Experimental Fundamental Frequencies Research Articles

F12+EOM Quartic Force Fields for Rovibrational Predictions of Electronically Excited States.

Quartic force fields (QFFs) constructed using a sum of ground-state CCSD(T)-F12b energies with EOM-CCSD excitation energies are proposed for computation of spectroscopic properties of electronically excited states. This is dubbed the F12+EOM approach and is shown to provide similar accuracy to previous methodologies at lower computational cost. Using explicitly correlated F12 approaches instead of canonical CCSD(T), as in the corresponding (T)+EOM approach, allows for 70-fold improvement in computational time. The mean percent difference between the two methods for anharmonic vibrational frequencies is only 0.10%. A similar approach is also developed herein which accounts for core correlation and scalar relativistic effects, named F12cCR+EOM. The F12+EOM and F12cCR+EOM approaches both match to within 2.5% mean absolute error of experimental fundamental frequencies. These new methods should help in clarifying astronomical spectra by assigning features to vibronic and vibrational transitions of small astromolecules when such data are not available experimentally.

Analytical harmonic vibrational frequencies with VV10-containing density functionals: Theory, efficient implementation, and benchmark assessments.

VV10 is a powerful nonlocal density functional for long-range correlation that is used to include dispersion effects in many modern density functionals, such as the meta-generalized gradient approximation (mGGA), B97M-V, the hybrid GGA, ωB97X-V, and the hybrid mGGA, ωB97M-V. While energies and analytical gradients for VV10 are already widely available, this study reports the first derivation and efficient implementation of the analytical second derivatives of the VV10 energy. The additional compute cost of the VV10 contributions to analytical frequencies is shown to be small in all but the smallest basis sets for recommended grid sizes. This study also reports the assessment of VV10-containing functionals for predicting harmonic frequencies using the analytical second derivative code. The contribution of VV10 to simulating harmonic frequencies is shown to be small for small molecules but important for systems where weak interactions are important, such as water clusters. In the latter cases, B97M-V, ωB97M-V, and ωB97X-V perform very well. The convergence of frequencies with respect to the grid size and atomic orbital basis set size is studied, and recommendations are reported. Finally, scaling factors to allow comparison of scaled harmonic frequencies with experimental fundamental frequencies and to predict zero-point vibrational energy are presented for some recently developed functionals (including r2SCAN, B97M-V, ωB97X-V, M06-SX, and ωB97M-V).

Model Chemistry Recommendations for Scaled Harmonic Frequency Calculations: A Benchmark Study

Despite the widespread popularity of scaled harmonic frequency calculations to predict experimental fundamental frequencies in chemistry, sparse benchmarking is available to guide users on the appropriate level of theory and basis set choices (model chemistry) or deep understanding of expected errors. An updated assessment of the best approach for scaling to minimize errors is also overdue. Here, we assess the performance of over 600 popular, contemporary, and robust model chemistries in the calculation of scaled harmonic frequencies, evaluating different scaling factor types and their implications in the scaled harmonic frequencies and model chemistry performance. We can summarize our results into three main findings: (1) Using model-chemistry-specific scaling factors optimized for three different frequency regions (low (<1,000 cm-1), mid (1,000-2,000 cm-1), and high (>2,000 cm-1)) results in substantial improvements in the agreement between the scaled harmonic and experimental frequencies compared to other choices. (2) Larger basis sets and more robust levels of theory generally lead to superior performance; however, the particular model chemistry choice matters and poor choices lead to significantly reduced accuracies. (3) Outliers are expected in routine calculations regardless of the model chemistry choice. Our benchmarking results here do not consider the intensity of vibrational transitions; however, we draw upon previous benchmarking results for dipole moments that highlight the importance of diffuse functions (i.e., augmented basis sets) in high-quality intensity predictions. In terms of specific recommendations, overall, the highest accuracy model chemistries are double-hybrid density functional approximations with a non-Pople augmented triple-ζ basis set, which can produce median frequency errors down to 7.6 cm-1 (DSD-PBEP86/def2-TZVPD), which is very close to the error in the harmonic approximation, i.e., the anharmonicity error. Double-ζ basis sets should not be used with double-hybrid functionals as there is no improvement compared to hybrid functional results (unlike for double-hybrid triple-ζ model chemistries). Note that 6-311G* and 6-311+G* basis sets perform like a double-ζ basis set for vibrational frequencies. After scaling, all studied hybrid functionals with non-Pople triple-ζ basis sets will produce median errors of less than 15 cm-1, with the best result of 9.9 cm-1 with B97-1/def2-TZVPD. Appropriate matching of double-ζ basis sets with hybrid functionals can produce high-quality results, but the precise choice of functional and basis set is more important. The B97-1, TPSS0-D3(BJ), or ωB97X-D hybrid density functionals with 6-31G*, pc-1, or pcseg-1 are recommended for fast routine calculations, all delivering median errors of 11-12 cm-1. Note that dispersion corrections are not easily available for B97-1; given its strong performance here, we recommend these be added to major programs in coming updates.

VIBFREQ1295: A New Database for Vibrational Frequency Calculations.

High-throughput approaches for producing approximate vibrational spectral data for molecules of astrochemistry interest rely on harmonic frequency calculations using computational quantum chemistry. However, model chemistry recommendations (i.e., a level of theory and basis set pair) for these calculations are not yet available and, thus, thorough benchmarking against comprehensive benchmark databases is needed. Here, we present a new database for vibrational frequency calculations (VIBFREQ1295) storing 1295 experimental fundamental frequencies and CCSD(T)(F12*)/cc-pVDZ-F12 ab initio harmonic frequencies from 141 molecules. VIBFREQ1295's experimental data was complied through a comprehensive review of contemporary experimental data, while the ab initio data was computed here. The chemical space spanned by the molecules chosen is considered in-depth and is shown to have good representation of common organic functional groups and vibrational modes. Scaling factors are routinely used to approximate the effect of anharmonicity and convert computed harmonic frequencies to predicted fundamental frequencies. With our experimental and high-level ab initio data, we find that a single global uniform scaling factor of 0.9617(3) results in median differences of 15.9(5) cm-1. A far superior performance with a median difference of 7.5(5) cm-1 can be obtained, however, by using separate scaling factors (SFs) for three regions: frequencies less than 1000 cm-1 (SF = 0.987(1)), between 1000 and 2000 cm-1 (SF = 0.9727(6)), and above 2000 cm-1 (SF = 0.9564(4)). This sets a lower bound for the performance that could be reliably obtained using scaling of harmonic frequency calculations to predict experimental fundamental frequencies. VIBFREQ1295's most important purpose is to provide a robust database for benchmarking the performance of any vibrational frequency calculations. VIBFREQ1295 data could also be used to train machine-learning models for the prediction of vibrational spectra and as a reference and data starting point for more detailed spectroscopic modeling of particular molecules. The database can be found as part of the Supporting Information for this paper or in the Harvard DataVerse at https://doi.org/10.7910/DVN/VLVNU7.

Meta‐analysis of uniform scaling factors for harmonic frequency calculations

AbstractVibrational frequency calculations performed under the harmonic approximation are widespread across chemistry. However, it is well‐known that the calculated harmonic frequencies tend to systematically overestimate experimental fundamental frequencies; a limitation commonly overcome with multiplicative scaling factors. In practice, multiplicative scaling factors are derived for each individual model chemistry choice (i.e., a level of theory and basis set pair), where performance is judged by, for example, the root‐mean square error (RMSE) between the predicted scaled and experimental frequencies. However, despite the overwhelming number of scaling factors reported in the literature and model chemistry approximations available, there is little guidance for users on appropriate model chemistry choices for harmonic frequency calculations. Here, we compile and analyze the data for 1495 scaling factors calculated using 141 levels of theory and 109 basis sets. Our meta‐analysis of this data shows that scaling factors and RMSE approach convergence with only hybrid functionals and double‐zeta basis sets, with anharmonicity error already dominating model chemistry errors. Noting inconsistent data and the lack of independent testing, we can nevertheless conclude that a minimum error of 25 cm−1—arising from insufficiently accurate treatment of anharmonicity—is persistent regardless of the model chemistry choice. Based on the data we compiled and cautioning the need for a future systematic benchmarking study, we recommend ωB97X‐D/def2‐TZVP for most applications and B2PLYP/def2‐TZVPD for superior intensity predictions. With a smaller benchmark set, direct comparison prefers ωB97X‐D/6‐31G* to B3LYP/6‐31G*.This article is categorized under: Electronic Structure Theory &gt; Density Functional Theory Theoretical and Physical Chemistry &gt; Spectroscopy

Empirical Formulation for Estimating the Fundamental Frequency of Slender Masonry Structures

ABSTRACTThe fundamental frequency of a structure enables better assessment of its seismic demand for an efficient design and planning of its maintenance and retrofit strategy. The frequency is independent of the type of external loads, however, depends on structural stiffness, mass, damping and boundary conditions. In the case of slender masonry structures such as towers, minarets chimneys, and pagoda temples, it is influenced by mass and stiffness distribution, connection to adjacent structures, material properties, aspect ratio and slenderness ratio. In this present article, the data collected from various literature reviews on the slender masonry structures regarding dynamic, geometrical, and mechanical characteristics have been correlated to identify the major parameters influencing the fundamental frequency of such structures. The database has been used for developing an empirical formulation for predicting the fundamental frequency of such structures. The comparison between the experimental fundamental frequencies and the estimated fundamental frequencies are carried out in order to define reliability and accuracy of these empirical formulae.

Anharmonic vibrational analyses for the 1-silacyclopropenylidene molecule and its three isomers

The global minimum among possible structures of SiC2H2 has been experimentally and theoretically determined to be 1-silacyclopropenylidene (1S). In 1994 Maier and Reisenauer reported the generation of 1-silacyclopropenylidene and its three isomers (2S–4S) by pulsed-flash pyrolysis followed by matrix-spectroscopic identification. Reliable quartic force fields for 1-silacyclopropenylidene and its three isomers are determined employing ab initio coupled-cluster theory with single, double, and perturbative triple excitations [CCSD(T)] and the correlation-consistent core-valence quadruple zeta (cc-pCVQZ) basis set. Second-order vibrational perturbation theory (VPT2) has been utilized to determine equilibrium and zero-point vibration corrected rotational constants, centrifugal distortion constants, and harmonic and anharmonic vibrational frequencies. The distances between the average nuclear positions (r α ) are also determined. The predicted rotational constants, centrifugal distortion constants, and anharmonic frequencies for the four lowest-lying isomers (1S-4S) of SiC2H2, as well as their 13C and deuterated isotopologues, agree well with available experiments. Excluding the CH and CD stretching modes, the mean absolute deviation between theoretical anharmonic and experimental fundamental frequencies for isomer 1-silacyclopropenylidene (1S) is 4.1 cm−1 (5 isotopologues, 25 modes). The corresponding deviation for ethynylsilanediyl (2 S) is 4.9 cm−1 (7 isotopologues, 38 modes) without the SiH and SiD stretching modes, while it is 8.6 cm−1 (5 isotopologues, 22 modes) for silacyclopropyne (4S) without the SiC s-stretching, SiH2 a-stretching and SiD2 wagging modes. By comparing the theoretical harmonic and anharmonic with the experimental fundamental vibrational frequencies for the four isomers (1S-4S), it is demonstrated that the anharmonic effects greatly improve the harmonic results. This theoretically derived spectroscopic data should aid in the experimental detection of the transitions that have yet to be observed, particularly for the vinylidensilanediyl isomer. †Present address: Department of Chemistry, Tsinghua University, Beijing, P. R. China 100084

Analytical approach for the excited-state Hessian in time-dependent density functional theory: Formalism, implementation, and performance

The paper presents the formalism, implementation, and performance of the analytical approach for the excited-state Hessian in the time-dependent density functional theory (TDDFT) that extends our previous work [J. Liu and W. Z. Liang, J. Chem. Phys. 135, 014113 (2011)] on the analytical Hessian in TDDFT within Tamm-Dancoff approximation (TDA) to full TDDFT. In contrast to TDA-TDDFT, an appreciable advantage of full TDDFT is that it maintains the oscillator strength sum rule, and therefore yields more precise results for the oscillator strength and other related physical quantities. For the excited-state harmonic vibrational frequency calculation, however, full TDDFT does not seem to be advantageous since the numerical tests demonstrate that the accuracy of TDDFT with and without TDA are comparable to each other. As a common practice, the computed harmonic vibrational frequencies are scaled by a suitable scale factor to yield good agreement with the experimental fundamental frequencies. Here we apply both the optimized ground-state and excited-state scale factors to scale the calculated excited-state harmonic frequencies and find that the scaling decreases the root-mean-square errors. The optimized scale factors derived from the excited-state calculations are slightly smaller than those from the ground-state calculations.

Experimental and Theoretical Study of the Vibrational Spectra of Free 12-Crown-4

The Raman and IR spectra of free 12-crown-4 (12c4) were measured in the solid, liquid, and solution phases. In the three phases, IR active modes were Raman inactive and IR inactive modes were Raman active. According to the exclusion rule, this is consistent with a conformation with a center of inversion. This indicates that 12c4 in the above-mentioned three phases exists in the C(i) conformation. Harmonic force fields were calculated for five of the lowest energy conformations of 12c4 of C(i), S(4), C(4), C(2), and C(s) symmetries at the corresponding optimized geometries at the B3LYP/6-31+G level. The five force fields were scaled using a six-scale-factor scaling scheme. The scale factors were varied to minimize the difference between the calculated and experimental fundamental frequencies, except that corresponding to the C-H stretching mode that was held fixed. The root-mean-square (rms) deviation of the experimental to the calculated vibrational frequencies was 6.2, 12.0, 10.8, 13.2, and 13.5 cm(-1), for C(i)(), S(4), C(4), C(2), and C(s) conformations, respectively. This supports the above conclusion that 12c4 in the solid, liquid, and considered solution phases exists in the C(i) conformation.

Coupled‐cluster study of the equilibrium geometry and harmonic vibrational frequencies of the methyl radical

AbstractAccurate equilibrium geometries, harmonic vibrational frequencies, and equilibrium rotational constants of the methyl radical were computed using coupled‐cluster theory in combination with the cc‐pVnZ and cc‐pCVnZ basis sets of Dunning. Valence and all electron calculations were carried out. Using all electron correlations and CBS extrapolation, the re value is equal to 1.0758 Å, which is 0.0032 Å shorter than the experimental value of r0 (1.079 Å). The harmonic (ω1, ω2, ω3, and ω4) frequencies calculated using the CBS extrapolation differ from the experimental fundamental frequencies, respectively, by +142, +125, +21, and –82 cm−1. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

핵연료 조사시험용 캡슐 구조물의 좌굴 및 진동특성

The vibration and buckling characteristics of the capsule for fuel irradiation test are studied. The natural frequencies of the capsule in air and under water are obtained by modal testing and finite element (FE) analysis using ANSYS program, and accelerations with flow are measured to estimate the compatibility with the operation requirement of the HANARO reactor. The experimental fundamental frequencies of the capsule in the x and z direction are 8.5 Hz and 8.75 Hz in air, and 7.5 Hz and 7.75 Hz under water, respectively. The maximum amplitude of accelerations under the normal operating condition is measured as 11.0 m/s<TEX>$^2$</TEX> that is within the allowable vibrational limit(18.99 m/s<TEX>$^2$</TEX>) of the reactor structure. Also, the maximum displacement at 100％ flow is calculated as 0.13 mm which is not interference with other nearby structures. FE analysis results show that the natural frequencies are found to be similar to those of the modal testing when three supporting parts are considered as simply supported conditions. From the buckling analysis, when the loading tool is applied, the critical buckling load of the capsule is 233 N.

A new method to evaluate force constants from experimental spectral data

AbstractAn iterative procedure is proposed to facilitate the determination of molecular vibrational force constants from the experimental fundamental frequencies. Proper restrictions are introduced to the force constants based on physical considerations for getting reasonable results. The experimental data of Coriolis coupling coefficients and isotopic frequency shifts are utilized to reduce the uncertainty of the calculated force constants when they are available. A series of various kinds of molecules have been calculated by this method and the results are satisfactory.

Theoretical study of the highly vibrationally excited states of FHF−: Ab initio potential energy surface and hyperspherical formulation

A three-dimensional description of vibrationally highly excited linear molecules is formulated in hyperspherical coordinates, based on a successive adiabatic reduction scheme. The method is applied to the low-lying and highly excited vibrational states of FHF−, a prototype of symmetric bihalide anions, which has attracted spectroscopic interest due to its peculiar vibrational anharmonicity. Ab initio potential energy surfaces (PESs) which cover the ground-state potential well of FHF− and/or its dissociation to the F−+HF channel have been obtained by using the coupled electron pair approach (CEPA) method. An hyperspherical calculation using the ab initio PES of the sixth-order Simons–Parr–Finlan analytical form has correctly reproduced the experimental fundamental frequencies. Specifically, the vibrationally highly excited FHF− above the dissociation threshold is proposed as a candidate for transition state spectroscopy (TSS) of unimolecular dissociation reactions without barrier.

An empirical rule relating fundamental to harmonic frequencies

An empirical approach to estimate harmonic frequencies from experimental fundamental vibrational frequencies is presented. It is based on a partitioning of the eigenvectors of each normal mode, s = 1, 2, 3,…, 3 N − 6, into internal motion components involving bond stretching, B s , angle bending, A s , torsional motion, T s , and out-of-plane motion, O s . This is called the BATO method. An empirical function, f s = (1250B s + 2000A s + 1300T s + 1600O s) A ̊ , is developed to assign relative anharmonic contributions to each kind of internal motion by relating harmonic frequencies, ω s , to the anharmonic or fundamental frequencies, ν s , with ν s = ω s + 2 χ s = ω s (1 − f s ω s ). The anharmonic states are assumed to satisfy a Morse expression, ν sn = ω s(n s + 1 2 ) + χ s(n s + 1 2 ) 2 , where n s = 0, 1, 2, ⋯ denote the vibrational states of mode s. The coefficients of f s are calibrated with the available experimental data for ν s exp and ω s exp with isotopes of 19 molecules which contain bonds and groups required for biochemical applications. The objective is to use the BATO method to estimate empirical harmonic frequencies, ω s emp, from experimental fundamental frequencies, ν s exp, in a self-consistent procedure in cases where ω s emp are not known. Of the 394 states in 19 molecules studied, the method yields results within 10 cm −1 for 232 states. 11–20 cm −1 for 86 states, 21–30 cm −1 for 45 states, 31–40 cm −1 for 19 states, 41–60 cm −1 for 9 states, and 61–81 cm −1 for 3 states tested. Detailed results are presented for the molecule containing the most common isotope. The maximum errors occur in C 2H 6 and C 3H 8, for which the method tends to overestimate the CH stretch in several cases where the comparison is against Dennison's rule. All molecules studied satisfy the product and summation rules well.

Theoretical prediction of vibrational spectra. Scaled quantum mechanical (SQM) force field for fluorobenzene

The complete harmonic force field of fluorobenzene has been determined from ab initio Hartree-Fock calculations using the 4–21 Gaussian basis set. As force constants are systematically overestimated at this level of theory, the directly calculated force field was scaled by empirical factors taken over from benzene and methylfluoride. Except for a slight overestimation of the CF stretching frequency, the scaled quantum mechanical (SQM) force field obtained in this way reproduces the experimental fundamental frequencies of the parent molecule and two deuterated isotopomers within 20 cm −1 (with mean deviations below 12 cm −1), and experimental assignments are analyzed on this basis. Theoretical i.r. intensities reproduce the main features of the spectra fairly well.

The harmonic force field and vibrational spectra of pyrrole

The complete harmonic vibrational force field of pyrrole has been calculated by the ab initio gradient method at the Hartree-Fock level using the 4–21 basis set. The force field was then scaled with a set of six factors transferred from benzene, and the vibrational spectrum of pyrrole was calculated. This a priori prediction, made with no reference to observations on pyrrole, agreed with the known experimental fundamental frequencies with a mean deviation of 12 cm-1 for in-plane modes and 20 cm-1 for out-of-plane modes except for the NH wagging and NH stretch. A new set of ten scale factors was next obtained by direct fitting of the computed force field to the observed pyrrole spectrum, producing the best force field obtainable by combined use of the theoretical and experimental information. This force field reproduced the entire pyrrole spectrum with mean deviations of 4·2 cm-1 (in-plane) and 5·9 cm-1 (out-of-plane). The spectra of three deuterated forms of pyrrole were also computed. Infrared absorption intensities were calculated and proved very useful in examining assignments of the two ring torsional modes and the CH stretching modes.