Description of Molecular Distortions III. Trigonally-Planar XY3 Molecules

Second-rank tensors were used to calculate the degree of distortion for the NO3 ions in a number of crystalline compounds. All NO3 ions appear to be strictly planar, as found in a previous study1. A significant correlation, between the main components of the tensor and the wavenumbers of the components of the antisymmetric stretching vibration (r3) of the NO3 ion, was found

A Tensorial Approach to the Description of Molecular Distortions I. Tetrahedral Molecules

The inclusion of a tetrahedral XY4 molecule (or ion) in a crystal is, very often, followed by a lowering of its symmetry. In order to describe the apparent distortions of the tetrahedron, second-rank tensors were constructed. It was shown that the characteristic surface of such a tensor is always an ellipsoid. The relative lengths of the axes of the ellipsoid and their position with respect to the symmetry elements of the XY4 group can be used to determine the effective symmetry of the molecule, as well as the degree of its distortion. Some of the spectral properties of the studied compounds can also be predicted. 36 S04 ions with accurately refined structures were investigated and the results obtained by this method were compared with the results\u27 obtained by other-= methods. A correlation of rather high significance (r2 = = 0.97) was found between the main components of the tensor and the frequencies of the components of the antisymmetric stretching vibration (V3) of the molecule

An Iterative Perturbation Theory-Based Algorithm for Calculation of Diagonal Force Constants of the ν3 Modes in Quasi-Tetrahedral Systems with Low Anharmonicity

A novel, stationary perturbation theory – based iterative algorithm for calculation of the diagonal force constants of the ν3 modes in tetrahedral and quasi-tetrahedral systems with low anharmonicity is developed. The diagonal elements in the potential energy expression are calculated on the basis of experimentally measured fundamental and second-order transition wavenumbers, in a self-consistent manner. Perturbation corrections up to the second order are included in the model. The procedure is rapidly convergent, simple, and may be easily implemented within computer programs. The calculated diagonal force constants are consistent with the measured fundamental and second-order transition wavenumbers. The proposed model is applied to several isomorphously isolated (distorted-tetrahedral) sulfate impurities in selenate and chromate matrices. Experimental data for these systems were obtained by FT-IR spectroscopy

An Iterative Perturbation Theory-Based Algorithm for Calculation of Diagonal Force Constants of the ν3 Modes in Quasi-Tetrahedral Systems with Low Anharmonicity

A novel, stationary perturbation theory – based iterative algorithm for calculation of the diagonal force constants of the ν3 modes in tetrahedral and quasi-tetrahedral systems with low anharmonicity is developed. The diagonal elements in the potential energy expression are calculated on the basis of experimentally measured fundamental and second-order transition wavenumbers, in a self-consistent manner. Perturbation corrections up to the second order are included in the model. The procedure is rapidly convergent, simple, and may be easily implemented within computer programs. The calculated diagonal force constants are consistent with the measured fundamental and second-order transition wavenumbers. The proposed model is applied to several isomorphously isolated (distorted-tetrahedral) sulfate impurities in selenate and chromate matrices. Experimental data for these systems were obtained by FT-IR spectroscopy

Dependence of the Crystal Structure Parameters on the Size of the Structural Units in Some Isomorphous / Isostructural Series

Previous studies of kieserite analogues and also other systems (Tutton’s salts, alums and β-K2SO4 isomorphs) have shown that in a series of isostructural/isomorphous compounds the unit-cell parameters and volumes vary linearly with the eflfective ionic radii (the \u27size\u27) of the structural units. This is in line with the results of Shannon (Acta Cryst. A32 (1976) 751-767). Somewhat unexpectedly, other important parameters of the crystal structure, such as fractional atomic co-ordinates, exhibit systematic variations as well. An attempt is made to reveal the reasons that are at the origin of this finding

Comment on the »Calculation of the Thermodynamic Functions of Tetrafluorodiphosphine«

In a recent note,1 the thermodynamic functions of tetrafluorodiphos- phine (P2F4) were calculated, employing the ideal-gas rigid-rotor harmonic- oscillator approximation. Several points concerning the assignment of the bands in the vibrational spectra, the calculation of the thermodynamic functions and the stability of the compound are discussed

Crystal Structures of Members in Isostructural Series: Prediction of the Crystal Structure of Cs2MnO4 - a p-K2SO4 Type Isomorph

Recent studies have shown that the crystal structure of members within a group of isostructural compounds may be successfully predicted. The P-K2SO4 group isomorphs of the general formula M2XO4 were chosen as a family of closely related compounds for which accurately refined crystal structures exist. It is shown that the unit cell parameters and the fractional atomic coordinates exhibit systematic variations with both cation and anion size, as well as with the Mulliken charge of the oxygen atom in the tetrahedral anion. It is thus possible to predict the crystal structure of a given member in the series only on the basis of its chemical composition. The structure of Cs2MnO4 predicted in this way is compared with the one refined earlier by X-ray diffraction. The agreement be- tween the structural parameters in both structures is very good

A Quantum Theoretical Basis for Some Spectra – Structure Correlations in Crystalline Hydrates

Starting with the perturbation theory and the Hellmann-Feynmann theorem, an attempt was made to derive a fundamental theoretical basis for some frequency – structure correlations in crystalline hydrates. It was found that within a few reasonable approximations, a satisfactory theoretical background may be found for the νOH(OD) versus RO⋅⋅⋅O, as well as for the –2XOH(OD) versus νOH(OD) correlations (νOH(OD) is the spectroscopically measured wavenumber of the OH(OD) stretching vibration, RO⋅⋅⋅O is the hydrogen bond distance, and X is the anharmonicity of vibration). The OH(OD) oscillators were treated as mixed cubic – quartic anharmonic systems. The influence of hydrogen bonding on these oscillators was introduced through the changes in the harmonic diagonal force constants (as proposed by Sceats and Rice1), the other diagonal terms in the potential energy expression being regarded as practically unchanged in the course of the hydrogen bonding. The parameters obtained by empirical correlations, within the proposed model, describe the dependence of the intramolecular potential of the uncoupled OH(OD) species on the hydrogen bond strength

A Quantum Theoretical Basis for Some Spectra – Structure Correlations in Crystalline Hydrates

Starting with the perturbation theory and the Hellmann-Feynmann theorem, an attempt was made to derive a fundamental theoretical basis for some frequency – structure correlations in crystalline hydrates. It was found that within a few reasonable approximations, a satisfactory theoretical background may be found for the νOH(OD) versus RO⋅⋅⋅O, as well as for the –2XOH(OD) versus νOH(OD) correlations (νOH(OD) is the spectroscopically measured wavenumber of the OH(OD) stretching vibration, RO⋅⋅⋅O is the hydrogen bond distance, and X is the anharmonicity of vibration). The OH(OD) oscillators were treated as mixed cubic – quartic anharmonic systems. The influence of hydrogen bonding on these oscillators was introduced through the changes in the harmonic diagonal force constants (as proposed by Sceats and Rice1), the other diagonal terms in the potential energy expression being regarded as practically unchanged in the course of the hydrogen bonding. The parameters obtained by empirical correlations, within the proposed model, describe the dependence of the intramolecular potential of the uncoupled OH(OD) species on the hydrogen bond strength

The Vibrational Stark Shifts of Sulfate Internal Modes in SO42- Doped Potassium, Rubidium and Cesium Selenates. A Quantum Model for Measurement of Crystalline Fields

A quantum theoretical basis for the experimentally observed vibrational Stark shifts of the sulfate internal modes in SO42- doped K2SeO4, Rb2SeO4 and Cs2SeO4 is presented. Analytical first order perturbation theoretical expressions are derived for the field-dependent wavenumbers of the 1 ← 0 and 2 ← 0 transitions, harmonic wavenumbers, as well as for the Stark tuning rate and the electrostatic field strength at the Cs sites of the host lattices. It is shown that the local field differences are the factor dominating over the differences in the anharmonicities of the guest anions in various host lattices, and are thus responsible for the experimentally observed trends. The proposed method allows calculation of the local crystalline field strength if the anharmonic potential energy parameters of the dopant anions are known. The calculated values for the studied series of matrices range from 229 to 259 V nm-1, which are approximately 3 times larger than those reported for water molecule sites in several clathrate hydrates and for the N2O adsorbed in the NaA zeolite cavities. The model also successfully explains the greater bond length distortions than the angular ones of the dopant anions observed in ali studied cases