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

Scaling-up VPT2: A feasible route to include anharmonic correction on large molecules

Vibrational analysis plays a crucial role in the investigation of molecular systems. Though methodologies like second-order vibrational perturbation theory (VPT2) have paved the way to more accurate simulations, the computational cost remains a difficult barrier to overcome when the molecular size increases. Building upon recent advances in the identification of resonances, we propose an approach making anharmonic simulations possible for large-size systems, typically unreachable by standard means. This relies on the fact that, often, only portions of the whole spectra are of actual interest. Therefore, the anharmonic corrections can be included selectively on subsets of normal modes directly related to the regions of interest. Starting from the VPT2 equations, we evaluate rigorously and systematically the impact of the truncated anharmonic treatment onto simulations. The limit and feasibility of the reduced-dimensionality approach are detailed, starting on a smaller model system. The methodology is then challenged on the IR absorption and vibrational circular dichroism spectra of an organometallic complex in three different spectral ranges.

Unbiased Comparison between Theoretical and Experimental Molecular Structures and Properties: Toward an Accurate Reduced-Cost Evaluation of Vibrational Contributions

The tremendous development of hardware and software is constantly increasing the role of quantum chemical (QC) computations in the assignment and interpretation of experimental results. However, an unbiased comparison between theory and experiment requires the proper account of vibrational averaging effects. In particular, high-resolution spectra in the gas phase are now available for molecules containing up to about 50 atoms, which are too large for a brute-force approach with the available QC methods of sufficient accuracy. In the present paper, we introduce hybrid approaches, which allow the accurate evaluation of vibrational averaging effects for molecules of this size beyond the harmonic approximation, with special attention being devoted to rotational constants. After the validation of new tools for relatively small molecules, the β-estradiol hormone and a prototypical molecular motor have been considered to witness the feasibility of accurate computations for large molecules.

Comprehensive quantum chemical analysis of the (ro)vibrational spectrum of thiirane and its deuterated isotopologue

The (ro)vibrational spectra of thiirane, c-C2H4S, and its fully deuterated isotopologue, c-C2D4S, have been studied by means of vibrational configuration interaction theory, VCI, its incremental variant, iVCI, and subsequent variational rovibrational calculations, RVCI, which rely on multidimensional potential energy surfaces of coupled-cluster quality including up to four-mode coupling terms. Accurate geometrical parameters, fundamental vibrational transitions and first overtones, rovibrational spectra and rotational spectroscopic constants have been determined from these calculations and were compared with experimental results whenever available. A number of tentative misassignments in the vibrational spectra could be resolved and most results for the deuterated thiirane are high-level predictions, which may guide experiments to come. Besides this, a new implementation of infrared intensities within the iVCI framework has been tested for the transitions of the title compounds and are compared with results obtained from standard VCI calculations.

An approximation to the vibrational coupled-cluster method for CH-stretching of large molecules: application to naphthalene and anthracene

We propose an approximation to the vibrational coupled-cluster method (VCCM) to describe the CH-stretching region of the vibrational spectrum of large molecules. The vibrational modes of a molecule are divided into two sets: the target set and the bath set. The target set includes the CH stretches and the modes that are strongly coupled with the CH stretches and/or involve strong Fermi resonances with a CH stretch fundamental. The rest of the modes are in the bath set. First, the effective harmonic oscillator (EHO) approximation is invoked for the whole system to obtain the zeroth-order frequencies and modified potentials. The effects of interaction between the bath set and the target sets are included in the modified potential from the EHO calculation. The VCCM equations are constructed with the modified potential from the EHO calculations and for the target set only. The transition energies and intensities are calculated using such a truncated VCCM approximation. The proposed method is applied to calculate the IR spectra of naphthalene and anthracene. The results with three different criteria for selecting the modes in the target set are compared with the experimental IR spectra.

Vibrational resonance analysis of linear molecules using resummation of divergent Rayleigh-Schrödinger perturbation theory series

The large order Rayleigh-Schrödinger perturbation theory (RSPT) was applied for calculating vibrational states of linear molecules. Two molecules (CO₂ and C₂H₂) were used as test cases with using of isomorphic Watson Hamiltonian and quartic force fields. For CO₂ the Sayvetz condition can remove all degeneracies for purely vibrational states and the non-degenerate perturbation theory can be applied. However, an existence of two degenerate modes in C₂H₂ requires using the upgraded degenerate version of RSPT that was employed in this context for the first time. The dominating divergent behavior of such series requires the resummation technique that mimics the multivalued nature of the underlying solutions, and the applied quartic Padé-Hermite approximants (QPHA) provided full solution of the problem. Moreover, some mathematical properties of QPHA proved to be an efficient tool for studying resonance effects through the Katz theorem that controls the singular points of the eigenvalues on the complex plane. In the case of C₂H₂, not only all earlier observed classical resonances were confirmed and quantified, but also subtle interpolyad resonances (K_2/55,K_3/4555), proposed recently by Herman (2011) were described as well. Following the analysis, we found several novel resonances, of which we proposed one independent interpolyad resonance K_2/4444. The complete analysis of such critical points provided the full resonance picture of all studied molecules.

Etalon-Assisted Determination of the Complex Index of Refraction of a Solution for the Study of Strong Cavity-Vibrational Coupling of PF6- in Acetonitrile

A new method is established using an etalon cavity to assist in the determination of the wavelength-dependent complex index of refraction of a solution throughout the mid-infrared range. The results are used to study the cavity-vibration polaritons of PF₆^- in acetonitrile. Mixed states are formed by placing solution inside a pair of parallel plate mirrors with a wavelength-scale spacing, i.e., within an etalon, such that there are cavity states that are angle-tuned into resonance with the strong P-F vibrations. The dominant ν₃ vibrations of PF₆^- consist of nearly triply degenerate oscillations of the partial-positively charged phosphorous against antisymmetric concerted motions of different sets of fluorine atoms with partial negative charges. These vibrations are dominant even though the solute is 29 times less concentrated than the solvent on a molar basis. The first part of the paper describes the method of determining the complex index of refraction of the solution from a combination of etalon transmission maxima and the attenuated total reflection (ATR) absorption spectrum of the solution. The results are presented as an analytical function including a sum of 37 vibrational contributions. Absolute integrated isolated band intensities were determined to be 463 ± 4, 462 ± 7, and 266 ± 4 km/mol for the three ν₃ PF₆^- vibrations at 841.4, 847.4, and 854.0 cm^-1, respectively, which sum to 1191 ± 9 km/mol for the ν₃ band. Then, the results are used to simulate the measured etalon transmission using the transfer matrix (TM) method with and without the ν₃ target vibrations. The etalon transmission simulations reconstruct the position of cavity modes in the absence of target vibrations. They provide input data for the testing of simple quantum mechanical models for the interaction of vibrations with cavity modes and the interactions of vibrations with other vibrations within the molecule and between solute and solvent. The model shows that the nearly degenerate ν₃ vibrations interact with each other with a vibration-vibration coupling of 33 ± 5 cm^-1. This is comparable to the cavity-vibration coupling of 30.4 ± 2.9 cm^-1 of the two strongest vibrations of PF₆^-.

Matrix-isolation and cryosolution-VCD spectra of α-pinene as benchmark for anharmonic vibrational spectra calculations

The inclusion of anharmonicity in vibrational spectral analyis remains associated to small molecular systems with up to a dozen of atoms, with half a dozen of non-hydrogen atoms, typically thesize of propylene oxide. One may see two reasons for this: first of all, larger systems are often thought to be computationally too demanding (high computational costs) for a full anharmonic vibrational analysis. Second, the identification of resonances and their correction is often considered something only expert theoreticians could address because of the lack of unequivocal criteria. In this contribution, we illustrate that resonances can indeed become a complex problem, which can be handled almost transparently thanks to recent advances in vibrational perturbation theory (VPT2). The study also emphasizes the importance and the central role played by experiment in benchmarking novel theoretical approaches. In fact, we herein provide the currently highest resolution VCD spectra available for α- and β-pinene obtained under matrix-isolation conditions and in liquid Xenon as solvent. They are interpreted by VPT2 with novel tests for the identification of resonances. Hence, the study demonstrates the mutual stimulation of advances in both experimental techniques and computational models.

Anharmonic Vibrational Raman Optical Activity of Methyloxirane: Theory and Experiment Pushed to the Limits

Combining Raman scattering and Raman optical activity (ROA) with computer simulations reveals fine structural and physicochemical properties of chiral molecules. Traditionally, the region of interest comprised fundamental transitions within 200-1800 cm^-1. Only recently, nonfundamental bands could be observed as well. However, theoretical tools able to match the observed spectral features and thus assist their assignment are rather scarce. In this work, we present an accurate and simple protocol based on a three-quanta anharmonic perturbative approach that is fully fit to interpret the observed signals of methyloxirane within 150-4500 cm^-1. An unprecedented agreement even for the low-intensity combination and overtone transitions has been achieved, showing that anharmonic Raman and ROA spectroscopies can be valuable tools to understand vibrations of chiral molecules or to calibrate computational models.

Comparison of body definitions for incremental vibrational configuration interaction theory (iVCI)

Within incremental vibrational configuration interaction theory (iVCI), the vibrational state energy is determined by means of a many-body expansion, i.e., it is a sum of terms of increasing order, which allow for an embarrassingly parallel evaluation. The convergence of this expansion depends strongly on the definition of the underlying bodies, which essentially decompose the correlation space into fragments. The different definitions considered here comprise mode-based bodies, excitation level-based bodies, and energy-based bodies. An analysis of the convergence behavior revealed that accounting for resonances within these definitions is mandatory and leads to a substantial improvement of the convergence, that is, the expansions can be truncated at lower orders. Benchmark calculations and systematic comparisons of the different body definitions for a small set of molecules, i.e., ketene, ethene, and diborane, have been conducted to study the overall performance of these iVCI implementations with respect to accuracy and central processing unit time.

Fundamental studies of vibrational resonance phenomena by multivalued resummation of the divergent Rayleigh-Schrödinger perturbation theory series: deciphering polyad structures of three H216O isotopologues

A fundamental quantitative study of the vibrational resonances of three H₂¹⁶O H,D-isotopologues with a quartic Watson Hamiltonian was carried out using the resummation of the high-order (∼λⁿ, n ≤ 203) divergent Rayleigh-Schrödinger perturbation theory (RSPT) series by quartic Padé-Hermite multivalued diagonal approximants PH^[m,m,m,m,m], m ≤ 40. The resonance condition between a pair of states is formulated as the existence of a common complex energy solution branch point inside the unit circle: |E(λ_j)|,  Re(λ_j)² + Im(λ_j)² ≤ 1. For the matrix formulation of the vibrational problem (VCI), the existence of common branch points is governed by the Katz theorem and they can be found as roots of discriminant polynomials. The main branches of the Padé-Hermite approximants typically reproduce VCI energies with high accuracy while alternative branches often fit nearby resonant states. The resummation of the RSPT series for H₂O and D₂O (up to the tenth polyad) revealed not only Fermi and Darling-Dennison resonances, but also unusual (0,2,-5) and (5,-2,0) resonance effects matching the (5,2,5) polyad structure, while the (3,2,1) structure was rigorously confirmed for HDO. It is demonstrated that the (5,2,5) polyad structure ensures good organization of high-energy resonating states and breaks down the classic (2,1,2) structure. The advocated methodology of quantitative description of resonance phenomena and revealing polyad structures is suitable for larger molecules and can be adapted to linear molecules and symmetric tops. Its application ensures rigorous classification of vibrational states and can be used in quantitative vibration-rotation spectroscopy.

General Perturb-Then-Diagonalize Model for the Vibrational Frequencies and Intensities of Molecules Belonging to Abelian and Non-Abelian Symmetry Groups

In this paper, we show that the standard second-order vibrational perturbation theory (VPT2) for Abelian groups can be used also for non-Abelian groups without employing specific equations for two- or threefold degenerate vibrations but rather handling in the proper way all the degeneracy issues and deriving the peculiar spectroscopic signatures of non-Abelian groups (e.g., l -doubling) by a posteriori transformations of the eigenfunctions. Comparison with the results of previous conventional implementations shows a perfect agreement for the vibrational energies of linear and symmetric tops, thus paving the route to the transparent extension of the equations already available for asymmetric tops to the energies of spherical tops and the infrared and Raman intensities of molecules belonging to non-Abelian symmetry groups. The whole procedure has been implemented in our general engine for vibro-rotational computations beyond the rigid rotor/harmonic oscillator model and has been validated on a number of test cases.

Anharmonic vibrational analysis of s-trans and s-cis conformers of acryloyl fluoride using numerical-analytic Van Vleck operator perturbation theory

A new gas-phase infrared (IR) spectrum of acryloyl fluoride (ACRF, CH₂CHCFO) with a resolution of 0.1cm^-1 in the range 4000-450cm^-1 was measured. Theoretical ab initio molecular structures, full quartic potential energy surfaces (PES), and cubic surfaces of dipole moments and polarizability tensor components (electro-optical properties, EOP) of the s-trans and s-cis conformers of the ACRF were calculated by the second-order Møller-Plesset electronic perturbation theory with a correlation consistent Dunning triple-ζ basis set. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory was employed for predicting anharmonic IR and Raman scattering (RS) spectra of ACRF. To improve the anharmonic predictions, harmonic frequencies were replaced by their counterparts evaluated with the higher-level CCSD(T)/cc-pVTZ model, to form a "hybrid" PES. The original operator representation of the Hamiltonian is analytically reduced to a quasi-diagonal form, integrated in the harmonic oscillator basis and diagonalized to account for strong resonance couplings. Double canonical transformations of EOP expansions enabled prediction of integral intensities of both fundamental and multi-quanta transitions in IR/RS spectra. Enhanced band shape analysis reinforced the assignments. A thorough interpretation of the new IR experimental spectra and existing matrix-isolation literature data for the mixture of two conformers of ACRF was accomplished, and a number of assignments clarified.

Accurate Vibrational-Rotational Parameters and Infrared Intensities of 1-Bromo-1-fluoroethene: A Joint Experimental Analysis and Ab Initio Study

The medium-resolution gas-phase infrared (IR) spectra of 1-bromo-1-fluoroethene (BrFC═CH₂, 1,1-C₂H₂BrF) were investigated in the range 300-6500 cm^-1, and the vibrational analysis led to the assignment of all fundamentals as well as many overtone and combination bands up to three quanta, thus giving an accurate description of its vibrational structure. Integrated band intensity data were determined with high precision from the measurements of their corresponding absorption cross sections. The vibrational analysis was supported by high-level ab initio investigations. CCSD(T) computations accounting for extrapolation to the complete basis set and core correlation effects were employed to accurately determine the molecular structure and harmonic force field. The latter was then coupled to B2PLYP and MP2 computations in order to account for mechanical and electrical anharmonicities. Second-order perturbative vibrational theory was then applied to the thus obtained hybrid force fields to support the experimental assignment of the IR spectra.

The Virtual Multifrequency Spectrometer: a new paradigm for spectroscopy

On going developments of hardware and software are changing computational spectroscopy from a strongly specialized research area to a general tool in the inventory of most researchers. Increased interactions between experimentally-oriented users and theoretically-oriented developers of new methods and models would result in more robust, flexible and reliable tools and studies for the systems of increasing complexity, which are of current scientific and technological interest. This is the philosophy behind this review, which presents the development of a so-called virtual multi-frequency spectrometer (VMS) including state-of-the-art approaches in a user-friendly frame. The current status of the VMS tool will be illustrated by a number of case studies with special reference to infrared and UV-vis regions of the electro-magnetic spectrum including also chiral spectroscopies. Only the basic theoretical background will be provided avoiding explicit equations as far as possible, and pointing out the most recent advancements beyond the standard rigid-rotor harmonic-oscillator model coupled to vertical electronic excitation energies.

Anharmonic Effects on Vibrational Spectra Intensities: Infrared, Raman, Vibrational Circular Dichroism, and Raman Optical Activity

The aim of this paper is 2-fold. First, we want to report the extension of our virtual multifrequency spectrometer (VMS) to anharmonic intensities for Raman optical activity (ROA) with the full inclusion of first- and second-order resonances for both frequencies and intensities in the framework of the generalized second-order vibrational perturbation theory (GVPT2) for all kinds of vibrational spectroscopies. Then, from a more general point of view, we want to present and validate the performance of VMS for the parallel analysis of different vibrational spectra for medium-sized molecules (IR, Raman, VCD, ROA) including both mechanical and electric/magnetic anharmonicity. For the well-known methyloxirane benchmark, careful selection of density functional, basis set, and resonance thresholds permitted us to reach qualitative and quantitative agreement between experimental and computed band positions and shapes. Next, the whole series of halogenated azetidinones is analyzed, showing that it is now possible to interpret different spectra in terms of mass, electronegativity, polarizability, and hindrance variation between closely related substituents, chiral spectroscopies being particular effective in this connection.

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.

CC/DFT Route toward Accurate Structures and Spectroscopic Features for Observed and Elusive Conformers of Flexible Molecules: Pyruvic Acid as a Case Study

The structures and relative stabilities as well as the rotational and vibrational spectra of the three low-energy conformers of pyruvic acid (PA) have been characterized using a state-of-the-art quantum-mechanical approach designed for flexible molecules. By making use of the available experimental rotational constants for several isotopologues of the most stable PA conformer, Tc-PA, the semiexperimental equilibrium structure has been derived. The latter provides a reference for the pure theoretical determination of the equilibrium geometries for all conformers, thus confirming for these structures an accuracy of 0.001 Å and 0.1 deg for bond lengths and angles, respectively. Highly accurate relative energies of all conformers (Tc-, Tt-, and Ct-PA) and of the transition states connecting them are provided along with the thermodynamic properties at low and high temperatures, thus leading to conformational enthalpies accurate to 1 kJ mol(-1). Concerning microwave spectroscopy, rotational constants accurate to about 20 MHz are provided for the Tt- and Ct-PA conformers, together with the computed centrifugal-distortion constants and dipole moments required to simulate their rotational spectra. For Ct-PA, vibrational frequencies in the mid-infrared region accurate to 10 cm(-1) are reported along with theoretical estimates for the transitions in the near-infrared range, and the corresponding infrared spectrum including fundamental transitions, overtones, and combination bands has been simulated. In addition to the new data described above, theoretical results for the Tc- and Tt-PA conformers are compared with all available experimental data to further confirm the accuracy of the hybrid coupled-cluster/density functional theory (CC/DFT) protocol applied in the present study. Finally, we discuss in detail the accuracy of computational models fully based on double-hybrid DFT functionals (mainly at the B2PLYP/aug-cc-pVTZ level) that avoid the use of very expensive CC calculations.

Generalized Vibrational Perturbation Theory for Rotovibrational Energies of Linear, Symmetric and Asymmetric Tops: Theory, Approximations, and Automated Approaches to Deal with Medium-to-Large Molecular Systems

Models going beyond the rigid-rotor and the harmonic oscillator levels are mandatory for providing accurate theoretical predictions for several spectroscopic properties. Different strategies have been devised for this purpose. Among them, the treatment by perturbation theory of the molecular Hamiltonian after its expansion in power series of products of vibrational and rotational operators, also referred to as vibrational perturbation theory (VPT), is particularly appealing for its computational efficiency to treat medium-to-large systems. Moreover, generalized (GVPT) strategies combining the use of perturbative and variational formalisms can be adopted to further improve the accuracy of the results, with the first approach used for weakly coupled terms, and the second one to handle tightly coupled ones. In this context, the GVPT formulation for asymmetric, symmetric, and linear tops is revisited and fully generalized to both minima and first-order saddle points of the molecular potential energy surface. The computational strategies and approximations that can be adopted in dealing with GVPT computations are pointed out, with a particular attention devoted to the treatment of symmetry and degeneracies. A number of tests and applications are discussed, to show the possibilities of the developments, as regards both the variety of treatable systems and eligible methods.

Nonempirical anharmonic vibrational perturbation theory applied to biomolecules: free-base porphin

Anharmonic vibrational frequencies and intensities (infrared and Raman) of an isolated free-base porphin molecule are predicted from the quantum mechanical (QM) geometry, the "semi-diagonal" quartic force field, and dipole moment and polarizability surfaces. The second-order vibrational perturbation theory plus the numerical diagonalization of the Hamiltonian matrix containing off-diagonal Fermi and Darling-Dennison resonance couplings (VPT2+WK) was used. The QM calculations were carried out with the Becke-Lee-Yang-Parr composite exchange-correlation functional (B3LYP) and with the 6-31+G(d,p) basis set. The harmonic force field for the equilibrium configuration was transformed into nonredundant local symmetry internal coordinates, and normal coordinates were defined. The semi-diagonal quartic rectilinear normal coordinate potential energy surface (PES), as well as the cubic surfaces of dipole moment (p) and polarizability (α) components, needed for the VPT2+WK calculation, were constructed by a five-point finite differentiation of Hessians (for PES) and of the values and first derivatives of p and α. They were obtained at the point of equilibrium and for 432 displaced configurations. This theoretical approach provides very good agreement between the predicted and experimental frequencies and intensities. However, the favorable result can be partly attributed to error cancellation within the B3LYP/6-31+G(d,p) QM model, as observed in earlier studies. Reassignments of some observed bands are proposed.