Statistical thermodynamics of internal rotation in a hindering potential of mean force obtained from computer simulations

A method of statistical estimation is applied to the problem of one-dimensional internal rotation in a hindering potential of mean force. The hindering potential, which may have a completely general shape, is expanded in a Fourier series, the coefficients of which are estimated by fitting an appropriate statistical-mechanical distribution to the random variable of internal rotation angle. The function of reduced moment of inertia of an internal rotation is averaged over the thermodynamic ensemble of atomic configurations of the molecule obtained in stochastic simulations. When quantum effects are not important, an accurate estimate of the absolute internal rotation entropy of a molecule with a single rotatable bond is obtained. When there is more than one rotatable bond, the "marginal" statistical-mechanical properties corresponding to a given internal rotational degree of freedom are reduced. The method is illustrated using Monte Carlo simulations of two public health relevant halocarbon molecules, each having a single internal-rotation degree of freedom, and a molecular dynamics simulation of an immunologically relevant polypeptide, in which several dihedral angles are analyzed.     

Conformational studies by far infrared and Raman spectra of gases

One of the major goals of conformational analysis is the calculation of the energy difference between two or more conformers, ΔE, as well as the energy necessary for interconversion. The calculation of these energy quantities is facilitated by using a potential function which describes the vibrational motion, or internal rotation (torsion), as a function of the dihedral angle, α. The potential function is called asymmetric because both the frame and top portions of the molecule have no symmetry element higher than a plane. The most common type of potential function where at least one of the minima coincides with the plane of symmetry is of the type: V(α) = $\text{1}\text{2}$ΣV_i(1 - cos iα). The kinetic energy term, F(α), is extremely complicated. In general, if the only data being used to calculate the potential function are torsional transitions, and if one continues within the boundary conditions of a one-dimensional problem, then a cosine expansion of F(α) should be adequate: F(α) = F₀ + ΣF_i cos iα. For those systems where there is an equilibrium between a planar form and two non-planar forms, V₃ is usually the predominant term. This is because V₃ represents a three maxima/three minima potential per 2π (360°) internal rotation. In a similar fashion, V₂ is found to be the predominant term in the potential function for a system consisting of two equivalent non-planar conformers. Several examples of our most recent studies are given where the potential function for interconversion of two conformers has been determined.     

Effective internal rotation potential energy function of acryloyl fluoride,CH2  CH ? CF O

The geometry of acryloyl fluoride was optimized completely at the RHF/6-31G* level of theory at 10 points on the theoretical potential energy curve for internal rotation. The energies obtained were used to determine a six term cosine expansion of the torsional potential energy function. This function was then refined using the experimental torsional transition frequencies in the s-trans and s-cis wells in conjunction with the geometrical parameters optimized at the RHF/6-31G* level. The effective potential function obtained is compared with previous results. The necessity of accounting for relaxation of the geometry upon internal rotation is stressed. © 1992 by John Wiley & Sons, Inc.     

On the internal rotations in p-cresol in its ground and first electronically excited states

The overall rotation and internal rotation of p-cresol (4-methyl-phenol) has been studied by comparison of the microwave spectrum with accurate ab initio calculations using the principal axis method in the electronic ground state. Both internal rotations, the torsions of the methyl and the hydroxyl groups relative to the aromatic ring, have been investigated. The internal rotation of the hydroxyl group can be approximately described as the motion of a symmetrical rotor on an asymmetric frame. For the methyl group it has been found that the potential barrier hindering its internal rotation is very small with the first two nonvanishing Fourier coefficients of the potential V(3) and V(6) in the same order of magnitude. Different splittings of b-type transitions for the A and E species of the methyl torsion indicate a top-top interaction between both internal rotors through the benzene ring. An effective coupling potential for the top-top interaction could be estimated. The hindering barriers of the hydroxyl and methyl rotation have been calculated using second-order Moller-Plesset perturbation theory and the approximate coupled-cluster singles-and-doubles model (CC2) in the ground state and using CC2 and the algebraic diagrammatic construction through second order in the first electronically excited state. The results are in excellent agreement with the experimental values.     

Harmonic oscillator tensors. IV. A tensorial approach to internal rotations in molecules

A mapping of 2×2 matrices into the space of single boson operators is shown to lead to the angular momentum operators that give rise to irreducible tensors for the harmonic oscillator. The mapping may also be used to define an axis of quantization. A rotation about this axis induces a wave function and Hamiltonian that may be applied to the study of internal rotations in molecules. The example of a molecule containing two coaxial symmetric tops is presented as a case in point. The case of a potential with a high barrier leads to the approximation of an internal rotation as a torsional oscillator and, consequently, to torsional oscillator tensors whose properties are the same as those of the harmonic oscillator. The possibility of studying more complex potentials is discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 305–315, 1997     

Nearest-neighbor nonparametric method for estimating the configurational entropy of complex molecules

A method for estimating the configurational (i.e., non-kinetic) part of the entropy of internal motion in complex molecules is introduced that does not assume any particular parametric form for the underlying probability density function. It is based on the nearest-neighbor (NN) distances of the points of a sample of internal molecular coordinates obtained by a computer simulation of a given molecule. As the method does not make any assumptions about the underlying potential energy function, it accounts fully for any anharmonicity of internal molecular motion. It provides an asymptotically unbiased and consistent estimate of the configurational part of the entropy of the internal degrees of freedom of the molecule. The NN method is illustrated by estimating the configurational entropy of internal rotation of capsaicin and two stereoisomers of tartaric acid, and by providing a much closer upper bound on the configurational entropy of internal rotation of a pentapeptide molecule than that obtained by the standard quasi-harmonic method. As a measure of dependence between any two internal molecular coordinates, a general coefficient of association based on the information-theoretic quantity of mutual information is proposed. Using NN estimates of this measure, statistical clustering procedures can be employed to group the coordinates into clusters of manageable dimensions and characterized by minimal dependence between coordinates belonging to different clusters.     

Discovery of the gauche form in dichloroethane

Our study of internal rotation was carried out under the guidance of Professor San—ichiro Mizushima. Dipole moment measurements of 1,2-dichloroethane in non-polar solvents revealed a temperature dependence of the moment, which was interpreted as an indication of the existence of hindered rotation about the central CC bond, contrary to the original assumption of free rotation by van't Hoff and Le Bel. Raman Spectroscopy confirmed that the solid phase consists of the trans form only, whereas in the liquid and in the gas phases a second stable isomer coexists with the trans form. This second form was called > gauche< by Mizushima and its geometrical structure was established by electron diffraction. The potential function for internal rotation is considered with regard to its experimental determination and its interpretation by ab initio calculations.     

Excited state radiationless decay process with Duschinsky rotation effect: formalism and implementation

Duschinsky rotation effect is a simple and effective way to characterize the difference between the ground state and excited state potential energy surfaces. For complex molecules, harmonic oscillator model is still the practical way to describe the dynamics of excited states. Based on the first-order perturbation theory a la Fermi golden rule, the authors have applied the path integral of Gaussian type for the correlation function to derive an analytic formalism to calculate the internal conversion rate process with Duschinsky rotation effect being taken into account. The validity of their formalism is verified through comparison with previous work, both analytically for the case of neglecting Duschinsky rotation and numerically for the ethylene molecules with two-mode mixing. Their expression is derived for multimode mixing.     

SCF molecular orbital studies of internal rotation in the monofluorobenzaldehydes

The potential function for internal rotation of the aldehyde group in the orthometa- and para-fluorobenzaldehydes is calculated by CNDO/2, INDO and STO/6G methods, assuming invariant bond lengths. The INDO and STO/6G calculations indicate the equilibrium geometry to be essentially planar and the barrier to internal rotation to be the potential energy of the molecule when the plane of the aldehyde group is perpendicular to the ring plane; CNDO/2 leads to different conclusions, however. Relative stabilities of O-cis and O-trans rotamers of the planar configuration are discussed. Comparisons are made with the V barriers and the enthalpies of activation as determined by the dielectric absorption method using a matrix technique.     

Studies of Molecular Motions of Ocotillol-Type Saponins in Solution by ~(13)C Nuclear Magnetic Relaxation

In this paper, the fully anisotropicoverall tumbling motions and side groups internal rotation of ocotillol-type saponins separated from the leaves of Panax Quidquefolium L. are investigated by ~(13)C nuclear magnetic relaxation. The fully anisotropic overall tumbling motion model with methyl conformation jumps internal rotation among three equivalent sites is presented, and the spectral density function of this model is derived. The rotation rates for overall tumbling motions to ocotillol-type saponins (OTS) are computed by Woessner's fully anisotropic overall tumbling motion model, and the internal rotation rate and barrier for side groups in OTS are calculated using free diffusion internal rotation model, restriction diffusion internal rotation model and conformation jumps internal rotation model, respectively.     

Internal dynamics features in the free jet rotational spectrum of the acetaldehyde-Kr molecular complex

The structure and the dynamics of internal motions in the complex formed between acetaldehyde and Kr are studied by free jet absorption microwave spectroscopy performed in the range 60-78 GHz. The fourfold structure of each rotational line is evidence of the vibration-rotation coupling between the overall rotation of the complex, a tunneling motion of the Kr atom between two equivalent positions and the internal rotation of the methyl group in the acetaldehyde moiety. The four sets of transitions could be fitted with a coupled Hamiltonian which allows for the Coriolis interaction obtaining the energy separation between the vibrational energy levels related to the tunneling motion, while the observed splittings due to the methyl group internal rotation were analyzed independently with an appropriate model. The potential energy barriers for the tunneling motion and the internal rotation of the methyl group have been calculated and the interaction of the rare gas atom with the acetaldehyde moiety is reflected in the change of the V(3) barrier to internal rotation in going from the molecule to the weakly bound complex.     

Pulsed field ionization-ZEKE spectroscopy of cresoles and their aqueous complexes: internal rotation of methyl group and intermolecular vibrations

Pulsed field ionization-ZEKE photoelectron spectroscopy and (1 + 1) R2PI spectroscopy have been applied to the cis- and trans-m-cresol.H2O clusters. The internal rotational structure in the S1 state has been re-assigned, and the potential curve has been determined for the cluster. The PFI-ZEKE spectra of the cis- and trans-isomers show low-frequency bands up to 1000 cm-1 above the adiabatic ionization potential IP0. The low-frequency bands are assigned to the internal rotation of the methyl group, the intermolecular stretching and their combination bands in the m-cresol.H2O cluster cation. Level energies and relative transition intensities are reproduced well by a one-dimensional rotor model with a three-fold axis potential. Potential curves for the internal rotation have been determined for both cis- and trans-isomers of m-cresol.H2O cations. The effect of the cluster formation upon the internal methyl rotation, and the interaction between the methyl rotation and the intermolecular vibration are discussed.     

Internal rotation and rotational isomerism in substituted ethanes

The types of potential energy curves of internal rotation in substituted ethanes were studied. Formulas for the calculation of the energy differences in rotational isomers and of the potential barriers to internal rotation in these derivatives were obtained by Bernstein's method. As a rule, the calculated results are in agreement with experiment.     

Calculation of argon trimer rovibrational spectrum

Rovibrational spectra of Ar3 are computed for total angular momenta up to J=6 using row-orthonormal hyperspherical coordinates and an expansion of the wave function on hyperspherical harmonics. The sensitivity of the spectra to the two-body potential and to the three-body corrections is analyzed. First, the best available semiempirical pair potential (HFDID1) is compared with our recent ab initio two-body potential. The ab initio vibrational energies are typically 1-2 cm-1 higher than the semiempirical ones, which is related to the slightly larger dissociation energy of the semiempirical potential. Then, the Axilrod-Teller asymptotic expansion of the three-body correction is compared with our newly developed ab initio three-body potential. The difference is found smaller than 0.3 cm-1. In addition, we define approximate quantum numbers to describe the vibration and rotation of the system. The vibration is represented by a hyper-radial mode and a two-degree-of-freedom hyperangular mode, including a vibrational angular momentum defined in an Eckart frame. The rotation is described by the total angular momentum quantum number, its projection on the axis perpendicular to the molecular plane, and a hyperangular internal momentum quantum number, related to the vibrational angular momentum by a transformation between Eckart and principal-axes-of-inertia frames. These quantum numbers provide a qualitative understanding of the spectra and, in particular, of the impact of the nuclear permutational symmetry of the system (bosonic with zero nuclear spin). Rotational constants are extracted from the spectra and are shown to be accurate only for the ground hyperangular mode.     

Microwave and theoretical investigation of the internal rotation in m-cresol

The microwave spectrum of m-cresol (3-methylphenol) has been investigated using a molecular beam Fourier transform microwave spectrometer in the frequency range from 3 to 26.5 GHz. The rotation of the hydroxy group into two different unequal energetic minima leads to different spectra for the syn- and anticonformers. Because of a high potential barrier both conformers can be analyzed independently. The methyl group is undergoing an almost free internal rotation which is only hindered by small barriers and splits the vibrational ground state in two states of internal rotation denoted as A and E species. The spacing between the species is found to be up to 10 GHz. The potential for the internal rotation can be determined from the spectra and analyzed in terms of the Fourier components V3 and V6. For syn-m-cresol these parameters were determined as V3=673(3) GHz and V6=-335(24) GHz and for anti-m-cresol V3=95(5) GHz and V6=-416(46) GHz. The barriers to internal rotation were furthermore calculated with second-order Moller-Plesset perturbation theory and second-order coupled-cluster singles- and-doubles model (CC2) in the electronic ground state and with CC2 in the first excited state. The CC2 method is found to be an appropriate method to calculate potential barriers in electronic excited states of such compounds.     

Vibrational spectrum and internal rotation in 2,5-dimethylpyrazine

The infrared and Raman spectra of 2,5-dimethylpyrazine have been recorded and assigned on the basis of a C_2h molecular geometry previously determined by MINDO/3. The potential energy function corresponding to the internal rotation of both methyl groups has been used to solve the Schrödinger equation, and to obtain the energy levels of that motion on the basis of a molecular symmetry G₃₆. The rotation of each substituent is found to be almost independent of the other.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.