Hamiltonian for rovibrational spectra of “hot” molecules

A variational method based on the determination of rotation as a state with a constant angular momentum has been proposed for calculating rovibrational energy levels of a polyatomic molecule. By using this method, energy level values can be determined for any vibrational state with arbitrary values of vibrational quantum numbers. This makes it possible to calculate the rovibrational energy levels of the “hot” molecules.     

General methods of analysing molecular vibrations

An acute need may arise to develop for the complete analysis of molecular vibrations practically convenient general methods based on coordinates other than “chemical coordinates”. One reason is the proven proposition: Among independent internal coordinates corresponding to a molecule, there cannot be one which describes a small displacement of a chemical group as a whole relative to a certain molecular plane, provided this group contains more than two linearly or three non-linearly arranged atoms. Two methods are presented in some detail. The first is based on the use of X⁰_δ coordinates which are components of “bond vectors” in the “sown” (for each “bond”) Cartesian coordinate system. The second method utilizes X⁰ coordinates, i.e. the components of atomic displacements in the “down” (for each atom) Cartesian coordinate system. Computation of the torsional vibration of transdichloroethane is given as an example illustrating the first method. The Mayants treatment of the symmetry of a molecule, proceeding from elementary considerations which do not use the group theory explicitly and are valid for any coordinates, is expounded in a somewhat improved version. The peculiarities arising when considering the mean-square amplitude matrix, Σ, in X⁰_δ and X⁰ coordinates are also discussed.     

A simple method for derivation of the rovibrational Hamiltonian: application to orthogonal radau coordinate systems as special cases

The molecular Hamiltonian of polyatomic molecules has been obtained. A general choice of internal coordinates depending on external parameters was considered. The rovibrational Hamiltonian for this set of coordinate system was derived in general terms as a function of the external parameters a and b. This procedure is also applicable to various kinds of internal coordinates in a straightforward way. The rovibrational Hamiltonian of triatomic molecules is considered as an application of this general formulation. In addition, orthogonal Radau coordinates are considered as cases of this new approach     

Two alternative criteria for labelling approximate treatments for molecules as “fully quantal” ones

If the confinement of a trial wavefunction on a subspace of the full Hilbert space of a systems is taken as a “constraint” that prevents an approximate method to be labelled as a “fully quantum” one, then it can be shown that practically all known molecular approximate method are not rigorously quantal ones. Another weaker (and already known) criterion used by nuclear physicists for such a labelling is discussed and illustrated for the generator coordinate method (in this case a quantum approach) and Hartree—Fock method (a semiclassical approach. On the other hand, although Lathouwers' treatment of molecular spectra may be labelled “fully quantum” (for certain molecules at least) under this softer criterion, it will be remarked that the loss of the variational upper-bound property attenuates much of its theoretical attractiveness.     

An algorithm for R matrix calculations for atom–diatom reactive scattering

Some aspects of R matrix theory that derive from a finite element method implementation of a hyperspherical coordinate formulation of the quantum mechanical three-body problem are analyzed with a view of designing suitable algorithms. Propagation of the R matrix from a hyperspherical surface to asymptotically appropriate wave fronts is given special attention.     

On the “New possibility of chemical bonding”: Anti-resonance phenomena

A chemical bond in the absence of a local minimum in the potential surface has been obtained recently by Pollak, Manz, Meyer and Römelt for the I-HI complex. We associate this “new chemical bond” with Simon's proof that a discrete spectrum can be obtained even for a quantum Hamiltonian for which the volume |(P, q)|P² + V(q) ? El is infinite and therefore an infinite number of classical trajectories lead to dissociation (an “anti-resonance” phenomenon). A simple model Hamiltonian which yields for any given E an infinite number of classical dissociative trajectories and also has a discrete spectrum in quantum mechanics, is presented.     

Location of energy barriers. VII. Sudden and gradual late-energy-barriers

In the earlier papers of this series it was noted that barriers of “type II” (“Late barriers”, associated with substantially endothermic reaction) exhibited a large cross section if a given reagent energy was present predominantly as vibration in the bond under attack. It was found, however, that it was advantageous to retain in the reagents an amount of translation sufficient to surmount that part of the endothermic energy barrier which lay in the coordinate of approach; this could be identified with the repulsive part of the energy-release in the reverse, exothermic, reaction. In the present 3D classical trajectory study we compare S_r(V′)_T′+V′, for two endothermic surfaces with almost identical barrier heights and late barrier-crest locations, but differing fractions of the barrier in the coordinate of approach. For the surface IIS with a “sudden” rise to the barrier crest (implying substantial attractive energy-release in the reverse direction), S_r(V′_T′+V′ increased continually with V′, despite very low T′. By contrast surface IIG with “gradual” rise to the barrier crest (highly repulsive in the reverse direction), exhibited an S_r(V′_T′+V′ that passed through a maximum. Two further surfaces were investigated; surface IIHS resembling IIS but with a substantially higher barrier, and surface I, IIG with an intermediate barrier-location and hence a large fraction of the barrier along the coordinate of approach. The dynamics exhibited S-type (sudden) behaviour on IIHS, and G-type (gradual) behaviour on I, IIG. The presence of a significant fraction of the barrier on G-type surfaces along the endothermic coordinate of approach correlated with a more gradual curvature of the repulsive wall. This is likely to contribute to the greater availability of translation for barrier-crossing on G-type surfaces. Increased reagent rotation, R′, increased the reactive cross-section for endothermic reaction; the indications are that the mechanism by which rotation is effective involves vibration-rotation interaction.     

Substituent effect on the chemiluminescence quantum efficiency of some acridan derivatives

Eight 9-benzylidene-N-methylacridan derivatives were synthesized and their chemiluminescence quantum yields on reaction with NaOCl-H₂O₂ or O₃ inN,N-dimethylformamide were plotted versusHammett's “σ” values. A good linear fit was obtained, the higher quantum yields being associated with negative “σ” values. Two acridans did not fit in the plot and a novel treatment of chemiluminescence quenching is outlined to account for the discrepancy.     

On the origin of the occurrence of “magic numbers” in cluster size distributions of xenon in the compressed gas phase

The formation of clusters in compressed argon and xenon gases is studied with the molecular dynamics simulation technique using two-body Lennard-Jones and three-body Axilrod-Teller potentials. It is suggested that the occurrence of clusters corresponding to the “magic number” n = 13 for xenon and the absence of such stable clusters for argon is due to the dispersion forces that result from triple-dipole interactions.     

Resonance by the complex coordinate method with hermitean hamiltonian

Using the complex coordinate method, a new hermitean hamiltonian is formulated, in which the resonance position and width are the “natural” perturbation strength parameters. If the resonance position is known the resonance width obtained by the presented theory is a lower bound to the exact value.     

Molecular shapes and molecular structures

The first part is a response to E.B. Wilson's remarks about molecular structure at the 1979 Sanibel Symposium; his proposed extension of the notion of “potential energy surface” is a convention, not a part of fundamental molecular quantum theory. An energy surface has no essential role in a non-adiabatic theory which, both in principle and practice, can account for the spectroscopic term formulae with greater accuracy than models based on PE surfaces. Results obtained from the generator coordinate method are cited as confirmation of this conclusion. Claverie and Diner's distinction between “quantum structure” and “classical molecular structure” is discussed; it is the latter that is problematic in molecular quantum theory. The use of molecular structure is optional, and determined by utility, rather than essential in chemical physics; photoelectron spectroscopy is an example where this appreciation has proved fruitful.     

On the description of high-resolution experiments in molecular physics

We argue that all high-resolution experiments on small molecules some of which currently “belong” to chemistry and others to physics, should be brought together and discussed in a unified way using the best theory available, namely quantum mechanics, and that preconceptions about “molecular structure” should be avoided in this area of physical science.     

Theoretical temperature dependence of relaxation rate between widely spaced levels,due to gas or liquid fluctuations

In describing relaxation of a small system whose quantum level spacing ?_μ1, exceeds k_BT, one must treat the heat bath quantally even when it is a “classical” fluid. A simple gas model illustrates this, with possible implications for liquids.     

Selection rules for classical reactive trajectories: Dynamical allowedness and forbiddenness

A simple model for a particle climbing an inclined plane limited by rigid walls, towards an exit symbolizing the entrance of a product valley, demonstrates the general existence of reactive and unreactive bands. The system can be described by a single parameter η. The dynamically “allowed” or “forbidden” outcome of the reaction is shown to depend on the dynamical number N which characterizes the number of rebounds of the particle on the rigid walls. For given total energy, each value of N defines a reactive band which is accessible only for certain intervals of the initial energy in the direction perpendicular to the reaction coordinate. Depending on the value of the parameter η, the onset energy for reactive bands can vary either quadratically with N or linearly with N.     

Classical trajectory study of three-body direct photofragmentation of Cd(CH3)2. Comparison of sampling methods

The classical trajectory method has been used to investigate product-state symmetry in the three-body direct photofragmentation of Cd(CH₃)₂. Photon absorption was assumed to satisfy the Franck principle and relative weights for trajectory initial conditions were assigned by either the Wigner or the classical (300 K) distribution function. In qualitative agreement with earlier “quasi-classical” sampling studies of Kellman et al. for photolysis of the Cd(CH₃)₂ vibrational ground state, product-state asymmetry is found to increase with increasing photolysis wavelength. Contrary to the “quasi-classical” sampling results, however, sampling with the classical distribution function indicates that product-state symmetry is unaffected by vibrational excitation of Cd(CH₃)₂ prior to photolysis. For sampling with the Wigner distribution function, in agreement with the “quasi-classical” sampling studies, excitation of the Cd(CH₃)₂ asymmetric stretch prior to photolysis is found to increase product-state asymmetry.     

Quantum effects in semiclassical treatment of molecular reactions

By the use of the known Coulomb wavefunctions, “quantal” Coulomb streamlines are defined for the eikonal of the system. Comparisons of such streamlines with the classical Coulomb trajectories are made. Use of such “quantum” Coulomb streamlines is proposed for reaction systems containing the Coulomb interaction U. The derived equations of motion are of particular use for cases where the residual potential V₁ (V₁ = V ? U) has either no classical forbidden region or has relatively small effects on trajectories.     

The semiclassical and the quasiclassical partition function of a quartic anharmonic oscillator

After a substitution a known Laplace-type integral is used to derive quantum corrections to the classical partition function of a quartic anharmonic oscillator in the framework of the Wigner—Kirkwood perturbation expansion. By straightforward calculations results are given in a closed form allowing the analytical formulation of the thermodynamic functions H, E, S, C_υ. The numerical results agree for arbitrary anharmonicity and for high and intermediate temperatures with the numerical partition function calculated from the Hioe—Montroll eigenvalues. Furthermore, the same integral type is used for the analytical calculation of a “quasiclassical” partition function and of “quasiclassical” moments. In the trace formulation of the partition function all commutators are neglected. The harmonic oscillator density matrix is applied to the evaluation of the truncated trace expressions. The “quasiclassical” partition function is an exact upper bound and lies always below the classical partition function.     

FR-IR spectroscopic studies of polyurethanes—Part II. Ab initio quantum chemical studies of the relative strengths of “carbonyl” and “ether” hydrogen-bonds in polyurethanes

Ab initio quantum chemical computations were carried out on (a) dimethyl ether, (b) N-methyl formamide, (c) dimethyl ether-N-methyl formamide complex, and (d) N-methyl formamide dimer to compute the strengths of hydrogen bonds (H-bonds) between the NH groups and CO and ether COC groups. The basis set used was the 3-21G set of the GAUSSIAN 80 program obtained from QCPE, Bloomington, IN. Variations in the strengths of these two H-bonds with the N . . O distance (where O is either carbonyl or ether group oxygen) were studied and found to be similar in behavior. The strength of the “ether” hydrogen bond is computed to be 10.32 kcal mol⁻¹, which is quite significant compared to the value of 10.11 kcal mol⁻¹ for the more accepted “carbonyl” hydrogen bond. The “ether” hydrogen bond is found to be directional, specific and non-negligible. Work with two more basis sets has indicated that the results so obtained are not dependent on their choice. Possible importance of such a hydrogen bond in polyurethanes, inhalation anesthetics, and depsi-peptides is indicated.     

Zur Definition von Selektivität,Spezifität und Empfindlichkeit von Analysenverfahren

The three metric, functional concepts named in the title can be defined without arbitrariness and their values may be calculated for any multicomponent-analytical procedure from the relation of the measurable physical quantities x _ito the contents c _kwhich are to be determined. This relation is mathematically a “mapping” (achieved by systems of functions). The system of “analytical functions” (x _i→ c _k) is the inverse of the system of “calibration functions” (c _k→ x _i) which alone can be directly gained by experiments. The ‘calibration matrix” (γik) whose elements γ_ik are the “partial sensitivities” \(\tfrac{{\partial x_i }}{{\partial c_k }}\) represents the system in first approximation but only locally, i.e. for the respective constitution of the sample. The “sensitivity” of the analytical procedure as a whole, is numerically given by the determinant of this matrix; the “selectivity” is derived from the condition that the inversion of the calibration system to the analytical system (of functions) shall be possible by an iteration process. “Specificity” is defined in analogy to selectivity. In the appendix it is explained why selectivity is the strongest means to reduce the expenditure for the complete calibration of multicomponent analyses to a realistic and tolerable degree.     

Rovibrational interactions in linear triatomic molecules: a theoretical study in curvilinear vibrational coordinates

Variational solution of the rovibrational problem in curvilinear vibrational coordinates has been implemented and used to investigate the nuclear motions in several linear triatomic molecules, like HCN, OCS, and HCP. The dependence of the rovibrational energy levels on the rotational quantum numbers and the l-doubling has been studied. Two approximations to the rovibrational Hamiltonian have been examined, depending on the level of truncation of the potential energy operator. It turns out that the truncation after the fifth order in the potential is sufficient to produce vibrational energies of high accuracy. An interesting feature of the present formulation of the problem in terms of the curvilinear vibrational coordinates is the explanation of the l-doubling of the rovibrational levels, which in this picture is interpreted as the result of the inequivalency of the average rotational constants in mutually perpendicular planes, rather than as the effect of the Coriolis-type interactions between the vibrational and rotational motions. The present theoretical results are compared with the available experimental data from high-resolution spectroscopy, as well as with other ab initio calculations.