Internal Rotation In Molecules Research Articles

The torsional states of methyl hydroperoxide molecule calculated using anharmonic zero point vibrational energy

The 2D surfaces of potential energy, kinematic coefficients, components of the dipole moment, the heights of potential barriers, the energies of stationary torsional states, and the tunneling frequencies of hydroxyl and methyl groups in the methyl hydroperoxide molecule were calculated at MP2/CBS and CCSD(T)/Aug-cc-pVTZ levels of theory. Additionally, calculations of the 2D surface of zero point vibrational energy of the molecule in the harmonic and anharmonic approximations were performed at MP2/Aug-cc-pVTZ level of theory. The zero point vibrational energy calculated in two approximations is summed up with the potential energy of the methyl hydroperoxide molecule, calculated at two levels of theory, and the resulting four outcomes of the refined potential energy are used to calculate the energies of stationary torsional states and tunneling frequencies. The results obtained are compared with the experimental and theoretical data presented in the literature to evaluate the efficiency of taking into account the zero point vibrational energy when examining the internal rotation in molecules.

Spectroscopy of Interstellar Internal Rotors: An Important Tool for Investigating Interstellar Chemistry

This paper is a review of molecules with internal rotors detected or tentatively detected with upper limits in the interstellar medium. Internal rotor molecules containing one or two methyl groups are present at each of the different phases leading to stellar and planetary formation, from the molecular clouds to the star and its planets. They are also found in comets and asteroids. They represent a ubiquitous hydrogen-rich substituent in many complex organic molecules. This paper shows how high resolution spectra (mainly in the microwave, millimeter and sub-millimeter wave range) are analyzed in the laboratory and what theoretical methods can be used to properly reproduce the spectra. We also present a range of examples of important methyl rotors studied and how they can be used as tools to study the chemistry of interstellar medium.

Kinematic anharmonicity of internal rotation of molecules

The methods of analysis the strongly coupled vibrations are proposed for a number of molecules of aromatic and heterocyclic carbonyl (and some others) compounds. The qualitative principles are formulated for molecular systems with a significant kinematic anharmonicity.

Potential and matrix elements of the hamiltonian of internal rotation in molecules in the basis set of Mathieu functions

The advantages of the orthonormal basis set of 2π-periodic Mathieu functions compared to the trigonometric basis set in calculations of torsional states of molecules are substantiated. Explicit expressions are derived for calculating the Hamiltonian matrix elements of a one-dimensional torsional Schrodinger equation with a periodic potential of the general form in the basis set of Mathieu functions. It is shown that variation of a parameter of Mathieu functions allows the rotation potential and the structural function to be approximated with a good accuracy by a small number of series terms. The conditions for the best choice of this parameter are specified, and approximations are obtained for torsional potentials of n-butane upon rotation about the central C-C bond and of its univalent radical n-butyl C2H5C·H2 upon rotation of the C·H2 group. All algorithms are implemented in the Maple package.

Sensitivity of transitions in internal rotor molecules to a possible variation of the proton-to-electron mass ratio

Recently, methanol was identified as a sensitive target system to probe variations of the proton-to-electron mass ratio $\ensuremath{\mu}$ [Jansen et al., Phys. Rev. Lett. 106, 100801 (2011)]. The high sensitivity of methanol originates from the interplay between overall rotation and hindered internal rotation of the molecule; that is, transitions that convert internal rotation energy into overall rotation energy, or vice versa, have an enhanced sensitivity coefficient, ${K}_{\ensuremath{\mu}}$. As internal rotation is a common phenomenon in polyatomic molecules, it is likely that other molecules display similar or even larger effects. In this paper we generalize the concepts that form the foundation of the high sensitivity in methanol and use this to construct an approximate model which makes it possible to estimate the sensitivities of transitions in internal rotor molecules with ${C}_{3v}$ symmetry, without performing a full calculation of energy levels. We find that a reliable estimate of transition sensitivities can be obtained from the three rotational constants ($A$, $B$, and $C$) and three torsional constants ($F$, ${V}_{3}$, and $\ensuremath{\rho}$). This model is verified by comparing obtained sensitivities for methanol, acetaldehyde, acetamide, methyl formate, and acetic acid with a full analysis of the molecular Hamiltonian. Of the molecules considered, methanol is by far the most suitable candidate for laboratory and cosmological tests searching for a possible variation of $\ensuremath{\mu}$.

An analysis of the vibrational structure of high-resolution UV spectra and long-wave IR Fourier transform spectra in studies of internal rotation in molecules

The methods for analyzing the vibrational structure of high-resolution UV spectra and long-wave IR Fourier transform spectra in studies of internal rotation in α,β-unsaturated carbonyl compounds R4R3C=CR2-COR1 (R1 = F, Cl; R2 = R3 = R4 = H, CH3) are compared. These methods were found to give different experimental values for systems of torsional vibration energy levels up to high quantum numbers, torsional frequencies (0–1 transitions), and anharmonicity coefficients x 11 for trans and cis isomers of the same molecules in the ground electronic state (S 0). It was shown that the experimental technique for analyzing the vibrational structure of UV spectra excludes the hydrolysis of compounds under study. Taking into account Fermi resonance and numerous Deslandres tables constructed for trans and cis isomers provides reliable determination of values necessary for the construction of internal rotation potential functions, because they are multiply repeated in various Deslandres tables. An analysis of the vibrational structure of UV spectra gives more reliable V n internal rotation potential function parameters. The V n parameter values were substantiated by quantum-mechanical calculations performed by other authors.

ChemInform Abstract: Internal Rotation of Molecules with Two Equivalent Tops: The Microwave Spectrum of 2-Fluoropropane.

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Calculations of partition functions for internal rotation of molecules

A simple numerical method for calculations of torsional energy for barriers of arbitrary forms was suggested. Within the framework of the Wigner-Kirkwood approximation, an analytic equation was derived for the partition function of a hindered top, which was used for interpretation and a comparison with numerically exact results.

Determining Conformational Preference for Molecules by Electron Momentum Spectroscopy

Many molecules can rotate internally around one or more bonds so that during 360°'s rotation, they will change between unstable and relative stable conformers. The internal rotation of molecules can lead to much different electron density distributions for certain molecular orbitals (MOs). By comparing the measured electron momentum profiles with the calculated ones, the relative stability of different conformers of 1-butene has been determined, demonstrating that the electron momentum spectroscopy (EMS) is a feasible experimental method to study the conformational preference of molecules.

Energy level promotion in the correlation from the tunnelling-doubled harmonic oscillator to the bi-rotor: application to internal rotation in molecules

A surprisingly rich variety of phenomena are revealed in the energy level correlation between the limits of a tunnelling doubled harmonic oscillator and a bi-rotor. Some levels are found to have their vibrational quantum number "promoted" upon removal of the barrier to rotation, other levels, which we dub "invariant", are found to be completely independent of the barrier, while yet other levels exhibit a smooth transition between these limits. The general nature of these features can be understood in terms of the different degeneracies of the limiting cases. The elucidation of these effects aids the understanding of the rotational-vibrational energy levels of molecules having two internal rotor moieties.

Computer calculation of the Van Vleck second moment for materials with internal rotation of spin groups

This paper describes m2rc3, a program that calculates Van Vleck second moments for solids with internal rotation of molecules, ions or their structural parts. Only rotations about C 3 axes of symmetry are allowed, but up to 15 axes of rotation per crystallographic unit cell are permitted. The program is very useful in interpreting NMR measurements in solids. Program summary Title of the program: m2rc3 Catalogue number: ADUC Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUC Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland License provisions: none Computers: Cray SV1, Cray T3E-900, PCs Installation: Poznań Supercomputing and Networking Center ( http://www.man.poznan.pl/pcss/public/main/index.html) and Faculty of Physics, A. Mickiewicz University, Poznań, Poland ( http://www.amu.edu.pl/welcome.html.en) Operating system under which program has been tested: UNICOS ver. 10.0.0.6 on Cray SV1; UNICOS/mk on Cray T3E-900; Windows98 and Windows XP on PCs. Programming language: FORTRAN 90 No. of lines in distributed program, including test data, etc.: 757 No. of bytes in distributed program, including test data, etc.: 9730 Distribution format: tar.gz Nature of physical problem: The NMR second moment reflects the strength of the nuclear magnetic dipole–dipole interaction in solids. This value can be extracted from the appropriate experiment and can be calculated on the basis of Van Vleck formula. The internal rotation of molecules or their parts averages this interaction decreasing the measured value of the NMR second moment. The analysis of the internal dynamics based on the NMR second moment measurements is as follows. The second moment is measured at different temperatures. On the other hand it is also calculated for different models and frequencies of this motion. Comparison of experimental and calculated values permits the building of the most probable model of internal dynamics in the studied material. The program described in this paper calculates the second moment for solids with rotation of different groups of spins with C 3 symmetry. Method of solution: The rotation of molecules or their parts, for example CH 3 groups, is simulated as a random walk process by rotating each individual group of spins about its symmetry axis by an angle allowed by the type of symmetry. It is not a continuous rotation, but is in the form of jumps between consecutive positions allowed by the symmetry of the rotating group. Such a model of rotation fulfills assumptions on which theoretical equations used in NMR are derived. The value of Van Vleck's second moment averaged by this rotation is evaluated. The degree of averaging depends on the number of rotational jumps simulated during calculation. This number is then expressed in terms of the frequency of rotation and finally into the temperature. As a result we obtain simulated values of the NMR second moment as a function of temperature. Restrictions on the complexity of the problem: The only restriction is the number of spins for which calculations can be performed in a reasonable amount of CPU time. This restriction is therefore a combination of the number of spins in the unit cell, number of unit cells included in the calculation, and the speed of the computer used. The tested version of the program was compiled for a maximum number of 6250 spins, arranged in 125 unit cells. There are 15 axes of rotation allowed per unit cell. Any of these restrictions can be overcome by increasing the dimensions of the appropriate arrays in the program. The dimensions given in the program are sufficient for analysis of most of the NMR data which one can find in the scientific literature. This is due to the fact that the magnetic dipole–dipole interaction decreases with the third power of distance between spins, and calculations including spins up to a distance of about 2.0 nm give a final accuracy of the second moment equal to about 1%, while experimental values are determined with 5% accuracy or even worse. The program was designed to handle any combination of complex rotations, but only about C 3 axes. Overcoming this restriction by introducing the possibility of C 6 or C 4 rotations would require some changes in the program. They may be quite easily introduced by an experienced programmer.

Ground and first excited torsional states of acetamide

We present a first global study involving rotational levels in the lowest three torsional states of acetamide (CH 3CONH 2). New measurements of this spectrum, consisting of approximately 1600 lines and involving torsion–rotation transitions with J up to 20 and K a up to 11, have been carried out between 49 and 149 GHz using the millimeter-wave spectrometer in Kharkov. After removing the observed quadrupole hyperfine splittings, the new data were combined with previously published measurements and fitted using a rho-axis-method (RAM) torsion–rotation Hamiltonian in conjunction with a new computer-automated v t and K labeling algorithm. The final fit used 48 parameters to give an overall weighted standard deviation of 0.72 for 759, 587, and 265 lines belonging, respectively, to the ground, first, and second excited torsional states and 95 lines corresponding to Δ v t =1 transitions falling in the millimeter wave range. Separate root-mean-square (rms) deviations for the A (25 kHz) and E (27 kHz) species, as well as for the v t =0 state (26 kHz), v t =1 (25 kHz), v t =2 (22 kHz), and Δ v t =1 transitions (35 kHz) indicate a similar quality of the fit for the two symmetry species and for the three torsional states. The RAM Hamiltonian was found to have rather good predictive ability when fits of only v t =0 (or only v t =0 and 1) transitions were used to calculate v t =1 (or v t =2) lines. In addition, the combination of this Hamiltonian and the computer-automated v t and K labeling algorithm seems to provide a rather powerful tool for the precise assignment and fitting of torsion–rotation transitions in C 3 v internal-rotor molecules with low torsional potential barriers.

Structure and Internal Rotation of Molecules in Chlorodifluoroacetic Acid

The potential energy profile of an isolated CF2ClCOOH molecule with a CF2Cl group rotating around the C–C bond was determined by the Hartree–Fock method using the 6-31G(d) basis set. Barriers to internal rotation were estimated for this molecule; its geometrical parameters were found for the equilibrium and transition states that are due to the torsion potential with unequal wells. Crystal effect on CF2Cl reorientations in solid chlorodifluoroacetic acid has been evaluated.

Torsional Angle Definitions and Linear and Quadratic Force Field Variations along the Torsional Coordinate for CH3OH and CH3CHO

This paper concerns itself primarily with three questions: (i) What is the range of commonly used definitions for the internal rotation angle, and how do variations within this range affect intercomparisons among results from various laboratories? (ii) Do the force-constant and projected-frequency variations with internal rotation angle delivered by present commercial ab initio packages have sufficient precision to allow meaningful extraction of higher-order vibration–torsion–rotation interaction terms? (iii) What fraction of the extensive dynamics literature on vibrational frequencies along the chemical reaction path can be easily adapted for use in high-resolution spectroscopic studies of small-amplitude vibrations in methyl top internal rotor molecules? Aspects of these questions are discussed both from an algebraic and from a numerical computation point of view.

τ-Clustering for molecules with large torsional splittings: the ν7+ ν8 and 2 ν8 combination and overtone bands of CH 3OH

Line clustering into two groups of differing τ torsional symmetries is considered for parallel infrared vibrational bands of 3-fold internal rotor molecules with large torsional splittings, such as methanol. In bands involving large changes in torsional parameters, the τ-clustering produces a distinctive pair of Q-branch peaks in the low-resolution spectrum. The origin of the τ-clustering and the formation of the twin Q-branch peaks are discussed together with new subband assignments in Fourier transform spectra of the ν 7+ ν 8 CH 3-rock/CO-stretch combination and 2 ν 8 CO-stretching overtone bands of CH 3OH, which are good representatives of large and small changes in torsional parameters, respectively.

The Ground and First Excited Torsional States of Acetic Acid

A global fit of microwave and millimeter-wave rotational transitions in the ground and first excited torsional states (vt = 0 and 1) of acetic acid (CH3COOH) is reported, which combines older measurements from the literature with new measurements from Kharkov, Lille, and NIST. The fit uses a model developed initially for acetaldehyde and methanol-type internal rotor molecules. It requires 34 parameters to achieve a unitless weighted standard deviation of 0.84 for a total of 2518 data and includes A- and E-species transitions with J ≤ 30 and Ka ≤ 15. While these results represent a significant improvement over past fitting attempts, it should be cautioned that the present data set is dominated by vt = 0 transitions, and no direct infrared measure of the vt = 1 ← 0 torsional interval is available.

Dark state illuminated: infrared spectrum and inverted torsional structure of the nu(11) out-of-plane CH3-rocking mode of methanol

The nu(11) out-of-plane CH3-rocking band of methanol, one of the last fundamentals remaining en route to full vibrational description of this prototype internal rotor, has been identified in the Fourier transform spectrum of CH3OH in the 1150 cm(-1) region. The nu(11) torsional energy pattern is found to be inverted, the first such discovery for a bending state and an important pointer to potential generality of this phenomenon for the whole class of threefold internal rotor molecules. Level-crossing resonances giving transfer channels for intramolecular vibrational energy redistribution (IVR) have been found, and new insights are reported for far-infrared laser emission involving nu(11) levels.

A precessing rotor model for structural flexing during torsional motion: is there evidence from the internal rotation kinetic energy coefficient F?

A precessing rotor model is used to mimic the structural flexing which is predicted by ab initio calculations to accompany internal rotation in molecules. The internal rotation kinetic energy coefficient F( τ) is found to vary with τ, and to have an average value F 0 significantly less than the constant F rig calculated for a rigid model. Analysis of published torsional data for a range of substituted toluenes shows that F 0 obs/F rig ≈0.97, consistent with the occurrence of precession with a tilt angle θ≈1.5°. In fitting experimental spectra, it was found that the Fourier coefficients F 3 and F 6 are highly correlated with the potential constants V 3 and V 6, and cannot be separately determined. Thus the predicted variation of F with τ could not be confirmed. The effect of structural flexing on torsion–rotation interaction constants is also examined.

Improvement of the standard method for solving the problem of internal rotation in molecules with symmetrical tops

Drawbacks of the standard method for solving the problem of internal rotation (IR) in molecules with symmetrical tops were examined. The drawbacks can be eliminated by taking into account the higher harmonics of the IR potentials. A new calculation procedure based on the approximation of the lower portion of the potential curve by the equationV(φ)=1/2V n * (1-cosn φ), which describes adequately the arrangement of the torsion levels, was proposed. The successfull implementation of the proposed procedure showed that the procedures used currently are incorrect as well as the Herschbach procedure, which includes a systematic error at each iteration step.

Determination of the energy levels of total and internal rotation of arbitrary polyatomic molecules with a single internal rotation axis

An algorithm is developed for calculating the eigenvalues of the Hamiltonian operator in the case of asymmetric-asymmetric molecules with a common axis of internal rotation, taking into account the total and internal rotation of these molecules. The algorithm is based on the Ritz variational method. A program is written for reducing a symmetric band matrix to tridiagonal form with subsequent diagonalization of the resulting matrix.