y + 13% 5p σ Y13 σ
50% 5p σ Y + 36% 4d σ Y 39% 5p σ Y + 53% 4d σ Y 34% 5p σ Y + 60% 4d σ Y5 π +( − )
94% 2p π O 95% 2p π O 96% 2p π O6 π +( − )
67% 5p π Y + 32% 4d π Y 46% 5p π Y + 52% 4d π Y 34% 5p π Y + 63% 4d π Y2 δ +( − ) δ Y 100% 4d δ Y 99% 4d δ Y a See footnote to Table 1 for designations. able 3 Theoretical spin-free equilibrium constants of YO in its low-lying doublet states: adiabatic excitation energies, T e – cm -1 , bond lengths, r e – Å,vibrational frequencies, ω e – cm -1 , and dipole moments µ e – D. The relevant experimental data are listed in Table 4. X Σ + A (cid:48) ∆ A Π B Σ + C Π D Σ + T e XMS-CASPT2/awCVTZ 0 16183 16210 20521 20198 24139MRCI/aVTZ 0 16370 16287 17029EOM-CCSD/TZ D a r e XMS-CASPT2/awCVTZ 1.822 1.851 1.824 1.858 2.107 1.991MRCI/aVTZ 1.830 1.864 1.833 2.149EOM-CCSD/TZ D a ω e XMS-CASPT2/awCVTZ 796 726 763 696 592 696MRCI/aVTZ 777 693 750 542EOM-CCSD/TZ D a
881 822 847 804 606 648MCPF
855 785 832MRCI
866 801 834 789 638 µ e MRCI/aVTZ b D c a EOM-IP for C Π and D Σ + , EOM-EA for the remaining states. b calculated for each electronic state at the respective CCSD(T)/CBS1 equilibrium bond length. c calculated at a bond length of 1.7932 Å. Figure 6
The CCSD/awCV5Z calculated T diagnostics of YO An insight into the reliability of the CCSD(T) PECs over the en-tire bond length range explored, and for all electronic states con-sidered, including those not yet characterized experimentally, canbe provided by using the MR diagnostic criteria commonly em-ployed to identify the suitability of single reference wavefunction-based methods: T , the Frobenius norm of the coupled clusteramplitude vector related to single excitations, and D , the ma-trix norm of the coupled cluster amplitude vector arising fromcoupled cluster calculations. The utility of different MR diag-nostics has been examined in earlier studies on various 3dand 4d TM species. The following criteria have been proposed as a gauge for the latter to predict the possible need to employmultireference wavefunction-based methods while describing en-ergetic and spectroscopic molecular properties: T ≥ D ≥ % TAE[(T)] ≥ % . The symbol % TAE[(T)] denotes herethe percent total atomization energy corresponding to a relation- ship between energies determined with CCSD and CCSD(T) calcu-lations . Obviously, the % TAE[(T)] diagnostic is applicable forjudging the SR/MR character of the electronic ground state only.For the YO molecule, the CCSD/awCV5Z calculated % TAE[(T)] of5.6 % is well below the proposed MR threshold. This fact providesfurther evidence for single reference character of the X Σ + wave-function in the near-equilibrium region of Y–O bond lengths. Figure 7
RHF-UCCSD/awCV5Z wave function spin contamination in thelow-lying doublet electronic states of YO.
The CCSD/awCV5Z T plots vs. Y–O bond length are shown inFig. 6. The similar D plots are illustrated in Fig. S1 of the supple-mentary material. At shorter bond lengths, the diagnostics amountto 0.02–0.03 ( T ) and 0.05–0.12 ( D ), remaining below the MRthresholds down to 1.4 Å for most states. Upon bond stretch, T and D rapidly increase, typically exceeding the MR threshold at2.1–2.2 Å. The behaviour of these diagnostics for the C Π state is a7 able 4 CCSD(T) spin-free equilibrium constants of YO in its low-lying electronic states: The dissociation energy D (eV) referring to the groundelectronic state X Σ + , the excitation energies T e (cm -1 ) of the A (cid:48) ∆ , A Π , B Σ + , C Π D Σ + , and a Π states, bond length r e (Å), spectroscopicconstants ω e (cm -1 ), ω e x e (cm -1 ) and α e (cm -1 ), and dipole moment µ e (D). X Σ + A (cid:48) ∆ A Π B Σ + C Π D Σ + a Π D , T e awCVTZ 7.060 28924awCVQZ 7.207 14809 16555 20893 21423 23261 29296awCV5Z 7.260 14712 16538 20898 21700 23528 29465CBS1 7.298 14633 16526 20901 21925 23745 29603CBS2 7.298 14629 16525 20897 21917 23741 29592CBS3 7.289 29564expt. 7.290(87) r e awCVTZ 1.7978 2.0902awCVQZ 1.7927 1.8201 1.7971 1.8268 2.0408 1.9345 2.0841awCV5Z 1.7905 1.8177 1.7950 1.8244 2.0384 1.9323 2.0817CBS1 1.7887 1.8157 1.7932 1.8225 2.0365 1.9306 2.0797CBS2 1.7890 1.8160 1.7935 1.8228 2.0368 1.9308 2.0799CBS3 1.7892 2.0802expt. 1.7882 ω e awCVTZ 855.2 546.0awCVQZ 861.4 794.0 822.5 780.6 601.8 661.2 550.6awCV5Z 864.2 797.1 825.1 783.2 603.3 662.3 552.1CBS1 866.5 799.7 827.1 785.3 604.6 663.1 553.3CBS2 866.2 799.4 826.8 785.1 604.6 663.1 553.3CBS3 865.8 552.9expt. 861.5 ω e x e awCVTZ 2.79 2.52awCVQZ 2.78 3.06 3.17 2.94 2.58 2.60 2.53awCV5Z 2.79 3.05 3.18 2.98 2.57 2.60 2.57CBS1 2.79 3.04 3.19 3.01 2.57 2.61 2.60CBS2 2.79 3.03 3.18 3.01 2.57 2.61 2.60CBS3 2.79 2.59expt. 2.84 α e · awCVTZ 1.70 1.83awCVQZ 1.68 1.85 1.90 1.86 1.79 1.85 1.83awCV5Z 1.68 1.85 1.91 1.87 1.80 1.86 1.83CBS1 1.68 1.85 1.91 1.87 1.80 1.87 1.83CBS2 1.68 1.84 1.91 1.87 1.80 1.87 1.83CBS3 1.68 1.83expt. 1.73 µ e awCVTZ 4.615 7.595 3.711 1.749 2.059 1.256 3.605awCVQZ 4.614 7.620 3.724 1.764 2.082 1.275 3.615awCV5Z 4.611 7.626 3.728 1.770 2.090 1.281 3.618CBS1 4.609 7.630 3.730 1.775 2.097 1.287 3.621CBS2 4.609 7.630 3.731 1.777 2.097 1.287 3.621CBS3 4.609 7.629 3.729 1.774 2.095 1.285 3.620expt. 4.45(7) T and D remainwell below the MR threshold throughout the bond length rangestudied. The D Σ + state is also noteworthy: its T and D diagnos-tics are indicative of the CCSD D Σ + wave function retaining itsSR character in much narrower range of bond lengths comparedto the other doublet states under study.The relative importance of SR/MR character of YO can also beguessed with the use of spin contamination appearing in RHF-UCCSD calculations as a result of unrestricted spin at the CCSDlevel. According to Jiang et al. , spin contamination with < S − S z − S z > greater than 0.1 in an RHF-UCCSD wave function canbe viewed as a strong indication of nondynamical correlation in anopen-shell system. Plotting spin contamination vs. bond length,Fig. 7, clearly indicates the severe admixture of higher spin statesin the CCSD A (cid:48) ∆ , A Π , B Σ + and D Σ + wave functions at bondlengths beyond 2.2–2.3 Å. Greater extent of spin contaminationat longer internuclear distances can obviously be associated withlarger values of T and D (Fig. 6 and Fig. S1 of the supplementarymaterial) exceeding the established MR thresholds.It is instructive to compare the MR diagnostics discussed abovewith the weights of the principal configurations, C , in the MRCIwavefunctions of YO (see Table 1). At shorter bond lengths, the C values amount to ∼ B Σ + state and 0.79 – 0.83for the remaining doublet electronic states under study. These val-ues are smaller than the threshold, C ≥ C of the entire MRCI wavefunction differs from that of theCASSCF reference function due to the contributions of externalconfigurations, which make C a smaller number.Upon the YO bond stretch, there is a gradual decrease in theweights of the configurations serving as a reference for the cou-pled cluster treatment of the X Σ + , A (cid:48) ∆ , and A Π states. Thisindicates greater multireference character of the respective wavefunctions at longer bond distances, as do the CC-based MR diag-nostics. The reference configuration for the C Π state has approx-imately the same weight, C ∼ = We have studied five low-lying quartet electronic states of YO atthe CASSCF, CASPT2, CASPT3, and MRCI levels of theory usingthe aVTZ basis set. The results are shown in Table 5 together withthe earlier theoretical findings. The lowest quartet, a Π , wasalso studied at the CCSD(T) level (Table 4). At larger internucleardistances, e.g., at r = . Å, all the quartets feature similar orbital
Table 5
Theoretical spin-free equilibrium constants of YO in its low-lyingquartet states: adiabatic excitation energy T e – cm -1 , bond length r e –Å, vibrational frequency ω e – cm -1 . The aVTZ basis set has been usedthroughout. Π Φ Σ + ∆ Σ − T e a CASSCF 20460 68 1062 1825 2542CASPT2 24140 56 1432 2158 2785CASPT3 23772 53 1450 2175 2788MRCI 25046 55 1394 2128 2748MRCI+Q 26274 59 1357 2089 2699CASSCF b,c
52 1933 2664 3359 r e CASSCF 2.141 2.143 2.110 2.110 2.114CASPT2 2.197 2.198 2.192 2.191 2.196CASPT3 2.210 2.211 2.207 2.208 2.214MRCI 2.213 2.214 2.209 2.209 2.214MRCI+Q 2.218 2.219 2.218 2.218 2.222CASSCF b b ω e CASSCF 517 516 522 521 519CASPT2 515 515 517 494 477CASPT3 519 518 524 494 474MRCI 507 507 501 495 492MRCI+Q 506 506 501 496 493CASSCF
543 543 524 522 520MCPF b a The energies of the Φ , Σ + , ∆ and Σ − states are given here withrespect to the Π state which is the lowest-lying quartet state of YO. b In Ref. 53, the symmetry of the lowest quartet state of YO was re-ported to be Φ rather than Π . c In Ref. 53, the adiabatic excita-tion energy of the lowest quartet state was obtained from the MCPFcalculations, and the relative energies of various quartet states weredetermined at the CASSCF level. character corresponding to the Y 5 s d , O 2p electron configu-ration consistent with the Y + O − bonding (see Table 6).The results for the YO quartet states obtained in our work agreewith those of Langhoff and Bauschlicher , Table 5, except for thesymmetry of the lowest quartet state that was reported to be Φ rather than Π .The single-reference CCSD(T) method is expected to yield quiteaccurate results for the a Π state of YO since its MR diagnostics, C = 0.90, T = 0.024, and D = 0.078, indicate essentially SRcharacter of the a Π wave function in the vicinity of the a Π PECminimum, 2.00 – 2.25 Å. The very large CCSD(T)/CBS excitationenergy of the a Π state, 29600 cm -1 , suggests that the quartetstates in YO are too high in energy to significantly affect the spec-troscopy of its low-lying doublet states. Spin–orbit coupling effects were studied in a perturbative fashionat the MRCI level and more rigorously at the 4c-EOM-CCSD levelof theory including spin from the outset. The 4c-EOM-CCSD cal-culated spin–orbit coupling effects on the spectroscopic constantsof YO are shown in Table 7. The theoretical spin–orbit couplingconstants, SOCCs ( A SO ), can be compared with the relevant ex-perimental numbers for the A (cid:48) ∆ and A Π electronic states of YOreported previously. The A SO ( A (cid:48) ∆ ) SOCCs of 336 cm -1 and313 cm -1 obtained at the MRCI SI-SO and 4c-EOM-CCSD levels, re-spectively, agree well with each other and with the experimentalnumber of 339 cm -1 determined by Chalek and Gole . The cal-culation results for A SO ( A Π ), 346 cm -1 (MRCI SI-SO) and 438cm -1 (4c-EOM-CCSD), are also in reasonable agreement with therespective experimental value of 428 cm -1 . As the Y–O dis-tance reaches the avoided crossing point between A Π and C Π ,9 able 6 Main configurations in the low-lying quartet electronic states of YO derived from analyzing the MRCI/aVTZ wave function at a bond length of2.19 Å. Weight of each configuration is ∼ % . Configurations a State 10 σ σ σ σ π + π − π + π − δ + δ − Π ( b ) 2 2 + 0 + 2 0 0 + 02 2 + 0 2 + 0 0 0 + Φ ( b ) 2 2 + 0 + 2 0 0 + 02 2 + 0 2 + 0 0 0 + Σ + ∆ ( a ) 2 2 + 0 + 2 + 0 0 02 2 + 0 2 + 0 + 0 0 Σ − a See footnote to Table 1 for designations.
Figure 8 D potential energy curves for the spin–coupled components of the A Π and C Π electronic states of YO in theavoided crossing region of bond length values. the A SO ( C Π ) and A SO ( A Π ) values change their sign: the A Π / spin component of the A Π state becomes lower in energy than its A Π / counterpart, and vice versa for the spin–coupled compo-nents of the C Π state (see Fig. 8). There is also a change in the ab-solute values of A SO ( A Π ) and A SO ( C Π ): at r < r ac , | A SO ( A Π )|is much lower in magnitude than | A SO ( C Π )| and vice versa at r > r ac . However, the numerical values of A SO ( A Π ) at r > r ac de-termined with the MRCI SI-SO and 4c-EOM-CCSD methods, e.g., −
45 cm -1 and −
186 cm -1 , respectively, at a bond length of 2.04 Å,are in less satisfactory agreement with each other than those at r < r ac .The SOC matrix elements between various doublet states of YO,which also accurately account for the corresponding phases, areshown in Fig. 9 as a function of r (Y–O). The relative phases of thecouplings are important when used for solving the nuclear motionproblem as part of the coupled Schrödinger equation, see, for ex-ample, discussion by Patrascu et al. The full details of the abinitio coupling curves including the magnetic quantum numbersare provided as part of the supplementary data.
The CCSD(T)/CBS dipole moment of 4.61 D for the X Σ + stateof YO (Table 4) is in agreement with the respective values ob-tained experimentally by Steimle and Shirley in a molecularbeam-optical Stark study, 4.45(7) D, and by Suenram et al. fromthe more precise microwave measurement, 4.524(7) D. For thespin–orbit components Ω = 1/2 and Ω = 3/2 of the A Π state,the dipole moment values, µ e ( A Π / ) = 4.185 D and µ e ( A Π / )= 4.125 D, were obtained at the 4c-EOM-EA-CCSD/TZ D level oftheory at the respective CCSD(T)/CBS equilibrium bond lengthsof 1.7937 Å and 1.7929 Å, estimated by applying the 4c-EOM-EA-CCSD ∆ SO r e spin–orbit corrections (from Table 7) to the spin-free CCSD(T)/CBS1 bond length, r e = 1.7932 Å. The 4c-EOM-EA-CCSD dipole moments are overestimated by 0.4–0.5 D comparedto the spin-free CCSD(T)/CBS µ e ( A Π ) value of 3.73 D, the lat-ter being in good agreement with the µ e ( A Π / ) = 3.68(2) Dmeasured by Steimle and Shirley . However, the experimentalwork reports the dipole moment µ e ( A Π / ), 3.22(8) D, to belower than µ e ( A Π / ). This result is not supported by our ab initio calculations. Steimle and Shirley compared the dipole momentsin the A Π spin–orbit components of YO to those of the valence-isoelectronic molecule ScO , where µ e ( A Π / ) > µ e ( A Π / ),and proposed an explanation of the different order in YO. Accord-ing to Steimle and Shirley , the reason for µ ( A Π / ) being largerthan µ ( A Π / ) in YO in contrast to ScO is the smaller energy gapof the A Π and A (cid:48) ∆ states, which results in mixing between the Ω = 3/2 spin–orbit components of these states. This idea is, how-ever, based on low-level ab initio computations by Langhoff andBauschlicher , which predicted the A (cid:48) ∆ state in YO to lie 200cm -1 higher than A Π , whereas for ScO the analogous calcula-tions yielded a difference of about 1900 cm -1 , with A (cid:48) ∆ beinglower in energy. In fact, the A (cid:48) ∆ state lies around 1800 cm -1 lowerthan A Π in YO and 1500 cm -1 lower in ScO, as evidenced by ex-perimental works of Chalek and Gole , i.e., the A Π – A (cid:48) ∆ energy gap in YO exceeds that in ScO. Also, at high levels of the-ory including SO coupling, the PECs for the Ω = 3/2 componentsof A Π and A (cid:48) ∆ lie quite far apart (see the excitation energiesin Table 7), and their mixing is almost negligible. Furthermore,it is worth noting that for another analogous molecule, LaO, theexperimental data also indicate that µ e ( A Π / ) > µ e ( A Π / ).To shed more light on the alleged different order of the dipolemoment values for the spin–orbit components of the A Π state inYO compared to ScO and LaO, we performed additional 4c-EOM-10 igure 9 The MRCI/aVTZ SI-SO calculated (above) and (below) spin–orbit matrix elements of YO. EA-CCSD/TZ D computations for the two latter molecules at the ex-perimental equilibrium bond lengths of 1.6826 Å (ScO) and1.8400 Å (LaO). These resulted in the following values: µ ( A Π / )= 4.543 D, µ ( A Π / ) = 4.532 D (ScO), µ ( A Π / ) = 3.011 D, µ ( A Π / ) = 2.907 D (LaO), i.e., the ab initio predicted differ-ence between the two spin–orbit components monotonically in-creases on passing in the series ScO → YO → LaO: 0.01 → → are: µ ( A Π / ) = 4.43(2) D, µ ( A Π / ) = 4.06(3) D (ScO), µ ( A Π / )= 2.44(2) D, µ ( A Π / ) = 1.88(6) D (LaO). Since the 4c-EOM-EA-CCSD dipole moments are expected to be overestimated by at least0.5 D, one can consider the theoretical results to be in reasonableagreement with experiment. In light of our results, the experimen-tal dipole moments for the two Ω components of the A Π stateof YO need to be revisited.The MCPF dipole moments obtained by Langhoff andBauschlicher , 3.976 D ( X Σ + ), 7.493 D ( A (cid:48) ∆ ) and 3.244 D( A Π ), are systematically smaller than our CCSD(T) (Table 4),MRCI, and EOM-EA-CCSD (Table 3) results.The MRCI DMCs and TDMCs of YO are shown in Fig. 10. TheEAMCs are shown in Fig. S2 of the supplementary material. Allthese curves as well as the SOC ones (Fig. 9) exhibit irregular be-haviour at bond lengths around r ∼ A Π , B Σ + , C Π and D Σ + wave functions over the avoidedcrossing region. D UO calculations For D UO calculations, we selected the following set of curves repre-senting our highest level of theory: the CCSD(T)/CBS PECs shownin Fig. 5 and MRCI SOCs, (T)DMCs, and EAMCs shown in Figs. 9and 10, as well as Fig. S2 of the supplementary material. Due to limitations of single reference CCSD(T), the CCSD(T) curves forthe A Π , C Π and B Σ + , D Σ + states do not exhibit avoided cross-ings and hence are not consistent with the MRCI property curves.To alleviate this deficiency, these four PECs were transformed bysimply switching the corresponding energy points between A and C ( Π states) as well as those between B and D ( Σ + states) at r > r ac . We have decided to apply this rather simplistic procedurebecause it has marginal effect on the overall accuracy of our modeland is sufficient for the goal of this pure ab initio study, not aimingat spectroscopic accuracy. A proper diabatic representation of theYO electronic states will be, however, important when refining the ab initio curves by fitting to experiment, which is a goal of futurework. In this study we work directly with the ab initio data in thegrid representation without representing ab initio curves analyti-cally. We do not perform any diabatizations here, which is oftenuseful for representing the variation of the data with respect to thebond length in an intuitive and more compact form. One of theartifacts of this choice to use the ab initio curves directly is thatthe crossing points of the MRCI PECs hence the points of drasticchange in the MRCI property curves differ by a few hundredths ofÅ from the crossing points of the CCSD(T)/CBS PECs. Again, thishas small impact on the overall agreement of the current modelwith the experiment. However, a more accurate study will requirea more consistent treatment of the crossing points. Our preferredchoice would be to use the CCSD(T)/CBS values of the correspond-ing crossing points.11 able 7 D molecular properties of YO in its low-lying spin-coupled electronic states (EOM-IP for C Π and EOM-EA for the remainingstates): bond length r e – Å, vibrational frequency ω e , adiabatic excitation energy T e – cm -1 . The respective spin–orbit effects, ∆ SO , are provided as well. r e ∆ SO r e ω e ∆ SO ω e T e ∆ SO T e X Σ + / A (cid:48) ∆ / − − A (cid:48) ∆ / − A Π / − − A Π / − B Σ + / − C Π / − − C Π / − Figure 10
MRCI/aVTZ electric dipole moment curves of YO: diagonal(above), non-diagonal µ x (middle) and non-diagonal µ z (below). The D UO rovibronic wavefunctions of YO in conjunction withthe ab initio TDMCs were then used to produce Einstein A coef- ficients for all rovibronic transitions between states considered inthis work covering the wavenumber range from 0 to 40000 cm -1 and J ≤ . . These Einstein A coefficients and the correspondingenergies from the lower and upper states involved in each tran-sition were organized as a line list using the ExoMol format. This format uses a two file structure with the energies includedinto the States file (.states) and Einstein coefficients appearing inthe Transitions file (.trans). This ab initio line list is available from . The ExoMol format has the advantage of be-ing compact and compatible with our intensity simulation programE XO C ROSS (see below).
In Table 8 we compare our computed vibrational excitations at J = . and J = . (as proxy for vibrational band centres) of Y Owith the experimentally derived values. Based on this comparison,as an ad hoc improvement we applied the following shifts to PECsof the excited states: +9.509 cm -1 ( A Π ), +81.096 cm -1 ( A (cid:48) ∆ ), − -1 ( B Σ + ), and +358.626 cm -1 ( D Σ + ). These shiftsare small compared to the observed minus calculated differencesoften encountered in calculations of electronic term values fortransition metal oxides. We also scaled the SOC of A Π by 1.14in order to increase the SO splitting of v = by about 33 cm -1 .Even though we are not targeting fully quantitative accuracy inthis work, without such empirical shifts it would be difficult to re-produce band heads even qualitatively.To allow for a direct comparison with the observed spectra of Y O, we generated a line list covering rotational excitations upto J = and the energy/wavenumber range up to 40,000 cm -1 ,with a lower state energy cutoff of 16,000 cm -1 . The partition function of Y O computed using our ab initio linelist is shown in Fig. 11, which is compared to that recently reportedby Barklem and Collet . Since Y has a nuclear spin degener-acy of two, we have multiplied Barklem and Collet’s partition func-tion by a factor of two to compensate for the different conventionsused; we follow HITRAN and include the full nuclear spin inour partition functions. The partition function of YO was also re-ported by Vardya , which is shown in Fig. 11. All three partitionfunctions are almost identical for their ranges of validity.
Using the ab initio Y O line list, spectral simulations wereperformed with our code E XO C ROSS . E XO C ROSS is an opensource Fortran 2003 code with the primary use to producespectra of molecules at different temperatures and pressures inthe form of cross sections using molecular line lists as input.12 able 8
Comparison of our ab initio and experimentally derived term val-ues of Y O in cm -1 . The ab initio PECs were shifted by +9.509 cm -1 ( A Π ), +81.096 cm -1 ( A (cid:48) ∆ ), − -1 ( B Σ + ) and +358.626 cm -1 ( D Σ + ). The SOC of A Π was scaled by 1.1376. The ‘Obs’ values of A , A (cid:48) , B and D were derived using spectroscopic constants from the corre-sponding works with the help of the PGOPHER program . The X state‘Obs.’ values are represented by the corresponding band centers (limit J = ). υ J Ω D UO Obs.[109] [13] X a A [13]0 0.5 0.5 16295.492 16295.4530 1.5 1.5 16724.499 16724.5411 1.5 1.5 17117.400 17109.8451 1.5 1.5 17545.141 17538.4592 1.5 1.5 17931.536 17916.8802 1.5 1.5 18358.848 18345.7683 1.5 1.5 18740.505 18716.6743 1.5 1.5 19169.245 19146.5934 1.5 1.5 19547.495 19510.0644 1.5 1.5 19964.360 19940.4885 1.5 1.5 20350.928 20296.6365 1.5 1.5 20732.561 20727.595[39] B D [37]0 0.5 0.5 23969.916 23969.9401 0.5 0.5 24659.745 24723.766 a Band centers. P a r t i t i on f un c t i on T, K This work [ Barklem and Collet (2016)]x2 [Vardya (1970)]x2
Figure 11
Partition functions of YO: Solid line is from this work computedusing the energies of the six lowest electronic states; filled circles repre-sent the partition function values by Vardya generated using spectro-scopic constants of 3 lowest electronic states X , A and B (multiplied by afactor of 2 to account for the nuclear statistics); open squares representvalues by Barklem and Collet (times the factor 2). Here we use the YO line list generated with D UO in the Ex-oMol format, the description of which can be found, e.g., inYurchenko et al. or Tennyson et al. . E XO C ROSS can beaccessed via http://exomol.com/software/ or directly at https://github.com/exomol . Amongst other features, E X - O C ROSS can generate spectra for non-local thermal equilibriumconditions characterized with different vibrational and rotationaltemperatures, lifetimes, Landé g -factors, partition and coolingfunctions.An overview of the YO absorption spectra in the form of crosssections at the temperature T = -1 was used. This figure shows contributions fromeach electronic band originating from the ground electronic state.The strongest bands are A Π – X Σ + and B Σ + – X Σ + . The visible A – X band is known to be important for the spectroscopy of coolstars. The C state is of the same symmetry as A , however, the cor-responding band C – X is much weaker due to the small Franck-Condon effects. The A (cid:48) ∆ – X Σ + band is forbidden and barelyseen in Fig. 12, however, it is strong enough to be experimentallyknown. Figure 13 shows a simulated emission spectrum of the strongestorange system YO ( A Π – X Σ + , (0,0)), which is compared to theexperiment of Badie and Granier (from the plume emission closeto the liquid Y O surface). It is remarkable that even pure ab ini-tio calculations (after modest adjustment of the corresponding T e value by +9.509 cm -1 ) provide very close reproduction of exper-iment. It shows that our line list at the current, ab initio qualityshould be useful for modelling spectroscopy of exoplanets and coolstars in the visible region.13 .20.40.60.81.01.2 I n t en s i t y , a r b . un i t s Experiment -10 -11 -11 -11 -11 Theory (T = 3000 K) e r g c m /[ s t r ad m o l e c u l e ] wavelength, mm Figure 13
Comparison of the computed A Π – X Σ + orange band with theobservations of Badie and Granier . Our simulations assume T = Kand Gaussian line profile of HWHM = 1 cm -1 . -24 -22 -20 -18 -16 -14 A D B C A’ X c r o ss - s e c t i on s , c m / m o l e c u l e wavenumber, cm -1 wavelength, mm Figure 12
An overview of a theoretical absorption spectrum of YO at T = K for different electronic bands, designated by their upper state. Thespectrum was computed using our ab initio line list for YO assuming aGaussian profile with a half-width-at-half-maximum (HWHM) of 5 cm -1 . Figure 14 illustrates the A (cid:48) ∆ – X Σ + (0,0) forbidden band inemission simulated for T = K compared to the experimentalspectrum of Simard et al. . Here, a shift of +81.096 cm -1 was ap-plied to the T e value of the A (cid:48) ∆ state. In spectral simulations, thisregion appears to be contaminated by the dipole-allowed hot A – X transitions, which are not necessarily very accurate in this region.We therefore applied a filter to select the A (cid:48) ∆ – X Σ + transitionsonly. The difference in shape of the spectra can be attributed eitherto the non-LTE (Local Thermal Equilibrium) effects present in theexperiment or broadening effects, which we have not attempted tomodel properly. This figure is only to illustrate the generally goodagreement of the positions of the rovibronic lines in this band. Experiment -17 -17 -17 -18
Theory (T = 77 K) e r g c m /[ s t r ad m o l e c u l e ] wavenumber, cm -1 Figure 14
Comparison of the computed emission A (cid:48) ∆ – X Σ + (0,0) bandwith the measurements of Simard et al. at T = K and Gaussian lineprofile of HWHM = 0.1 cm -1 . Figure 15 shows a series of absorption bands compared to themeasurements of Zhang et al. who observed bands in boththe B Σ + – X Σ + and D Σ + – X Σ + systems in a heavily non-thermal environment where the vibrations were hot and the rota-tions cooled to liquid nitrogen temperatures. In this case of multi-band system it was important to include at least some non-LTEeffects by treating it using two temperatures, vibrational and ro-tational, assuming that the corresponding degrees of freedom arein LTE. The rotational temperature T rot = K was set to valuespecified by Zhang et al. , while the vibrational temperature wasadjusted to T vib = 2000 K to better reproduce the experimentalspectrum. The spectrum is divided into five spectroscopic win-dows (I–V) which are also detailed in Table 9. In order to matchthe positions of the vibronic bands in the experiment, some of thewindows were shifted. For example, the D Σ + – X Σ + (1,0) bandwas shifted by about 76.5 cm -1 . This shift is an indication of the in-accuracy of our model to reproduce the vibrationally excited statesof D Σ + . This is not surprising considering the complexity of thequantum-chemistry part of these systems as well as of the nuclearmotion part. The avoided crossing with the B Σ + state leads tovery complex shapes of the D Σ + PEC and of the SO and electronicangular momentum coupling curves with the A and C states. Thecorresponding SOCs of the B and D states with the nearby state C are also relatively large, ∼
30 cm -1 and 80 cm -1 , respectively(see Fig. 9), and therefore important. Besides, the D PEC is rathershallow with the equilibrium in the vicinity of the avoided crossingpoint, which also complicates the solution. An accurate descrip-tion of the B and D curves would require diabatic representationsbefore attempting any empirical refinement by fitting to the exper-iment. In all cases our simulations, while not perfect, show strikingagreement with the observed spectra.14 able 9 Five spectroscopic windows (cm -1 ) used to compare five vibronicbands of YO ( B and D ) in Fig. 15. Experiment is by Zhang et al. whileTheory is from this work. Experiment Theory BandI 20714.5 - 20753.5 20715 - 20754 B (0,0)II 23078.5 - 23117 23073 - 23112 D (0,1)III 23837.5 - 23874.5 23769 - 23806 D (1,1)IV 23934.5 - 23973 23934.5 - 23973 D (0,0)V 24689 - 24730 24625.5 - 24666 D (1,0) D(0,1)
IVIIIII E x pe r i m en t a l i n t en s i t i e s -14 -15 -15 -15 -15 D(1,0)D(0,0)D(1,1)B(0,0) x2 V T heo r e t i c a l c r o ss s e c t on s , c m - wavenumber, cm -1 x2 x10 I Figure 15
Comparison of our computed emission spectra to the measure-ments of Zhang et al. Our simulations assumed a cold rotational tem-perature of T rot = 77 K and a hot vibrational temperature of T vib = 2000 K.The Gaussian line profile of HWHM = 0.1 cm -1 was used. The lifetimes of Y O in the A Π and B Σ + states ( v ≤ ) weremeasured by Liu and Parson using laser fluorescence detectionof nascent product state distributions in the reactions of Y with O ,NO, and SO . Some lifetimes were also measured by Zhang et al. and computed ab initio by Langhoff and Bauschlicher . Table 10presents a comparison of these results with our calculations withD UO , showing value of the states with corresponding lowest J and the positive parity. It can be seen that our A Π state lifetimesappear to be shorter than the observed ones. This suggests thatthe A Π – X Σ + transition dipoles may be slightly too large. Goodagreement is obtained for the lifetimes of the B Σ + states, whilethe D Σ + state lifetimes are underestimated by a factor of 2 indi-cating that the corresponding transition dipole moments D – B and D – X , or at least one of them might be too large. Table 10
Lifetimes of Y O states in ns: comparison with the measure-ments of Liu and Parson and Zhang et al. , and the ab initio calcula-tions of Langhoff and Bauschlicher . State v [12] [37] [53] This work A Π / . ± . .
71 36 . ± . . A Π / . ± . .
91 30 . ± . .
32 33 . ± . .
76 41 . ± . . B Σ + . ± . ± .
71 32 . ± . . D Σ + ± . D Σ + ± . In this work, a composite approach to accurate first-principlesdescription of the spectroscopy of open-shell TM-containing di-atomics is proposed and its high efficiency is demonstrated tak-ing the example of the yttrium oxide molecule. The approach isbased on the combined use of single reference coupled cluster andmultireference methods of electronic structure theory, accompa-nied with a thorough joint analysis of the SR/MR character of themolecular wave function. A full set of potential energy, (transi-tion) dipole moment, spin–orbit, and electronic angular momentacurves for the lowest 6 electronic states of YO was produced ab ini-tio using a combination of the CCSD(T)/CBS and MRCI methods.These curves were then used to solve the fully coupled Schrödingerequation for the nuclear motion using the D UO program. Given thecomplexity of the system under study, the results show remarkablygood agreement with the experiment. Our ultimate goal is to pro-duce an accurate, empirical line list for Y O for applications inmodelling the spectroscopy of atmospheres of exoplanets and coolstars. This will require a refinement of the ab initio curves by fit-ting to the experimental data in the diabatic representation as wellas inclusion of the non-adiabatic coupling effects and will be ad-dressed in future work. The A Π band of YO has strong absorptionin the visible region, i.e. where the stellar radiation usually peaks.Such systems are known to cause the temperature inversion in at-mospheres of exoplanets, similar to the inversion caused by TiOand VO in giant exoplanets . Opacities of such species are cru-cial in modelling the degree of temperature inversion in giant exo-planets. YO is yet to be detected in exoplanetary atmospheres andthis work is meant to provide the necessary spectroscopic data.YO is one of the few molecules with the strong potential forlaser-cooling applications, which have widely ranging applica-tions, from quantum information and chemistry to searches fornew fundamental physics. The results of this work will help tomodel the cooling properties of YO and thus will be important fordesigning and implementing laser-cooling experiments.The ab initio curves of YO obtained in this study are providedas part of the supplementary material to this paper along with ourspectroscopic model in a form of a D UO input file. The computedline list can be obtained from . This is givenin the ExoMol format which also includes state-dependent life-times. The line list can be directly used with the E XO C ROSS pro-gram to simulate the spectral properties of YO.
Acknowledgements
This work was supported by STFC Projects No. ST/M001334/1and ST/R000476/1. The authors acknowledge the use of the15CL Legion High Performance Computing Facility (Legion@UCL),and associated support services, in the completion of this work,along with the Cambridge COSMOS SMP system, part of the STFCDiRAC HPC Facility supported by BIS National E-infrastructurecapital grant ST/P002307/1 and ST/R002452/1 and STFC op-erations grant ST/R00689X/1. DiRAC is part of the National e-Infrastructure. A. N. S. and V. G. S. acknowledge support from theMinistry of Science and Higher Education of the Russian Federa-tion (Project No. 4.3232.2017/4.6).
References [1] P. F. Bernath,
International Reviews in Physical Chemistry ,2009, , 681–709.[2] L. K. McKemmish, S. N. Yurchenko and J. Tennyson, Mol.Phys. , 2016, , 3232–3248.[3] S. A. Cotton,
Scandium, Yttrium & the Lanthanides: Inor-ganic & Coordination Chemistry , John Wiley & Sons, Ltd.,2006.[4] E. Anders and N. Grevesse,
Geochim. Cosmochim. Acta ,1989, , 197–214.[5] P. S. Murty, Astrophys. Lett. Comm. , 1982, , 7–9.[6] P. S. Murty, Astrophys. Space Sci. , 1983, , 295–305.[7] V. P. Goranskii and E. A. Barsukova, Astron. Rep. , 2007, ,126–142.[8] T. Kaminski, M. Schmidt, R. Tylenda, M. Konacki andM. Gromadzki, Astrophys. J. Suppl. , 2009, , 33–50.[9] J. M. Badie, L. Cassan and B. Granier,
Eur. Phys. J.-Appl.Phys , 2005, , 111–114.[10] J. M. Badie, L. Cassan and B. Granier, Eur. Phys. J.-Appl.Phys , 2005, , 61–64.[11] J. M. Badie, L. Cassan and B. Granier, Eur. Phys. J.-Appl.Phys , 2007, , 177–181.[12] K. Liu and J. M. Parson, J. Chem. Phys. , 1977, , 1814–1828.[13] A. Bernard and R. Gravina, Astrophys. J. Suppl. , 1983, ,443–450.[14] M. T. Hummon, M. Yeo, B. K. Stuhl, A. L. Collopy, Y. Xia andJ. Ye, Phys. Rev. Lett. , 2013, , 143001.[15] M. Yeo, M. T. Hummon, A. L. Collopy, B. Yan, B. Hemmer-ling, E. Chae, J. M. Doyle and J. Ye,
Phys. Rev. Lett. , 2015, , 223003.[16] A. L. Collopy, M. T. Hummon, M. Yeo, B. Yan and J. Ye,
NewJ. Phys , 2015, , 055008.[17] G. Quéméner and J. L. Bohn, Phys. Rev. A , 2016, ,012704.[18] A. L. Collopy, S. Ding, Y. Wu, I. A. Finneran, L. Anderegg,B. L. Augenbraun, J. M. Doyle and J. Ye, Phys. Rev. Lett. ,2018, , 213201.[19] J. B. Shin and R. W. Nicholls,
Spectr. Lett. , 1977, , 923–935. [20] C. Linton, J. Mol. Spectrosc. , 1978, , 351–364.[21] A. Bernard, R. Bacis and P. Luc, Astrophys. J. , 1979, ,338–348.[22] K. Liu and J. M. Parson,
J. Phys. Chem. , 1979, , 970–973.[23] T. Wijchers, H. A. Dijkerman, P. J. T. Zeegers and C. T. J.Alkemade, Spectra Chimica Acta B , 1980, , 271–279.[24] S. P. Bagare and N. S. Murthy, Pramana , 1982, , 497–499.[25] T. Wijchers, H. A. Dijkerman, P. J. T. Zeegers and C. T. J.Alkemade, Chem. Phys. , 1984, , 141–159.[26] W. J. Childs, O. Poulsen and T. C. Steimle, J. Chem. Phys. ,1988, , 598–606.[27] T. C. Steimle and J. E. Shirley, J. Chem. Phys. , 1990, ,3292–3296.[28] R. Dye, R. Muenchausen and N. Nogar, Chem. Phys. Lett. ,1991, , 531–536.[29] D. Fried, T. Kushida, G. P. Reck and E. W. Rothe,
J. Appl.Phys. , 1993, , 7810–7817.[30] C. E. Otis and P. M. Goodwin, J. Appl. Phys. , 1993, ,1957–1964.[31] J. M. Badie and B. Granier, Chem. Phys. Lett. , 2002, ,550–555.[32] J. M. Badie and B. Granier,
Eur. Phys. J.-Appl. Phys , 2003, , 239–242.[33] T. Kobayashi and T. Sekine, Chem. Phys. Lett. , 2006, ,54–57.[34] J. M. Badie, L. Cassan, B. Granier, S. A. Florez and F. C.Janna,
J. Sol. Energy Eng. Trans.-ASME , 2007, , 412–415.[35] A. Bernard and R. Gravina,
Astrophys. J. Suppl. , 1980, ,223–239.[36] J. W. H. Leung, T. M. Ma and A. S. C. Cheung, J. Mol. Spec-trosc. , 2005, , 108–114.[37] D. Zhang, Q. Zhang, B. Zhu, J. Gu, B. Suo, Y. Chen andD. Zhao,
J. Chem. Phys. , 2017, , 114303.[38] C. L. Chalek and J. L. Gole,
J. Chem. Phys. , 1976, , 2845–2859.[39] B. Simard, A. M. James, P. A. Hackett and W. J. Balfour, J.Mol. Spectrosc. , 1992, , 455–457.[40] U. Uhler and L. Akerlind,
Arkiv For Fysik , 1961, , 1–16.[41] T. C. Steimle and Y. Alramadin, Chem. Phys. Lett. , 1986, , 76–78.[42] J. Hoeft and T. Torring,
Chem. Phys. Lett. , 1993, , 367–370.[43] P. H. Kasai and W. Weltner, Jr.,
J. Chem. Phys. , 1965, ,2553.1644] T. C. Steimle and Y. Alramadin, J. Mol. Spectrosc. , 1987, , 103–112.[45] R. D. Suenram, F. J. Lovas, G. T. Fraser and K. Matsumura,
J. Chem. Phys. , 1990, , 4724–4733.[46] L. B. Knight, J. G. Kaup, B. Petzoldt, R. Ayyad, T. K. Ghantyand E. R. Davidson, J. Chem. Phys. , 1999, , 5658–5669.[47] T. C. Steimle and W. Virgo,
J. Mol. Spectrosc. , 2003, ,57–66.[48] D. M. Manos and J. M. Parson,
J. Chem. Phys. , 1975, ,3575–3585.[49] C. L. Chalek and J. L. Gole, Chem. Phys. , 1977, , 59–90.[50] J. Tennyson, K. Hulme, O. K. Naim and S. N. Yurchenko, J.Phys. B: At. Mol. Opt. Phys. , 2016, , 044002.[51] A. G. Gaydon, Dissociation Energies and Spectra of DiatomicMolecules , Chapman and Hall, Ltd., London, 3rd edn, 1968.[52] R. J. Ackermann and E. G. Rauh,
J. Chem. Phys. , 1974, ,2266–2271.[53] S. R. Langhoff and C. W. Bauschlicher, J. Chem. Phys. , 1988, , 2160–2169.[54] P. Sriramachandran and R. Shanmugavel, New Astron. ,2011, , 110–113.[55] J. Tennyson and S. N. Yurchenko, Mon. Not. R. Astron. Soc. ,2012, , 21–33.[56] L. Lodi, S. N. Yurchenko and J. Tennyson,
Mol. Phys. , 2015, , 1559–1575.[57] L. K. McKemmish, S. N. Yurchenko and J. Tennyson,
Mon.Not. R. Astron. Soc. , 2016, , 771–793.[58] L. K. McKemmish, T. Masseron, J. Hoeijmakers, V. V. Pérez-Mesa, S. L. Grimm, S. N. Yurchenko and J. Tennyson,
Mon.Not. R. Astron. Soc. , 2019, , 2836–2854.[59] J. F. Stanton and J. Gauss,
Adv. Chem. Phys. , 2003, ,101–146.[60] J. Tennyson, L. Lodi, L. K. McKemmish and S. N. Yurchenko,
J. Phys. B: At. Mol. Opt. Phys. , 2016, , 102001.[61] ˇ L . Horn ´ y , A. Paul, Y. Yamaguchi and H. F. Schaefer III, J.Chem. Phys. , 2004, , 1412–1418.[62] C. Puzzarini and K. A. Peterson,
J. Chem. Phys. , 2005, ,084323.[63] A. Paul, Y. Yamaguchi, H. F. Schaefer III and K. A. Peterson,
J. Chem. Phys. , 2006, , 034310.[64] A. Paul, Y. Yamaguchi and H. F. Schaefer III,
J. Chem. Phys. ,2007, , 154324.[65] S. N. Yurchenko, L. Lodi, J. Tennyson and A. V. Stolyarov,
Comput. Phys. Commun. , 2016, , 262–275. [66] H.-J. Werner and P. J. Knowles,
J. Chem. Phys. , 1988, ,5803–5814.[67] P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. , 1988, , 514–522.[68] P. J. Knowles and H.-J. Werner,
Theor. Chem. Acc. , 1992, ,95–103.[69] H.-J. Werner and P. J. Knowles, J. Chem. Phys. , 1985, ,5053–5063.[70] P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. , 1985, , 259 – 267.[71] K. A. Peterson, D. Figgen, M. Dolg and H. Stoll,
J. Chem.Phys. , 2007, , 124101.[72] R. A. Kendall, T. H. Dunning and R. J. Harrison,
J. Chem.Phys. , 1992, , 6796–6806.[73] T. Shiozaki, W. Gy˝orffy, P. Celani and H.-J. Werner, J. Chem.Phys. , 2011, , 081106.[74] K. A. Peterson and T. H. Dunning,
J. Chem. Phys. , 2002, , 10548.[75] A. Berning, M. Schweizer, H.-J. Werner, P. J. Knowles andP. Palmieri,
Mol. Phys. , 2000, , 1823–1833.[76] L. Visscher, Theor. Chem. Acc. , 1997, , 68–70.[77] K. G. Dyall, J. Chem. Phys. , 1994, , 2118–2127.[78] K. G. Dyall,
Theor. Chem. Acc. , 2007, , 483–489.[79] K. G. Dyall,
Theor. Chem. Acc. , 2016, , 128.[80] A. Shee, T. Saue, L. Visscher and A. S. P. Gomes,
J. Chem.Phys. , 2018, , 174113.[81] A. Kramida, Yu. Ralchenko, J. Reader and and NIST ASDTeam, NIST Atomic Spectra Database (ver. 5.6.1), [Online].Available: https://physics.nist.gov/asd [2019,April 13]. National Institute of Standards and Technology,Gaithersburg, MD., 2019.[82] C. Hampel, K. A. Peterson and H.-J. Werner,
Chem. Phys.Lett. , 1992, , 1–12.[83] J. D. Watts, J. Gauss and R. J. Bartlett,
J. Chem. Phys. , 1993, , 8718–8733.[84] P. J. Knowles, C. Hampel and H. J. Werner, J. Chem. Phys. ,1993, , 5219–5227.[85] J. M. L. Martin, Chem. Phys. Lett. , 1996, , 669–678.[86] A. Karton and J. M. L. Martin,
Theor. Chem. Acc. , 2006, ,330–333.[87] K. A. Peterson, D. Woon and T. H. Dunning,
J. Chem. Phys. ,1994, , 7410–7415.[88] J. L. Dunham,
Phys. Rev. , 1932, , 721–731.1789] H. J. Werner, P. J. Knowles, G. Knizia, F. R. Manby,M. Schütz, P. Celani, W. Gy˝orffy, D. Kats, T. Korona,R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar,T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L.Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll,C. Hampel, A. Hesselmann, G. Hetzer, T. Hrenar, G. Jansen,C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. Mc-Nicholas, W. Meyer, M. E. Mura, A. Nicklass, D. P. O’Neill,P. Palmieri, D. Peng, K. Pflüger, R. Pitzer, M. Reiher, T. Sh-iozaki, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinssonand M. Wang, MOLPRO, version 2015.1, a package of ab ini-tio programs
Intern. J. Quantum Chem. ,2017, , 92–103.[92] A. T. Patrascu, C. Hill, J. Tennyson and S. N. Yurchenko,
J.Chem. Phys. , 2014, , 144312.[93] S. N. Yurchenko, I. Szabo, E. Pyatenko and J. Tennyson,
Mon. Not. R. Astron. Soc. , 2018, , 3397–3411.[94] H.-Y. Li, J. Tennyson and S. N. Yurchenko,
Mon. Not. R. As-tron. Soc. , 2019, , 2351–2365.[95] T. J. Lee and P. R. Taylor,
Intern. J. Quantum Chem. , 1989, , 199–207.[96] C. L. Janssen and I. M. B. Nielsen, Chem. Phys. Lett. , 1998, , 423–430.[97] W. Jiang, N. J. DeYonker and A. K. Wilson,
J. Chem. TheoryComput. , 2012, , 460–468.[98] J. Wang, S. Manivasagam and A. K. Wilson, J. Chem. TheoryComput. , 2015, , 5865–5872. [99] A. Karton, E. Rabinovich, J. M. L. Martin and B. Ruscic, J.Chem. Phys. , 2006, , 144108.[100] A. Karton, S. Daon and J. M. L. Martin,
Chem. Phys. Lett. ,2011, , 165–178.[101] J. Shirley, C. Scurlock and T. Steimle,
J. Chem. Phys. , 1990, , 1568–1575.[102] C. W. Bauschlicher and S. R. Langhoff, J. Chem. Phys. , 1986, , 5936–5942.[103] T. C. Steimle and W. Virgo, J. Chem. Phys. , 2002, ,6012–6020.[104] R. Stringat, C. Athénour and J. L. Féménias,
Can. J. Phys. ,1972, , 395–403.[105] A. Bernard and J. Vergés, J. Mol. Spectrosc. , 2000, ,172–174.[106] J. Tennyson, S. N. Yurchenko, A. F. Al-Refaie, E. J. Barton,K. L. Chubb, P. A. Coles, S. Diamantopoulou, M. N. Gor-man, C. Hill, A. Z. Lam, L. Lodi, L. K. McKemmish, Y. Na,A. Owens, O. L. Polyansky, T. Rivlin, C. Sousa-Silva, D. S.Underwood, A. Yachmenev and E. Zak,
J. Mol. Spectrosc. ,2016, , 73–94.[107] S. N. Yurchenko, A. F. Al-Refaie and J. Tennyson,
Astron.Astrophys. , 2018, , A131.[108] C. M. Western,
J. Quant. Spectrosc. Radiat. Transf. , 2017, , 221–242.[109] R. R. Reddy, Y. Nazeer Ahammed, K. Rama Gopal, P. Ab-dul Azeem and S. Anjaneyulu,
Astrophys. Space Sci. , 1998, , 223–240.[110] P. S. Barklem and R. Collet,
Astron. Astrophys. , 2016, ,A96.[111] R. R. Gamache, C. Roller, E. Lopes, I. E. Gordon, L. S. Roth-man, O. L. Polyansky, N. F. Zobov, A. A. Kyuberis, J. Ten-nyson, S. N. Yurchenko, A. G. Császár, T. Furtenbacher,X. Huang, D. W. Schwenke, T. J. Lee, B. J. Drouin, S. A.Tashkun, V. I. Perevalov and R. V. Kochanov,
J. Quant. Spec-trosc. Radiat. Transf. , 2017, , 70–87.[112] M. S. Vardya,
Astron. Astrophys. , 1970, , 162–&.[113] N. Madhusudhan and S. Gandhi, Mon. Not. R. Astron. Soc. ,2017,472