Dynamics,hyperfine structure and symmetry of nonrigid molecules

The transformation properties of the Born—Oppenheimer problem and of two probability densities for electronic singlet states of semirigid molecules are reviewed, based on the isometric group concept. A definition of sets of equivalent nuclei of semirigid molecules is put forward, which will be used to derive general transformation formulae for vector operators associated with nuclear spin operators of equivalent sets. Furthermore transformation formulae for general tensor operators and for phenomenological field gradient and spin—rotation coupling tensors of semirigid molecules will be presented and illustrated by examples. The phenomenological relations will be verified by considering the transformation properties of some of the relevant terms of the molecular hamiltonian.     

Symmetry of nonrigid molecules and isomerization processes

Starting from the concept of a totally flexible molecule, rigidity is introduced by restricting progressively the allowed permutations. Each nonrigid isomer is then characterized by its Longuet-Higgins group ? of allowed permutation inversions. For a given ? all nonrigidity types correspond to all possible choices of the symmetry subgroup ? of the skeleton. The structure of ? also allows a characterization of the isomerization processes studied in chemical kinetics. In both situations, isomerization mechanisms may be associated with generators and are, in the simplest situations, represented by Cayley graphs.     

Application of the theory of symmetry of nonrigid molecules to the calculation of the conformational dependences of the torsional potential and dipole moment of acetone-related molecules

We have investigated the conformational dependences of the torsional potential and dipole moment of double-rotor molecules related to acetone, using semiempirical and ab initio calculations and expressing the results in terms of limited Fourier-series expansions. The use of the isodynamic operations of nonrigid molecules to obtain symmetry-adapted quasianalytic forms for the various properties helps compute the representative surfaces with a minimum number of points. Potential surfaces have been calculated for planar ground-state acetone (CNDO /2, STO /3G, and STO /4-31G) and both pyramidal excited-triplet acetone and ground-state dimethylamine (CNDO /2). For groundstate acetone STO /4-31G brings the results obtained with STO /3G closer to those from CNDO /2 and from experiment. The potential surface of excited-triplet acetone appears intermediate between those of ground-state acetone and dimethylamine. For dipole moments the convergence of the harmonic expansions of the vector components is slower than that of the torsional potential whereas that of the vector magnitude is faster.     

Rotational recurrences in thermal ensembles of nonrigid molecules

An approach has been developed to calculate the periods of rotational recurrences (RRs) in thermal ensembles of nonrigid molecules. Starting from the standard nonrigid rotor Hamiltonian with centrifugal distortion (CD) terms, the explicit analytical formulas were derived for CD-induced corrections to the periods of J, H, and K transients for symmetric top molecules. The corrections are shown to be proportional to the CD constants and the temperature. In addition, they depend considerably on the rotational constants of the molecule. Generally, the CDs are demonstrated to induce a shift in the periods of the RRs, which depends on the order (n=1,2,…) of RRs. Only the first few RRs exhibit n-independent shifts, and this is the situation in which our theory is exact. For higher order RRs, our approach starts to exaggerate slightly the values of the CD-induced corrections, giving their upper boundary.     

Continuous symmetry numbers and entropy

Traditionally, entropy changes are corrected for rotational permutability only if the molecule is perfectly rotationally symmetric. By this approach, only a small fraction of all known molecules must be evaluated in terms of symmetry numbers, while all other molecules are totally exempt of these considerations. A general approach which encompasses all molecules, symmetric or not, is proposed here. It is based on introducing the notion of continuity to symmetry numbers and on allowing noninteger values. In the first part of the account, we provide arguments as to why continuity is needed and what difficulties one may encounter by adopting the "black-or-white" approach to symmetry. In the second part, we provide a working methodology of how to evaluate the symmetry number content of any molecule, symmetric or not. Finally, in the third part, we demonstrate the implications of this approach on entropy issues involving melting points, Jahn-Teller distortions (of fullerene) upon ionization, molecular distortion due to overcrowdedness, permutability of isotopes, and the structure of proton sponges. It is shown that continuous symmetry numbers provide entropy values, which better agree with experimental observations, and that they are capable of identifying correlations between symmetry and physical/chemical measurables.     

Relativistic double group spinor representations of nonrigid molecules

The character theory of relativistic double group spinor representations is developed in order to represent the total rovibronic states of nonrigid molecules. It is shown that the double groups can be represented in terms of wreath products and powerful matrix cycle type generators that are used to construct their character tables. It is shown that these tables are of use when spin-orbit coupling is included in the Hamiltonian even for molecules containing lighter atoms. Applications to nonrigid molecules such as Tl2H4/Tl2H4+ are considered. It is shown that the tunneling splittings and the nuclear spin statistical weights can be obtained for such species using the character tables thus constructed. The spinor double groups of several other molecules such as hexamethyl dilead and heavy weakly bound clusters such as (PoH2)4 are also considered.     

Potential energy surface determinations for nonrigid molecules: Application to acetone

In the present work, the problem of the determination of the potential energy surface for nonrigid molecules is examined in the case of the double rotation of the methyl groups in acetone. From the symmetry adapted functional form for the potential, the minimum number of configurations to be calculated is deduced in order to have a reliable surface. With this consideration in mind, the potential energy surface of acetone is determined in some Hartree–Fock semiempirical (CNDO /2) and ab initio procedures with different standard basis sets. In addition, ab initio calculations are performed using different sets of floating Gaussian functions in order to introduce some polarization effects in the wave function. Finally, the influence of the electronic correlation effects in the barrier height, and the role of the possible relaxation of the structure during the rotation is discussed.     

The inversion eigenvalues of non-σ states of diatomic molecules,expressed in terms of quantum numbers

By adopting the phase convention of Condon and Shortley for spherical harmonics, it is shown how to determine in a standard way phases for electronic wave functions of a diatomic molecule. The method proposed is compared with the united atom method. On the basis of the phase convention introduced, the inversion eigenvalue of a non- state in a diatomic molecule is expressed in terms of quantum numbers characterizing the state in question. The inversion eigenvalue expressions are used to give an extended definition of Mulliken's c and d notation.

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Zusammenfassung Mit der Phasenkonvention von Condon u. Shortley für die Kugelfunktionen als Grundlage wird eine Standardmethode angegeben, um die Phasenfaktoren der elektronischen Wellenfunktion eines zweiatomigen Moleküls zu bestimmen. Diese Methode wird mit der Methode des vereinigten Atoms verglichen. Auf der Basis der eingeführten Phasenkonvention wird der Inversionswert eines Zustandes in einem zweiatomigen Molekül, der nicht vom -Typ ist, durch die Quantenzahlen ausgedrückt, die den vorliegenden Zustand charakterisieren. Die Ausdrücke für die Inversionseigenwerte werden benutzt, um eine erweiterte Definition von Mullikens c- und d-Bezeichnungen zu geben.

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Résumé En adoptant la convention de phase de Condon et Shortley pour les harmoniques sphériques, on montre comment déterminer d'une manière standard les phases des fonctions d'onde électroniques d'une molécule diatomique. Sur la base de la convention de phase ainsi introduite la valeur propre d'inversion d'un état non dans une molécule diatomique est exprimée en termes des nombres quantiques caractérisant l'état en question. Les expressions des valeurs propres d'inversion sont utilisées pour donner une définition étendue des notations c et d de Mulliken.

     

Spatial symmetry holes in many-electron atoms and molecules

When a many-electron system has a spatial symmetry, it is shown that there exist spatial symmetry holes, which imply that two or more electrons are prohibited from being at certain spatial positions simultaneously. Inversion holes, rotation holes, and reflection holes, which result from inversion, twofold rotation, and reflection symmetries, respectively, are discussed in detail. The electron-electron counterbalance hole reported in literature is a particular case of the inversion hole. The spatial symmetry holes are illustrated for simple atoms and molecules.     

Internal dynamics of nonrigid molecules. I. Application to acetone

The group theory for nonrigid molecules is used for studying the internal dynamics of the two equivalent C_3v rotor “bent” molecules. Special emphasis is given to the deduction of the symmetry basis vectors which represent in box form the Hamiltonian operator. It is shown that these basis vectors may be advantageously employed in order to simplify the resolution of the two-rotor equation. The procedure is applied to the acetone molecule. It is found that the lowest solutions are clustered into groups of four. The four lowest levels are related to vibrational states, the upper 64 to vibro–rotational states, in which the rotors are rotating in a restricted manner. Only few states show some cogwheel effect. Internal rotation contributions to the principal thermodynamic parameters of acetone are also computed.     

Isometric groups and chirality of nonrigid molecules: a generalization of Kelvin's theorem

A generalization of Kelvin's theorem for the chirality of rigid molecules to nonrigid molecules is given. It is based on the concept of the isometric group of nonrigid systems which allows a formulation of the chirality criterion of nonrigid systems in close analogy with Kelvin's original formulation. A number of examples, in particular such as investigated by Mislow, are given.     

Virtual geometrical symmetry group of a nonrigid molecule with energetically nonequivalent equilibrium configurations

The notion of a virtual geometrical symmetry group of a nonrigid molecule with energetically nonequivalent equilibrium configurations is introduced. Due to this, methods of qualitative analysis of intramolecular effects for such molecules using the concept of symmetry group chains are significantly generalized. Insitute of Applied Physics, Novgorod. Translated fromZhumal Struckturmoi Khimii, Vol. 37, No. 3, pp. 432–439, May–June, 1996.     

Internal symmetry groups of non-rigid molecules

Hougen has established, for quasi-rigid molecules, the relationship between permutationinversions acting on the molecular Hamiltonians written in Cartesian co-ordinates and permutation-rotations (perrotations) of symmetry acting on nuclear equilibrium configurations. We extend these relations to the case of non-rigid molecules. For this, we introduce kinetic perrotations which act on nuclear equilibrium configurations in the same way as do Altmann's isodynamic operators. We show that isodynamic operators do not always form a group. Moreover, their action cannot be extended to the electrons. They cannot be used for the classification of molecular wave functions. This classification is achieved by using the group of Longuet-Higgins and the group of the corresponding feasible perrotations.