Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations

In a symmetrical hyperspherical framework, the internal coordinates for the treatment of N‐body systems are conveniently broken up into kinematic invariants and kinematic rotations. Kinematic rotations describe motions that leave unaltered the moments of the inertia of the N‐body system and perform the permutation of particles. This article considers the corresponding expansions of the wave function in terms of hyperspherical harmonics giving explicit examples for the four‐body case, for which the space of kinematic rotations (the “kinetic cube”) is the space SO(3)/V₄ and then the related eigenfunctions will provide a basis on such manifold, as well as be symmetrical with respect to the exchange of identical particles (if any). V₄ is also denoted as D₂. The eigenfunctions are obtained studying the action of projection operators for V₄ on Wigner D‐functions. When n of the particles are identical, the exchange symmetry can be obtained using the projection operators for the S_n group. This eigenfunction expansion basis set for kinematic rotations can be also of interest for the mapping of the potential energy surfaces. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

On the Role of Symmetry in Vibrational Strong Coupling: The Case of Charge‐Transfer Complexation

It is well known that symmetry plays a key role in chemical reactivity. Here we explore its role in vibrational strong coupling (VSC) for a charge‐transfer (CT) complexation reaction. By studying the trimethylated‐benzene–I₂ CT complex, we find that VSC induces large changes in the equilibrium constant K_DA of the CT complex, reflecting modifications in the ΔG° value of the reaction. Furthermore, by tuning the microfluidic cavity modes to the different IR vibrations of the trimethylated benzene, ΔG° either increases or decreases depending only on the symmetry of the normal mode that is coupled. This result reveals the critical role of symmetry in VSC and, in turn, provides an explanation for why the magnitude of chemical changes induced by VSC are much greater than the Rabi splitting, that is, the energy perturbation caused by VSC. These findings further confirm that VSC is powerful and versatile tool for the molecular sciences.     

Etude de composés d'addition d'acides de Lewis,XXXIV [1] Composés d'addition de benzoquinone-1, 4 avec TiCl4

The adduct 1,4-benzoquine · TiCl₄ has been prepared in CH₂Cl₂ solution at about ?60°. Its IR. spectrum has been recorded at the same temperature. The experimental study of the vibrational frequencies has been completed by the calculation of the fundamental vibrations in the molecular plane, using Wilson's FG method, with slightly simplified models of 1,4-benzoquinone · TiCl₄ (13 masses) and 1,4-benzoquinone · 2 TiCl₄ (14 masses); analysis by use of internal and symmetry coordinates. An assignment of most of the observed bands is proposed and the conclusion is reached that the complex, when solid, is (1,4-benzoquinone · TiCl₄)_n. The force constants F(C? O) are 9,85 · 10⁵ dyne/cm for the quinone and 8,8 · 10⁵ dyne/cm for the disturbed carbonyl bond of the polymerized complex in the model proposed. At ordinary temperature in benzene solution of the components the adduct 1,4-benzoquinone. TiCl₄ · benzene precipitated; with the help of the models, the fundamental vibrations of its IR. spectrum have been assigned.     

Normal coordinate analysis and mean amplitudes of bis(2,2,6,6-tetramethylheptane-3,5-dionato)uranyl

A normal coordinate analysis has been performed for bis(2,2,6,6-tetramethylheptane-3,5-dionato)uranyl according to two simplified models: (i) a 35-atom model consisting of one tetramethylheptanedionate ligand attached to uranyl; symmetry C_2V and (ii) a 31-particle model of the whole complex, where the methyl groups are taken as point masses; symmetry D_2h. The construction of independent symmetry coordinates by the “method of fragments” is described. Calculated vibrational frequencies are reported along with mean amplitudes (l- values) and perpendicular amplitude coefficients (K- values) for selected interatomic distances.     

On the Role of Symmetry in Vibrational Strong Coupling: The Case of Charge-Transfer Complexation

It is well known that symmetry plays a key role in chemical reactivity. Here we explore its role in vibrational strong coupling (VSC) for a charge-transfer (CT) complexation reaction. By studying the trimethylated-benzene–I₂ CT complex, we find that VSC induces large changes in the equilibrium constant K_DA of the CT complex, reflecting modifications in the ΔG° value of the reaction. Furthermore, by tuning the microfluidic cavity modes to the different IR vibrations of the trimethylated benzene, ΔG° either increases or decreases depending only on the symmetry of the normal mode that is coupled. This result reveals the critical role of symmetry in VSC and, in turn, provides an explanation for why the magnitude of chemical changes induced by VSC are much greater than the Rabi splitting, that is, the energy perturbation caused by VSC. These findings further confirm that VSC is powerful and versatile tool for the molecular sciences.     

Vibrational spectrum of BiH3: Six-dimensional variational calculations on high-level ab initio potential energy surfaces

We report a theoretical study of the ground electronic state of BiH₃. The potential energy surface (PES) is obtained from coupled cluster CCSD(T) calculations with a large basis set (289 contracted Gaussian functions). The previously available quartic force field (P4) is extended by adding the dominant quintic and sextic stretching terms to yield improved potential functions in symmetry coordinates (P6) and Morse-type coordinates (M4). Second-order rovibrational perturbation calculations on the P4-PES and full variational calculations on the P6-PES and M4-PES yield almost identical vibrational term values which is rationalized by considering the local mode behavior of BiH₃ and the Morse-type character of the M4-PES. The remaining deviations between the computed and observed vibrational term values must thus be caused by imperfections in the CCSD(T) surface. A refinement of this ab initio surface by a restrained fit to experimental data allows an essentially perfect reproduction of the observed vibrational term values. Variational calculations on this refined surface provide predictions for several overtone and combination bands that have not yet been observed. Dedicated to Hermann Stoll on the occasion of his 60th birthday     

Calculation of partition functions of tetra-atomic molecules

A calculation of partition functions of tetra-atomic molecules performed on the basis of the date obtained by means of the ab initio quantum mechanical method is proposed. The symmetry coordinates are used for partitioning the vibrational secular problem. The results for N₂H₂ and C₂H₂ molecules are presented. The accuracy is found to be comparable with those of results obtained using the experimentally derived structural parameters.     

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.     

Normal coordinate analysis of H8Si8O12

Normal coordinate analysis of the fundamental vibrations of H₈Si₈O₁₂ has been carried out. Because of the octahedral symmetry, the 78 vibrational degrees of freedom lead to 33 different vibrations, six of which are infrared active, 13 are Raman active and 14 are inactive. From the internal coordinates one gets 116 symmetry coordinates. We describe a straightforward method for determining the internal symmetry coordinates of any molecular system. Internal coordinates, symmetry force constants, the full set of orthonormal symmetry coordinates as well as the 38 redundant orthonormal symmetry coordinates of H₈Si₈O₁₂ are tabulated. The potential energy distribution analysis shows that most of the fundamental vibrations can be very well interpreted in terms of the internal vibrations ν(SiH), ν(SiO), δ(SiH), δ(OSiO) and δ(SiOSi) which makes it easy to compare them with vibrations observed in other silsesquioxanes and similar silicon compounds.     

Vibrational spectra and IR dichroism of poly(tetramethyl-p-silphenylene-siloxane) and its chain structure

Raman spectra of poly(tetramethyl-p-silphenylene-siloxane) were investigated in solution with depolarization ratios and in the solid state. Polarized infrared (IR) spectra were studied on oriented polymer samples. Assignments of the vibrational bands are given and IR dichroic and Raman data are in agreement with tetragonal P4₁ space group symmetry. These data provide information on the arrangement of the Si? C₆H₄? Si groups in the unit cell. On the basis of vibrational spectra the dynamic interaction between adjacent benzene rings is negligibly small and the rings preserve their local D_2h Pseudosymmetry.     

Moleküldynamik und Coriolis-Kopplungskoeffizienten einiger Hexafluoride oktaedrischer Symmetrie

The hexafluorides of sulfur, selenium, tellurium, molybdenum, technetium, ruthenium, rhodium, tungsten, rhenium, osmium, iridium, platinum, uranium, neptunium, and plutonium possessing an octaedral symmetry with the symmetry point group O_h have been briefly analyzed for the molecular structural data and the infrared absorption andRaman spectra. On the basis of group theoretical considerations, potential energy constants have been evaluated by employing a general harmonic force field. The L matrix elements connecting the internal coordinates and normal coordinates have been derived from the symmetrized force constants, G matrix elements, and vibrational frequencies. The Coriolis coupling coefficients of vibration-rotation have also been, in terms of the symmetrized force constants, G matrix elements, and vibrational frequencies, computed for these fifteen hexafluorides, and compared with the available values of experimental investigations. A brief discussion of the results follows.

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Diese Arbeit stellt einen Teil der Dissertation dar, die vonD. C. Brinkley zur Erlangung eines Master of Science-Grads an der Graduate School des Valdosta State College eingereicht wurde.     

C—H...π interactions in cocrystals of bis(trimethylsilyl)acetylene and diphenylacetylene with benzene

We present here the crystal structures of two acetylene derivatives cocrystallized with benzene, namely bis(trimethylsilyl)acetylene benzene solvate, C₈H₁₈Si₂·C₆H₆, (I), and diphenylacetylene benzene solvate, C₁₄H₁₀·C₆H₆, (II). In (I), both molecules belong to the symmetry point group C_2h and are located about special positions with site symmetry 2/m. In (II), both molecules show crystallographic inversion symmetry. In both structures, there are C—H...π contacts between aromatic H atoms and the π‐electrons of the triple bond. In addition to these, in (II) there are C—H...π contacts between aromatic H atoms and the π‐electron cloud of the benzene molecules.     

Separation of the vibrational,rotational, and translational motions of polyatomic molecules using curvilinear vibrational coordinates

A new method is suggested for separating the vibrational, rotational, and translational motions of polyatomic molecules using curvilinear vibrational coordinates that are linear with respect to the natural vibrational coordinates. It is shown that, in this case, Coriolis interactions between the vibrational and rotational motions are absent. The solutions of the anharmonic vibrational-rotational problems in the curvilinear and linear vibrational coordinates are compared. The absence of Coriolis interactions between the vibrational and rotational motions in the curvilinear vibrational coordinates is proved numerically. The same conclusion is additionally supported by calculations of the anharmonic vibrational energy levels for the H₂O, H₂S, NO₂, SO₂, and ClO₂ molecules in the linear and curvilinear vibrational coordinates using the Hamiltonian designed in the curvilinear vibrational coordinates with and without Coriolis vibrational-rotational interactions. Volgograd Pedagogical University. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 239–254, March–April, 1995. Translated by I. Izvekova     

Algorithm for variational calculations of molecular symmetry in solving anharmonic vibrational problems

A universal program for variational calculations of molecular symmetry in solving anharmonic vibrational problems, realized by the author, is described. The program uses the group-theoretical method. Symmetrized basis wave functions are constructed with the aid of the generalized KJebsch-Gordan series suggested by the author. The method of constructing symmetrized basis wave functions and the program for adequate calculations of molecular symmetry were verified for many molecules of different symmetry groups: O_h, O, T_d, T_h, T, D_∞h, C_t8v, D_nd, D_nh, D_n, C_nv, C_nh, S_2n, C_n, C_i, C_s, and C₁ where 2 ≤n ≤6. It was confirmed that the program provides correct results and high-speed operation. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 6, pp. 1146–1153, November–December, 1997.     

Selectivity of vibronic coupling in complex molecules with benzene fragments

This paper deals with the mechanisms of localization of Franck-Condon vibronic coupling of πσ^*- or πlπ^*-orbital type in a few vibrational modes, (LVM) in excited electronic states of polyatomic molecules. The analysis of vibronic coupling uses highly symmetric basis sets (for representing MO structures and normal coordinates ξ_R) as well as simplified models that relate the shift Δ_R of the electron potential minima along the normal coordinates to the MO structure and to ξ_R in the form of analytical expressions. The modes that are active in LVM are determined from the experimental luminescence spectra. These ideas about approximately high local symmetry of vibronic coupling in benzene fragments as well as the estimates of Δ_R depending on variations in the MO structure explain the experimental results. L. Ya. Karpov Physicochemical Scientific Research Institute. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 286–291, March–April, 1995. Translated by L. Smolina