Verification labels for rovibronic quantum-state energy uncertainties.

Transition wavenumbers contained in line-by-line rovibronic databases can be compromised by errors of various nature. When left undetected, these errors may result in incorrect quantum-state energies, potentially compromising a large number of derived spectroscopic data. Spectroscopic networks treat the complete set of line-by-line spectroscopic data as a large graph, and through a least-squares refinement the measured line positions are converted into empirical quantum-state energies. Spectroscopic networks also offer a highly useful framework to develop mathematical tools helping to identify possible errors and conflicts within the dataset. For example, wavenumber errors can be detected by checking for violations of the law of energy conservation. This paper describes a new graph-theory tool, which results in so-called verification labels for the quantum states. Verification labels help to express the vulnerability of a calculated empirical energy value and its uncertainty against possible wavenumber errors, providing complementary information to simple statistical uncertainties.

MARVEL analysis of high-resolution rovibrational spectra of 13 C 16 O 2 .

A set of empirical rovibrational energy levels, obtained through the MARVEL (measured active rotational-vibrational energy levels) procedure, is presented for the 13 C 16 O 2 isotopologue of carbon dioxide. This procedure begins with the collection and analysis of experimental rovibrational transitions from the literature, allowing for a comprehensive review of the literature on the high-resolution spectroscopy of 13 C 16 O 2 , which is also presented. A total of 60 sources out of more than 750 checked provided 14,101 uniquely measured and assigned rovibrational transitions in the wavenumber range of 579-13,735 cm - 1 . This is followed by a weighted least-squares refinement yielding the energy levels of the states involved in the measured transitions. Altogether 6318 empirical rovibrational energies have been determined for 13 C 16 O 2 . Finally, estimates have been given for the uncertainties of the empirical energies, based on the experimental uncertainties of the transitions. The detailed analysis of the lines and the spectroscopic network built from them, as well as the uncertainty estimates, all serve to pinpoint possible errors in the experimental data, such as typos, misassignment of quantum numbers, and misidentifications. Errors found in the literature data were corrected before including them in the final MARVEL dataset and analysis.

On the 12C2H2 near-infrared spectrum: absolute transition frequencies and an improved spectroscopic network at the kHz accuracy level.

Lamb dips of twenty lines in the P, Q, and R branches of the ν1 + ν3 + ν41 vibrational band of 12C2H2, in the spectral window of 7125-7230 cm-1, have been measured using an upgraded comb-calibrated frequency-stabilized cavity ring-down spectrometer, designed for extensive sub-Doppler measurements. Due to the large number of carefully executed Lamb-dip experiments, and to the extrapolation of absolute frequencies to zero pressure in each case, the combined average uncertainty of the measured line-center positions is 15 kHz (5 × 10-7 cm-1) with a 2-σ confidence level. Selection of the twenty lines was based on the theory of spectroscopic networks (SN), ensuring that a large number of transitions, measured previously by precision-spectroscopy investigations, could be connected to the para and ortho principal components of the SN of 12C2H2. The assembled SN contains 331 highly precise transitions, 119 and 121 of which are in the ortho and para principal components, respectively, while the rest remain in floating components. The para- and ortho-12C2H2 energy-level lists, determined during the present study, contain 82 and 80 entries, respectively, with an accuracy similar to that of the lines. Based on the newly assembled lists of para- and ortho-12C2H2 empirical energy levels, a line list, called TenkHz, has been generated. The TenkHz line list contains 282 entries in the spectral range of 5898.97-7258.87 cm-1; thus far, only 149 of them have been measured directly via precision spectroscopy. The TenkHz line list includes 35 intense lines that are missing in the HITRAN2020 database.

Analysis of measured high-resolution doublet rovibronic spectra and related line lists of 12CH and 16OH.

Detailed understanding of the energy-level structure of the quantum states as well as of the rovibronic spectra of the ethylidyne (CH) and the hydroxyl (OH) radicals is mandatory for a multitude of modelling efforts within multiple chemical, combustion, astrophysical, and atmospheric environments. Accurate empirical rovibronic energy levels, with associated uncertainties, are reported for the low-lying doublet electronic states of 12CH and 16OH, using the Measured Active Rotational-Vibrational Energy Levels (MARVEL) algorithm. For 12CH, a total of 1521 empirical energy levels are determined in the primary spectroscopic network (SN) of the radical, corresponding to the following seven electronic states: X 2Π, A 2Δ, B 2Σ-, C2 Σ+, D 2Π, E 2Σ+, and F 2Σ+. The energy levels are derived from 6348 experimentally measured and validated transitions, collected from 29 sources. For 16OH, the lowest four doublet electronic states, X 2Π, A 2Σ+, B 2Σ+, and C 2Σ+, are considered, and a careful analysis and validation of 15 938 rovibronic transitions, collected from 45 sources, results in 1624 empirical rovibronic energy levels. The large set of spectroscopic data presented should facilitate the refinement of line lists for the 12CH and 16OH radicals. For both molecules hyperfine-resolved experimental transitions have also been considered, forming SNs independent from the primary SNs.

Selecting lines for spectroscopic (re)measurements to improve the accuracy of absolute energies of rovibronic quantum states.

Improving the accuracy of absolute energies associated with rovibronic quantum states of molecules requires accurate high-resolution spectroscopy measurements. Such experiments yield transition wavenumbers from which the energies can be deduced via inversion procedures. To address the problem that not all transitions contribute equally to the goal of improving the accuracy of the energies, the method of Connecting Spectroscopic Components (CSC) is introduced. Using spectroscopic networks and tools of graph theory, CSC helps to find the most useful target transitions and target wavenumber regions for (re)measurement. The sets of transitions suggested by CSC should be investigated by experimental research groups in order to select those target lines which they can actually measure based on the apparatus available to them. The worked-out examples, utilizing extensive experimental spectroscopic data on the molecules H[Formula: see text]O, [Formula: see text]S[Formula: see text]O[Formula: see text], H[Formula: see text]C[Formula: see text]O, and [Formula: see text]NH[Formula: see text], clearly prove the overall usefulness of the CSC method and provide suggestions how CSC can be used for various tasks and under different practical circumstances.

Spectroscopic-network-assisted precision spectroscopy and its application to water.

Frequency combs and cavity-enhanced optical techniques have revolutionized molecular spectroscopy: their combination allows recording saturated Doppler-free lines with ultrahigh precision. Network theory, based on the generalized Ritz principle, offers a powerful tool for the intelligent design and validation of such precision-spectroscopy experiments and the subsequent derivation of accurate energy differences. As a proof of concept, 156 carefully-selected near-infrared transitions are detected for H216O, a benchmark system of molecular spectroscopy, at kHz accuracy. These measurements, augmented with 28 extremely-accurate literature lines to ensure overall connectivity, allow the precise determination of the lowest ortho-H216O energy, now set at 23.794 361 22(25) cm-1, and 160 energy levels with similarly high accuracy. Based on the limited number of observed transitions, 1219 calibration-quality lines are obtained in a wide wavenumber interval, which can be used to improve spectroscopic databases and applied to frequency metrology, astrophysics, atmospheric sensing, and combustion chemistry.

Accurate empirical rovibrational energies and transitions of H216O.

Several significant improvements are proposed to the computational molecular spectroscopy protocol MARVEL (Measured Active Rotational-Vibrational Energy Levels) facilitating the inversion of a large set of measured rovibrational transitions to energy levels. The most important algorithmic changes include the use of groups of transitions, blocked by their estimated experimental (source segment) uncertainties, an inversion and weighted least-squares refinement procedure based on sequential addition of blocks of decreasing accuracy, the introduction of spectroscopic cycles into the refinement process, automated recalibration, synchronization of the combination difference relations to reduce residual uncertainties in the resulting dataset of empirical (MARVEL) energy levels, and improved classification of the lines and energy levels based on their accuracy and dependability. The resulting protocol, through handling a large number of measurements of similar accuracy, retains, or even improves upon, the best reported uncertainties of the spectroscopic transitions employed. To show its advantages, the extended MARVEL protocol is applied for the analysis of the complete set of highly accurate H216O transition measurements. As a result, almost 300 highly accurate energy levels of H216O are reported in the energy range of 0-6000 cm-1. Out of the 15 vibrational bands involved in accurately measured rovibrational transitions, the following three have definitely highly accurate empirical rovibrational energies of 8-10 digits of accuracy: (v1v2v3) = (0 0 0), (0 1 0), and (0 2 0), where v1, v2, and v3 stand for the symmetric stretch, bend, and antisymmetric stretch vibrational quantum numbers. The dataset of experimental rovibrational transitions and empirical rovibrational energy levels assembled during this study, both with improved uncertainties, is considerably larger and more accurate than the best previous datasets.

Small Molecules-Big Data.

Quantum mechanics builds large-scale graphs (networks): the vertices are the discrete energy levels the quantum system possesses, and the edges are the (quantum-mechanically allowed) transitions. Parts of the complete quantum mechanical networks can be probed experimentally via high-resolution, energy-resolved spectroscopic techniques. The complete rovibronic line list information for a given molecule can only be obtained through sophisticated quantum-chemical computations. Experiments as well as computations yield what we call spectroscopic networks (SN). First-principles SNs of even small, three to five atomic molecules can be huge, qualifying for the big data description. Besides helping to interpret high-resolution spectra, the network-theoretical view offers several ideas for improving the accuracy and robustness of the increasingly important information systems containing line-by-line spectroscopic data. For example, the smallest number of measurements necessary to perform to obtain the complete list of energy levels is given by the minimum-weight spanning tree of the SN and network clustering studies may call attention to "weakest links" of a spectroscopic database. A present-day application of spectroscopic networks is within the MARVEL (Measured Active Rotational-Vibrational Energy Levels) approach, whereby the transitions information on a measured SN is turned into experimental energy levels via a weighted linear least-squares refinement. MARVEL has been used successfully for 15 molecules and allowed to validate most of the transitions measured and come up with energy levels with well-defined and realistic uncertainties. Accurate knowledge of the energy levels with computed transition intensities allows the realistic prediction of spectra under many different circumstances, e.g., for widely different temperatures. Detailed knowledge of the energy level structure of a molecule coming from a MARVEL analysis is important for a considerable number of modeling efforts in chemistry, physics, and engineering.

Promoting and inhibiting tunneling via nuclear motions.

Accurate, experimental rotational-vibrational energy levels determined via the MARVEL (Measured Active Rotational-Vibrational Energy Levels) algorithm and published recently for the symmetric-top (14)NH3 molecule in J. Quant. Spectrosc. Radiat. Transfer, 2015, 116, 117-130 are analyzed to unravel the promoting and inhibiting effects of vibrations and rotations on the tunneling splittings of the corresponding symmetric (s) and antisymmetric (a) rovibrational energy level pairs. The experimental transition data useful from the point of view of the present analysis cover the range 0.7-7000 cm(-1), sufficiently detailed rovibrational energy sets worth analyzing are available for 20 vibrational bands. The highest J value, where J stands for the rotational quantum number, within the experimental dataset employed is 30. Coupling of the "umbrella" motion of (14)NH3 with other vibrational degrees of freedom has only a minor effect on the a-s tunneling splitting characterizing the ground vibrational state, 0.79436(70) cm(-1). In the majority of the cases rotation around the C3 axis increases, while rotation around the two perpendicular axes decreases the tunneling splittings. For example, for the pair of vibrational ground states, 0(+) and 0(-), the tunneling splitting basically disappears at around J = 25 for the (J,K) = (J,1) states, where K = |k| is the usual quantum number characterizing the projection of the rotational angular momentum on the principal axis. The tunneling splittings, defined as energy differences E(a) - E(s) of corresponding energy level pairs, as a function of J and K show a very regular behavior for the ground state (GS) and the nν2 bands. For the other bands investigated exceptions from a regular behavior do occur, especially for bands characterized by degenerate vibrations, and occasionally the data available are not sufficient to arrive at definitive conclusions. The most irregular behavior is observed for rotational states characterized by the k - l = 3n rule (l is the vibrational angular momentum quantum number), with n = 0, 1, 2,… High-quality, variationally computed rovibrational data support all the conclusions of this study based on experimental energy levels.

Zero-Cost Estimation of Zero-Point Energies.

An additive, linear, atom-type-based (ATB) scheme is developed allowing no-cost estimation of zero-point vibrational energies (ZPVE) of neutral, closed-shell molecules in their ground electronic states. The atom types employed correspond to those defined within the MM2 molecular mechanics force field approach. The reference training set of 156 molecules cover chained and branched alkanes, alkenes, cycloalkanes and cycloalkenes, alkynes, alcohols, aldehydes, carboxylic acids, amines, amides, ethers, esters, ketones, benzene derivatives, heterocycles, nucleobases, all the natural amino acids, some dipeptides and sugars, as well as further simple molecules and ones containing several structural units, including several vitamins. A weighted linear least-squares fit of atom-type-based ZPVE increments results in recommended values for the following atoms, with the number of atom types defined in parentheses: H(8), D(1), B(1), C(6), N(7), O(3), F(1), Si(1), P(2), S(3), and Cl(1). The average accuracy of the ATB ZPVEs is considerably better than 1 kcal mol(-1), that is, better than chemical accuracy. The proposed ATB scheme could be extended to many more atoms and atom types, following a careful validation procedure; deviation from the MM2 atom types seems to be necessary, especially for third-row elements.

Simple molecules as complex systems.

For individual molecules quantum mechanics (QM) offers a simple, natural and elegant way to build large-scale complex networks: quantized energy levels are the nodes, allowed transitions among the levels are the links, and transition intensities supply the weights. QM networks are intrinsic properties of molecules and they are characterized experimentally via spectroscopy; thus, realizations of QM networks are called spectroscopic networks (SN). As demonstrated for the rovibrational states of H2(16)O, the molecule governing the greenhouse effect on earth through hundreds of millions of its spectroscopic transitions (links), both the measured and first-principles computed one-photon absorption SNs containing experimentally accessible transitions appear to have heavy-tailed degree distributions. The proposed novel view of high-resolution spectroscopy and the observed degree distributions have important implications: appearance of a core of highly interconnected hubs among the nodes, a generally disassortative connection preference, considerable robustness and error tolerance, and an "ultra-small-world" property. The network-theoretical view of spectroscopy offers a data reduction facility via a minimum-weight spanning tree approach, which can assist high-resolution spectroscopists to improve the efficiency of the assignment of their measured spectra.

MARVEL analysis of the rotational-vibrational states of the molecular ions H2D+ and D2H+.

Critically evaluated rotational-vibrational line positions and energy levels, with associated critically reviewed labels and uncertainties, are reported for two deuterated isotopologues of the H3(+) molecular ion: H2D(+) and D2H(+). The procedure MARVEL, standing for Measured Active Rotational-Vibrational Energy Levels, is used to determine the validated levels and lines and their self-consistent uncertainties based on the experimentally available information. The spectral ranges covered for the isotopologues H2D(+) and D2H(+) are 5.2-7105.5 and 23.0-6581.1 cm(-1), respectively. The MARVEL energy levels of the ortho and para forms of the ions are checked against ones determined from accurate variational nuclear motion computations employing the best available adiabatic ab initio potential energy surfaces of these isotopologues. The number of critically evaluated, validated and recommended experimental (levels, lines) are (109, 185) and (104, 136) for H2D(+) and D2H(+), respectively. The lists of assigned MARVEL lines and levels and variational levels obtained for H2D(+) and D2H(+) as part of this study are deposited in the ESI to this paper.

The fourth age of quantum chemistry: molecules in motion.

Developments during the last two decades in nuclear motion theory made it possible to obtain variational solutions to the time-independent, nuclear-motion Schrödinger equation of polyatomic systems as "exact" as the potential energy surface (PES) is. Nuclear motion theory thus reached a level whereby this branch of quantum chemistry started to catch up with the well developed and widely applied other branch, electronic structure theory. It seems to be fair to declare that we are now in the fourth age of quantum chemistry, where the first three ages are principally defined by developments in electronic structure techniques (G. Richards, Nature, 1979, 278, 507). In the fourth age we are able to incorporate into our quantum chemical treatment the motion of nuclei in an exact fashion and, for example, go beyond equilibrium molecular properties and compute accurate, temperature-dependent, effective properties, thus closing the gap between measurements and electronic structure computations. In this Perspective three fundamental algorithms for the variational solution of the time-independent nuclear-motion Schrödinger equation employing exact kinetic energy operators are presented: one based on tailor-made Hamiltonians, one on the Eckart-Watson Hamiltonian, and one on a general internal-coordinate Hamiltonian. It is argued that the most useful and most widely applicable procedure is the third one, based on a Hamiltonian containing a kinetic energy operator written in terms of internal coordinates and an arbitrary embedding of the body-fixed frame of the molecule. This Hamiltonian makes it feasible to treat the nuclear motions of arbitrary quantum systems, irrespective of whether they exhibit a single well-defined minimum or not, and of arbitrary reduced-dimensional models. As a result, molecular spectroscopy, an important field for the application of nuclear motion theory, has almost black-box-type tools at its disposal. Variational nuclear motion computations, based on an exact kinetic energy operator and an arbitrary PES, can now be performed for about 9 active vibrational degrees of freedom relatively straightforwardly. Simulations of high-resolution spectra allow the understanding of complete rotational-vibrational spectra up to and beyond the first dissociation limits. Variational results obtained for H(2)O, H, NH(3), CH(4), and H(2)CCO are used to demonstrate the power of the variational techniques for the description of vibrational and rotational excitations. Some qualitative features of the results are also discussed.

Variational quantum mechanical and active database approaches to the rotational-vibrational spectroscopy of ketene, H2CCO.

A variational quantum mechanical protocol is presented for the computation of rovibrational energy levels of semirigid molecules using discrete variable representation of the Eckart-Watson Hamiltonian, a complete, "exact" inclusion of the potential energy surface, and selection of a vibrational subspace. Molecular symmetry is exploited via a symmetry-adapted Lanczos algorithm. Besides symmetry labels, zeroth-order rigid-rotor and harmonic-oscillator quantum numbers are employed to characterize the computed rovibrational states. Using the computational molecular spectroscopy algorithm presented, a large number of rovibrational states, up to J = 50, of the ground electronic state of the parent isotopologue of ketene, H(2) (12)C=(12)C=(16)O, were computed and characterized. Based on 12 references, altogether 3982 measured and assigned rovibrational transitions of H(2) (12)C=(12)C=(16)O have been collected, from which 3194 were validated. These transitions form two spectroscopic networks (SN). The ortho and the para SNs contain 2489 and 705 validated transitions and 1251 and 471 validated energy levels, respectively. The computed energy levels are compared with energy levels obtained, up to J = 41, via an inversion protocol based on this collection of validated measured rovibrational transitions. The accurate inverted energy levels allow new assignments to be proposed. Some regularities and irregularities in the rovibrational spectrum of ketene are elucidated.

From a network of computed reaction enthalpies to atom-based thermochemistry (NEAT).

A simple and fast, weighted, linear least-squares refinement protocol and code is presented for inverting the information contained in a network of quantum chemically computed 0 K reaction enthalpies. This inversion yields internally consistent 0 K enthalpies of formation for the species of the network. The refinement takes advantage of the fact that the accuracy of computed enthalpies depends strongly on the quantum-chemical protocol employed for their determination. Different protocols suffer from different sources of error; thus, the reaction enthalpies computed by them have "random" residual errors. Since it is much more natural for quantum-chemical energy and enthalpy results, including reaction enthalpies, to be based on the electronic ground states of the atoms and not on the historically preferred elemental states, and since these two possible protocols can be converted into each other straightforwardly, it is proposed that first-principles thermochemistry should employ the ground electronic states of atoms. In this scheme, called atom-based thermochemistry (AT), the enthalpy of formation of a gaseous compound corresponds simply to the total atomization energy of the species; it is always positive, and it reflects the bonding strength within the molecule. The inversion protocol developed and based on AT is termed NEAT, which represents the fact that the protocol proceeds from a network of computed reaction enthalpies toward atom-based thermochemistry, most directly to atom-based enthalpies of formation. After assembling a database that consisted of 361 ab initio reactions and reaction enthalpies involving 188 species, collected from 31 literature sources, the following dependable 0 K atom-based enthalpies of formation, Delta(f)${H{{{\rm AT}\hfill \atop 0\hfill}}}$, all in kJ mol(-1), have been obtained by means of NEAT: H(2)=432.07(0), CH=334.61(15), NH=327.69(25), OH=425.93(21), HF=566.13(31), CO=1072.08(28), O(2)=493.51(34), CH(2)=752.40(21), H(2)O=918.05(20), HO(2)=694.53(32), CO(2)=1597.77(40), CH(3)=1209.64(29), NH(3)=1157.44(33), C(2)H(2)=1625.78(40), and CH(4)=1641.68(40), in which the uncertainty values given in parentheses represent 95 % confidence intervals. The average deviation of these values from the well-established active thermochemical tables (ATcT) values is a mere 0.25 kJ mol(-1), with a maximum deviation of 0.7 kJ mol(-1). This shows that the use of a large number of ab initio reaction enthalpies within a NEAT-type protocol has considerable advantages over the sequential utilization of the ab initio information.

Use of a nondirect-product basis for treating singularities in triatomic rotational-vibrational calculations.

A technique has been developed which in principle allows the determination of the full rotational-vibrational eigenspectrum of triatomic molecules by treating the important singularities present in the triatomic rotational-vibrational kinetic energy operator given in Jacobi coordinates and the R(1) embedding. The singular term related to the diatom-type coordinate, R(1), deemed to be unimportant for spectroscopic applications, is given no special attention. The work extends a previous [J. Chem. Phys., 2005, 122, 024101] vibration-only approach and employs a generalized finite basis representation (GFBR) resulting in a nonsymmetric Hamiltonian matrix [J. Chem. Phys., 2006, 124, 014110]. The basis set to be used is obtained by taking the direct product of a 1-D DVR basis, related to R(1), with a 5-D nondirect-product basis, the latter formed by coupling Bessel-DVR functions depending on the distance-type coordinate causing the singularity, associated Legendre polynomials depending on the Jacobi angle, and rotational functions depending on the three Euler angles. The robust implicitly restarted Arnoldi method within the ARPACK package is used for the determination of a number of eigenvalues of the nonsymmetric Hamiltonian matrix. The suitability of the proposed approach is shown by the determination of the rotational-vibrational energy levels of the ground electronic state of H(3)(+) somewhat above its barrier to linearity. Convergence of the eigenenergies is checked by an alternative approach, employing a Hamiltonian expressed in Radau coordinates, a standard direct-product basis, and no treatment of the singularities.

On equilibrium structures of the water molecule.

Equilibrium structures are fundamental entities in molecular sciences. They can be inferred from experimental data by complicated inverse procedures which often rely on several assumptions, including the Born-Oppenheimer approximation. Theory provides a direct route to equilibrium geometries. A recent high-quality ab initio semiglobal adiabatic potential-energy surface (PES) of the electronic ground state of water, reported by Polyansky et al. [ ibid. 299, 539 (2003)] and called CVRQD here, is analyzed in this respect. The equilibrium geometries resulting from this direct route are deemed to be of higher accuracy than those that can be determined by analyzing experimental data. Detailed investigation of the effect of the breakdown of the Born-Oppenheimer approximation suggests that the concept of an isotope-independent equilibrium structure holds to about 3 x 10(-5) A and 0.02 degrees for water. The mass-independent [Born-Oppenheimer (BO)] equilibrium bond length and bond angle on the ground electronic state PES of water is r(e) (BO)=0.957 82 A and theta e (BO)=104.48(5) degrees , respectively. The related mass-dependent (adiabatic) equilibrium bond length and bond angle of H2 (16)O is r(e) (ad)=0.957 85 A and theta e (ad)=104.50(0) degrees , respectively, while those of D2 (16)O are r(e) (ad)=0.957 83 A and theta e (ad)=104.49(0) degrees . Pure ab initio prediction of J=1 and 2 rotational levels on the vibrational ground state by the CVRQD PESs is accurate to better than 0.002 cm(-1) for all isotopologs of water considered. Elaborate adjustment of the CVRQD PESs to reproduce all observed rovibrational transitions to better than 0.05 cm(-1) (or the lower ones to better than 0.0035 cm(-1)) does not result in noticeable changes in the adiabatic equilibrium structure parameters. The expectation values of the ground vibrational state rotational constants of the water isotopologs, computed in the Eckart frame using the CVRQD PESs and atomic masses, deviate from the experimentally measured ones only marginally, especially for A0 and B0. The small residual deviations in the effective rotational constants are due to centrifugal distortion, electronic, and non-Born-Oppenheimer effects. The spectroscopic (nonadiabatic) equilibrium structural parameters of H2 16O, obtained from experimentally determined A'0 and B'0 rotational constants corrected empirically to obtain equilibrium rotational constants, are r(e) (sp)=0.957 77 A and theta e (sp)=104.48 degrees .

Treating singularities present in the Sutcliffe-Tennyson vibrational Hamiltonian in orthogonal internal coordinates.

Two methods are developed, when solving the related time-independent Schrodinger equation (TISE), to cope with the singular terms of the vibrational kinetic energy operator of a triatomic molecule given in orthogonal internal coordinates. The first method provides a mathematically correct treatment of all singular terms. The vibrational eigenfunctions are approximated by linear combinations of functions of a three-dimensional nondirect-product basis, where basis functions are formed by coupling Bessel-DVR functions, where DVR stands for discrete variable representation, depending on distance-type coordinates and Legendre polynomials depending on angle bending. In the second method one of the singular terms related to a distance-type coordinate, deemed to be unimportant for spectroscopic applications, is given no special treatment. Here the basis set is obtained by taking the direct product of a one-dimensional DVR basis with a two-dimensional nondirect-product basis, the latter formed by coupling Bessel-DVR functions and Legendre polynomials. With the basis functions defined, matrix representations of the TISE are set up and solved numerically to obtain the vibrational energy levels of H3+. The numerical calculations show that the first method treating all singularities is computationally inefficient, while the second method treating properly only the singularities having physical importance is quite efficient.