Accurate Spectroscopic Models for Methane Polyads Derived from a

Article pubs.acs.org/JPCA

Accurate Spectroscopic Models for Methane Polyads Derived from a Potential Energy Surface Using High-Order Contact Transformations Vladimir Tyuterev,*,† Sergei Tashkun,‡ Michael Rey,† Roman Kochanov,†,‡ Andrei Nikitin,†,‡ and Thibault Delahaye† †

GSMA, UMR CNRS 7331, University of Reims, BP 1039, 51687 Reims Cedex 2, France LTS, V.E. Zuev Institute of Atmospheric Optics, Academician Zuev square 1, 634021, Tomsk, Russia

‡

S Supporting Information *

ABSTRACT: A new spectroscopic model is developed for theoretical predictions of vibration−rotation line positions and line intensities of the methane molecule. Resonance coupling parameters of the effective polyad Hamiltionians were obtained via high-order contact transformations (CT) from ab initio potential energy surface. This allows converging vibrational and rotational levels to the accuracy of best variational calculations. Average discrepancy with centers of 100 reliably assigned experimental bands up to the triacontad range was 0.74 cm−1 and 0.001 cm−1 for GS rotational levels up to J = 17 in direct CT calculations without adjustable parameters. A subsequent “fine tuning” of the diagonal parameters allows achieving experimental accuracy for about 5600 Dyad and Pentad line positions, whereas all resonance coupling parameters were held fixed to ab initio values. Dipole transition moment parameters were determined from selected ab initio line strengths previously computed from a dipole moment surface by variational method. New polyad model allows generating a spectral line list for the Dyad and Pentad bands with the accuracy ∼10−3 cm−1 for line positions combined with ab initio predictions for line intensities. The overall integrated intensity agreement with Hitran-2008 empirical database is of 4.4% for the Dyad and of 1.8% for the Pentad range.

1. INTRODUCTION Precise knowledge of methane spectra and of related molecular properties is important in numerous fields of science. Acting as a greenhouse gas of the earth atmosphere, CH4 is also a significant constituent of various planetary atmospheres, like those of the Giant Planets (Jupiter, Saturn, Uranus, and Neptune)1 and of Titan (Saturn’s main satellite).2,3 Accurate calculations of the CH4 opacity are essential for the modeling of recent observations for brown dwarfs, exoplanets and for other astrophysical applications.4−6 Infrared spectroscopy is one of the best diagnostic tools to study CH4 by remote sensing methods. Because of a large optical depth for atmospheric and planetary applications, accurate knowledge of line positions and line intensities is required even for very weak transitions. Methane is an active absorber in the entire infrared range with only few transparency windows. Various spectroscopic databases7−14 report several hundred thousands methane vibration−rotation transitions, only a small fraction of line intensities being truly measured because of well-known © 2013 American Chemical Society

difficulties of experimental intensity determinations in dense spectra with overlapping lines. A large part of information on methane transitions available in databases rely on calculations and extrapolations using theoretical analyses. The analysis of excited vibration−rotation energy levels and transitions of the methane molecule is a difficult problem due to complex structures of vibrational polyads, numerous resonance couplings, and relatively large number of degrees of freedom for the nuclear motion. In general, among various methods and theoretical models for calculation of high-resolution molecular spectra, the most widespread types are the following: (A) Spectroscopic empirical models, based on effective Hamiltonians (Heff) for sets of nearby vibrational states (polyads) Special Issue: Terry A. Miller Festschrift Received: August 13, 2013 Revised: October 14, 2013 Published: October 16, 2013 13779

dx.doi.org/10.1021/jp408116j | J. Phys. Chem. A 2013, 117, 13779−13805

The Journal of Physical Chemistry A

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initio methane DMS and methane spectra at T = 1000 K using the MULTIMODE program.44 Cassam-Chenai and Lievin46 computed a third-order dipole moment normal mode expansion and calculated line intensities for rotational transitions of methane within its vibrational ground state. Recently new potential and dipole moment functions of methane have been constructed by Nikitin, Rey and Tyuterev47,48 based on extended ab initio calculations on 19882 nuclear configurations up to ∼20000 cm−1. Their ab initio PES has been subsequently optimized47 using an empirical scaling of four quadratic parameters to the fundamental frequencies. These surfaces (hereafter referred to as NRT PES47 and DMS48) have been used for variational predictions49,50 of infrared spectra of methane isotopologues up to 9300 cm−1 giving an average accuracy for band centers of the order of ∼1 cm−1 or better. The major issue for variational approach (B) is a convergence for high V, J quantum numbers related to large dimensions of matrices involved. On the other hand, a target accuracy of ∼0.001 cm−1 for high-resolution applications would hardly be achievable on this way. The aim of this work is to combine the advantages of effective (A) and global (B) approaches by bringing to spectroscopic models the missing information from ab initio PES and DMS via an accurate implementation of the contact transformation (CT) method. It is well known that Heff can be derived from the full Hamiltonian by CT or alternative forms of the perturbation theory. Following Van Vleck,51 successive CTs have been widely used to simplify the eigenvalue problem by transforming the Hamiltonian matrix to a block-diagonal form. In this way certain spectral ranges can be approximately separated. In some cases, this is mathematically equivalent to a full or to a partial separation of variables. In molecular physics and spectroscopy the CT method has been applied for vibration−rotation interactions in series of works by Nielsen, Amat, and co-workers52,53 (and references therein), Chedin and Cihla,54 Aliev and Watson,55 Birss and Choi,56 Camy-Peyret and Flaud57 and many other studies58−63 (the list being nonexhaustive). Primas64 and Tyuterev and Makushkin65−68 have investigated mathematical aspects of transformations using the “superoperators” technique. Applications to nonrigid molecules have been considered in refs 69−71. Bunker and Moss72 and Schwenke73,74 have applied CTs to a separation of electronic and nuclear variables in order to account for nonadiabatic contributions. Tyuterev and Perevalov75 have generalized the equations of CTs allowing to account for multiple accidental resonances and to obtain other forms of perturbation expansions as particular cases. Time-dependent CT has been considered67,76 under the same general scheme (note also related works77,78 though using somewhat different techniques). Implementations of the CT method in a form of computerassisted formal calculations have been reported in refs 54, 63, and 79−81, extended up high orders of the perturbation theory for diatomics.81 Although formally 20th order has been technically achieved,79,81 a big remaining issue is an investigation of the convergence of the power series PES representation, which could become phenomenological above certain limit. An extension of CTs to four and five atomic molecules has been achieved by Sibert and co-workers82−84 using both the Watson and the Meyer−Gü nthard−Pickett Hamiltonian representations with applications to H2CO and to CHD3. McCoy et a.l85,86 have investigated vibration−rotation mixtures in excited states of three and four atomic molecules using

accounting for strong rovibrational resonance interactions within these polyads; line intensities being usually calculated from empirically fitted band transition moments. (B) Global variational type methods, which allow in principle calculating an entire set of rovibrational states and transitions up to the dissociation threshold from a potential energy surface (PES) being used to calculate energies and line positions and dipole moment surfaces (DMS) being used to calculate line strengths and band intensities. There exist also extended empirical effective Hamiltonian models using vibration extrapolation scheme for successive polyads15−18 that allow simultaneous fitting of spectral data in various ranges. Such models represent in a sense an intermediate case between approaches (A) and (B). They are often referred to as “global effective models” and indeed allow certain types of useful extrapolations. However, the term “global” has to be taken with caution in this case. The reliability of this approach depends on the extent of the underlying set of experimentally available data. Effective and variational approaches in high-resolution molecular spectroscopy have their specific advantages, and shortcomings and can be considered as complementary ones. Effective models (A) have been most widely used in highresolution spectroscopy for metrological applications because this allows achieving a good accuracy for low and medium quantum numbers using relatively fast computations. In order to account for the high symmetry of the methane molecule (Td point group) a symmetry adapted formalism based on irreducible tensor operators (ITO) has been developed in the Dijon group where the most of first high-resolution spectra analyses for lower polyads have been carried out (Champion, Pierre, Hilico, Loete, et al15,16,19−22) as reviewed and generalized for other point groups by Boudon et al.16,23 and for rovibronic problems by Rey et al.123 The analyses have been further extended in collaboration with Tomsk and Reims groups for higher wavenumber ranges and for methane isotopologues.24−28 A wealth of experimental spectra recorded in laboratories of Pasadena,10 Zurich,17,29 Grenoble,30 Prague,31 Reims,27,28 Moscow32 and in other groups have been used or are currently under analyses. All related references on methane spectroscopy can be found in the recent review paper by Brown et al.,10 which also discusses several issues that face an empirical approach. Number of adjusted parameters increase very rapidly with vibrational excitations. Also missing information for the “dark” (experimentally invisible)29 states perturbing observed transitions via resonance interactions prevent from a reliable parameter determination from spectra. One of major problems is a characterization of resonance coupling parameters involved in Heff . These parameters are usually poorly determined33−35 from line positions fit but play a key role in the intensity transfer among strong and weak bands. An improvement of ab initio methods for electronic structure calculations have enabled a considerable progress for global (B)-type approach in case of methane. Low order ab initio force constants of methane have been calculated by Lee et al.36 while Marquardt and Quack37,38 have computed a global methane PES and have determined its analytical representation by adjustments to an ab initio data set under special consideration of additional experimental constraints. Schwenke and Partridge39,40 carefully investigated the role of various approximations in electronic structure calculations to the methane PES. The PES of Schwenke and Partridge was used by Carrington and Wang41−43 to study the convergence of vibration−rotation calculations as well as by Carter, Bowman et al.44 within the MULTIMODE approach. Warmbier et al.45 calculated an ab 13780



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different coordinate choices. Ramesh and Sibert87,88 have studied the convergence of CT using fourth and sixth orders of CT in comparison with variational calculations applied to CHBr3 and CHF3. Wang and Sibert89 have applied the fourthorder Van Vleck perturbation theory in order to optimize the methane force field using experimental band centers of nine isotopic species. They have also extended calculations to low-J values90 and have shown that both rectilinear and curvilinear representations of normal coordinates yield similar errors. These results have been recently used for the assignment of 13 CH4 spectra recorded by Niederer et al.91 Joyeux and Sugny92 have applied a classical version of canonical transformations using Gustafson technique to study the vibrational dynamics. Earlier work on CTs has been reviewed in refs 53, 55, and 67, and more recent studies were reviewed by Sarka and Demaison93 and by Krasnoshchekov et al94,95 who have also applied Van Vleck perturbation theory to vibrational calculations for six-atomic molecules. A primary purpose of our work is to address the main issue for effective spectroscopic models of methane polyads concerning the ambiguity of the vibration−rotation resonance coupling parameters. This aims at a precise computation of Heff parameters from the molecular PES up to all orders required for the high-resolution spectra analyses in the ITO representation. To this end, as a first step, we use our previously implemented MOL_CT (“Molecular Contact Transformations”) program suite,96 which is extended in this work to higher orders. The corresponding transformations for the dipole moment were previously implemented in MOL_CT for triatomic molecules in ref 97. As all structural constants of the vibration−rotation Lie algebra are exactly computed, our target is converging the solution of the perturbation theory implemented using the MOL_CT software to the accuracy of the best variational calculations. Both vibrational and rotational levels are concerned. As a second step we fix all intrapolyad resonance coupling parameters to ab initio CT values and optimize only the parameters of diagonal vibrational block using experimental line positions. This gives much more robust spectroscopic polyad models with fewer adjustable parameters. As a third step, we combine variational predictions for ab initio line intensities with our model. At the end, this approach provides an experimental accuracy for about 5600 ro-vibrational line positions for all Dyad and Pentad bands of methane combined with ab initio values for line intensities. The paper is structured as follows. In Section 2 we briefly recall transformations of the full Eckart-frame rovibrational Hamiltonian to rectilinear normal coordinates. Section 3 outlines the algorithm of CTs based on our previously developed formalism65−68,75 as well as the standard representations implemented in MOL_CT for vibrational and rotational operators. Section 4 is devoted to vibrational predictions up to 18000 cm−1 (polyad 12) and to comparison with experimental data and with variational calculations using NRT PES.47 Rotational and centrifugal distortion constants are computed in Section 5 followed by some examples of vibration−rotation calculations for lower polyads. In Section 6 we describe key tests for the reliability of our predictions for the resonance coupling parameters by fitting experimental line positions and line intensities whereas coupling parameters remain constrained to ab initio values. In Section 7 we consider the impact of resonance parameters on energy redistribution due to the strong state mixing with some qualitative quantum and quasi-classical illustrations. In Section 8, a combined “CT/variational”

model for ab initio line intensities is described and the strategy of this new approach is discussed in Section 9.

2. EXPANSION OF ROVIBRATIONAL HAMILTONIANS AND COMPUTATIONAL TOOLS As a starting point, we use the Eckart frame Hamiltonian of the molecular nuclear motion expressed in normal mode rectilinear coordinates and the total angular momenta as derived by Wilson−Howard, Darling−Dennison, and simplified by Watson98 to the well-known concise form: H vr 1 = hc 2

3N − 6

∑ k=1

1 − 8

ωkpk2 +

1 2

∑ (Jα − πα)μαβ (Jβ − πβ) α ,β

∑ μαα + U (q) α

(1)

All notations are standard and explained in detail elsewhere:55,98 α = x, y, z; Jα and πα are dimensionless molecular frame components of the total and vibrational angular momenta, respectively, and μαβ is the reciprocal inertia tensor. U(q) is the potential function in terms of normal coordinates qi, and −∑αμαα/8 is a small mass-dependent contribution introduced by Watson, which is a purely quantum mechanical term of kinetic origin. Apart from dimensionless elementary operators, all quantities (ωk, μαβ, U) in the right-hand side of eq 1 as well as their expansion coefficients are in wavenumber units (cm−1). For convenience of the comparison with spectroscopic data we will also give eigenvalues E ⇒ E/hc = ν̃ in the same units. The MOL_CT package, which is extended in this work to highly symmetric molecules, contains several routines implemented for various steps of calculations. They serve to build rovibrational effective normal Hamiltonians in the Eckart embedding from an arbitrarily defined molecular PES. The resulting Heff is constructed in a form that could be directly employed for analyses of experimental high-resolution spectra. We recall here just a brief outline, as many steps of calculations have been previously described. Given an ab initio or empirically defined PES, the following major tasks can be distinguished among various steps required for spectroscopic parameters calculations: (i). Preparation of a Full Rovibrational Hamiltonian Expansion in a Suitable Representation. This involves various well-known procedures. First the energy minimum is sought and a PES is expanded in power series in terms of bond length/bond angle internal coordinates r = {rm} around the equilibrium configuration resulting to a standard set of force constants up to Nth order. The second order force constants Fml = (∂2U/∂rm∂rl)e are used to determine the set of normal coordinates {qm} via GF Wilson99 formalism. In the present work, we use the rectilinear normal coordinates {qm} as these coordinates are involved in the Watson Hamiltonian by construction of eq 1. An alternative way would be to use nonrectilinear normal coordinates as described by Sibert et al.82−90 that require another form of the full Hamiltonian. The latter could offer certain advantages82−90 but analytical expansions for the kinetic energy operator would not be feasible. As we stay here with the Watson form of the full Hamiltonian (1) the set of q={qm} coordinates is related by a standard linear L-matrix transformation with rectilinear components of internal coordinates: {r}rect = Lq. However a PES U(r) is usually expressed as a function of true internal coordinates {rm} which are in 13781



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corresponds to full expansions accounting for D2 symmetries of rotational operators only (see Section 3 for more details). For this reason we use fully analytical transformations mainly for validations (check with available formulas) and for general theoretical considerations, whereas for practical applications at high orders an optimal option for the present approach is a “mixed analytical-numerical form”: analytical form for operators with numerical values for parameters calculated from an accurate molecular PES. Though the general algorithm65−68,75 does not have limitations in orders, the current 64bit implementation96 is in principle limited by N = 24, and a maximum number of atoms is limited by eight. In practical terms the feasibility of CT calculations is of course limited by accessible memory and CPU time. Currently 14th-order calculations for triatomics and eighth-order calculations for pentatomics are technically feasible on a standard desktop workstation. Note again that convergence of expansions is an important issue and has to be checked in each case particularly for highly excited rovibrational states. Usually available PES representations are not sufficiently reliable for using very high-order expansions beyond a certain reasonable limit. In this work we have found that on the sample of currently assigned methane data, the CT calculations are converged to the precision of the best available variational calculation at order six or somewhat better at order eight. This is valid under the condition that the corresponding rovibrational algebra is precisely described without arbitrary truncations and omissions. (iii). Effective Dipole Transition Moment Operator and Determination of Transition Moment Parameters for Individual Ro-vibrational Bands. The rotational algebra for the CTs of the dipole moment operator is much more complicated than for the step (ii) because the commutators and anticommutators of {Jα} with the direction cosine angular operators are involved. The exact structural constants of rotational Lie algebra for the dipole moment terms have been computed97 up to the power 8. This part also requires a transformation from geometrically defined body frame to the Eckart frame.104 Up to now the DMS transformations have been implemented in full up to order 5 for triatomics.97 For the methane molecule, only a preliminary implementation for low orders of the effective transition moment operator μ⃗ eff is currently available. For this reason in the present study we determined the transition moment parameters involved in μ⃗ eff from their fit to the variationally computed ab initio line intensities.49,50 The latter ones have been generated from the same PES47 and from the ab initio DMS48 as previously described in refs 49 and 50. (iv). Account for Symmetry Properties, Conversion to the ITO Formalism, Computation of Energy Levels, Eigenfunctions, and Spectral Transitions. As soon as the Heff is built at the step (5) to a required accuracy, a calculation of ro-vibrational levels is no more a major computational problem: a solution of the eigenvalue problem becomes thus much less demanding than for variational methods. Anyway, an explicit account for the molecular symmetry properties in Heff and in μ⃗ eff is crucially important for obvious reasons: further reduction of the dimensions by symmetry blocks, accounting for selection rules and transitions assignment. Another important reason is a compatibility with the software traditionally employed for the spectra analyses. In the case of symmetric and spherical top molecules, the ITO formalism has been used for many years for this purpose,15−17,19−29 but mostly for

general nonrectilinear. Consequently the nonlinear relations between {rm} and {qm} have to be established up to the orders needed for CT-calculations of spectroscopic constants with a requested precision: rm =

∑ mLiqi + ∑ mL′ij qiqj + ··· + ∑ mL′′ijk qiqjqk + ··· i

ij

ijk···

+ ··· + O(q N )

(2)

The importance of nonlinear contributions has been emphasized by Hoy, Mills, and Strey100 who have derived expressions for lower-order terms in eq 2. In MOL_CT, a general algorithm was implemented to give U (rm , rl , ...) ⇒ U ′(qi , qj , ...) ⇒ U ″(ai+ , aj+ , ...; ai , aj , ...) + ··· + O(λ N )

(3)

a+i

where and aj are creation and annihilation operators for normal mode vibrational quanta. In the case of analytical Taylor type expansions, this PES representation is referred to as qtrepresentation. For symmetric and spherical top molecules this procedures is described in [101, 102]. For methane molecule our code allows generating all terms up to N = 14. Another option implemented in the code is to establish exact relations between the grids of {qm} points and its image grid of {rm} points in the nuclear configuration space following the suggestion of Ermler et al.103 In this case we do not need Taylor series expansions, the parameters of the potential function U(q) being fitted to the values of the original PES U(r) on the related grids. This PES representation of U(q) is referred to as qf-representation. This latter procedure can be applied to numerically defined PES, as, for example, ab initio PES generated with spline approximations. A similar development is applied to the kinetic energy operator T (rm , Jα ) ⇒ T ′(μ(qi), πα , Jβ ) ⇒ T ″(ai+ , aj+ , ...; ai , aj , ...; J± , Jz ) + ··· + O(λ N )

(4)

with the same maximum power N = 14 for the implemented code in case of pentatonic molecules. An extension in a number of atoms or in orders would be formally straightforward but much more demanding. (ii). Contact Transformations to Effective Hamiltonians up to a Required Order under Given Resonance Conditions. This involves perturbative calculations analytical in rotational and vibrational operators implemented in ref 96 and outlined in the following Section 3. H(ai+ , aj+ , ...; ai , aj , ...; J± , Jz ) ⇒ CT ⇒ H eff (ai+ , aj+ , ...; ai , aj , ...; J± , Jz ) + ··· + O(λ N )

(5)

Various ordering schemes to optimize the CT convergence are available. The simplification is due to the fact that Heff is defined on the polyad subspaces only with finite dimensional matrices (Section 3). The major part of the transformations was implemented both in analytical and numerical forms. Because of extremely rapid increase of the number of terms, full analytical transformations seem to be practicable up to medium orders only. For example up to the order N = 6 in the case of fiveatomic molecules the full vibrational Hamiltonian H in the representation (5) contains ∼4 × 105 terms and the full initial rovibrational Hamiltonian contains ∼9 × 10 5 terms. This 13782



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Figure 1. The scheme of successive Contact Transformations for methane polyads with the ω1 ≈ ω3 ≈ 2ω2 ≈ 2ω4 resonance conditions: P0 = Ground vibration state, P1 = Dyad, P2 = Pentad, etc.

purely empirical models. Here we convert the Heff computed from the PES to the ITO representation H

eff

(ai+ ,

aj+ ,

...; ai , aj , ...; J± , Jz ) ⇒ H

eff

′(V (mΓ,)u

eff H ⇒ ...T2+T1+(H )TT = H0 + λΔH1eff + λ 2ΔH2eff 1 2... = H

+ ... + λ nΔHneff + ...

Ω(k , Γ)

⊗R

The unitary operators are usually chosen as exponentials Tk = exp(−iλkSk), where generators Sk are hermitian operators and k stands for the number of CT’s. In the conventional notations of previous works,68,75 the effective Hamiltonian reads

(6) Ω(k,Γ) in terms of vibrational V(Γ) tensor m,u and rotational R operators (Section 4.1). For the methane-type molecule the TENSOR code50,102 allows a conversion (6) up to 10th order as described in detail in refs 101, 102 that is sufficient for our applications. In this work, we use the coupling scheme of symmetrized powers of vibrational operators as defined by Nikitin et al.24 With this definition our model derived by CT from PES is fully compatible with the MIRS computational code26 for spectra analyses, which has been recently extended105 for variational calculations as well. Calculations of methane energy levels, mixing coefficients, line positions, and line intensities using our polyad CT-model presented in this work were carried out with the last version of the MIRS code.105

nmax ) H eff = H0(0) + λH1(1) + λ 2H2(2) + ... + λ nmax Hn(max

(9)

The scheme illustrating a simplification of the Hamiltonian matrices due to CT is given in the diagram of Figure 1. To describe a general CT scheme it is convenient to use two quite simple operations hereafter denoted as ⟨...⟩ and 1 (...). + The notation ⟨Y⟩ stands for the block-diagonal part of X with respect to the polyads structure. Here X could be a rovibrational operator or a corresponding matrix in an appropriately chosen basis set. The operation 1 (...) is used to formalize a + solution of a commutator equation [H0,X] = Y, where Y is known and X is unknown, which could be written in terms of Lie algebra notations as + (X) = Y. Under the condition ⟨Y⟩ = 0, a formal particular solution thus reads X = 1 (Y). In the + context of molecular CT, these operations are discussed in some details in refs 64−68 and 71−76 and in more general form in Lie algebra handbooks, but for the purpose of this study we do not need explicit definitions, which would allow calculating operations ⟨Y⟩ and 1 (Y) in a general case. This is because + with an appropriate choice of a representation, one could reduce these operations just to simple multiplications by constants depending of a vibrational resonance conditions as described below. Taking into account the relations following from the extension of Wigner theorem for CT68,75 one can express k-times transformed term via results of previous transformations:

3. CONTACT TRANSFORMATIONS As mentioned above, the major advantage of the effective Hamiltonians derived by CT is that this approach allows one to reduce an extent of calculations by focusing on a certain group of vibrational states “localized” within a limited energy range. The latter is supposed to be of interest for an interpretation of a concrete experimental spectrum within a given wavenumber range. For the spectroscopic models obtained in this way, the dimension of matrices is dramatically reduced and computations become much simpler. Consequently, a metrological accuracy of calculations is, in principle, feasible. 3.1. General Algorithm of CT. For the computer implementation,96 we applied the generalized formulation of Contact Tranformations suggested in previous works.65−68,75 Here we shall only briefly remind the general procedure. Consider an initial untransformed molecular rovibrational Hamiltonian sorted according to orders of the perturbation theory H = H0(0) + λH1(0) + λ 2H2(0) + ... + λ nmax Hn(0) max

(8)

)

Hk(k) = ⟨Hk(k − 1)⟩ = ... = ⟨H⌊kk /2⌋⟩

(7)

(10)

Here ⌊k/2⌋ is the greatest integer less than or equal to k/2. Equation 10 can be written as

where the subscript n = 0, 1,..., nmax denotes orders of the perturbation theory according to a certain small formal parameter λ, and the superscript denotes the number of successive contact transformations. The goal of the method CT is to build a simpler effective Hamiltonian Heff by applying successive unitary transformations

H eff = ⟨H0(0)⟩ + λ⟨H1(0)⟩ + λ 2⟨H2(1)⟩ + λ 3⟨H3(1)⟩ + λ 4⟨H4(2)⟩ + ... + λ n⟨H⌊nn /2⌋⟩ 13783

(11)



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Figure 2. Convergence of methane vibration levels using successive CT orders. Discrepancies between calculations using (order 1 and order 8), then (order 2 and order 8), and (order 6 and order 8) are given versus band centers ν̃ = Evib/hc. Note a different vertical scale at the left-hand side panel.

Expressions for commutators of the operators Vεub are given in refs 67 and 96. The action of the operations ⟨...⟩ and 1 (...) of + the CT method on the elementary vibrational operator (eq 15) reads

Thus to calculate Heff, it is necessary to perform ⌊nmax/2⌋ successive contact transformations. Each term H(k) n is a linear combination of elementary rovibartional operators hi Hn(k) =

∑ cihi

(12)

H(k) n

H(k−1) n

ε ε ⟨V ub ⟩ = ΔubV ub

A recursive derivation from requires rovibrational commutators. The generator iSk of the kth transformation is calculated as65,75 iSn =

1 (Hn(n − 1)) +

(16)

Here ω = (ω1,ω2,ω3,...) is a formal vector composed of harmonic frequencies and ω·(u − b) = ω1(u1 − b1) + ω2(u2 − b2) + .... denotes a usual scalar product. The constant Δub plays the role of an extended Kronecker symbol taking the values 0 or 1 only. In a nondegenerate or pure degenerate case, Δub =1 if ω·u = ω· b, and Δub = 0 otherwise. The extension to a near-degenerate case75 reads:

(13)

3.2. Elementary Operators (“Operator Basis Set”) for the Vibrational Algebra. The methane molecule has a quite deep potential well and can be considered as a semirigid molecule for the energy range of available experimental spectroscopic data. The methane spectra under current analyses, though being extremely complicated, correspond to vibrational energies much below the dissociation energy of approximately 35 000 cm−1. This makes it possible to expand a full rovibrational Hamiltonian in a power series of normal coordinates corresponding to relatively small amplitude vibrations as outlined in Section 2. With this approach all operators H(k) n of CT can be expressed as finite linear combinations of polynomials in rovibrational operators. Such elementary terms in these combinations form an “operator basis set” in the Lie algebra of the CT method.68 It had been shown56,58 that powers of creation and annihilation operators of normal vibrations form a “canonical vibrational operator basis set”67 for CTs. This set was defined as follows ε V ub =

1 {Wub + ε(Wub)+ } 2

Δub = 1 if ω·u ≈ ω·b ,

Δub = 0 otherwise

(17)

In this way rather general resonance conditions can be easily accounted for. This part of the algorithm is a general one for polyatomic molecules. In the case of high symmetry molecules for an unambiguous labeling of degenerate vibration modes, it is necessary to indicate types of an intermediate tensor coupling. An ITO formulation is the most suitable for this purpose. Note that general properties (eqs 16,17) with respect to CT are the same for all semirigid polyatomic molecules and keep unaltered in tensorial formulations as well.67 The extension to nonrigid molecules is discussed in ref 71. 3.3. Elementary operators (“operator basis set”) for the rotational algebra. Let Jx, Jy, Jz be the Eckart frame angular momentum components which satisfy the following commutation relations

(14)

where the superindex ε distinguishes hermitian (ε = +1) and antihermitian (ε = −1) operators. The superscript “+” stands for hermitian conjugation. The operator Wub is the normalordered product of the elementary creation a+ and annihilation a vibrational operators Wub = (a1+)u1 (a 2+)u2 (a3+)u3 ...(a1)b1 (a 2)b2 (a3)b3 ...

1 − Δub −ε 1 ε (V ub )= V ub + ω · (u − b )

and

[Jα , Jβ ]− = −ieαβγ Jγ

(18)

where eαβγ is the antisymmetric unitary tensor. The molecular fixed ladder components are defined as J± = Jx ∓ iJy. For an analytical function f(x) the shift-relations67

(15)

J±n f (Jz ) = f (Jz ∓ n)J±n

The components of formal vectors u = {u1, u2, u3...} and b = {b1, b2, b3...} are integer powers of creation and annihilation operators. From eqs 14 and 15 one has the following hermicity property: (Vεub)+ = ε(Vεub). All vibrational operators (eq 14) are invariant under the time reversal and have real matrices in the normal mode harmonic oscillator basis set |v1,v2 ....⟩.

and

f (Jz )J±n = J±n f (Jz ± n)

(19)

play a key role to derive algebraic properties of Hrot terms. For the generality of the approach it is appropriate to classify rotational operators according to the irreps A,Bx,By,Bz of the D2 point group {E,Cx2,Cy2,Cz2} composed of three orthogonal axes of second orders. The full symmetry properties of the molecule 13784


Γ

Dyad 1 F2 1 E Pentad 2 A1 2 F2 2 E 2 F2 2 F1 2 A1 2 F2 2 A1 2 E Octad 3 F2 3 A1 3 F1 3 F2 3 E 3 F1 3 A1 3 F2 3 E 3 A2 3 F2 3 F2 3 E 3 F1 3 A1 3 F2 3 F1 3 F2 3 E 3 F1 3 F2 3 E 3 A2 3 A1 Tetradecad 4 A1 4 F2 4 E 4 F2

Pn

1310.81 1533.41

2587.28 2614.31 2624.81 2830.64 2846.20 2916.37 3019.47 3064.00 3065.35

3870.84 3909.17 3920.66 3931.23 4102.03 4129.23 4133.55 4143.12 4151.56 4162.14 4223.57 4319.41 4322.49 4322.74 4323.02 4349.46 4364.11 4379.35 4435.23 4537.75 4544.08 4592.83 4595.67 4595.98

5122.75 5144.04 5167.99 5210.86

2 2 2 3 1 3 4 4 3

5 5 2 6 4 3 6 7 5 1 8 9 6 4 7 10 5 11 7 6 12 8 2 8

9 13 9 14

ν̃

1 1

N

13785

0.674 −0.803 0.753 0.831

0.709 −0.967 −0.978 0.742 0.877 −0.945 −0.942 −0.971 −0.921 −0.999 0.942 0.965 0.945 −0.941 −0.924 0.668 0.967 0.713 0.958 −0.959 −0.936 −0.992 0.999 0.999

0.967 −0.990 −0.999 0.970 1.000 0.963 −0.963 −0.995 0.999

(0004) (0004) (0004) (0004)

(0003) (0003) (0003) (0003) (0102) (0102) (0102) (0102) (0102) (0102) (1001) (0011) (0011) (0011) (0011) (0201) (0201) (0201) (1100) (0110) (0110) (0300) (0300) (0300)

(0002) (0002) (0002) (0101) (0101) (1000) (0010) (0200) (0200)

1.000 (0001) 1.000 (0100)

c1 V(1) norm

0.595 −0.434 −0.541 −0.450

(0004) (0004) (0004) (0004)

(0003) (0011) (0011) (0003) (0102) (0011) (0011) (0011) (0102) (0300) (0003) (0003) (0102) (0102) (0102) (0201) (0110) (0201) (0102) (0201) (0201) (1100) (0102) (0102)

(0002) (0101) (1000) (0002)

0.254 −0.239 0.098 −0.006 0.623 −0.239 −0.195 −0.655 0.303 0.269 0.298 −0.163 0.343 0.011 0.245 −0.179 0.242 −0.292 −0.325 0.656 −0.246 −0.698 0.255 −0.253 0.236 0.128 0.011 0.011

(1000) (0010) (0200) (0010)

−0.253 −0.134 −0.006 −0.233

c2 V(2) norm

(0003) (0102) (0102) (0003) (0003) (0110) (0102) (1001) (0300) (0102) (0201) (0011)

0.183 −0.153 0.221 0.168 0.200 0.306 0.064 0.057 0.122 0.112 0.226 −0.009

−0.396 0.336 −0.342 0.246

(1002) (1002) (1002) (0012)

0.002 (0011)

(1001) (0102) (0102) (0011) (0011) (0110) (0003) (0110) (0011)

−0.303 0.085 0.067 −0.124 −0.285 −0.136 −0.157 −0.146 0.155

0.095 (0200) 0.122 (0002) −0.001 (0002)

0.068 (0002)

−0.026 (0200) 0.037 (0101)

c3 V(3) norm Icosad 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Pn

N

(continued) A1 22 A2 6 F1 24 E 27 F2 40 A1 23 F1 25 F2 41 F2 42 F1 26 E 28 A1 24 F2 43 F1 27 F2 44 F2 45 E 29 F2 46 F1 28 A2 7 F1 29 A1 25 F2 47 E 30 E 31 A1 26 F2 48 F1 30 F1 31 F2 49 F1 32 E 32 F2 50 F1 33 A1 27 F2 51 A2 8 F2 52 E 33 F1 34 F2 53 E 34

Γ 6738.60 6747.32 6756.17 6767.13 6769.78 6809.68 6822.75 6833.96 6859.86 6863.53 6863.63 6864.37 6872.58 6892.14 6898.22 6907.72 6909.33 6910.95 6915.82 6919.31 6922.40 6923.23 6925.79 6926.79 6939.84 6941.78 6941.97 6946.93 6951.21 6963.60 6973.68 6990.95 6993.71 7021.27 7025.10 7035.82 7057.38 7086.05 7046.48 7086.10 7098.65 7108.34

ν̃ 0.834 0.779 0.907 −0.591 0.596 −0.866 0.881 −0.639 0.617 0.613 0.662 −0.646 −0.465 −0.626 −0.764 −0.503 −0.889 −0.798 0.852 −0.920 −0.669 0.678 0.639 −0.664 0.851 0.832 −0.520 −0.580 −0.823 0.672 −0.569 0.732 0.607 0.842 0.829 0.874 −0.912 −0.561 0.824 −0.787 0.553 −0.646

(0104) (0104) (0104) (0104) (1003) (1003) (1003) (1003) (0013) (0013) (0013) (0013) (0203) (0203) (0013) (0203) (0013) (0013) (0013) (0013) (0013) (0013) (0013) (0013) (0203) (0203) (0203) (0203) (0203) (0203) (0203) (1102) (0203) (1102) (1102) (1102) (1102) (2001) (1102) (0112) (0112) (0112)

c1 V(1) norm

Table 1. Example of vibrational energy levels computed with 8-th order CTs and normal mode contributions for the Hef f eigenfunctions

0.304 0.511 −0.241 0.489 0.485 −0.304 0.282 0.598 0.554 0.566 0.544 −0.510 −0.458 −0.513 0.275 0.490 0.250 −0.258 0.220 −0.383 0.590 −0.563 −0.574 0.575 0.326 0.281 −0.447 −0.563 −0.289 −0.438 −0.559 −0.324 −0.585 −0.276 −0.317 0.301 −0.350 0.450 −0.314 −0.316 0.518 0.388

(0104) (0104) (0112) (0104) (1003) (0005) (0005) (1003) (0013) (0013) (0013) (0013) (0203) (0203) (0013) (0203) (0104) (0013) (0104) (0104) (0013) (0013) (0013) (0013) (0112) (0112) (0203) (0203) (0112) (0203) (0203) (2100) (0203) (1011) (1011) (0104) (0104) (0112) (1102) (0021) (2001) (0112)

c2 V(2) norm 0.281 −0.246 0.174 0.422 −0.403 0.213 −0.192 −0.303 −0.306 −0.325 −0.316 0.313 −0.318 0.280 −0.197 0.300 0.235 −0.220 −0.202 −0.084 −0.219 0.311 0.352 −0.239 0.266 0.261 0.391 −0.241 0.243 0.315 0.420 0.286 −0.336 0.269 0.276 0.185 −0.108 −0.387 0.304 0.246 −0.272 −0.297

(0112) (0013) (0104) (0104) (2001) (0104) (0005) (0005) (1011) (1011) (1011) (0005) (0203) (1201) (0021) (0112) (0021) (0104) (0013) (0112) (0013) (0104) (0104) (0104) (0211) (0211) (0203) (0203) (0203) (0203) (0203) (0104) (0203) (0104) (0104) (1011) (0302) (0021) (0104) (1110) (0112) (0021)

c3 V(3) norm

The Journal of Physical Chemistry A Article


Γ

E F1 A1 F2 F1 E F2 F1 F2 F1 A1 F2 E F2 A1 A1 E F2 F1 E F1 F2 F2 E F1 A2 F2 A1 E F2 F1 A1 F2 F1 E A1 E A2 F2 F1 F2 F2 F1

Pn

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

10 7 10 15 8 11 16 9 17 10 11 18 12 19 12 13 13 20 11 14 12 21 22 15 13 3 23 14 16 24 14 15 25 15 17 16 18 4 26 16 27 28 17

N

Table 1. continued

5229.17 5230.94 5240.97 5371.95 5390.68 5425.37 5430.82 5437.95 5445.18 5463.35 5493.05 5521.69 5533.81 5588.58 5605.82 5614.33 5614.59 5615.65 5616.21 5619.85 5626.65 5628.00 5643.62 5655.35 5656.68 5664.76 5669.43 5682.28 5691.73 5727.30 5745.24 5789.51 5823.29 5826.10 5832.99 5835.62 5843.29 5843.56 5844.68 5847.80 5861.11 5868.48 5880.78

ν̃ 0.782 −0.970 0.729 −0.680 −0.762 −0.913 −0.659 −0.874 −0.635 −0.781 0.832 −0.914 −0.919 0.897 −0.689 0.631 −0.629 0.890 0.908 0.718 −0.900 0.910 −0.643 0.690 0.924 −0.934 −0.681 0.847 0.724 0.892 0.937 0.835 0.567 −0.800 0.658 0.808 0.704 −0.934 0.756 −0.755 −0.694 −0.685 0.800

(0004) (0004) (0004) (0103) (0103) (0103) (0103) (0103) (0103) (0103) (1002) (1002) (1002) (0012) (0012) (0012) (0202) (0012) (0012) (0012) (0012) (0012) (0202) (0202) (0202) (0202) (0202) (0202) (0202) (1101) (1101) (2000) (0111) (0111) (0111) (0111) (0111) (0111) (0111) (0111) (1010) (0301) (0301)

c1 V(1) norm 0.565 0.233 −0.655 −0.550 −0.459 −0.253 −0.454 0.334 −0.598 0.522 −0.348 −0.301 −0.333 −0.268 0.493 0.603 −0.563 0.267 −0.291 −0.494 −0.265 0.321 0.619 0.482 −0.272 0.358 −0.655 −0.484 −0.625 −0.254 0.259 0.363 −0.521 −0.559 0.521 −0.319 −0.605 −0.358 0.518 0.537 −0.394 0.383 0.481

(0004) (0012) (0004) (0103) (0103) (0111) (0103) (0103) (0103) (0103) (2000) (0004) (0004) (1010) (0202) (0202) (0012) (0103) (0012) (0202) (0103) (0103) (0202) (0202) (0111) (0111) (0202) (0202) (0202) (1010) (0103) (0020) (0111) (0111) (0111) (0202) (0111) (0202) (0111) (0111) (0301) (0301) (0301)

c2 V(2) norm (0012) (0103) (0012) (1101) (1101) (0012) (0103) (0012) (0103) (0103) (0004) (1010) (0004) (0103) (0111) (0202) (0111) (0004) (0103) (0103) (0012) (0103) (0111) (0202) (0210) (0111) (0111) (0202) (0103) (0103) (1002) (0020) (0103) (0202) (0020) (0202) (0202) (0202) (0111) (1010) (0210)

0.227 −0.067 −0.174 0.297 0.283 0.243 0.427 0.195 0.355 −0.265 0.330 −0.124 0.120 0.206 0.320 0.305 −0.314 −0.222 0.253 0.265 −0.257 −0.151 −0.348 −0.360 0.147 0.230 0.155 −0.252 0.218 0.174 0.332 −0.415 0.159 −0.309 −0.283 −0.263 0.268 −0.326 −0.376 0.348 0.326

c3 V(3) norm 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Pn A2 F2 F1 E A1 F2 F1 F2 A1 E A2 F1 F2 F1 F1 A1 F2 E F1 E F2 A1 F1 F2 E E A2 F2 F1 F2 F2 E F1 E A1 F2 F1 F2 F2 F1 A1 E F2

Γ 9 54 35 35 28 55 36 56 29 36 10 37 57 38 39 30 58 37 40 38 59 31 41 60 39 40 11 61 42 62 63 41 43 42 32 64 44 65 66 45 33 43 67

N 7115.89 7117.08 7118.88 7120.92 7122.49 7122.72 7131.89 7132.23 7133.74 7135.01 7135.73 7140.31 7142.65 7152.08 7156.02 7156.99 7158.22 7165.23 7166.68 7169.81 7170.14 7178.22 7181.52 7192.77 7193.31 7218.85 7222.57 7226.61 7246.92 7251.23 7270.24 7295.04 7296.57 7297.16 7300.27 7300.30 7326.99 7332.08 7338.44 7338.88 7343.43 7343.61 7347.04

ν̃ 0.939 0.688 −0.789 0.658 −0.540 −0.666 −0.773 0.791 0.530 0.640 −0.815 −0.608 0.619 0.615 0.627 0.666 −0.725 −0.703 −0.700 −0.627 −0.752 −0.719 0.759 0.795 −0.833 0.681 −0.931 0.606 0.882 −0.619 0.624 0.771 0.795 0.750 −0.781 0.747 −0.485 −0.505 0.853 −0.845 0.612 −0.544 0.848

(0112) (0112) (0112) (0302) (0302) (0112) (0112) (0112) (0112) (0112) (0302) (0112) (0112) (0112) (0302) (1011) (1011) (1011) (1011) (0302) (0302) (0302) (0302) (0302) (0302) (0302) (0302) (1201) (1201) (0021) (1201) (2100) (0021) (0021) (0021) (0021) (0211) (0211) (0021) (0021) (0211) (0211) (0211)

c1 V(1) norm −0.256 −0.343 −0.281 0.305 0.441 −0.391 −0.357 0.272 0.446 0.584 0.401 0.479 −0.421 0.543 −0.407 −0.507 −0.315 −0.275 −0.305 0.460 −0.393 −0.534 0.474 −0.423 −0.320 0.554 −0.320 0.469 −0.277 −0.568 −0.489 0.337 −0.366 −0.320 0.379 −0.356 0.447 −0.498 0.247 −0.281 0.419 −0.356 −0.277 (0104) (0021) (0203) (0302) (0112) (0112) (0112) (0112) (0302) (0112) (0211) (0112) (0112) (0112) (0302) (0112) (0112) (0112) (1102) (0302) (0302) (0302) (0302) (0302) (0302) (0302) (0302) (1201) (1110) (1201) (1201) (0120) (1011) (1011) (1011) (1011) (0211) (0211) (0112) (0112) (0211) (0211) (0120)

c2 V(2) norm 0.149 0.297 0.262 −0.301 0.399 0.339 0.285 0.266 0.331 −0.279 0.250 −0.363 −0.294 0.316 −0.337 0.287 −0.310 0.249 −0.234 −0.317 0.343 0.333 −0.201 0.205 −0.276 −0.383 0.101 −0.347 0.230 0.232 0.409 0.285 −0.218 −0.308 0.278 −0.311 −0.432 0.335 −0.205 0.212 0.397 −0.319 −0.239

(0120) (0203) (0112) (0211) (0302) (0112) (0203) (0203) (0021) (0203) (0302) (0112) (0112) (0302) (0112) (0021) (0112) (0302) (0112) (0211) (0211) (0211) (0211) (0211) (0211) (0302) (1102) (0021) (0203) (2001) (0021) (0021) (0211) (2100) (0013) (0211) (0120) (0120) (0211) (0211) (0302) (0302) (1110)

c3 V(3) norm


13786


13787

33 20 20 34 23 35 21 36 24 22 21 37 25 5 38 23 26 39

F2 A1 F1 F2 E F2 F1 F2 E F1 A1 F2 E A2 F2 F1 E F2

6379.72 6407.55 6430.65 6451.50 6507.77 6508.02 6530.34 6539.91 6620.47 6641.18 6658.37 6659.14 6682.57 6684.69 6719.24 6722.94 6731.07 6734.38

5895.32 5910.06 5939.98 5952.86 5968.10 6004.70 6043.97 6055.23 6061.27 6066.49 6118.46 6119.98 6124.87

ν̃ (0301) (0301) (1200) (1200) (0020) (0020) (0020) (0210) (0210) (0210) (0400) (0400) (0400) (0005) (0005) (0005) (0005) (0005) (0005) (0005) (0005) (0104) (0104) (0104) (0104) (0104) (0104) (0104) (0104) (0104) (0104)

−0.844 −0.853 0.739 −0.954 0.655 −0.819 −0.935 −0.669 0.929 −0.663 0.985 0.987 0.998 −0.564 −0.875 0.673 −0.654 0.916 0.706 −0.715 0.664 −0.548 −0.778 0.786 −0.705 −0.667 0.718 0.673 0.810 0.649 −0.831

c1 V(1) norm

−0.439 0.352 −0.577 0.442 −0.284 −0.465 −0.616 −0.514 −0.541 0.322 −0.363 −0.494 0.436 −0.517 −0.447 −0.283 0.366 0.291

−0.457 0.513 −0.548 −0.259 0.586 0.386 −0.222 0.653 −0.304 −0.615 −0.168 −0.144 −0.063 (0005) (1003) (0005) (0005) (0013) (0005) (0005) (0005) (0104) (1102) (0013) (0104) (0104) (0104) (0104) (0104) (0104) (0104)

(0301) (0301) (0020) (0202) (1200) (1010) (0111) (0210) (0301) (0210) (1200) (1200) (0400)

c2 V(2) norm

−0.345 −0.277 −0.363 0.399 −0.185 0.265 −0.189 −0.453 0.357 −0.309 −0.334 0.322 0.375 −0.301 −0.271 0.274 −0.324 0.207

−0.201 0.091 0.261 −0.138 0.304 −0.241 −0.207 −0.225 −0.160 −0.323 −0.020 0.064 0.016 (0005) (0013) (1003) (0005) (0013) (0005) (0013) (0005) (1102) (0104) (1102) (1102) (0104) (0013) (0104) (0112) (0104) (0112)

(0210) (1101) (0202) (0400) (0111) (0111) (0111) (0301) (0301) (0301) (0111) (0400) (0202)

c3 V(3) norm 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Pn F1 A2 E A1 E F2 F1 F1 F2 F2 F1 F2 F1 F2 E A2 A1 E F2 F1 A2 E A1 F1 F2 F1 F2 E A2 A1 E

Γ 46 12 44 34 45 68 47 48 69 70 49 71 50 72 46 13 35 47 73 51 14 48 36 52 74 53 75 49 15 38 50

N 7347.54 7350.27 7353.51 7362.18 7363.02 7366.60 7370.12 7373.46 7375.35 7385.85 7396.17 7409.97 7424.03 7438.21 7448.95 7469.08 7469.42 7484.50 7511.10 7513.16 7546.92 7552.97 7559.86 7570.94 7577.26 7582.32 7586.32 7642.30 7644.94 7646.22 7652.45

ν̃ −0.711 −0.833 −0.570 −0.699 −0.612 −0.636 −0.618 0.748 0.566 −0.507 −0.657 −0.808 0.704 0.700 −0.682 0.951 0.951 −0.634 −0.800 −0.776 −0.925 −0.907 0.886 −0.808 0.735 −0.816 0.692 0.979 −0.987 0.987 −0.999 (0211) (0211) (0211) (0211) (0211) (0211) (0211) (1110) (1110) (0401) (0401) (0401) (0401) (0401) (1300) (1300) (1300) (1300) (0120) (0120) (0120) (0120) (0120) (0310) (0310) (0310) (0310) (0500) (0500) (0500) (0500)

c1 V(1) norm 0.468 −0.375 0.558 0.588 0.483 0.392 −0.600 −0.395 0.515 −0.456 −0.599 −0.447 −0.638 −0.596 0.564 0.264 0.264 −0.583 0.381 0.343 −0.328 0.293 −0.303 0.405 0.532 −0.378 −0.537 −0.191 0.156 −0.155 0.049

(0211) (0302) (0211) (0211) (0211) (0211) (0211) (0211) (0211) (0401) (0401) (0401) (0401) (0401) (0120) (0302) (0302) (0120) (1110) (1110) (0211) (0211) (0211) (0310) (0310) (0310) (0310) (1300) (1300) (1300) (0500)

c2 V(2) norm −0.256 0.332 0.405 −0.257 0.473 −0.290 −0.275 0.268 −0.283 −0.354 0.311 −0.317 0.230 0.372 −0.280 0.147 0.146 0.285 −0.253 0.285 0.172 0.223 −0.268 −0.216 −0.298 −0.315 −0.298 0.050 −0.021 −0.021 0.015

(0120) (0120) (0302) (0302) (0211) (1110) (0302) (0211) (0401) (1110) (0310) (0310) (0310) (0401) (0302) (0500) (0500) (2100) (0211) (0310) (0112) (0211) (0211) (0120) (0401) (0401) (0401) (0500) (0211) (0211) (0302)

c3 V(3) norm

Pn: polyad; Γ: symmetry species for Td group; N: global ranking number for given Γ; ν̃ = Evib/hc [cm−1]: term values directly calculated by CT from NRT PES47 without any adjustable parameters; ci: expansion coefficient of Heff eigenfunctions in the normal mode basis set V(i) norm with the corresponding vibrational quantum numbers (v1, v2, v3, v4).

29 18 17 19 18 30 20 31 19 32 19 21 22

F2 F1 A1 E A1 F2 E F2 F1 F2 A1 E E

4 4 4 4 4 4 4 4 4 4 4 4 4 Icosad 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

N

Γ

Pn

Table 1. continued




Article

Figure 3. The overall view on vibrational polyads of methane up to ν̃ = 18000 cm−1. At the main panel n bending quanta of ν4 are subtracted for Pn polyads. Various colors give an idea of the importance of anharmonic vibrational coupling among the sublevels of the same symmetry. For each level, the major mixing coefficient is given in red, the next one in green, then blue, etc.

will be accounted for at the final step of the Heff transformations. An elementary homogeneous polynomial can be characterized using the following labeling: powers of rotational components J±, Jz, and of J2 (below n,m,l); symmetry species Γ; hermicity index ε distinguishing hermitian (ε = +1) and antihermitian (ε = −1) operators; time reversal126 index τ with τ = +1 for invariant and τ = −1 for anti-invariant terms. R-Operators Basis Set in the D2 Point Group. This type of operator basis set67,96 has been often employed for a modeling of high-resolution spectra of low symmetry molecules. Here we use a definition adapted to the above explained hermicity and symmetry labeling κ R mε ,,Γn ,2l = (Z+mn + ε( −)Γ Z −mn)J2l (20) 2 where κ is a phase factor, J = + diagonal” operators are defined as 2

Z+mn = J+m (Jz + m /2)n

J2x

and

J2y

+

J2z ,

There are two convenient choices of the phase factor κ in the definition of elementary rotation operators: (α) the simplest choice for which all Rε,Γ m,n,2l have real matrix elements in the standard |J,k⟩ basis set: κ = 1 ⇒ τ = ( −)Γ

However, in this case, rotational operators of Γ″ symmetry type are anti-invariant under the time reversal and consequently corresponding Hamiltonian parameters would take imaginary values. ε,Γ (β) the choice for which all Rm,n,2l are invariant under the time reversal operation ⎧1, for Γ = Γ′ τ = +1 ⇒ κ = ⎨ ⎩ i , for Γ = Γ″

and “one-rotational-

Z −mn = (Jz + m /2)n J−m (21)

being hermitian conjugate one to the other: (Z+mn) = Z−mn and (Z−mn)+ = Z+mn. As mentioned above for the sake of generality we first consider the symmetry classification on the D2 point group independently of the molecular point group. Elementary rotational operators (eq 20) fall in two subsets corresponding to the symmetry species Γ′ = {A,By} and Γ″ = {Bx,Bz} for which the signs in eq 20 are defined as follows: and

( −)Γ = −1, if Γ ∈ Γ″ (22)

By definition the hermicity index ε verifies the relation ε,Γ + (Rε,Γ m,n,2l) = ε(Rm,n,2l) and is unambiguously determined by the powers m, n of the J±, Jz components:

ε = ( −1)m + n

(25)

As the full Hamiltonian H is also invariant under the time reversal, all parameters involved in the Hamiltonian expansion ε,Γ remain real. Though matrix elements of Rm,n,2l ″ in the standard |J,k⟩ basis set are imaginary with this choice, it is possible to convert the Hamiltonian matrix in a real form by an appropriate transformation of the basis set functions.101 The exact expressions for commutators and anticommutators ε,Γ given in ref 96 for Γ′ symmetry species of the operators Rm,n,2l could be easily extended to a general case. 3.4. Elementary Operators for the Ro-vibrational Algebra. Because the initial and transformed Hamiltonians have to be totally symmetric, all terms H(k) n involved in CT are expressed as a linear combination of totally symmetric rovibrational contributions. For this reason they contain products of vibrational V and rotational R factors of the same symmetry species ΓV = ΓR = Γ

+

( −)Γ = 1, if Γ ∈ Γ′

(24)

Hn(k) =

∑ hubr ,

ε hubr = tubrV ub R rε , Γ

(26)

where t is a numerical parameter corresponding to the rovibrational contribution. In order to apply the latter condition, one

(23) 13788



Article

Table 2. Comparison of Vibrational Levels Computed from PES with Empirical Values for the Methane Polyads (All Values in cm−1) Pn

vib

Dyad 1 0001 1 0100 Pentad 2 0002 2 0002 2 0002 2 0101 2 0101 2 1000 2 0010 2 0200 2 0200 Octad 3 0003 3 0003 3 0003 3 0003 3 0102 3 0102 3 0102 3 0102 3 0102 3 0102 3 1001 3 0011 3 0011 3 0011 3 0011 3 0201 3 0201 3 0201 3 1100 3 0110 3 0110 3 0300 3 0300 3 0300 Tetradecad 4 0004 4 0004 4 0004 4 0004 4 0004 4 0004 4 0004 4 0103 4 0103 4 0103 4 0103 4 0103 4 0103 4 0103 4 1002 4 0012 4 0012 4 0012 4 0202 4 0012 4 0012

Γ

N

m

CT o8

CT conv o6-o8

CT o6

emp

ref

emp-CT o8

Var-CT o8

F2 E

1 1

1 1

1310.81 1533.41

0.01 0.05

1310.80 1533.36

1310.76 1533.33

−0.05 −0.08

−0.05 −0.08

A1 F2 E F2 F1 A1 F2 A1 E

2 2 2 3 1 3 4 4 3

1 1 1 2 1 2 3 3 2

2587.28 2614.31 2624.81 2830.64 2846.20 2916.38 3019.47 3064.00 3065.35

−0.12 −0.02 −0.07 −0.03 0.00 0.34 0.28 0.06 0.06

2587.41 2614.33 2624.89 2830.67 2846.20 2916.04 3019.19 3063.94 3065.28

2587.04 2614.26 2624.62 2830.32 2846.07 2916.48 3019.49 3063.64 3065.14

−0.24 −0.05 −0.20 −0.32 −0.13 0.11 0.03 −0.35 −0.20

−0.16 −0.07 −0.10 −0.16 −0.16 0.11 0.03 −0.21 −0.18

F2 A1 F1 F2 E F1 A1 F2 E A2 F2 F2 E F1 A1 F2 F1 F2 E F1 F2 E A2 A1

5 5 2 6 4 3 6 7 5 1 8 9 6 4 7 10 5 11 7 6 12 8 2 8

1 1 1 2 1 2 2 3 2 1 4 5 3 3 3 6 4 7 4 5 8 5 2 4

3870.84 3909.17 3920.66 3931.23 4102.03 4129.23 4133.55 4143.12 4151.56 4162.14 4223.57 4319.41 4322.49 4322.74 4323.02 4349.46 4364.11 4379.35 4435.23 4537.75 4544.08 4592.83 4595.67 4595.98

−0.32 0.07 −0.26 −0.21 −0.34 −0.21 −0.21 −0.11 −0.18 −0.24 0.39 0.57 0.34 0.33 0.39 −0.19 −0.11 −0.11 0.54 0.59 0.43 −0.05 −0.02 0.05

3871.16 3909.10 3920.91 3931.44 4102.36 4129.44 4133.77 4143.23 4151.73 4162.38 4223.18 4318.84 4322.15 4322.42 4322.64 4349.65 4364.22 4379.46 4434.69 4537.16 4543.65 4592.88 4595.68 4595.92

3870.49 3909.20 3920.50 3930.92 4101.39 4128.77 4132.88 4142.86 4151.20 4161.84 4223.46 4319.21 4322.18 4322.58 4322.72 4348.72 4363.62 4378.94 4435.13 4537.55 4543.76 4592.04 4595.28 4595.52

−0.35 0.03 −0.15 −0.31 −0.64 −0.47 −0.67 −0.26 −0.36 −0.30 −0.11 −0.20 −0.30 −0.16 −0.31 −0.74 −0.49 −0.41 −0.10 −0.20 −0.33 −0.79 −0.39 −0.46

−0.08 0.02 −0.13 −0.01 −0.24 −0.23 −0.17 −0.19 −0.17 −0.19 0.06 −0.04 0.18 −0.08 0.01 −0.33 −0.28 −0.28 −0.01 −0.12 −0.14 −0.40 −0.29 −0.31

A1 F2 E F2 E F1 A1 F2 F1 E F2 F1 F2 F1 A1 F2 A1 A1 E F2 F1

9 13 9 14 10 7 10 15 8 11 16 9 17 10 11 19 12 13 13 20 11

1 1 1 2 2 1 2 3 2 3 4 3 5 4 3 7 4 5 5 8 5

5122.75 5144.04 5167.99 5210.86 5229.17 5230.94 5240.97 5371.95 5390.68 5425.37 5430.82 5437.95 5445.18 5463.35 5493.05 5588.58 5605.82 5614.33 5614.59 5615.66 5616.21

−0.79 −0.49 −1.00 −0.26 −0.66 −0.52 −0.44 −0.88 −0.69 −0.16 −0.59 −0.60 −0.34 −0.54 0.02 0.21 0.69 −0.18 −0.46 0.37 0.59

5123.55 5144.53 5169.00 5211.12 5229.82 5231.46 5241.41 5372.83 5391.37 5425.53 5431.41 5438.56 5445.51 5463.89 5493.02 5588.37 5605.13 5614.51 5615.05 5615.29 5615.62

5121.76 5143.36 5167.20 5210.74 5228.74 5230.58 5240.45 5370.49 5389.73 5424.78 5429.87 5437.30 5444.80 5462.90 5493.16 5587.97 5604.47 5613.85 5614.59 5615.99 5615.41

−0.99 −0.68 −0.79 −0.12 −0.42 −0.36 −0.52 −1.46 −0.96 −0.59 −0.95 −0.65 −0.38 −0.44 0.11 −0.62 −1.34 −0.48 0.00 0.34 −0.80

0.34 0.19 0.39 1.06 2.05 0.85 0.58 0.11 −0.17 0.50 0.67 0.31 0.25 0.29 0.51 0.22 0.23 0.11 −0.03 0.56 0.24

13789



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Table 2. continued Pn

vib

Tetradecad 4 0012 4 0012 4 0012 4 0202 4 0202 4 0202 4 0202 4 0202 4 0202 4 0202 4 0111 4 0111 4 0111 4 0111 4 0111 4 0111 4 0111 4 0111 4 1010 4 0301 4 0301 4 0301 4 0301 4 1200 4 0020 4 0020 4 0210 4 0210 4 0210 4 0400 4 0400 Icosad 5 0005 5 0005 5 0005 5 0005 5 0005 5 0005 5 0005 5 0005 5 1011 5 0120 Triacontad 6 1012 6 0030 6 0030

Γ

N

m

CT o8

CT conv o6-o8

CT o6

emp

E F1 F2 F2 E F1 A2 F2 A1 E F2 F1 E A1 E A2 F2 F1 F2 F2 F1 F2 F1 E F2 E F2 F1 F2 E E

14 12 21 22 15 13 3 23 14 16 25 15 17 16 18 4 26 16 27 28 17 29 18 19 30 20 31 19 32 21 22

6 6 9 10 7 7 1 11 6 8 13 9 9 8 10 2 14 10 15 16 11 17 12 11 18 12 19 13 20 13 14

5619.86 5626.65 5628.00 5643.62 5655.35 5656.68 5664.76 5669.43 5682.28 5691.73 5823.29 5826.10 5832.99 5835.62 5843.29 5843.56 5844.68 5847.80 5861.11 5868.48 5880.78 5895.32 5910.06 5952.86 6004.70 6043.97 6055.23 6061.27 6066.49 6119.98 6124.87

−0.46 0.30 0.26 −0.70 −0.53 −0.39 −0.38 −0.37 −0.40 −0.55 0.69 0.81 0.10 0.19 0.53 0.63 0.61 0.46 0.64 −0.51 −0.43 −0.31 −0.42 0.72 0.97 0.85 0.90 0.66 0.42 −0.35 −0.11

5620.32 5626.35 5627.73 5644.32 5655.89 5657.08 5665.14 5669.79 5682.68 5692.29 5822.60 5825.29 5832.89 5835.44 5842.75 5842.93 5844.08 5847.34 5860.47 5868.99 5881.20 5895.63 5910.48 5952.14 6003.73 6043.12 6054.33 6060.62 6066.07 6120.33 6124.98

5618.24 5626.11 5627.34 5641.86 5654.54 5655.76 5664.07 5668.35 5681.29 5691.04 5823.10 5825.42 5832.02 5834.82 5842.56 5843.20 5844.03 5847.38 5861.49 5867.53 5879.76 5894.39 5908.71 5952.44 6004.62 6043.82 6054.61 6060.62 6065.59 6118.90 6124.11

F2 A1 F1 F2 E F2 F1 F2 F2 F2

33 20 20 34 23 35 21 36 58 73

1 1 1 2 1 3 2 4 26 41

6379.72 6407.55 6430.65 6451.50 6507.77 6508.02 6530.34 6539.91 7158.22 7511.10

−1.20 0.11 −1.88 −1.42 −0.40 −0.78 −1.17 −0.94 1.01 1.57

6380.93 6407.44 6432.53 6452.92 6508.17 6508.79 6531.51 6540.85 7157.20 7509.53

6377.53 6405.97 6429.24 6450.06 6507.39 6507.55 6529.78 6539.18 7156.72 7510.97

a a a a a a a a b c

−2.19 −1.58 −1.41 −1.44 −0.38 −0.47 −0.56 −0.73 −1.49 −0.13

F2 F2 F2

111 158 166

36 82 87

8421.60 8907.77 9046.67

−1.01 0.58 1.82

8422.61 8907.19 9044.85

8421.00 8907.30 9045.96

b b b

−0.60 −0.47 −0.71

dRMS/cm−1

0.60

ref

emp-CT o8

Var-CT o8

−1.62 −0.55 −0.65 −1.76 −0.81 −0.93 −0.69 −1.08 −0.99 −0.70 −0.18 −0.68 −0.97 −0.81 −0.73 −0.36 −0.66 −0.42 0.38 −0.95 −1.01 −0.93 −1.35 −0.43 −0.08 −0.15 −0.62 −0.65 −0.90 −1.08 −0.75

0.31 0.27 0.34 0.11 −0.02 −0.07 0.05 0.00 −0.06 −0.12 0.32 0.07 0.06 −0.02 0.05 −0.05 0.13 −0.04 0.45 −0.01 −0.27 −0.22 −0.37 −0.04 0.04 0.08 −0.23 −0.20 −0.30 −0.61 −0.38

0.74

0.35

Pn: polyad; vib: vibrational normal mode numbers v1, v2, v3, v4; Γ: symmetry irrep for Td group; N and m: global and polyads ranking numbers for given symmetry type; CT: Contact Transformations calculation from NRT PES;47 CT conv: convergence of CT method between successive orders o8 and o6. ″emp″: empirically derived values from experimental spectra (up to P4 in refs 28 and 29, (a) in ref 27, (b) in ref 127,and (c) in ref 128; Var-CT: discrepancy between variational calculations47 and CT−o8 using the same PES.

has to establish an isomorphism between the D2 group (see Section 3.3 above) and the molecular point group G or one of its subgroups. If it is not the case, one can work with a subgroup of D2. For a very large panel of molecules, one can use an isomorphism for subgroups Cs ∈ G and Cs ∈ D2. With an appropriate choice of axes we have Γ′ = A′ and Γ″ = A″ in eqs 20−26 where A′ and A″ are symmetry species of the Cs subgroup. Another approach would be

working in the ITO representation accounting for the full molecular symmetry. Also the hermicity of rotational and vibrational factors must be the same εV = εR = ε. As in Section 3.2, vector indices in eq 26 u = {u1, u2, u3...} and b = {b1, b2, b3...} represent powers of creation and annihilation operators for normal vibration modes and r = (n,m,2l) represents powers of rotational components as in Section 3.3. In a framework of the conventional perturbation 13790



Article

Table 3. Comparison of Rotational and Centrifugal Distortion Constants Computed from PES for the Methane Ground State with Empirical Values in the ITO Representation empirical from spectra fit

predicted from PESa rot tensors R R R R R R R R R R

CT−o4

2(0,0A1) 4(0,0A1) 4(4,0A1) 6(0,0A1) 6(4,0A1) 6(6,0A1) 8(0,0A1) 8(4,0A1) 8(6,0A1) 8(8,0A1)

b

5.241009354 −1.108053448 −3.029054292 5.948705517 −1.500109387 −5.921366200

CT−o6

ref 22, STDS

ref 29

5.241018319 −1.110654529 −3.043646124 6.351127219 −1.673328120 −5.988055814 −5.159251570 −7.852523937 −6.256790652 −5.159825888

5.241045999 −1.110233288 −3.036059582 6.368552465 −1.676644820 −6.151353108 −5.098836673 −8.237821519 −5.978781619 −4.105457644

5.2410445(13) −1.11018(11) −3.036065(35) 6.369(35) −1.6778(14) −6.1643(27) −5.22(39) −8.51(14) −6.204(43) −5.06(14)

× × × × × × × × ×

10−4 10−6 10−9 10−10 10−11 10−13 10−15 10−15 10−16

a

Using PES47 in the qt-representation (Section 2). bThe ordering scheme of the (A2) ansatz described in ref 108 was used. Parameter values are given in cm−1, standard errors for empirical determination29 being given in parentheses.

The CT transformed Hamiltonian takes a block-diagonal form in the zero order wave functions (Figure 1). Each block corresponds to an effective Hamiltonian “projected” onto the corresponding polyad subspace. The transformed Hamiltonian takes the form which in the concisest notations can be written as15−17,60

theory for a separation of molecular variables (vibrational zeroorder approximation), rotational factors behave like constant parameters with respect to operations of CT ⟨VR ⟩ = ⟨V ⟩R

and

1 1 (VR ) = (V )R + +

(27)

This means that eqs 16,17, and commutator relations are in principle sufficient to carry out computation up to a needed order. Note that the exact formal polynomial algebra for rovibrational operator involved in the ro-vibrational Hamiltonian transformations up to arbitrary power M has been implemented96 accounting for all contributions in commutators and anticommutators. These calculations are rather involved and are optionally available as a FORTRAN routine of MOL_CT or in the form of tables for Lie algebra structure constants readable by an external code.

H eff =

n

n

+ /{Octad} + /{Tetradecad} + ...

(29)

In this work, the CT transformed Hamiltonians were first computed in the representation (eqs 14−26) and then converted to the ITO representation on the O(3) ⊃ Td group chain as explained in Section 2: /{Pn} =

∑ tuΩu(K...,bΓ)b ...β( τVuΓ u ...b b .. ⊗ RΩ(k ,Γ))A1 1 2

1 2

1 2

1 2

(30)

τ Γ Vu1u2...b1b2..

4. POLYADS OF THE METHANE MOLECULE 4.1. Resonance Conditions and Polyad Effective Hamiltonian in the ITO Representation. The theory of spectra of spherical top molecule is known to be quite complex primary because of high symmetry giving rise to degeneracies and quasi-degeneracies and to vast and complicated resonance interactions. Four normal-mode frequencies of CH4 exhibit an approximate relation of stretching and bending frequencies with ω1 ≈ ω3 ≈ 2ω2 ≈ 2ω4, resulting in vibrational levels being grouped into polyads with levels of nearby energies. Each polyad Pn is defined by the integer polyad number n as n = 2(v1 + v3) + v2 + v4

∑ /{P } = /{GS} + /{Dyad} + /{Pentad}

Vibrational operators are constructed by recursive couplings of elementary creation (a+) and annihilation (a) operators for each normal mode, τ = ± 1 is the parity under time reversal, and Γ is the irreducible representation of the Td point group. The constant β was introduced15 to simplify a comparison with standard expressions for rotation constants. Different construction rules are possible.15,12,24 Here we use the coupling scheme suggested by Nikitin et al.,24 which is compatible with the MIRS software26,105 for high-resolution spectra calculations. Rotational tensors RΩ(k,Γ) were also built recursively from successive couplings of the elementary tensor R1(1) = 2J following Moret-Bailly106 and Zhilinskii.60,107 The upper indices indicate the rotational characteristics of the considered term: Ω is the rotational power with respect to the angular momentum components; k is the tensor rank in the full rotation group; Γ is the rotational symmetry type coinciding with the vibrational symmetry type to satisfy the invariance condition under the molecular point group operations. The procedure for the converting vibration−rotation terms from a standard representation (eqs 14,26) to the ITO representation (eq 30) for symmetric-top and spherical-top molecules has been described in detail in refs 60, 101, and 102. The lower case indices in vibrational terms τVΓu1u2...b1b2.. specific for each polyad belong to restricted subsets in a way that the coupling terms between different polyads vanish. The tΩ(K,Γ) u1u2...b1b2... parameters with the same powers {u1u2...} = {b1b2...} of creation (a+) and annihilation (a) normal mode operators are called

(28)

where the (v1,v2,v3,v4) are principal vibrational normal mode quantum numbers taking values 0,1,2, etc. Methane normal mode vibrations are labeled by the irreducible representations {A1, A2, E, F1, F2} of the Td point group: v1(A1), v3(F2) (stretching modes), v2(E) and v4(F2) (bending modes). The v1(A1) corresponds to a one-dimensional oscillator, while v2(E) is doubly degenerate and v3(F2) and v4(F2) are triply degenerate. The ground state (GS) polyad P0 contains only (0000) vibrational state, the Dyad P1 contains {(0100)/(0001)}, the Pentad P2 contains {(1000)/(0010)/(0200)/(0101)/(0002)}, and so on. Due to the degeneracy of three of the normal vibrations, each vibrational level contains a certain number of vibrational sublevels (example are given in Figure 1), whose symmetries can be found by means of group theoretical methods. 13791



Article

Table 4. Comparison of Rotational Energy Levels Computed from PES for the Methane Ground State with Empirical Values (in cm−1) J

Γ

n

CT o4 (cm−1)

emp-CT (cm−1)

J

Γ

n

CT o4 (cm−1)

emp-CT (cm−1)

J

Γ

n

CT o4 (cm−1)

emp-CT (cm−1)

0 1 2 2 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10

A1 F1 E F2 A2 F1 F2 A1 E F1 F2 E F1 F1 F2 A1 A2 E F1 F2 F2 A2 E F1 F1 F2 F2 A1 E E F1 F1 F2 F2 A1 A2 E F1 F1 F1 F2 F2 A1 A2 E

1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 2 1 2 1 1 2 1 2 1 2 1 1 1 1 2 3 1 2 1 1 1

0.0000 10.4816 31.4419 31.4422 62.8778 62.8754 62.8764 104.7722 104.7754 104.7741 104.7794 157.1363 157.1234 157.1380 157.1270 219.9440 219.9187 219.9123 219.9401 219.9139 219.9356 293.1528 293.1687 293.1216 293.1772 293.1252 293.1631 376.7289 376.7341 376.8196 376.7321 376.8031 376.7842 376.8246 470.8291 470.8710 470.7972 470.7153 470.8035 470.8531 470.7186 470.8632 575.2210 575.0539 575.0495

0.0000 0.0001 0.0002 0.0002 0.0004 0.0004 0.0004 0.0006 0.0006 0.0006 0.0006 0.0009 0.0009 0.0009 0.0009 0.0012 0.0011 0.0011 0.0012 0.0011 0.0011 0.0014 0.0014 0.0013 0.0014 0.0013 0.0014 0.0015 0.0015 0.0016 0.0015 0.0016 0.0016 0.0016 0.0018 0.0018 0.0017 0.0016 0.0017 0.0018 0.0016 0.0018 0.0018 0.0017 0.0016

10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14

E F1 F1 F2 F2 F2 A2 E E F1 F1 F1 F2 F2 F2 A1 A1 A2 E E F1 F1 F1 F2 F2 F2 A1 A2 E E F1 F1 F1 F1 F2 F2 F2 A1 A2 E E E F1 F1 F1

2 1 2 1 2 3 1 1 2 1 2 3 1 2 3 1 2 1 1 2 1 2 3 1 2 3 1 1 1 2 1 2 3 4 1 2 3 1 1 1 2 3 1 2 3

575.2699 575.1824 575.2578 575.0509 575.1682 575.2834 689.8605 689.8845 690.0378 689.7037 689.9550 690.0474 689.7060 689.8751 690.0157 814.6447 815.1420 815.0874 814.6475 814.9916 814.6465 814.8828 815.1299 814.8650 815.0065 815.1143 950.1525 950.3838 950.1281 950.5036 949.8404 950.1352 950.3035 950.5209 949.8419 950.3356 950.4859 1095.8278 1095.2523 1095.2505 1095.8938 1096.1561 1095.6312 1095.8683 1096.1321

0.0019 0.0018 0.0019 0.0016 0.0018 0.0019 0.0017 0.0017 0.0019 0.0016 0.0018 0.0019 0.0016 0.0017 0.0018 0.0014 0.0017 0.0017 0.0014 0.0016 0.0014 0.0016 0.0017 0.0016 0.0016 0.0017 0.0013 0.0014 0.0013 0.0014 0.0012 0.0013 0.0013 0.0014 0.0012 0.0014 0.0014 0.0009 0.0008 0.0008 0.0009 0.0009 0.0009 0.0009 0.0009

14 14 14 14 15 15 15 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 17 17 17 17 17 17

F2 F2 F2 F2 A1 A2 A2 E E F1 F1 F1 F1 F2 F2 F2 F2 A1 A1 A2 E E E F1 F1 F1 F1 F2 F2 F2 F2 A1 A2 E E E F1 F1 F1 F1 F1 F2 F2 F2 F2

1 2 3 4 1 1 2 1 2 1 2 3 4 1 2 3 4 1 2 1 1 2 3 1 2 3 4 1 2 3 4 1 1 1 2 3 1 2 3 4 5 1 2 3 4

1095.2511 1095.6184 1095.9805 1096.1701 1252.0019 1251.2911 1252.0614 1251.3075 1251.7796 1250.8360 1251.5903 1251.8070 1252.0276 1250.8369 1251.3017 1251.6465 1252.0462 1416.5540 1417.8658 1417.5799 1416.5550 1417.5000 1418.1196 1416.5547 1417.1297 1417.8079 1418.1000 1417.1203 1417.5202 1417.7536 1418.1382 1593.0577 1593.7862 1593.0463 1593.9301 1594.3695 1592.3605 1593.0500 1593.5247 1594.0279 1594.3856 1592.3610 1593.5635 1593.8803 1594.3473

0.0008 0.0009 0.0009 0.0009 0.0002 0.0003 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 0.0003 0.0003 0.0003 0.0002 −0.0004 −0.0006 −0.0006 −0.0004 −0.0004 −0.0009 −0.0004 −0.0004 −0.0006 −0.0009 −0.0004 −0.0005 −0.0005 −0.0009 −0.0014 −0.0015 −0.0014 −0.0017 −0.0023 −0.0014 −0.0014 −0.0014 −0.0017 −0.0023 −0.0014 −0.0015 −0.0017 −0.0024

No adjustable parameters were used in CT calculations, corresponding to parameters o4 of Table 3. Empirical values correspond to the parameters derived from the most recent global analyses of experimental data.29

“diagonal parameters”. For the matrix elements they correspond to “blue” terms of the CT-scheme of Figure 1. The terms with different powers {u1u2...} ≠ {b1b2...} having off-diagonal vibrational matrix elements within the polyads are called “resonance coupling terms” with the corresponding “resonance coupling parameters”. For the matrix elements they correspond to “orange” terms of the CT-scheme of Figure 1. Note that, according to the extrapolation scheme for methane polyads originally suggested by Champion et al.15 and because of the properties of a+i and aj operators, the terms specific to / {Pn}

also contribute to the matrix elements of the higher polyads Pn+1, Pn+2, etc. 4.2. Building Nonempirical Heff for High Vibrational Polyads. In this work, the spectroscopic polyad parameters tΩ(K,Γ) u1u2...b1b2... were numerically computed from the NRT methane PES47 in the qf-representation using the MOL_CT code up to eighth order. A good convergence of vibrational levels versus the CT orders is illustrated in Figure 2. A simplified extract from the full calculations including vibration levels and the expansion coefficients of the Heff eigenfunctions in the 13792



Article

Table 5. Statistics for the Line Positions Fit with Resonance Coupling Parameters Held Fixed to Ab Initio CT-Values vib

Figure 4. Errors of our ro-vibrational predictions using NRT PES47 via direct CT calculations for the CH4 Dyad: dRMS =0.1 cm−1 for rovibrational levels computed from empirical constants22 up to J = 30 and dRMS (obs-calc) = 0.06 cm−1 for the set of all observed Dyad transitions up to J = 22.

Γ

1 2 total

0100 0001

E F2

1 2 3 4 5 6 7 8 9 total

1000 0200 0200 0101 0101 0010 0002 0002 0002

A1 E A1 F1 F2 F2 A1 E F2

N obs linesa Dyad 354 1109 1463 Pentad 186 215 170 617 720 597 244 630 785 4164

Jmax

dRMS/cm−1

σ

19 23 23

0.00011 0.00016 0.00015

0.67 0.77 0.74

18 16 17 18 19 23 19 20 21 23

0.00112 0.00061 0.00063 0.00056 0.00052 0.00059 0.00041 0.00048 0.00046 0.00058

1.04 0.71 0.68 0.64 0.61 0.62 0.53 0.57 0.58 0.63

Using experimental data reviewed in refs 10, 19, 21, and 130; σ is the weighted standard deviation of the fit (dimensionless). Number of adjusted diagonal parameters = 169. a

harmonic normal mode basis is given in Table 1. Extended results are given in the Supporting Information. An overall view on vibrational polyads of methane up to P12 (corresponding to ν̃ up to 18 000 cm−1) is given in Figure 3. The vibrational energy levels of successive methane polyads are shown together with the mixing coefficients due to intrapolyad anharmonic resonances, which are defined here as square of wave function expansion coefficients |ci|2 in the normal mode basis (Table 1). It is clearly seen that for high polyads the normal mode vibrations are completely mixed. The highest range 12000−17000 cm−1 in Figure 3 (polyads P8−P12) correspond to methane spectra recorded by Kozlov et al.,32 which have not yet been analyzed. Table 2 gives a comparison of CT results with experimental levels and with variational calculations using exact kinetic energy operator and nonpolynomial NRT

PES representation in curvilinear coordinates.47 No any adjusted parameters are used in our CT-calculations. Empirically determined vibrational levels for the comparison are taken from refs 27, 28, 29, 127, and 128. We have only included in our comparison those vibration sublevels for which sub-band centers had been reliably assigned from rovibrational spectra analyses (for example, relatively uncertain values marked by * in ref 28 were excluded). Note that the uncertainty of empirical determinations generally increase with energy: for example, the values for the ν2 + 2ν2+3(F2) band center around 7511 cm−1 reported by Niederer127 and by Manca Tanner128 differ by 0.63 cm−1. Table 2 shows that the average difference between CT and variational calculation47 (which did not use kinetic and

Figure 5. Example of energy levels ν̃ and ro-vibrational mixing coefficients (shown in various colors) for the Pentad of CH4 as directly computed from the PES47 using CT method. 13793



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potential energy power series expansions) is on average smaller (dRMS (CT-Var) = 0.35 cm−1 up to the tetradecad) than the discrepancies with experimental levels. Note that variational predictions have not yet been published for Pn > 4. These calculations suggest that the CT convergence is achieved for the considered energy range (as the difference between order six and order eight) is smaller than the precision of the PES: dRMS (o8-o6) = 0.6 cm−1. The agreement with empirical data for CT of order eight is slightly better (dRMS (o8-obs) = 0.74 cm−1) than for CT of order six (dRMS (o6-obs) = 1.08 cm−1).

Table 6. Example of comparison for predicted and empirically fitted diagonal Heff parameters for low order terms of fundamental bands vib

5. DIRECT CALCULATION OF ROTATIONAL AND VIBRATION-ROTATION ENERGY LEVELS As a next step we have proceeded by rotational calculations using CT up to order 6. The results for rotational and centrifugal distortion constants computed from NRT PES (qt-representation was used in this example) for the methane ground state with empirical values in the ITO representation are given in Table 3. A comparison with empirical parameters22 (Table 3) included in the STDS information system12 and with the results of the most recent “global” analysis29 shows an excellent agreement with our direct calculations. With these constants we have computed rotational energy levels of CH4 (see also the work of Cassam-Chenai et al.108 for a comparison of our results with other versions of the perturbation theory). It is seen (Table 4) that our theoretical values are practically within the experimental accuracy at least up to J = 15. Again, no any adjustable parameters are used in our CT calculations. In this work we have also calculated vibration−rotation levels for excited degenerate and quasi-degenerate vibrational states. The illustrations are given here for the Dyad and the Pentad up to J = 30. Figure 4 shows the discrepancies between our CT calculations of order 6 and those obtained with empirical parameters determined by Roche and Champion22 from the analyses of experimental spectra for the Dyad of ν2(E)/ν4(F2) bands. The root-mean-squares deviation (dRMS) between our direct calculation (without adjustable parameters) and empirical ones up to Jmax = 30 is only of 0.1 cm−1. On the sample of all available experimental dyad transitions (up to J = 22), the error of our direct CT-calculation is dRMS =0.06 cm−1. Figure 5 blows up a central part of the Pentad energy range showing quite complicated patterns, clustering and crossings of various series. Ro-vibrational energy levels are given here together with the corresponding normal mode mixing coefficients in a similar way as was usually computed from empirical models.15−17,19−29 In our study this relies on spectroscopic constants directly computed from the methane PES47 via the CT method. The most important effect on the complicated ro-vibrational patterns of the polyads is due to the resonance coupling parameters which in our case are of the ab initio origin. The predicted values of the rovibrational resonance coupling parameters for Dyad and Pentad are given in the Supporting Information.

P

rot. tensor

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2

R R R R R R R R R R R R R R R

0(0,0A1) 2(0,0A1) 2(2,0E) 0(0,0A1) 1(1,0F1) 2(0,0A1) 2(2,0E) 2(2,0F2) 0(0,0A1) 2(0,0A1) 0(0,0A1) 1(1,0F1) 2(0,0A1) 2(2,0E) 2(2,0F2)

u

b

direct CT

fitted to obs

0100 0100 0100 0001 0001 0001 0001 0001 1000 1000 0010 0010 0010 0010 0010

0100 0100 0100 0001 0001 0001 0001 0001 1000 1000 0010 0010 0010 0010 0010

1.53341 −3.77888 −3.06316 1.31081 1.03424 −3.06281 −7.02944 −2.94644 2.89680 −3.76705 3.00196 6.58147 −3.50115 8.02191 −1.92085

1.53333 −3.82055 −3.02833 1.31076 1.03507 −3.06371 −6.77412 −2.92254 2.89694 −3.78527 3.00201 6.53224 −3.50639 8.18372 −1.98695

× × × × × × × × × × × × × × ×

103 10−3 10−2 103 101 10−3 10−3 10−2 103 10−2 103 10−1 10−2 10−3 10−3

All values are in cm−1.

Table 7. Statistics for Line Intensities Fit for the Pentad−GS Bands of 12CH4 with Resonance Coupling Parameters Fixed to Ab Initio CT Values 1 2 3 4 5 6 7 8 9 total

vib

Γ

N obs lines

Jmax

σ (%)

1000 0200 0200 0101 0101 0010 0002 0002 0002

A1 E A1 F1 F2 F2 A1 E F2

24 148 98 334 412 392 24 205 236 1873

17 16 17 15 16 20 11 14 14 20

6.7 5.7 4.8 6.8 4.7 3.2 4.2 8.5 7.0 5.8

Using experimental data reviewed in refs 10, 19, and 20; σ (%) is the weighted standard deviation of the fit; N effective dipole parameters = 29.

parameters from experimental data in case of degenerate levels. The situation is particularly complicated in the case of resonance interactions: empirically fitted coupling parameters appear to be strongly correlated among themselves and with diagonal band parameters resulting to the “colinearity” issues.33−35 On the theoretical basis, the ambiguity of effective Hamiltonians is well-known.109,65,110,67,75 In a somewhat simplified point of view a fitting of a polyad effective Hamiltonian to experimental energy levels (or to line positions) is equivalent to a determination of off-diagonal matrix elements from experimental eigenvalues. This is known to be a mathematically ill-defined problem. An analytical representation (eqs 26 and 30) of Heff imposes some constraints on this procedure. However, this does not fully remove a fundamental ambiguity whereas a number of “degrees of freedom” rapidly increases with the polyad dimension: it has been shown that various empirical fits of the same data could result in very different values of Heff parameters.33−35 Apart from questions related to a physically meaningful parametrization of spectra, in practical terms, a number of well-known difficulties occur in least-squares fits of model

6. ARE THE RESONANCE COUPLING TERMS RELIABLY PREDICTED? The last statement of the previous section raises an important question: What would be a most reliable physically meaningful way to determine the resonance coupling terms? 6.1. Problem of Ill-Defined Resonance Coupling Parameters for Empirical Polyad Models. A major problem for interacting-states effective empirical models is known to be a fundamental ambiguity of the determination of 13794



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Figure 6. Quantum energy levels and classical stationary axes versus J (left panel) with an example of a conical intersection of rotational energy surfaces (RES, right panel) for the methane Dyad as calculated from the ab initio molecular potential function47 via the CT method. This example of RES intersection corresponds to the crossing of the curves for C3 relative equilibria (blue curves at J ∼ 26) for vibrational components of triply degenerate (0001) (F2) vibrational mode.

the least-squares fit. The resonance coupling parameters tΩ(K,Γ) u1u2...b1b2... corresponding to different vibrational indices {u1u2...} ≠ {b1b2...} remained fixed to the theoretically predicted values. Note that with this definition corresponding to the CT-scheme of Figure 1, the coupling terms among sublevels of the same vibrational state belong to diagonal “blue” blocks and are empirically optimized. An important point is that the MOL_CT program suite accounts for the exact vibrational-rotation algebra of commutators/anticommutators. Therefore at every order, the computations were complete: no any additional approximations of so-called “leading contributions” and of related arbitrary truncations usually applied in the CT literature were introduced. These transformations remove small extra-polyad regular interactions without neglecting them: their effect is compensated by introducing equivalent terms with higher rotational powers in the polyad blocks. Finally, the transformed Hamiltonian was projected onto the polyad subspace to give a nonempirical spectroscopic effective Hamiltonian for the polyad Pn under study. We have recently tested a similar approach for the ozone triad that has allowed a considerable improvement of the spectra modeling.111 Table 5 shows that by a “fine tuning” of diagonal parameters we achieve the experimental accuracy of the fit with fewer adjusted parameters while all resonance coupling parameters (“orange” terms of the CT scheme of Figure 1) were held fixed to ab initio CT-values. An example of the comparison between initial and optimized values for low order terms is given in Table 6. For higher order terms the differences are of course significantly larger. Over 5600 line positions for the Dyad and the Pentad collected from various available spectra analyses reviewed in refs [10, 19, 20,130] were fitted simultaneously with the dRMS = 0.00015 cm−1 for the Dyad lines and dRMS = 0.00058 cm−1 for the Pentad lines. If resonance coupling parameters would be wrong, one could never be able to approach such accuracy. 6.3. Line Strengths Fit: A Key Test for the Resonance Intensity Transfer among Strongly Coupled Bands. Another crucial test is the intensity study. If our coupling parameters would be wrong, the intensity “borrowing” from

parameters to experimental data. This is appropriate not only for methane but for all molecules with quasi-degenerate vibrations. In the case of methane, due to the symmetry, the number of coupling terms for in Heff for high polyads could mount up to several hundred parameters or even more.15,29,102 Note that a change of sign for an off-diagonal term is not in general a small transformation and could spoil the Hamiltonian convergence. In a pure empirical approach one thus faces a huge number of combinations to test which could not be solved with gradient least-squares methods. A wrong choice for the coupling parameters could false the resonance intensity transfer between strong and weak bands and damage the modeling of line strengths. The important question is: could we bring ab initio information to Heff in order to make the spectroscopic models more robust? To answer this question quantitatively, we propose two crucial tests: one for the line position modeling, another one for the intensity fits. 6.2. Mixed “Half Ab Initio/Half Empirical” Spectroscopic Polyad Model: Fitting Line Positions up to the Pentad Range with Experimental Accuracy. If some of coupling parameters of a polyad model are not constrained, this would result in a poor convergence or divergence of the leastsquares fit to experimental energies. This is a consequence of strong correlations among fitted parameters35 (well-known “colinearity issue” in data reduction methods). Constraining them to zero is not always a good solution (even for higherorder terms) as this can degrade the dRMS of the fit. Their determination became a kind of “art” relying on a physical intuition and on a luck with the “trial and error” method. In this work we develop a new approach for building a robust polyad ro-vibrational model using a priori information from molecular PES. At the initial step, a full set of ro-vibrational terms, including higher-order ones, in effective polyad Hamiltonian are accurately derived from PES using the MOL_CT program suit. For polyatomic molecules, the accuracy of a PES is usually not sufficient to directly reach the experimental precision of high-resolution spectra. However, this step could provide a physically consistent initial set of parameters values for Heff. At the second step, some of diagonal parameters are “relaxed” through the empirical optimization, but only those of them which are well determined in 13795



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Figure 7. Quantum energy levels and classical stationary axes versus J (upper left panel) and the same ro-virational patterns blown up (right panels) with an example of a conical intersection of rotational energy surfaces (RES, lower left panel) for the methane Pentad as calculated from the ab initio molecular potential function47 via the CT method. This example of RES intersection corresponds to the crossing of the curves for C4 relative equilibria (red curves at J ∼13) between nondegenerate (1000) state and the upper vibrational components of the F2 triply degenerate substate of the (0002) mode. See also Figures 1 and 5 for the states assignments.

7. QUALITATIVE SEMI-CLASSICAL ANALYSIS Effective Hamiltonians have been widely used for the understanding of qualitative features in complicated vibration rotation level structures. A combination of quantum calculations with semiclassical methods proved to be very efficient112−122 for investigating “critical points” and of the asymptotic behavior of state series for high quantum numbers. In the case of methane, the pioneering work of Dorney and Watson112 has allowed the first interpretation of clusters (sets of closely lying rotational levels) in terms of classical rotations around equivalent stationary axes. The quasi-degeneracies for the corresponding cluster series become stricter with increasing J values. The impact of vibrational resonance interactions on qualitative changes in excited states using classical or quantum effective models has been evidenced by Child,113 Kellman,114 Sibert et al.,115 Jung et al.,116 Joyeux and Sugny,92 Herman117 (a list being not exhaustive) and in many other works. These

strong to weak bands through the resonance mixing of normal mode wave functions would also be corrupted. The statistics of the effective transition moment fit to observed line strengths given in Table 7 correspond to nearly experimental accuracy. The standard deviation of the fit was σ = 5.8% for 1873 measured intensities collected from experimental studies reviewed in refs 10, 12, and 20, with 29 adjusted transition moment parameters for the Pentad bands. This qualitative test allows validating our approach and suggests that the resonance coupling parameter derived by CT method from PES have physically meaningful values. By fixing these parameters to ab initio values, one could avoid well-known ambiguities in empirical resonance parameters as well as their correlations with diagonal ones. These correlations are known to be the origin of a poor fit convergence in many previous works that used purely empirical polyad models. Right combination of relative signs in predicted resonances parameters proved to be important for a correct intensity modeling. 13796



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Figure 8. Example of intensity comparison for strong Pentad lines between variational ab initio predictions and those fitted with the CT/polyads model using the approach described in Section 8.2. The major difference between two calculations concerns the line positions with dRMS (CT-Var) ∼ 0.08 cm−1 that is not visible in this “bird’s-eye view” . With the CT/polyads model, an experimental accuracy on line positions (with dRMS (obscalc) = 0.0005 cm−1) was achieved.

Figure 9. Overall view on the Dyad methane transitions in the log scale. Upper panel (black): empirical values of HITRAN_2008 spectroscopic database.7 Lower panel (red): predictions using combined “CT-polyad/variational” model with ab initio intensities as described in Sections 6 and 8. The marked intervals (a),(b),... correspond to spectral ranges blown up in the following figures.

where θ and ϕ are spherical angles, and |J| = (J(J + 1))1/2 is the amplitude of the angular momentum. Pavlichenkov and Zhilinskii119 have studied the bifurcations of stationary points (rotational relative equilibria) of EJ (θ,ϕ), which could occur with a variation of J. Further qualitative studies for vibration− rotation models of tetrahedral molecules have been reported by Sadovskii, Zhilinskii et al.,120,121,129 and by Van Hecke et al.122 This approach was also extended to multiplet electronic states.123 In Figures 6 and 7 we give two examples corresponding to conical intersections of RES for methane vibrational components by extending the definitions (eqs 31 and 32) to degenerate and quasi-degenerate states. Because the mixings of quantum states and classical relative equilibria for complicated vibration−rotation polyads are mostly determined by the

changes induced by resonance couplings of quantum states appear as counterparts of bifurcations in classical phase space. Well-known examples are normal-to-local mode transitions as well as fingerprints of saddle-node bifurcations in vibrational spectra.113−115 Harter and Patterson118 extensively developed the approach of stationary axes by introducing so-called “rotational energy surface” (RES) EJ (θ,ϕ), which has been obtained by replacing quantum angular momenta components by the corresponding classical variables (Jx , Jy , Jz ) ⇒ |J|(sin θ cos ϕ , sin θ sin ϕ , cos θ) eff

H (Jx , Jy , Jz ) ⇒ EJ (θ , ϕ)

rot

(31)

(32) 13797



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Figure 10. Comparison of calculated lines (combined “CT-polyad/variational” model: ab initio intensities in red) with HITRAN-2008 spectroscopic database7 (black) for the methane Dyad. The first panel gives a general view on intensity sticks for “middle size” lines. Panel (a): intensity sticks diagram for the strong ν4 band. Panel (b): the same for the region of weak ν2 band (P-branch) essentially borrowing the intensity from the strong ν4 band via the ν2/ν4 resonance coupling. Panel (c): simulated absorption spectrum (arbitrary units, resolution 0.001 cm−1) for the lower edge of the ν4 band corresponding to high J values. Panel (c): the same for the Q-branch of the ν2 band. Note a good agreement both for weak and strong lines.

8.1. Variational ab initio predictions. Recently new ab initio DMS of the methane molecules has been constructed48 using extended ab initio CCSD(T) calculations at VQZ and CVQZ levels on the grid of 19882 nuclear configurations. The DMS analytical representation was determined through an expansion in symmetry adapted products of internal nonlinear coordinates up to the sixth order. Using this DMS, we have previously reported variational predictions (Nikitin et al.48 and Rey et al.49,50) for line intensities in the infrared range up to 9300 cm−1. More than 1 million ab initio lines have been generated in this way for the methane isotopologues.48−50 These calculations use the same PES47 and the same normal mode representation of the full nuclear motion Hamitonian as in the present study. The difference with the present work was that variational calculations dealt with infinite matrices and that a basis convergence for high J values represented an enormous challenge for the 12-dimensional nuclear motion. New variational predictions are expected to be helpful as a starting point for the analyses, particularly for hot bands and for astrophysical applications. However, despite a substantial improvement in the PES accuracy, variational calculations could hardly approach in the near future the metrological experimental accuracy

resonance coupling terms, it appears important to analyze these qualitative features using reliable information deduced from accurate molecular potentials. We plan to extend in future this study for higher polyads where satisfactory analyses are not yet available. An understanding of qualitative effects of resonance interaction with increasing V and J quantum numbers (e.g., as required in hot spectra analyses) could help interpretation of experimental spectra currently in progress at very higher energy range.131,132

8. AB INITIO LINE INTENSITIES COMPUTED WITH COMBINED “CT/VARIATIONAL” MODEL As was mentioned in Section 2, a high-order MOL_CT implementation for the full sequence of DMS transformations involving rotational line strengths dependence is currently available for triatomic systems only.97 For methane we have only lower order results for the effective dipole transition operator μ⃗eff which can be used as an initial approximation. This is, however, sufficient to build up a combined “CT polyads/ variational” model for ab initio line intensities as described below. 13798



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Figure 11. Overall view on the Pentad methane transitions in the log scale. Upper panel (black): empirical values of HITRAN-2008 spectroscopic database.7 Lower panel (red): predictions using combined “CT-polyad/variational” model with ab initio intensities as described in Sections 6, and 8. The marked intervals (a),(b),... correspond to spectral ranges blown up in the following figures.

in line positions. The latter is on the order of ∼0.001 cm−1, which represents ∼10−6−10−7 of the relative error on the wavenumber scale. For line intensities, the error considerations are quite different: for many methane bands it was possible to predict line intensities48−50 up to 1−3% accuracy that is on the order of the experimental uncertainty or sometimes better. For some other lines, a disagreement could be much larger, but it is not always clear whether the reasons for the intensity disagreements are of experimental or of theoretical origin. The question is how to combine experimental accuracy ∼0.001 cm−1 on the line position with the best ab initio accuracy for intensities. 8.2. Fitting Variational Ab Initio Line Intensities with the Mixed CT-Polyads Model. We propose to address this issue using a combined “CT-polyad/variational” model for ab initio line intensities. To this end, we employ the CT-model for effective Hamiltonians using resonance coupling terms held fixed to ab initio values accurately deduced from the PES, with a subsequent “fine tuning” of the diagonal tΩ(K,Γ) polyad paramu;u eters as described in Section 6. Line intensity for an electro-dipole rovibrational transition is by definition determined by the DMS and by wave functions of lower and upper states. In cases of accidental resonances, the wave functions could be very sensitive to small errors in energy level calculations. Such small errors in line positions could produce large errors in line strengths particularly for weak bands, which “borrow” their intensities from stronger bands via a resonance intensity transfer. From previous accurate ab initio intensity calculations for triatomics,124,97,125 it is well-known that there are so-called “unstable” lines intensities125 which are extremely sensitive to accidental resonances and are practically impossible to control with variational calculations even for relatively simple molecules like water. For methane the situation is much more complicated because of very dense ro-vibrational level patterns and numerous resonances within the polyads as can be seen in Figures 3 and 5.

We hope that the approach described in Section 6 could allow removing this type of “unstability” because the energy levels are computed with the experimental accuracy, whereas the coupling terms remain ab initio by their origin. Our strategy is therefore as follows: - build as a first step first an accurate polyad model for line positions using ab initio coupling terms (Section 6); - select in the variational ab initio intensity predictions only “stable” intensities not corrupted by accidental perturbations for upper-state levels; - fit these “stable” ab initio variational line intensities with the μ⃗ eff model using theoretical initial values for the transition moment parameters. With this method we intend to precisely determine wave functions and ab initio transition moment values for each band, whereas effects of intensity transfer will be described by the resonance coupling parameters accurately computed from the molecular PES. The results of first calculations with this new combined polyad model are given in Figures 8−12 for the Dyad and Pentad bands. For the Dyad bands we have included in the effective dipole moment fit 5814 line intensities variationally computed50 from NRT PES47 and DMS48 up to J = 20 and with the Imin cutoff of 10−26 cm/mol. With 13 fitted μef f parameters the standard deviation was 1.8% . The results are given in the electronic Supporting Information. If we remove 49 outliers σ drops down to 0.9%. For the Pentad bands, the fit was somewhat less accurate. Among the 12814 variationally computed ab initio intensities included in the primary fit, we found 627 significant outliers mostly corresponding to weak lines. Using the remaining 11187 variationally computed intensities, we obtained σfit = 3.0% up to J = 20 with 104 μeff parameters. The results are also given in the electronic Supporting Information including line lists and effective dipole moment parameters. Strong lines are generally in very good agreement, as shown in Figure 8 and in Table 8. Note that for some very weak transitions, the discrepancies I (f itted-variational) could amount to an order of magnitude, though this does not 13799



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Figure 12. Comparison of calculated lines (combined “CT-polyad/variational” model: ab initio intensities in red) with HITRAN-2008 spectroscopic database7 (black) for the methane Pentad. The first panel gives a general view on intensity sticks for “middle size” lines. Panel (a): simulated absorption spectrum (arbitrary units, resolution 0.01 cm−1) for the weak 2ν4 band. Panel (b): the same for the Q-branch of the strong ν3 band. Note two orders of difference in the absorption scale between (a) and (b) panels. Panel (c): simulated absorption spectrum (resolution 0.01 cm−1) for the range involving 2ν4 and ν2+ν4 lines. Panel (d): the same for a very dense central range of the pentad involving strong ν3 and ν2+ν4 bands as well as a weak “forbidden” ν1 band. Panel (e): P and Q branches of the strongest ν3 band, the Q-branch being resolved in the panel (b). Panel (f): High wavenumber edge of the Pentad involving ν3 and 2ν2 lines. Note four orders of difference in the absorption scale among various panels.

“unstable” lines or due to difference of assignments in two methods.

affect integrated band intensities. Further studies are needed to understand whether these outliers are due to 13800


13801

2.16 2.15 1.72 1.60 1.30 1.29 1.29 1.29 1.20 1.19 1.15 1.07

× × × × × × × × × × × ×

I

10−19 10−19 10−19 10−19 10−19 10−19 10−19 10−19 10−19 10−19 10−19 10−19

10−19 10−19 10−20 10−20 10−20 10−20 10−21 10−21 10−21 10−21 10−21 10−21

4 3 6 3 4 4 3 3 5 5 2 6

3 4 4 3 6 6 14 14 14 14 14 14 our

A1 A2 A1 A2 F2 F1 F1 F2 F1 F1 F2 A1

low

A2 A1 A1 A2 A1 A2 F1 F1 F2 F2 F1 F1

low

1 1 1 1 1 1 1 1 2 1 1 1

1 1 1 1 1 1 2 1 1 2 1 2

upper

0000−0010 5 A2 4 A1 7 A2 3 A1 5 F1 5 F2 4 F2 4 F1 6 F2 6 F2 3 F1 5 A2

upper

0000−0001 4 A1 5 A2 4 A2 3 A1 6 A2 6 A1 14 F2 14 F2 14 F1 13 F1 13 F2 13 F2

7 7 10 4 20 18 16 15 23 24 13 8

1 1 1 1 1 1 6 5 4 9 10 9

−0.0076 −0.0065 −0.0043 −0.0007 −0.0069 −0.0073 −0.0069 −0.0067 −0.0068 −0.0077 −0.0059 0.0172

Δν̃ 0.2 0.3 0.6 0.2 0.2 0.5 0.0 0.0 0.5 0.7 0.7 0.8

ΔI

0.0212 −0.2 0.0241 0.0 0.0097 0.0 0.0073 0.0 0.0131 0.0 0.0141 0.0 0.0471 0.3 0.0466 0.0 0.0483 0.1 −0.0813 −0.1 −0.0806 0.5 0.1 −0.0730 variat - our

ΔI

−0.0001 0.0000 −0.0002 0.0005 −0.0001 0.0004 −0.0001 0.0002 0.0000 0.0000 0.0003 0.0002

Δν̃ −1.9 −1.9 −1.2 −1.9 −1.5 −1.6 −2.4 −2.4 −1.7 −0.8 −0.9 −1.0

ΔI

−0.0004 3.8 −0.0004 3.4 3.6 −0.0003 −0.0003 3.6 0.0000 3.3 −0.0001 3.5 −0.0001 1.7 0.0000 2.6 0.0004 2.6 −0.0001 2.7 0.0000 1.8 −0.0002 2.8 hit - our

Δν̃

Δν̃

ΔI

hit - our

variat - our

2828.0923 2919.1311 2900.1169 2953.1956 2878.3620 2847.7186 2966.1087 2945.9421 2818.5658 2916.0033 2918.7344 2989.0854

ν̃

1418.8478 1601.7669 1534.9020 1482.5227 1412.1877 1556.8641 1425.6986 1405.8656 1412.1527 1556.8238 1405.7954 1655.5082

ν̃

I 10−21 10−21 10−21 10−21 10−21 10−21 10−21 10−21 10−21 10−21 10−21 10−21

10−22 10−22 10−22 10−22 10−22 10−22 10−22 10−22 10−22 10−22 10−22 10−22

× × × × × × × × × × × ×

× × × × × × × × × × × ×

5.00 4.97 4.10 3.87 3.29 3.24 3.13 3.10 3.08 2.60 2.56 2.55

1.24 1.23 1.21 1.21 1.20 1.19 1.17 1.17 1.14 1.11 1.10 1.09

I

4 4 3 6 3 0 7 6 3 4 4 8

13 5 6 5 14 13 12 15 14 13 15 10 our

our

A1 A1 A2 A2 A2 A1 A2 A1 A2 F2 F1 A1

low

E F1 A1 F2 F2 F1 F2 A2 F1 A1 A1 F2

low

1 1 1 1 1 1 1 1 1 1 1 1

2 1 1 1 4 1 1 2 3 1 1 3

Lower and upper state assignment include J, Γ, n ; line positions ν̃ and discrepancies Δν̃ are given in cm−1; line intensities I are in cm × mol

3067.3002 3057.6873 3085.8323 3018.5283 3067.1642 3067.2611 3057.7606 3057.7264 3076.5497 3076.7252 3048.1531 2958.0171

ν̃

× × × × × × × × × × × ×

1327.0744 1332.7213 1303.7123 1306.1401 1306.2694 1302.0444 1298.1244 1290.7407 1279.6678 1216.6298 1216.3290 1210.7839

1.01 1.00 8.29 7.99 6.92 6.71 1.21 1.17 1.16 1.14 1.13 1.09

I

ν̃

our upper

−1

3 6 5 7 4 2 9 7 2 16 15 10

11 8 2 5 17 15 13 7 17 6 6 11

ΔI

0.0075 −0.0311 −0.0228 −0.0476 0.0070 −0.0055 −0.0520 −0.0368 0.0161 −0.0294 −0.0297 −0.0667

Δν̃

−0.1 0.3 0.1 0.3 0.0 0.1 0.1 0.4 −0.5 0.1 0.7 0.6

ΔI

−0.0162 0.2 −0.0061 0.4 0.0144 0.5 0.0032 0.6 −0.1 −0.0197 −0.0260 0.2 −0.0196 0.5 −0.6 −0.0239 −0.4 −0.0196 −0.0262 0.2 −0.0235 0.2 0.0192 1.1 variat - our

Δν̃

, and intensity discrepancies ΔI in %.

0000−0101 4 A2 5 A2 4 A1 7 A1 4 A1 1 A2 8 A1 7 A2 3 A1 5 F1 5 F2 9 A2

upper

0000−0100 12 E 6 F2 6 A2 4 F1 13 F1 13 F2 11 F1 14 A1 13 F2 13 A2 14 A2 11 F1

variat - our

ΔI

0.0003 0.0006 0.0006 0.0002 0.0003 0.0006 0.0007 0.0003 −0.0002 0.0005 0.0005 −0.0003

Δν̃

4.0 2.7 3.1 2.3 3.2 3.9 1.9 2.2 3.8 2.6 2.6 1.9

ΔI

0.0001 −0.8 0.0000 −2.5 0.0001 −1.7 0.0001 −2.5 −0.0001 0.0 0.0000 −1.7 0.0000 −1.7 0.0000 0.0 −0.0001 −0.8 0.0001 −0.9 0.0000 0.9 0.0000 0.9 hit - our

Δν̃

hit - our

Table 8. Example of Comparisons of Our Calculated 12CH4 Line List with Ab Initio Variational Calculations50 and with the HITRAN 2008 Data Base7 for Some Strong Dyad and Pentad Lines




Article

9. CONCLUSION The practical aim of this work is to address the question: How could one generate line lists for methane spectra containing line positions determined within experimental accuracy and ab initio intensities in a consistent way? A theoretical modeling of methane line positions is a major issue for atmospheric, planetological, and astrophysical applications. This is particularly important for line intensities because the corresponding measurements are quite laborious and delicate for dense spectra involving impurities and a tremendous number of weak lines, many of them being blended. An accurate experimental measurement of millions of line intensities under various temperature and pressure conditions is clearly beyond practical feasibility. In this work we have developed a combined spectroscopic model for methane polyads which involves all resonance terms very accurately derived from the molecular PES via high-order CT method. We refer to the corresponding parameters as “ab initio based coupling parameters” which are responsible for effects of resonance intensity transfer among various bands within a given polyads. Direct CT calculations have shown that we reach a variational accuracy with the same PES both for vibrational and for rotational levels (Sections 4 and 5). On the other hand the tests reported in Section 7 show that by keeping fixed the coupling parameters to ab initio values, we can achieve experimental accuracy, on average better than 10−3 cm−1 for about 5600 Dyad and Pentad line positions, by only slight empirical optimization (“fine tuning”) of the diagonal parameters. Dipole transition moment parameters of our model were determined from selected ab initio line strengths previously computed from a DMS48 by variational method.48−50 Overall we have an excellent agreement for ab initio intensities with pure variational line list.48−50 The relative discrepancies for integrated band intensities between our line list (provided in the Supporting Information) and the variational one50 up to J = 25 are very small (few examples for comparisons are given in Table 8 for strong lines of the fundamental bands), but there remain significant outliers for some weak lines. It is expected that a new combined model could be more robust with respect to the problem of “unstable” lines perturbed by accidental resonances, which is very difficult to control by variational methods. This is because our combined model can achieve much more accurate calculation of rovibrational energy levels and line positions (Section 6 and Figures 8−12), and thus the resonance intensity “borrowing” by weak lines could be described more reliably. This important issue certainly deserves more detailed studies. The overall integrated intensity agreement with Hitran-2008 empirical database is of 4.4% for the Dyad and of 1.8% for the Pentad. Comparisons for individual lines are given in Figures 8−12. Despite an excellent overall comparison, some discrepancies appear for weak lines and we have some disagreements both line positions and intensities for high J values. Note that accurate intensity measurements of weak transitions were quite seldom and Hitran line lists relied mostly on extrapolations for blended lines and for low absorption ranges. Calculations with this mixed “CT/ab initio” model provide more straightforward spectroscopic assignment than the variational ones. Technically they have nearly no limitations on J values because the polyads matrices have finite dimensions. However, practically we need to achieve a good convergence of

both methods for efficient calculations. We plan extending this work to higher polyads eventually combining with new qualitative semiclassical approaches under development131,132 particularly for Pn (n ≥ 5) for which a satisfactory model for the analyses had not yet been developed.

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ASSOCIATED CONTENT

S Supporting Information *

Supporting Information for the methane 12CH4 calculations: Tables containing directly computed levels of vibrational polyads up to 10700 cm−1 with simplified and full assignments for orders o6 and o8 of CT; rovibrational resonance coupling parameters of the effective Hamiltonian computed from PES using CT for Dyad and Pentad; predicted Dyad-GS transitions and predicted Pentad-GS transitions and corresponding effective dipole moment parameters fitted to ab initio intensities. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +33-326-91-33-80. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the SAMIA (Spectroscopie d’Absorption des Molécules d’Intérêt Atmosphérique) contract between CNRS (France) and RFBR (Russia). We acknowledge the support from IDRIS and CINES computer centers of France as well as the Clovis computer centre of ReimsChampagne-Ardenne. R.K. is indebted to the French Embassy in Moscow for the Ph.D. stipend and A.N. thanks Reims University for the invitation to work on the project.

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REFERENCES

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Accurate Spectroscopic Models for Methane Polyads Derived from a

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