Exact Kinetic Energy Operator Research Articles

CO2 inside sI clathrate-like cages: Automated construction of neural network/machine learned guest-host potential and quantum spectra computations.

We present new results on the underlying guest-host interactions and spectral characterization of a CO2 molecule confined in the cages of the sI clathrate hydrate. Such types of porous solids raise computational challenges, as they are of practical interest as gas storage/capture materials. Accordingly, we have directed our efforts toward addressing their modeling in a proper manner, ensuring the quality of the input data and the efficiency of the computational approaches. The computational procedure for spectral simulations, within the multi-configurational time-dependent Hartree framework, involves the development of a fully coupled Hamiltonian, including an exact kinetic energy operator and a many-body representation of the potential, along with dipole moment surfaces, both obtained through neural network machine learning techniques. The resulting models were automatically trained and tested on extensive datasets generated by PW86PBE-XDM calculations, following the outcome of previous benchmark studies. Our simulations enable us to explore various aspects of the quantized dynamics upon confinement of CO2@D/T, such as constrained rotational-translational quantum motions and the averaged position/orientation of the CO2 guest in comparison to the experimental data available. Particularly notable are the distinct energy patterns observed in the computed spectra for the confined CO2 in the D and T cages, with a considerably high rotational-translational coupling in the CO2@T case. Leveraging reliable computations has proved instrumental, highlighting the sensitivity of the spectral features to the shape and strength of the potential interactions, with the explicit description of many-body contributions being significant.

Computational Energy Spectra of the H2O@C70 Endofullerene.

A water molecule confined inside the C70 fullerene wasquantum-mechanically described using a computational approachwithin the MCTDH framework. Such procedure involvesthe development of a full-dimensional coupled hamiltonian, with an exact kinetic energy operator, including allrotational, translational and vibrational degrees of freedomof the endofullerene system. In turn, through an effectivepairwise potential model, the ground and rotationally excitedstates of the encapsulated H2O inside the C70 cagewere calculated, and traced back to the isotropic case ofthe H2O@C60 endofullerene in order to understand the nature and physical origin of the symmetry breaking observedexperimentally in the latter system. Moreover, the computationalscheme used here allows to study the quantizationof the translational movement of the encapsulated watermolecule inside theC70fullerene, and to investigate theconfinement effects in the vibrational energy levels of theH2O@C70 system.

ExoMol line lists – XLVIII. High-temperature line list of thioformaldehyde (H2CS)

ABSTRACT A comprehensive rotation–vibration (ro–vibrational) line list of thioformaldehyde (1H212C32S) that is applicable for elevated temperatures (${2000}{\, \mathrm{K}}$) is presented. The new MOTY line list covers the 0–8000 cm−1 range (wavelengths $\lambda \gt {1.3}{\, \mu \mathrm{m}})$ and contains around 43.5 billion transitions between 52.3 million states with rotational excitation up to J = 120. Line list calculations utilize a newly determined empirically refined potential energy surface (PES) – the most accurate H2CS PES to date – a previously published high-level ab initio dipole moment surface, and the use of an exact kinetic energy operator for solving the ro–vibrational Schrödinger equation. Post-processing of the MOTY line list is performed by replacing calculated energy levels with empirically derived values, vastly improving the accuracy of predicted line positions in certain spectral windows and making the line list suitable for high-resolution applications. The MOTY line list is available from the ExoMol data base at www.exomol.com and the CDS astronomical data base.

Ro-vibrational levels and their (e-f) splitting of acetylene molecule calculated from new potential energy surfaces

Ro-vibrational energy levels of acetylene are reported using variational nuclear motion calculations from new ab initio and empirically optimized full six-dimensional potential energy surfaces in the ground electronic state of the acetylene molecule. The calculations account for the triple, quadruple and quintuple excitations as well as relativistic and diagonal Born-Oppenheimer corrections. Variational nuclear motion calculations were performed using the exact kinetic energy operator in orthogonal coordinates. The convergence of energy levels calculations versus the size of the vibrational basis set functions was verified. Our best ab initio potential energy surface that includes the above-mentioned contributions provides the RMS (obs.-calc.) errors of 0.95 cm−1 for five fundamental energy levels. The largest contribution to the RMS error is caused primarily by a significant deviation of the ν4 fundamental frequency. Experimental values of 120 vibrational band origins were used to empirically adjust few lower-order parameters of the potential energy surface. The average error drops down to 0.45 cm−1 or 0.25 cm−1 for empirically optimized potential energy function with two or seven adjusted parameters corresponding to quadratic force field terms and one third order term. The splitting of (e – f) rovibrational doublets and their J dependence were calculated with high accuracy due to the full account of Coriolis interactions. Computed (e – f) splittings allow one to check the correctness of the assignment of empirical energy levels. The estimation of the accuracy for the calculated vibrational levels in an extended range up to 9500 cm−1 shows that the set of ab initio vibrational levels can be used for future assignments of empirically not observed ro-vibrational energy levels. The comparison of the calculated and experimental ro-vibrational energy levels of the C2D2 and 13C2H2 isotopologues is also reported.

Encapsulation of a Water Molecule inside C60 Fullerene: The Impact of Confinement on Quantum Features.

We introduce an efficient quantum fully coupled computational scheme within the multiconfiguration time-dependent Hartree (MCTDH) approach to handle the otherwise extremely costly computations of translational–rotational–vibrational states and energies of light-molecule endofullenes. Quantum calculations on energy levels are reported for a water molecule inside C60 fullerene by means of such a systematic approach that includes all nine degrees of freedom of H2O@C60 and does not consider restrictions above them. The potential energy operator is represented as a sum of natural potentials employing the n-mode expansion, along with the exact kinetic energy operator, by introducing a set of Radau internal coordinates for the H2O molecule. On the basis of the present rigorous computations, various aspects of the quantized intermolecular dynamics upon confinement of H2O@C60 are discussed, such as the rotational energy level splitting and the significant frequency shifts of the encapsulated water molecule vibrations. The impact of water encapsulation on quantum features is explored, and insights into the nature of the underlying forces are provided, highlighting the importance of a reliable first-principles description of the guest–host interactions.

Vibrational energy levels of the S 0 and S 1 states of formaldehyde using an accurate ab initio based global diabatic potential energy matrix

Low-lying vibrational energy levels of both the ground (S 0) and first excited singlet (S 1) states of formaldehyde have been determined using an exact kinetic energy operator and adiabatic potential energy surfaces derived from a recently developed full-dimensional global diabatic potential energy matrix (DPEM) based on a neural network representation of high level ab initio data. The agreement with available experimental band origins provides strong evidence in support of the accuracy of the DPEM, which can be used in the future for understanding the internal conversion of H2CO(S 1) and the subsequent roaming dynamics on its S 0 state.

Vibrational levels of formaldehyde: Calculations from new high precision potential energy surfaces and comparison with experimental band origins

Vibrational energy levels of H2CO are reported using variational nuclear motion calculations from new ab initio and empirically optimized full 6-dimensional ab initio potential energy surfaces in the ground electronic state of the formaldehyde molecule. Ab initio calculations were carried out using extended electronic structure coupled-cluster calculations accounting for dynamic electron correlations including triple and quadruple excitations as well as relativistic and diagonal Born-Oppenheimer corrections. Variational nuclear motion calculations are compared in different sets of coordinates with exact kinetic energy operator and in normal coordinates with Watson-Eckart kinetic energy operator. Our best ab initio potential energy surface including the above mentioned contributions provides the RMS (obs.-calc.) errors of 0.25 cm−1 for fifteen vibrational band origins without empirically adjusted parameters. The average error drops down to 0.08 cm−1 for an empirically optimized potential energy function with six adjusted parameters corresponding to quadratic force field terms. The estimation of the accuracy for the calculated vibrational levels in an extended range up to 4500 cm−1 shows that the set of ab initio vibrational levels could be more consistent than experimental levels obtained from earlier studies of low resolution spectra. The comparison of the calculated and experimental vibrational energy levels of the D2CO isotopologue is also reported.

Treating linear molecules in calculations of rotation-vibration spectra.

In this article, a numerical implementation of the exact kinetic energy operator (KEO) for triatomic molecules (symmetric of XY2-type and asymmetric of YXZ-type) is presented. The implementation is based on the valence coordinates with the bisecting (XY2-type molecules) and bond-vector (YXZ) embeddings and includes the treatment of the singularity at linear geometry. The KEO is represented in a sum-of-product form. The singularity caused by the undetermined angle at the linear configuration is resolved with the help of the associated Legendre and Laguerre polynomials used as parameterized bending basis functions in the finite basis set representation. The exact KEO implementation is combined with the variational solver theoretical rovibrational energies, equipped with a general automatic symmetry-adaptation procedure and efficient basis step contraction schemes, providing a powerful computational solver of triatomic molecules for accurate computations of highly excited ro-vibrational spectra. The advantages of different basis set choices are discussed. Examples of specific applications for computing hot spectra of linear molecules are given.

ExoMol line lists – XXXIX. Ro-vibrational molecular line list for CO2

ABSTRACT A new hot line list for the main isotopologue of CO2, 12C16O2 is presented. The line list consists of almost 2.5 billion transitions between 3.5 million rotation-vibration states of CO2 in its ground electronic state, covering the wavenumber range 0–20 000 cm−1 (λ > 0.5 µm) with the upper and lower energy thresholds of 36 000 cm−1 and 16 000 cm−1, respectively. The ro-vibrational energies and wavefunctions are computed variationally using the accurate empirical potential energy surface Ames-2. The ro-vibrational transition probabilities in the form of Einstein coefficients are computed using an accurate ab initio dipole moment surface with variational program TROVE. A new implementation of TROVE which uses an exact nuclear-motion kinetic energy operator is employed. Comparisons with the existing hot line lists are presented. The line list should be useful for atmospheric retrievals of exoplanets and cool stars. The UCL-4000 line list is available from the CDS and ExoMol data bases.

Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator.

We present the first use of curvilinear vibrational coordinates, specifically polyspherical coordinates, in combination with vibrational coupled cluster theory. The polyspherical coordinates are used in the context of both the adaptive density-guided approach to potential energy surface construction and in the subsequent vibrational coupled cluster calculations of anharmonic vibrational states. Results obtained based on the polyspherical coordinate parametrization are compared to results obtained with the use of rectilinear vibrational coordinates, namely, normal coordinates and hybrid optimized and localized coordinates for the formaldehyde molecule. This comparison is carried out with the full vibrational configuration interaction model, using the respective fully coupled potential energy surfaces and untruncated kinetic energy operators. The polyspherical coordinates are shown to facilitate an acceleration of convergence for truncated methods when compared to the use of normal coordinates. We furthermore report on calculations on the hydrogen peroxide molecule in the polyspherical coordinate parametrization. The polyspherical vibrational coordinates are shown to perform very well, even for truncated methods, especially when considering the difficulty that rectilinear vibrational coordinates can exhibit in treating complicated internal molecular motion.

Highly excited vibrational levels of methane up to 10 300 cm−1: Comparative study of variational methods

In this work, we report calculated vibrational energy levels of the methane molecule up to 10 300 cm-1. Two potential energy surfaces constructed in quite different coordinate systems with different analytical representations are employed in order to evaluate the uncertainty of vibrational predictions. To calculate methane energy levels, we used two independent techniques of the variational method. One method uses an exact kinetic energy operator in internal curvilinear coordinates. Another one uses an expansion of Eckart-Watson nuclear motion Hamiltonian in rectilinear normal coordinates. In the Icosad range (up to five vibrational quanta bands-below 7800 cm-1), the RMS standard deviations between calculated and observed energy levels were 0.22 cm-1 and 0.41 cm-1 for these two quite different approaches. For experimentally well-known 3v3 sub-levels, the calculation accuracy is estimated to be ∼1 cm-1. In the Triacontad range (7660-9188 cm-1), the average error of the calculation is about 0.5 cm-1. The accuracy and convergence issues for higher energy ranges are discussed.

Computing vibrational energy levels of CH4 with a Smolyak collocation method.

In this paper, we demonstrate that it is possible to apply collocation to compute vibrational energy levels of a five-atom molecule using an exact kinetic energy operator (with cross terms and coordinate-dependent coefficients). This is made possible by using (1) a pruned basis of products of univariate functions; (2) a Smolyak grid made from nested sequences of grids for each coordinate; (3) a collocation method that obviates the need to solve a generalized eigenvalue problem; (4) an efficient sequential transformation between the (nondirect product) grid and the (nondirect product) basis representations; and (5) hierarchical univariate functions that make it possible to avoid storing large intermediate vectors. The accuracy of the method is confirmed by computing 500 vibrational energy levels of methane.

Ro-vibronic transition intensities for triatomic molecules from the exact kinetic energy operator; electronic spectrum for the C̃ 1B2 ← X̃ 1A1 transition in SO2.

A procedure for calculating ro-vibronic transition intensities for triatomic molecules within the Born-Oppenheimer approximation is reported. Ro-vibrational energy levels and wavefunctions are obtained with the DVR3D suite, which solves the nuclear motion problem with an exact kinetic energy operator. Absolute transition intensities are calculated both with the Franck-Condon approximation and with a full transition dipole moment surface. The theoretical scheme is tested on C̃ 1B2 ← X̃ 1A1 ro-vibronic transitions of SO2. Ab initio potential energy and dipole moment surfaces are generated for this purpose. The calculated ro-vibronic transition intensities and cross sections are compared with the available experimental and theoretical data.

First fullyab initiopotential energy surface of methane with a spectroscopic accuracy

Full 9-dimensional ab initio potential energy surfaces for the methane molecule are constructed using extended electronic structure coupled-cluster calculations with various series of basis sets following increasing X cardinal numbers: cc-pVXZ (X = 3, 4, 5, 6), aug-cc-ACVXZ (X = 3, 4, 5), and cc-pCVXZ-F12 (X = 3, 4). High-order dynamic electron correlations including triple and quadrupole excitations as well as relativistic and diagonal Born-Oppenheimer breakdown corrections were accounted for. Analytical potential functions are parametrized as non-polynomial expansions in internal coordinates in irreducible tensor representation. Vibrational energy levels are reported using global variational nuclear motion calculations with exact kinetic energy operator and a full account of the tetrahedral symmetry of CH4. Our best ab initio surface including above-mentioned contributions provides the rms (obs.-calc.) errors of less than 0.11 cm−1 for vibrational band centers below 4700 cm−1, and ∼0.3 cm−1 for all 229 assigned experimentally determined vibrational levels up to the Icosad range <7900 cm−1 without empirically adjusted parameters. These results improve the accuracy of ab initio methane vibrational predictions by more than an order of magnitude with respect to previous works. This is an unprecedented accuracy of first-principles calculations of a five-atomic molecule for such a large data set. New ab initio potential results in significantly better band center predictions even in comparison with best available empirically corrected potential energy surfaces. The issues related to the basis set extrapolation and an additivity of various corrections at this level of accuracy are discussed.

Interpretation of the vibrational energy level structure of the astructural molecular ion H5 (+) and all of its deuterated isotopomers.

Variational nuclear motion computations, employing an exact kinetic energy operator and two different potential energy surfaces, are performed to study the first 60 vibrational states of the molecular ion H5 (+)≡ [H2-H-H2](+) and all of its deuterated isotopologues and isotopomers, altogether 12 species. Detailed investigation of the vibrational wavefunctions mostly results in physically intuitive labels not only for the fundamentals but also for the overtone and combination states computed. The torsional motion associated with the left and right diatomics appears to be well separated from the other vibrational degrees of freedom for all species. The unusual structure of the higher-lying bending states and the heavy mixing of the internal motions is partly due to the astructural character of all these molecular ions. The existence of distinct isotopomers in the H5-nDn (+), n = 1-4 cases, in the energy range studied, is confirmed. Two rules determine the stability order of the isotopomers: first, when possible, H prefers to stay in the middle of the ions rather than at the sides, and, second, the isotopomer with a homonuclear diatomic at the side is always lower in energy. The large number of precise vibrational energies of the present study, as well as the detailed assignment of the states, should serve as benchmarks for future studies by more approximate nuclear-motion treatments, such as diffusion Monte Carlo and multiconfiguration time-dependent Hartree.

Full-Dimensional Quantum Calculations of Vibrational Levels of NH4(+) and Isotopomers on An Accurate Ab Initio Potential Energy Surface.

Vibrational energy levels of the ammonium cation (NH4(+)) and its deuterated isotopomers are calculated using a numerically exact kinetic energy operator on a recently developed nine-dimensional permutation invariant semiglobal potential energy surface fitted to a large number of high-level ab initio points. Like CH4, the vibrational levels of NH4(+) and ND4(+) exhibit a polyad structure, characterized by a collective quantum number P = 2(v1 + v3) + v2 + v4. The low-lying vibrational levels of all isotopomers are assigned and the agreement with available experimental data is better than 1 cm(-1).

Numerical and exact kinetic energy operator using Eckart conditions with one or several reference geometries: Application to HONO.

For the computation of rovibrational levels and their spectroscopic intensities, the Eckart conditions are essential to achieve the optimal separation between rotation and vibration. Dymarsky and Kudin [J. Chem. Phys. 122, 124103 (2005)] proposed a procedure for a simplified calculation of the Eckart rotation matrix. In the present work, we have adapted their approach to obtain a kinetic energy operator in curvilinear coordinates using a numerical but exact procedure without resorting to finite differences. Furthermore, we have modified this approach for the study of molecular systems with several minima, for which several Eckart reference geometries are required. The HONO molecular system has been used to show the efficiency of our implementation. Using the Eckart conditions with multi-reference geometries allows for a calculation of the rotational levels as well as frequencies and intensities of the infrared spectra of both HONO isomers with a single calculation.

An efficient method for energy levels calculation using full symmetry and exact kinetic energy operator: tetrahedral molecules.

A simultaneous use of the full molecular symmetry and of an exact kinetic energy operator (KEO) is of key importance for accurate predictions of vibrational levels at a high energy range from a potential energy surface (PES). An efficient method that permits a fast convergence of variational calculations would allow iterative optimization of the PES parameters using experimental data. In this work, we propose such a method applied to tetrahedral AB4 molecules for which a use of high symmetry is crucial for vibrational calculations. A symmetry-adapted contracted angular basis set for six redundant angles is introduced. Simple formulas using this basis set for explicit calculation of the angular matrix elements of KEO and PES are reported. The symmetric form (six redundant angles) of vibrational KEO without the sin(q)(-2) type singularity is derived. The efficient recursive algorithm based on the tensorial formalism is used for the calculation of vibrational matrix elements. A good basis set convergence for the calculations of vibrational levels of the CH4 molecule is demonstrated.

Quantum Dynamics of Carbon Dioxide Encapsulated in the Cages of the sI Clathrate Hydrate: Structural Guest Distributions and Cage Occupation

We perform quantum multiconfiguration time-dependent Hartree calculations for the two cages of the sI CO2 clathrate hydrate, and we report, for first time, results on the translational, rotational, and vibrational states. The potential surface is constructed by the semiempirical SPC/E water model and pairwise additive interactions between the CO2 molecule and all water molecules forming each cage, while a spectroscopic-determined potential is used for the CO2 monomer. The exact kinetic energy operator is derived for the linear CO2 in the cage, and the potential energy operator is adapted to combined mode schemes, chosen adequately in order to obtain the desired product representation. The rotational ability of the CO2 is hindered in the small cavities (512), while in the large cage (51262) translational and rotational degrees of freedom are highly coupled. By analyzing the corresponding T-R eigenstates, we established a connection with experimental X-ray measurements on the orientation of the CO2 molecule in the two cages. Further, vibrational excitations of the fundamental symmetric and antisymmetric stretch modes, their overtones, and combination bands are computed from 7D quantum calculations. We found significant frequency shifts, which are comparable to those observed in the double-peak profile of the experimentally recorded infrared spectra, for the CO2 molecule trapped in the small and large cages of the sI clathrate hydrate. Such good agreement, with the data from recent spectroscopic studies of carbon dioxide clathrate hydrate at low temperatures, indicates a single occupancy of the sI structure cages, allowing an assessment of the theoretical approaches employed.

IUPAC critical evaluation of the rotational–vibrational spectra of water vapor. Part IV. Energy levels and transition wavenumbers for D216O, D217O, and D218O

This paper is the fourth of a series of papers reporting critically evaluated rotational–vibrational line positions, transition intensities, pressure dependences, and energy levels, with associated critically reviewed assignments and uncertainties, for all the main isotopologues of water. This paper presents energy level and transition data for the following doubly and triply substituted isotopologues of water: D216O, D217O, and D218O. The MARVEL (Measured Active Rotational–Vibrational Energy Levels) procedure is used to determine the levels, the lines, and their self-consistent uncertainties for the spectral regions 0–14016, 0–7969, and 0–9108cm−1 for D216O, D217O, and D218O, respectively. For D216O, D217O, and D218O, 53534, 600, and 12167 lines are considered, respectively, from spectra recorded in absorption at room temperature and in emission at elevated temperatures. The number of validated energy levels is 12269, 338, and 3351 for D216O, D217O, and D218O, respectively. The energy levels have been checked against the ones determined, with an average accuracy of about 0.03cm−1, from variational rovibrational computations employing exact kinetic energy operators and an accurate potential energy surface. Furthermore, the rovibrational labels of the energy levels have been validated by an analysis of the computed wavefunctions using the rigid-rotor decomposition (RRD) scheme. The extensive list of MARVEL lines and levels obtained is deposited in the Supplementary Material of this paper, in a distributed information system applied to water, W@DIS, and on the official MARVEL website, where they can easily be retrieved.