Analytical force constant calculation as a minimization problem

In this letter we show that the calculation of analytic second derivatives of variational potential energy surfaces with respect to nuclear coordinates is a minimization problem.     

The experimental design for computing the harmonic and anharmonic force constant matrices of polyatomic molecules

In the present work we propose a numerical approach to estimate the harmonic and anharmonic force constant matrices, supposing we are able to compute analytically the first order derivative vector of the potential energy surface with respect to the internal coordinates. We use a polynomial least square fit to interpolate this gradient in the stationary point region. The structure of the regression matrix shows that the harmonic force constant matrix may be obtained even for large molecules; the evaluation of the anharmonic contributions request slightly more labor but is possible for 5 to 7 atoms. The present work is applicable even at the CI level and the number of computations remains small. We use the experimental planification to select the geometries to be computed in order to improve the estimation of the regression coefficients i.e. this means to lower their variance.chercheur qualifié du Fonds National Belge de la Recherche Scientifique.     

Bending force constant formulas from a bond-charge model of triatomic molecules

A bond-charge point-dipole model of molecular electronic structure is used to derive a general relation between the bending force constant, K_θθ, and stretching force constant , K₁₁, of linear and non-linear triatomic molecules.     

Calculation of anharmonic force constants

Non-linear transformations from internal to cartesian displacements are considered for anharmonic calculation of vibrational frequencies. Force constants in normal coordinates, up to quartic terms, are related to Christoffel symbols. The latter are tabulated for valence angles or, for torsion and wagging out of plane, evaluated through expressions which avoid differentiation of explicit functional forms for each type of internal coordinate. The same symbols are used for cartesian tensors and redundancy coefficients up to the third order.     

Analytical probability distribution function for a one-dimensional anharmonic oscillator in thermal equilibrium

A simple closed formula is proposed for the probability distribution function of a one-dimensional anharmonic oscillator in thermal equilibrium. It is shown that this formula yields a plausible approximation for the distribution function over wide ranges of the parameter θ = hcωelkT and the degree of anharmonicity. A more general formula with extended limits of applicability is also considered.     

The anharmonic force constant K234 of CH3CN in different phases,solvents and complexes with Lewis acids

The Fermi coupling coefficient W and the anharmonic force constant K₂₃₄ of CH₃CN have been determined from i.r. and Raman studies of the v₂—v₃+v₄ Fermi doublet in liquid and gaseous phases, solutions in inert, basic and acidic solvents and in crystalline complexes with Lewis acids. The values of these parameters have been calculated by a modification of the Winther method. This method is also applied to gas phase data of the isotopic speices ¹³CH₃CN, CH₃ ¹³CN and CH₃C¹⁵N. Values of W are approximately 13±1 cm⁻¹ except for basic and inert solvents where they are approximately 18 cm⁻¹.     

Analytical formulas for Fermi resonance interactions in continuous distributions of states

Fermi resonance interaction between a distributed fundamental vibrational level and a distributed overtone is considered. The overtone is assumed to be spectrally inactive. Simple analytical expressions are derived for the resulting spectral profile in terms of the distribution functions for the fundamental and overtone. The formulas enable straightforward modeling of spectra with Fermi resonances.     

The formate anion: the harmonic and anharmonic force fields

Harmonic and anharmonic symmetry force constant matrices have been obtained for six isotopically substituted species of the formate anion. The valence force fields derived from the above matrices were used to recalculate the vibrational frequencies. These force fields are compared with others previously reported and obtained via other methods.     

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

The classical and Wigner—Kirkwood semiclassical distribution functions for single and double-well anharmonic oscillators with ? = p?²/2m + k₂x?² + k_2nx?²ⁿ are given in closed form. Applications in the theory of electron diffraction are discussed briefly.     

Update of the anharmonic force field parameters of the ozone molecule

A new set of spectroscopic constants of the ¹⁶O₃ molecule (ω_i, x_ij, y_ijk, γ^DD, _i^X, β_ij^X,…), which determine vibrational dependence of band centres and rotational parameters, is derived from recent accurate analysis of high-resolution experimental ro-vibrational spectra through the theoretical approach based on second-order perturbation expansions in normal coordinates accounting for Darling–Dennison resonance interactions. These values are used to update empirical values of anharmonic coefficients (k_ijl, k_ijlm) of the potential function expansion in normal coordinates. Quadratic f_rr, f_r, f_rr′, f as well as cubic f_rst and quartic f_rstl force constants in internal (bond lengths, bond angle) coordinates are also derived. A detailed discussion is devoted to the accuracy of parameter determination for each of four steps of calculations. It is emphasised that the conventional method based on the inversion of formulae of the perturbation theory gives the largest uncertainties at the last step of calculations: the determination of the anharmonic force field in internal coordinates.     

An assessment of the anharmonic force fields of the halogen cyanides

Anharmonic force fields in the ϱ representation (ϱ = Я/r) have been calculated using the matrix diagonalization technique. A simple model force field and comparison method are used to assess the force fields currently available for these molecules. From this analysis, the potential surfaces defined by the present force fields are estimated to be within 2–5% of the actual molecular potential.     

Thermal aspects of constant rate transformation

A mathematical model for the conjugate formulation of the working conditions of the Q-derivatograph is created. There are no simplifying assumptions about the heat transfer processes for reacting active media and for their thermal interaction with the inert surroundings in this model. The energetic significance of the key components of the transfer and the redistribution of the fluxes is calculated by means of special balance relations. The influence of the kinetic and thermal reaction parameters on the nature of the establishment of the constant transformation rate is demonstrated. The accompanying heat phenomena are studied in detail. The essential differences in the behaviour of the TA system during endo and exothermal transformations are discovered. The exothermal reactions introduce significant irregularities, up to sharp oscillations.

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Zusammenfassung Für die Arbeitsbedingungen des Q-Derivatographen wird ein mathematisches Modell aufgestellt. Dabei werden keine vereinfachenden Annahmen über die Wärmetransportprozesse für die reagierenden Stoffe und über ihre thermische Wechselwirkung mit der inerten Umgebung gemacht. Die Energien der entscheidenden Beiträge des Transports und der Verteilung der Ströme werden mit speziellen Bilanzbeziehungen berechnet. Der Einfluss der kinetischen und thermischen Reaktionsparameter auf die Einstellung einer konstanten Umwandlungsgeschwindigkeit wird gezeigt, die entsprechenden thermischen Vorgänge werden im einzelnen untersucht. Die wesentlichen Unterschiede im Verhalten des Systems bei endo- und exothermen Vorgängen werden aufgezeigt, exotherme Reaktionen bewirken erhebliche Schwankungen bis zu deutlichen Oszillationen.

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Analytical formulas for calculation of K X-ray production cross sections by alpha ions

In the present study, different procedures are followed to deduce the semi-empirical and the empirical K X-rayX-ray production cross sections induced by alpha ions from the available experimental data and the theoretical results of the ECPSSR model for elements with 20≤Z≤30. The deduced K X-ray production cross sections are compared with predictions from ECPSSR model and with other earlier works. Generally, the deduced K X-ray production cross sections obtained by fitting the available experimental data for each element separately give the most reliable values than those obtained by a global fit.     

Trans-1-chloro-2-fluoroethylene: microwave spectra and anharmonic force field

For the first time the millimeter-wave spectra of the trans-35ClHC=CHF and trans-37ClHC=CHF isotopomers have been observed in natural abundance. Many DeltaJ=0, +/-1 DeltaK(-1)=+1 transitions for 35ClHC=CHF and DeltaJ=0 DeltaK(-1)=+1 transitions for 37ClHC=CHF have been detected and assigned. This allowed us to accurately determine the vibrational ground-state rotational constants, quartic and some sextic centrifugal distortion constants, and nuclear quadrupole coupling constants for both 35Cl and 37Cl. The experimental investigation has been supported by highly accurate theoretical predictions. As far as ab initio computations are concerned, the complete set of cubic and quartic force constants have been evaluated by numerical differentiation of the analytic second-order M?ller-Plesset many-body perturbation theory/correlation consistent polarized valence triple zeta second derivatives. The anharmonic part of the force field completes the theoretical study on the equilibrium structure, dipole moment, chlorine quadrupolar tensor, and harmonic force field previously carried out by the same authors.     

Experimental designs for finding stationary points and for computing force constant matrix

We use an experimental design method for computing a local quadratic form which is adequate to find stationary points and to evaluate the force constant matrix on a k-dimensional potential energy hypersurface. Two types of designs are particularly discussed: the composite and the Doehiert's plane. We give examples for illustrating the methodology. Some thermodynamical properties are deduced from our results. Such a method may be used with every type of theoretical calculations (SCF or CI ) and with any program without modification.     

Application of the virial theorem to the calculation of anharmonic force constants for the water molecule

On the basis of the virial theorem, relationships have been found between the anharmonic force constants which make it possible to decrease the number of unknowns in the equations obtained from the general quantum-mechanical theory of the vibrations and rotations of molecules. The complete set of anharmonic force constants for the H₂O molecule have been used to calculate all the anharmonic constants X_SS' for the D₂O molecule.