A new “spectroscopic” potential energy surface for formaldehyde in its ground electronic state

Toward Accurate yet Effective Computations of Rotational Spectroscopy Parameters for Biomolecule Building Blocks

The interplay of high-resolution rotational spectroscopy and quantum-chemical computations plays an invaluable role in the investigation of biomolecule building blocks in the gas phase. However, quantum-chemical methods suffer from unfavorable scaling with the dimension of the system under consideration. While a complete characterization of flexible systems requires an elaborate multi-step strategy, in this work, we demonstrate that the accuracy obtained by quantum-chemical composite approaches in the prediction of rotational spectroscopy parameters can be approached by a model based on density functional theory. Glycine and serine are employed to demonstrate that, despite its limited cost, such a model is able to predict rotational constants with an accuracy of 0.3% or better, thus paving the way toward the accurate characterization of larger flexible building blocks of biomolecules.

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

On the separability of large-amplitude motions in anharmonic frequency calculations

Nuclear vibrational theories based upon the Watson Hamiltonian are ubiquitous in quantum chemistry, but are generally unable to model systems in which the wavefunction can delocalise over multiple energy minima, i.e. molecules that have low-energy torsion and inversion barriers. In a 2019 Chemical Reviews article, Puzzarini et al. note that a common workaround is to simply decouple these problematic modes from all other vibrations in the system during anharmonic frequency calculations. They also point out that this approximation can be "ill-suited", but do not quantify the errors introduced. In this work, we present the first systematic investigation into how separating out or constraining torsion and inversion vibrations within potential energy surface (PES) expansions affects the accuracy of computed fundamental wavenumbers for the remaining vibrational modes, using a test set of 19 tetratomic molecules for which high quality analytic potential energy surfaces and fully-coupled anharmonic reference fundamental frequencies are available. We find that the most effective and efficient strategy is to remove the mode in question from the PES expansion entirely. This introduces errors of up to +10 cm-1 in stretching fundamentals that would otherwise couple to the dropped mode, and ±5 cm-1 in all other fundamentals. These errors are approximately commensurate with, but not necessarily additional to, errors due to the choice of electronic structure model used in constructing spectroscopically accurate PES.

Simplified Representation of Multichannel Thermal Unimolecular Reactions. II. Refined Parametrization of Formaldehyde Dissociation

Simplified representations of branching fractions for thermal unimolecular two-channel reactions are discussed. The dissociation of formaldehyde serves as an illustrative example. Quantum-corrected classical trajectory calculations on an ab initio potential energy surface are combined with master equation calculations for collisional energy transfer. The treatment accounts for roaming atom dynamics. The dependence of the channel branching fractions on the bath gas pressure and temperature, on the collision efficiencies, and on the difference of channel threshold energies, are explored. It is discussed to what extent the derived simplified representations of channel branching fractions can be generalized.

Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3

In the present work, we report a detailed description of the symmetry propertiesof the eight-atomic molecule ethane, with the aim of facilitating the variational calculations ofrotation-vibration spectra of ethane and related molecules. Ethane consists of two methyl groupsCH3 where the internal rotation (torsion) of one CH3 group relative to the other is of large amplitudeand involves tunnelling between multiple minima of the potential energy function. The molecularsymmetry group of ethane is the 36-element group G36, but the construction of symmetrised basisfunctions is most conveniently done in terms of the 72-element extended molecular symmetrygroup G36(EM). This group can subsequently be used in the construction of block-diagonal matrixrepresentations of the ro-vibrational Hamiltonian for ethane. The derived transformation matricesassociated with G36(EM) have been implemented in the variational nuclear motion program TROVE(Theoretical ROVibrational Energies). TROVE variational calculations are used as a practical exampleof a G36(EM) symmetry adaptation for large systems with a non-rigid, torsional degree of freedom.We present the derivation of irreducible transformation matrices for all 36 (72) operations of G36(M)(G36(EM)) and also describe algorithms for a numerical construction of these matrices based on aset of four (five) generators. The methodology presented is illustrated on the construction of thesymmetry-adapted representations both of the potential energy function of ethane and of the rotation,torsion and vibration basis set functions.

Accuracy of Frequencies Obtained with the Aid of Explicitly Correlated Wave Function Based Methods

We asses the basis set convergence of harmonic frequencies using different explicitly correlated wave function based methods. All commonly available CCSD(T) variants as well as MP2-F12 and MP4(F12*) are considered, and a hierarchy of the different approaches is established. As for reaction and atomization energies, CCSD(F12*)(T*) is a close approximation to CCSD(F12)(T*) and clearly superior to the other tested approximations. The used scaling for the triples correction enhances the accuracy relative to CCSD(F12*)(T) especially for small basis sets and is very attractive since no additional computational costs are added. However, this scaling slightly breaks size consistency, and therefore we additionally study the accuracy of CCSD(F12*)(T*) and CCSD(F12*)(T) in the context of calculating anharmonic frequencies to check if this causes problems in the generation of the potential energy surface (PES). We find a fast basis set convergence for harmonic and anharmonic frequencies. Already in the cc-pVDZ-F12 basis, the RMSD to the CBS limit is only around 4-5 cm^-1.

Simulating electric field interactions with polar molecules using spectroscopic databases

Ro-vibrational Stark-associated phenomena of small polyatomic molecules are modelled using extensive spectroscopic data generated as part of the ExoMol project. The external field Hamiltonian is built from the computed ro-vibrational line list of the molecule in question. The Hamiltonian we propose is general and suitable for any polar molecule in the presence of an electric field. By exploiting precomputed data, the often prohibitively expensive computations associated with high accuracy simulations of molecule-field interactions are avoided. Applications to strong terahertz field-induced ro-vibrational dynamics of PH₃ and NH₃, and spontaneous emission data for optoelectrical Sisyphus cooling of H₂CO and CH₃Cl are discussed.

Resummation of divergent perturbation series: Application to the vibrational states of H2CO molecule

Large-order Rayleigh-Schrödinger perturbation theory (RSPT) is applied to the calculation of anharmonic vibrational energy levels of H2CO molecule. We use the model of harmonic oscillators perturbed by anharmonic terms of potential energy. Since the perturbation series typically diverge due to strong couplings, we apply the algebraic approximation technique because of its effectiveness shown earlier by Goodson and Sergeev [J. Chem. Phys. 110, 8205 (1999); ibid. 124, 094111 (2006)] and in our previous articles [A. D. Bykov et al. Opt. Spectrosc. 114, 396 (2013); ibid. 116, 598 (2014)]. To facilitate the resummation of terms contributing to perturbed states, when resonance mixing between states is especially strong and perturbation series diverge very quick, we used repartition of the Hamiltonian by shifting the normal mode frequencies. Energy levels obtained by algebraic approximants were compared with the results of variational calculation. It was found that for low energy states (up to ∼5000 cm(-1)), algebraic approximants gave accurate values of energy levels, which were in excellent agreement with the variational method. For highly excited states, strong and multiple resonances complicate series resummation, but a suitable change of normal mode frequencies allows one to reduce the resonance mixing and to get accurate energy levels. The theoretical background of the problem of RSPT series divergence is discussed along with its numerical analysis. For these purposes, the vibrational energy is considered as a function of a complex perturbation parameter. Layout and classification of its singularities allow us to model the asymptotic behavior of the perturbation series and prove the robustness of the algorithm.

Criteria for first- and second-order vibrational resonances and correct evaluation of the Darling-Dennison resonance coefficients using the canonical Van Vleck perturbation theory

The second-order vibrational Hamiltonian of a semi-rigid polyatomic molecule when resonances are present can be reduced to a quasi-diagonal form using second-order vibrational perturbation theory. Obtaining exact vibrational energy levels requires subsequent numerical diagonalization of the Hamiltonian matrix including the first- and second-order resonance coupling coefficients. While the first-order Fermi resonance constants can be easily calculated, the evaluation of the second-order Darling-Dennison constants requires more complicated algebra for seven individual cases with different numbers of creation-annihilation vibrational quanta. The difficulty in precise evaluation of the Darling-Dennison coefficients is associated with the previously unrecognized interference with simultaneously present Fermi resonances that affect the form of the canonically transformed Hamiltonian. For the first time, we have presented the correct form of the general expression for the evaluation of the Darling-Dennison constants that accounts for the underlying effect of Fermi resonances. The physically meaningful criteria for selecting both Fermi and Darling-Dennison resonances are discussed and illustrated using numerical examples.

Bend Excitation Is Predicted to Greatly Accelerate Isomerization of trans-Hydroxymethylene to Formaldehyde in the Deep Tunneling Region

Using a new potential energy surface, based on fitting around 33,000 CCSD(T)-F12/aug-cc-pVTZ energies, a robust set of predictions is made for mode-specific isomerization of trans-hydroxymethylene to formaldehyde in the deep tunneling region. The calculations make use of a recent projection model for mode-specific tunneling based on the rectilinear Q(im) path [ Wang , Y. ; Bowman , J. M. J. Chem. Phys. 2013 , 139 , 154303 ]. The most interesting prediction is a large decrease in the half-life from roughly 5 h for the ground vibrational state to roughly 1.5 min and 1 s by excitation of the fundamental and first overtone of the asymmetric bending normal mode, respectively. The properties of the new PES are described along with variational calculations of low-lying vibrational states of trans- and cis-hydroxymethylene.

Grid-based empirical improvement of molecular potential energy surfaces

A grid-based method designed to refine adiabatic potential energy surfaces (PES) of molecules via minimizing a suitable objective function is described. The objective function contains deviations from the reference (experimental) (ro)vibrational energy levels and is based on PES correction values determined at the grid points within a discrete-variable-representation nuclear-motion algorithm and first-order perturbation theory (PT). The proposed PES refinement technique is tested on the ground electronic state of the MgH molecule. The large number of numerical test results obtained suggest the following: (1) first-order PT is able to yield accurate correction values at the grid points representing the PES, and for practical cases there seems to be no need to go to higher orders of PT; (2) with the number of grid points greatly exceeding the number of experimental energy levels included in the refinement procedure, terms additional to the "obs-calc" term, including numerical first and second derivatives of the correction surface, are necessary in the objective function to arrive at a physically meaningful, "smooth" correction surface; (3) for a given J rotational quantum number, the corrected PES is able to reproduce experimental (ro)vibrational energies to within tenths of cm(-1) if they are included in the refinement or interpolated between states that are involved in the optimization, whereas extrapolated states tend to have somewhat larger remaining discrepancies; (4) the PES refined only for the J = 0 states introduces a minor systematic error for J > 0 states, with discrepancies growing with J; (5) when the number of experimental energies included in the refinement greatly exceeds the number of grid points upon which the PES is optimized, the systematic error of treating states with different J rotational quantum numbers can be reduced and an impressive average accuracy can be achieved for all rovibrational states; and (6) in the case of quasibound (also known as resonance) rovibrational states, energies can be computed to accuracies similar to those of the bound states and excellent lifetimes (widths) can also be determined. Changes in thermochemical functions upon inclusion of quasibound states during direct summation is discussed.

Variational calculations on the vibrational level structure of S0 HDCO

We perform converged high precision variational calculations to determine the frequencies of the vibrational levels in S0 HDCO, extending up to 5000 cm−1 of vibrational excitation energy. For these calculations we use our specific vibrational method (recently employed for studies on H2CO and D2CO), consisting of a combination of a search/selection algorithm and a Lanczos iteration procedure and based on the Martin, Lee, Taylor potential energy surface for formaldehyde. The calculated level structure is compared to the recently measured frequencies by Ellsworth et al. in order to improve their assignments and further clarify the vibrational mixing pattern and vibrational resonances in HDCO that are very different from the other more symmetric formaldehyde species H2CO and D2CO studied recently.

Variational study on the vibrational level structure and vibrational level mixing of highly vibrationally excited S₀ D₂CO

We perform converged high precision variational calculations to determine the frequencies of a large number of vibrational levels in S(0) D(2)CO, extending from low to very high excess vibrational energies. For the calculations we use our specific vibrational method (recently employed for studies on H(2)CO), consisting of a combination of a search/selection algorithm and a Lanczos iteration procedure. Using the same method we perform large scale converged calculations on the vibrational level spectral structure and fragmentation at selected highly excited overtone states, up to excess vibrational energies of ∼17,000 cm(-1), in order to study the characteristics of intramolecular vibrational redistribution (IVR), vibrational level density and mode selectivity.

Numerical-analytic implementation of the higher-order canonical Van Vleck perturbation theory for the interpretation of medium-sized molecule vibrational spectra

Anharmonic vibrational states of semirigid polyatomic molecules are often studied using the second-order vibrational perturbation theory (VPT2). For efficient higher-order analysis, an approach based on the canonical Van Vleck perturbation theory (CVPT), the Watson Hamiltonian and operators of creation and annihilation of vibrational quanta is employed. This method allows analysis of the convergence of perturbation theory and solves a number of theoretical problems of VPT2, e.g., yields anharmonic constants y(ijk), z(ijkl), and allows the reliable evaluation of vibrational IR and Raman anharmonic intensities in the presence of resonances. Darling-Dennison and higher-order resonance coupling coefficients can be reliably evaluated as well. The method is illustrated on classic molecules: water and formaldehyde. A number of theoretical conclusions results, including the necessity of using sextic force field in the fourth order (CVPT4) and the nearly vanishing CVPT4 contributions for bending and wagging modes. The coefficients of perturbative Dunham-type Hamiltonians in high-orders of CVPT are found to conform to the rules of equality at different orders as earlier proven analytically for diatomic molecules. The method can serve as a good substitution of the more traditional VPT2.

Variational study on the vibrational level structure and IVR behavior of highly vibrationally excited S0 formaldehyde

We perform large scale converged variational vibrational calculations on S(0) formaldehyde up to very high excess vibrational energies (E(v)), E(v)∼17,000cm(-1), using our vibrational method, consisting of a specific search/selection/Lanczos iteration procedure. Using the same method we investigate the vibrational level structure and intramolecular vibrational redistribution (IVR) characteristics for various vibrational levels in this energy range in order to assess the onset of IVR.