Vibrational Averaging Effects Research Articles

Accurate Vibrational and Ro-Vibrational Contributions to the Properties of Large Molecules by a New Engine Employing Curvilinear Internal Coordinates and Vibrational Perturbation Theory to Second Order.

The unbiased comparison between theory and experiment requires approaches more sophisticated than the basic harmonic-oscillator rigid-rotor model, for taking into account vibrational averaging effects and ro-vibrational couplings in molecules of increasing size. Second-order vibrational perturbation theory based on curvilinear internal coordinates (ICs) offers a remarkable compromise between accuracy and computational cost, thanks to the reduction of mode-mode couplings with respect to their counterparts based on Cartesian coordinates. Therefore, we have developed, implemented, and validated a general engine employing ICs, which allows the accurate evaluation of vibrational averages and ro-vibrational couplings for molecules containing up to about 50 atoms beyond the harmonic approximation. After validation of the new tool for relatively small molecules, the effectiveness of ICs has been demonstrated for some flexible and/or quite large molecular bricks of life.

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.

Accurate Geometries of Large Molecules by Integration of the Pisa Composite Scheme and the Templating Synthon Approach.

An effective yet reliable computational workflow is proposed, which permits the computation of accurate geometrical structures for large flexible molecules at an affordable cost thanks to the integration of machine learning tools and DFT models together with reduced scaling computations of vibrational averaging effects. After validation of the different components of the overall strategy, a panel of molecules of biological interest have been analyzed. The results confirm that very accurate geometrical parameters can be obtained at reasonable cost for molecules including up to about 50 atoms, which are the largest ones for which comparison with high-resolution rotational spectra is possible. Since the whole computational workflow can be followed employing standard electronic structure codes, accurate results for large-sized molecules can be obtained at DFT cost also by nonspecialists.

Hunting for Complex Organic Molecules in the Interstellar Medium: The Role of Accurate Low-Cost Theoretical Geometries and Rotational Constants.

A new approach to computation at affordable cost of accurate geometrical structures and rotational constants for medium-sized molecules in the gas phase is further improved and applied to a large panel of interstellar complex organic molecules. The most distinctive feature of the new model is the effective inclusion of core-valence correlation and vibrational averaging effects in the framework of density functional theory (DFT). In particular, a double-hybrid functional in conjunction with a quadruple-ζ valence/triple-ζ polarization basis set is employed for geometry optimizations, whereas a cheaper hybrid functional in conjunction with a split-valence basis set is used for the evaluation of vibrational corrections. A thorough benchmark based on a wide range of prototypical systems shows that the new scheme approaches the accuracy of state-of-the-art wave function methods with the computational cost of the standard methods (DFT or MP2) routinely employed in the interpretation of microwave spectra. Since the whole computational workflow involves the postprocessing of the output of standard electronic structure codes by a new freely available web utility, the way is paved for the accurate yet not prohibitively expensive study of medium- to large-sized molecules also by nonspecialists.

Interplay of stereo-electronic, vibronic and environmental effects in tuning the chiroptical properties of an Ir(III) cyclometalated N-heterocyclic carbene

Chiroptical spectra are among the most suitable techniques for investigating the ground and excited electronic states of chiral systems, but their interpretation is not straightforward and strongly benefits from quantum chemical simulations, provided that the employed computational model is sufficiently accurate and deals properly with stereo-electronic, vibrational averaging and environmental effects. Since the synergy among all these effects is only rarely accounted for, especially for large and flexible organometallic systems, the main aim of this contribution is to illustrate the latest developments of computational approaches rooted into the density functional theory for describing stereo-electronic effects and complemented by effective techniques to deal with vibrational modulation effects and solvatochromic shifts. In this connection, chiral iridium complexes offer an especially suitable case study in view of their bright phosphorescence, which is particularly significant for building effective light emitting diodes (OLEDs) and biomarkers and can be finely tuned by the nature of the metal ligands. For instance, a recently synthesized family of cycloiridiated complexes, KC and KD, bearing a pentahelicenic N-heterocyclic carbene (KB), has shown an enhanced long-lasting, bright phosphorescence. Deeper insights into the still unclear nature and origin of the enhancement could be gained by the interpretation of the chiroptical spectra, which is quite challenging in view of the presence of two sources of chirality, the chiral center on Ir and the chiral axis related to the helicene ligand, in addition to the relativistic effects related to the presence of the Ir center. At the same time, the large dimensions of KC and KD hamper the use of the most sophisticated (but prohibitively expensive) computational models, so that more approximate approaches must be validated on a suitable model compound. To this end, after optimizing the computational scheme on a model system devoid of the helicene moiety (KA), we have performed a comprehensive investigation of the KC and KD spectra, whose interpretation is further aided by novel graphical tools. The discussion and analysis of the results will not be focused on the theoretical background, but, rather, on practical details (specific functional, basis set, vibronic model, solvent regime) with the aim of providing general guidelines for the use of last-generation computational spectroscopy tools also by non-specialists.

CPL Spectra of Camphor Derivatives in Solution by an Integrated QM/MD Approach.

We extend a recently proposed computational strategy for the simulation of absorption spectra of semi-rigid molecular systems in condensed phases to the emission spectra of flexible chromophores. As a case study, we have chosen the CPL spectrum of camphor in methanol solution, which shows a well-defined bisignate shape. The first step of our approach is the quantum mechanical computation of reference spectra including vibrational averaging effects and taking bulk solvent effects into account by means of the polarizable continuum model. In the present case, the large amplitude inversion mode is explicitly treated by a numerical approach, whereas the other small-amplitude vibrational modes are taken into account within the harmonic approximation. Next, the snapshots of classical molecular dynamics computations are clusterized and one representative configuration from each cluster is used to compute a reference spectrum. In the present case, different clusters correspond to the two stable conformers of camphor in the S1 excited electronic state and, for each of them, to different numbers of strong solute-solvent hydrogen bonds. Finally, local fluctuation effects within each cluster are taken into account by means of the perturbed matrix model. The overall procedure leads to good agreement with experiment for absorption and emission spectra together with their chiral counterparts, thus allowing to analyze the role of different effects (stereo-electronic, vibrational, environmental) in tuning the overall experimental spectra.

The ONIOM/PMM Model for Effective Yet Accurate Simulation of Optical and Chiroptical Spectra in Solution: Camphorquinone in Methanol as a Case Study.

This paper deals with the development and first validation of a composite approach for the simulation of chiroptical spectra in solution aimed to strongly reduce the number of full QM computations without any significant accuracy loss. The approach starts from the quantum mechanical computation of reference spectra including vibrational averaging effects and taking average solvent effects into account by means of the polarizable continuum model. Next, the snapshots of classical molecular dynamics computations are clusterized and one reference configuration from each cluster is used to compute a reference spectrum. Local fluctuation effects within each cluster are then taken into account by means of the perturbed matrix model. The performance of the proposed approach is tested on the challenging case of the optical and chiroptical spectra of camphorquinone in methanol solution. Although further validations are surely needed, the results of this first study are quite promising also taking into account that agreement with experimental data is reached by just a couple of full quantum mechanical geometry optimizations and frequency computations.

Gas-phase structure of nickel dichloride. An electron-diffraction investigation augmented by ab initio and DFT calculations

The structure of nickel dichloride in the vapor at ca. 1050 K has been studied experimentally by electron diffraction and theoretically by ab initio and DFT molecular orbital calculations. The molecule is found to have a linear equilibrium structure, and from all levels of theory it is predicted to have a triplet ground state. Theory also predicts the existence of a singlet state about 30 kcal/mol less stable than the triplet where the molecule is triangular with a bond angle in the range of about 110°̊ – 117°̊. Two models of the experimental structure were investigated. Model A consisted of 11 “pseudoconformers” distributed at even intervals over the range 180°̊ ≤ ∠(ClNiCl) ≤ 130°̊ for which the main refined parameter was the distribution (relative weighting) of the pseudoconformers based on an assumed bending potential of the form V(Δθ) = V2(Δθ)2, where Δθ is the difference of the ClNiCl angle from linearity. In this model the pseudoconformational weighting led to averages for the bond- and nonbond distances. Model B was simpler, defined only by values of the bond distance and bond angle. The structural results for Model A are 〈rg(NiCl)〉 = 2.067(6) Å, 〈rg(Cl⋅⋅Cl)〉 = 4.037(28) Å, 〈lg(NiCl)〉 = 0.081(6) Å, and 〈lg(Cl⋅⋅Cl)〉 = 0.136(52) Å; in this case the rms amplitudes are for the frame (no contributions from the bending mode). For Model B they are 〈rg(NiCl)〉 = 2.059(5) Å, 〈rg(Cl⋅⋅Cl)〉 = 4.041(89) Å, 〈lg(NiCl)〉 = 0.081(6) Å, and 〈lg(Cl⋅⋅Cl)〉 = 0.159(60) Å; the uncertainties are estimated 2σ. The theoretical prediction of a linear ground state for nickel dichloride is confirmed by experimental values of about 156̊° for the bond angle ∠(ClNiCl) in each model, which differs from the theoretical 180̊° due to the effects of vibrational averaging that arise nearly entirely from the bending mode. The structure is discussed.

Configuration determination by residual dipolar couplings: accessing the full conformational space by molecular dynamics with tensorial constraints.

Residual dipolar couplings (RDCs) and other residual anisotropic NMR parameters provide valuable structural information of high quality and quantity, bringing detailed structural models of flexible molecules in solution in reach. The corresponding data interpretation so far is directly or indirectly based on the concept of a molecular alignment tensor, which, however, is ill-defined for flexible molecules. The concept is typically applied to a single or a small set of lowest energy structures, ignoring the effect of vibrational averaging. Here, we introduce an entirely different approach based on time averaged molecular dynamics with dipolar couplings as tensorial orientational restraints that can be used to solve structural problems in molecules of any size without the need of introducing an explicit molecular alignment tensor into the computation. RDC restraints are represented by their full 3D interaction tensor in the laboratory frame, for which pseudo forces are calculated using a secular dipolar Hamiltonian as the target. The resulting rotational averaging of each individual tensorial restraint leads to structural ensembles that best fulfil the experimental data. Using one-bond RDCs, the approach has been implemented in the force field procedures of the program COSMOS and extensively tested. A concise theoretical introduction, including the special treatment of force fields for stable and fast MD simulations, as well as applications regarding configurational analyses of small to medium-sized organic molecules with different degrees of flexibility, is given. The observed results are discussed in detail.

Tuning of NMR and EPR parameters by vibrational averaging and environmental effects: an integrated computational approach

Reliable computations of NMR and EPR spectroscopic parameters for flexible molecules in condensed phases require a proper account of stereo-electronic, dynamic, and environmental effects. In the framework of density functional theory methods, these effects can be introduced by second-order vibrational perturbation theory and polarisable continuum models. Two test studies illustrate potentialities of the integrated computational approach we have developed in the last years for these purposes. Further extensions to mixed discrete-continuum models and stochastic approaches for long-time dynamical effects are also sketched.

High resolution spectroscopy of several rovibronically excited bands of 5-cyanoindole – The effect of vibrational averaging

The rovibronic spectra of two bands of 5-cyanoindole at 348 and 884cm−1 have been measured and analyzed using a rigid rotor Hamiltonian. A vibrational assignment could be given on the basis of an anharmonic analysis of the vibrational spectrum of 5-cyanoindole making use of the information of the vibrationally averaged rotational constants. Strong vibronic mixing to a higher lying electronically excited state has been found.

Geometrical structure of benzene and naphthalene: Ultrahigh-resolution laser spectroscopy and ab initio calculation

Geometrical structures of the isolated benzene and naphthalene molecules have been accurately determined by using ultrahigh-resolution laser spectroscopy and ab initio calculation in a complementary manner. The benzene molecule has been identified to be planar and hexagonal (D(6h)) and the structure has been determined with accuracies of 2 × 10(-14) m (0.2 mÅ; 1 Å = 1 × 10(-10) m) for the C-C bond length and 1.0 × 10(-13) m (1.0 mÅ) for the C-H bond length. The naphthalene molecule has been identified to be symmetric with respect to three coordinate axes (D(2h)) and the structure has been determined with comparable accuracies. We discuss the effect of vibrational averaging that is a consequence of zero-point motions on the uncertainty in determining the bond lengths.

Interplay of stereo-electronic, environmental, and dynamical effects in determining the EPR parameters of aromatic spin-probes: INDCO as a test case

An integrated computational strategy for the evaluation of reliable structures and magnetic properties of spin probes and spin labels has been extended to aromatic species. From an electronic point of view, delocalization of the unpaired electron density over aromatic moieties reduces significantly the computed nitrogen isotropic hyperfine coupling constant (A(N)) with respect to values characteristic of aliphatic nitroxides. Solvent effects in not too high polarity media are quite small, but not negligible. At this stage computed A(N) are lower than their experimental counterparts by more than 1 G. Inclusion of vibrational averaging effects by molecular dynamics simulations with a new reliable force field restores full agreement with experiment pointing out the limits of static approaches irrespective of the sophistication of the electronic quantum mechanical method. The generality and computational effectiveness of the proposed integrated approach paves the route toward a reliable analysis of the interplay of stereo-electronic, environmental, and dynamical effects in tuning the properties of large flexible magnetic systems of biological and technological interest.

Accurate Determination of the Structure of Cyclooctatetraene by Femtosecond Rotational Coherence Spectroscopy and ab Initio Calculations

We combine femtosecond time-resolved rotational coherence spectroscopy with high-level ab initio theory to obtain accurate structural information for the nonpolar antiaromatic molecule 1,3,5,7-cyclooctatetraene (C8H8, COT) and its perdeuterated isotopomer COT-d8 (C8D8). We measure the rotational B0 and centrifugal distortion constants D(J), D(JK) of the v = 0 states of COT and COT-d8 to high accuracy, e.g. B0 (COT) = 2710.329(56) MHz, as well as B(v) for the v = 1 states nu6, nu11, nu17, nu22, and nu41/nu42 of COT. The experimental rotational constants are compared to those obtained from calculations at the coupled-cluster with single, double, and perturbative triples [CCSD(T)] level. The latter also take into account vibrational averaging effects of the ground and vibrationally excited states. Combining the experimental and calculated rotational constants with the calculated equilibrium bond lengths and angles allows us to determine accurate equilibrium structure parameters, e.g., r(e) (C-C) = 147.0 +/- 0.05 pm, r(e) (C=C) = 133.7 +/- 0.1 pm, and r(e) (C-H) = 107.9 +/- 0.1 pm. The equilibrium C-C and C=C bond lengths of COT are compared to those of 1,3-butadiene. The expected effect of decreased pi-electron delocalization due to the twisting of adjacent C=C double bonds in COT relative to butadiene is observed for the C-C bonds but not for the C=C bonds.

Structure and ESR features of a radiation-induced radical in α-glycine crystals

Extensive DFT calculations are reported for the carbon centered glycine radical embedded in α-glycine crystal. Combined use of periodic and embedded cluster models leads to accurate structural and magnetic properties. Environmental effects are not negligible for all properties, whereas vibrational averaging effects connected to inversion at the radical center are significant only for C α and H α atoms. The good agreement between computed and experimental ESR parameters confirms the nature of one of the radicals obtained upon irradiation of α-glycine crystals and the reliability of DFT computations properly including both vibrational averaging and environmental effects.

Effects of Zero-Point and Thermal Vibrational Averaging on Computed NMR Properties of a Model Compound for Purine Nucleosides.

A method for the incorporation of thermal averaging into the calculation of nuclear magnetic resonance properties is given. These properties are computed using density functional theory, anharmonic first-order perturbation corrections to a normal-mode analysis, and standard statistical mechanical averaging. The method is applied to the calculation of chemical shieldings and spin-spin coupling constants (J couplings) of 1'-imidazolyl-2'-deoxy-β-ribofuranose (IDR), a model compound for purine nucleosides, at the B3LYP/6-311++G(2d,p) level of theory. Thermal averaging causes substantial changes in the values of computed parameters. The calculated harmonic normal modes of IDR are also investigated; we find reasonable agreement with published results from vibrational spectroscopy on DNA fragments. Finally, the calculated magnetic and structural data regarding the reported hydrogen bond between H8 on the imidazole ring and O5' on the sugar ring are investigated; we find that such data do not strongly support the formation of a hydrogen bond between these two atoms.

Geometry and Cooperativity Effects in Adenosine−Carboxylic Acid Complexes

NMR experiments and theoretical investigations were performed on hydrogen bonded complexes of specifically 1- and 7-15N-labeled adenine nucleosides with carboxylic acids. By employing a freonic solvent of CDClF2 and CDF3, NMR spectra were acquired at temperatures as low as 123 K, where the regime of slow hydrogen bond exchange is reached and several higher-order complexes were found to coexist in solution. Unlike acetic acid, chloroacetic acid forms Watson-Crick complexes with the proton largely displaced from oxygen to the nitrogen acceptor in an ion pairing structure. Calculated geometries and chemical shifts of the proton in the hydrogen bridge favorably agree with experimentally determined values if vibrational averaging and solvent effects are taken into account. The results indicate that binding a second acidic ligand at the adenine Hoogsteen site in a ternary complex weakens the hydrogen bond to the Watson-Crick bound carboxylic acid. However, substituting a second adenine nucleobase for a carboxylic acid in the trimolecular complex leads to cooperative binding at Watson-Crick and Hoogsteen faces of adenosine.

Determining the Geometry of Hydrogen Bonds in Solids with Picometer Accuracy by Quantum‐Chemical Calculations and NMR Spectroscopy

The structure of multiply hydrogen-bonded systems is determined with picometer accuracy by a combined solid-state NMR and quantum-chemical approach. On the experimental side, advanced 1H-15N dipolar recoupling NMR techniques are capable of providing proton-nitrogen distances of up to about 250 pm with an accuracy level of +/-1 pm for short distances (i.e., around 100 pm) and +/-5 pm for longer ones (i.e., 180 to 250 pm). The experiments were performed under fast magic-angle spinning, which ensures sufficient dipolar decoupling and spectral resolution of the 1H resonance lines. On the quantum-chemical side, the structures of the hydrogen-bonded systems were computationally optimised, yielding complete sets of nitrogen-proton and proton-proton distances, which are essential for correctly interpreting the experimental NMR data. In this way, nitrogen-proton distances were determined with picometer accuracy, so that vibrational averaging effects on dipole-dipole couplings need to be considered. The obtained structures were finally confirmed by the complete agreement of computed and experimental 'H and '5N chemical shifts. This demonstrates that solid-state NMR and quantum-chemical methods ideally complement each other and, in a combined manner, represent a powerful approach for reliable, high-precision structure determination whenever scattering techniques are inapplicable.

Complete structural and magnetic characterization of biological radicals in solution by an integrated quantum mechanical approach: glycyl radical as a case study.

An integrated quantum mechanical approach for the structural and magnetic characterization of flexible free radicals in solution has been applied to a model of the glycyl radical engaged in peptidic chains. The hyperfine couplings computed using hybrid density functionals and purposely tailored basis sets are in good agreement with experiment when vibrational averaging effects from low frequency motions and solvent effects (both direct H bonding and bulk) are taken into the proper account. The g tensor shows a smaller dependence on the specific form of the density functional, the extension of the basis set over a standard double-zeta+polarization level, vibrational averaging, and bulk solvent effects. However, hydrogen bridges with solvent molecules belonging to the first solvation shell play a significant role. Together with their intrinsic interest, our results show that a comprehensive and reliable computational approach is becoming available for the complete characterization of open-shell systems of biological interest in their natural environment.

B3LYP Investigation of HPO2, trans-HOPO, cis-HOPO, and Their Radical Anions

The B3LYP density functional method was used to determine the optimized geometries, harmonic vibrational frequencies, and anharmonic vibrational frequencies of HPO2 and the trans- and cis-isomers of HOPO. Additionally, the isotropic hyperfine splitting (hfs) constants and the vibrationally averaged hfs were determined for the radical anions of each species. cis-HOPO is predicted to be the lowest energy isomer of the neutral systems, with trans-HOPO lying 3.0 kcal mol-1 higher and HPO2 lying 11.9 kcal mol-1 higher than that. The pyramidal HPO2- radical is the lowest energy anion, and the cis- and trans-isomers lie ∼5 and ∼8 kcal mol-1, respectively, above HPO2-. Higher lying σ radicals of HOPO- resemble PO2- and free H. The B3LYP/6-311+G(2d,2p) values of Aiso(1H) = 1211.98 MHz (trans) and 1199.90 MHz (cis) confirm that the 2A‘ HOPO anion dissociates to PO2- + H. H2PO was also investigated because of its similarity to HPO2-. The computed hfs constant for 1H (109.64 MHz) is fortuitously in excellent agreement with the experimental value, while that for 31P (950.29 MHz) differs significantly from experiment, and this deviation is attributed to the lack of a basis set for P that provides a good description of the P 1s core. Vibrational averaging affects the computed hfs in H2PO by 5−6 MHz for Aiso(31P) but only by ∼2 MHz in HPO2-. The effect of vibrational averaging is small (1−2 MHz) for Aiso(1H) but more significant (8−20 MHz) in HPO2-. Aiso(17O) is relatively unaffected by vibrational averaging in both H2PO and HPO2-. Solvent effects were evaluated using the conductor-like polarizable continuum model (CPCM), and these hfs values for HPO2- [Aiso(31P) = 1123.42 MHz, Aiso(1H) = 243.51 MHz] and H2PO [Aiso(31P) = 1003.54 MHz, Aiso(1H) = 109.27 MHz] are in closer agreement with experiment. Convergence of the Fermi contact terms for 31P computed using B3LYP with the aug-cc-pCVXZ and aug-cc-pV(X+d)Z families of basis sets is erratic and slow. On the other hand, with the same basis sets, CCSD(T) produces hfs constants for the P atom that are in close agreement with experiment.