Modelling rotations,vibrations, and rovibrational couplings in astructural molecules – a case study based on the H+5 molecular ion

One-dimensional (1D) and two-dimensional (2D) models are investigated, which help to understand the unusual rovibrational energy-level structure of the astronomically relevant and chemically interesting astructural molecular ion H⁺₅. Due to the very low hindering barrier characterising the 1D torsion-only vibrational model of H⁺₅, this model yields strongly divergent energy levels. The results obtained using a realistic model for the torsion potential, including the computed (near) degeneracies, can be rationalised in terms of the model with no barrier. Coupling of the torsional motion with a single rotational degree of freedom is also investigated in detail. It is shown how the embedding-dependent rovibrational models yield energy levels that can be rationalised via the 2D vibrational model containing two independent torsions. Insight into the complex rovibrational energy level structure of the models and of H⁺₅ is gained via variational nuclear motion and diffusion Monte Carlo computations and by the analysis of the wavefunctions they provide. The modelling results describing the transition from the zero barrier limit to the large barrier limit should prove to be useful for the important class of molecules and molecular ions that contain two weakly coupled internal rotors.     

Full-dimensional vibrational calculations for H5O2+ using an ab initio potential energy surface

We report quantum diffusion Monte Carlo (DMC) and variational calculations in full dimensionality for selected vibrational states of H(5)O(2) (+) using a new ab initio potential energy surface [X. Huang, B. Braams, and J. M. Bowman, J. Chem. Phys. 122, 044308 (2005)]. The energy and properties of the zero-point state are focused on in the rigorous DMC calculations. OH-stretch fundamentals are also calculated using "fixed-node" DMC calculations and variationally using two versions of the code MULTIMODE. These results are compared with infrared multiphoton dissociation measurements of Yeh et al. [L. I. Yeh, M. Okumura, J. D. Myers, J. M. Price, and Y. T. Lee, J. Chem. Phys. 91, 7319 (1989)]. Some preliminary results for the energies of several modes of the shared hydrogen are also reported.     

Characterizing excited states of CH5+ with diffusion Monte Carlo

The spectroscopy and dynamics of protonated methane have been of long-standing interest due to the unusual and highly fluxional behavior of CH5+. This reflects the fact that the ground-state wave function for CH5+ has nearly equal amplitude at the 120 equivalent minima and at the saddle points that connect these minima. While low-resolution spectra of CH5+ have been assigned, the nature of the couplings between the CH stretches and the low-frequency modes is not as well characterized. An understanding of this will be important in the interpretation of rotationally resolved spectra. In this work, fixed-node diffusion Monte Carlo techniques are used to calculate energies and probability amplitudes for several excited states. The calculated energies are shown to be in good agreement with previously reported vibrational configuration interactions calculations. Analysis of the 12-dimensional probability amplitudes shows that there are strong couplings between the high-frequency CH stretch and HCH bend motions and the low-frequency modes that lead to isomerization CH5+.     

Combined experimental/theoretical investigation of the He+ICl interactions. I. Rovibronic spectrum of He...ICl complexes in the ICl B-X, 3-0 region

Transitions of two different stereoisomers of the He...ICl(X,v" = 0) weakly bound complex, one with a T-shaped orientation and another that is most likely linear, have been observed in laser-induced fluorescence experiments performed in the ICl B-X region. Here we present experimental and theoretical results aimed at confirming the previous assignments and at gaining additional insights into the He+ICl interactions. High resolution action spectra were recorded in the same region to identify those features that could be attributed to transitions of the He...I35Cl(X,v" = 0) isomers and not to higher-order complexes, Hen...I35Cl, where n > or = 2, or I37Cl containing species. Calculations of the rovibronic spectra of the He...I35Cl complexes in the ICl B-X, 2-0 and 3-0 regions were performed using an ab initio potential energy surface for the He+ICl(X,v" = 0) ground state and two different pairwise additive potentials for the He+ICl(B,v' = 2,3) excited states. The rotation-vibration energies and wave functions for the He cdots, three dots, centered I35Cl complexes were obtained for all bound states with total angular momentum J < 10 using both of these potentials. Electronic spectra were generated using these results, assuming that the transition moment lies along the ICl bond and is not perturbed by the presence of the helium atom. The calculations qualitatively reproduce the He cdots, three dots, centered I35Cl action spectrum and strongly support the previous assignments. The calculations also indicate that some of the spectral congestion observed near the linear band may be attributed to transitions of the linear isomer to multiple intermolecular levels in the excited state. Coriolis coupling strongly mixes He cdots, three dots, centered ICl(B,v') states with rotational excitation, making simulations and assignments of the linear band observed in the experimental spectrum difficult.     

NONCOOPERATIVE MANAGEMENT OF THE NORTHEAST ATLANTIC COD FISHERY: A FIRST MOVER ADVANTAGE

The point of departure for this analysis is Bjørndal and Lindroos [2012], who developed an empirical bioeconomic model to analyze cooperative and noncooperative management of Northeast Atlantic cod. In their analysis, only constant strategies were analyzed for noncooperative games. In this paper, nonconstant strategies are considered. Moreover, the fishery in question is characterized by cooperative management. What may happen in the real world is that one nation breaks the cooperative agreement by fishing in excess of its quota. Often, it takes time for the other agent to detect this and respond. In this paper, we allow this kind of delayed response into a two‐agent noncooperative game so that, if country 2 exceeds its quota, there will be a time lag before this is detected by country 1; moreover, there may also be a delay until country 1 is able to respond. Results show that the outcome critically depends on the length of these two lags as well as initial conditions.     

Structures of protein-protein complexes are docked using only NMR restraints from residual dipolar coupling and chemical shift perturbations

NMR structures of protein-protein and protein-ligand complexes rely heavily on intermolecular NOEs. Recent work has shown that if no significant conformational changes occur upon complex formation residual dipolar coupling can replace most of the NOE restraints in protein-protein complexes, while restraints derived from chemical shift perturbations can largely replace intermolecular NOEs in protein-ligand structures. By combining restraints from chemical shift perturbations with orientation restraints derived from measurements of residual dipolar couplings, we show that the structure of the EIN-HPr complex can be calculated without NOE restraints. The final structure, built from the crystal structures of EIN and HPr in their uncomplexed form and docked only with NMR restraints, places HPr within 2.5 A of the position determined from the mean NMR structure of the complex.     

Unraveling anharmonic effects in the vibrational predissociation spectra of H5O2(+) and its deuterated analogues

The nature of anharmonic couplings in the H(5)O(2)(+) "Zundel" ion and its deuterated isotopologues is investigated through comparison of their measured and calculated vibrational spectra. This follows a recent study in which we reported spectra for H(5)O(2)(+), D(5)O(2)(+), and D(4)HO(2)(+) from ～600 to 4000 cm(-1), as well as H(4)DO(2)(+) in the OH and OD stretching regions [ J. Phys. Chem. B 2008 , 112 , 321 ]. While the assignments of the higher-energy transitions associated with the fundamentals of the exterior OH and OD motions are relatively straightforward, difficulties arise in the assignment of the lower-frequency regions that involve displacement of the bridging proton, especially for the isotopically mixed species. Here we revisit the Ar-tagged isotopomers, and report the low energy action spectrum of H(4)DO(2)(+) for the first time, as well as present significantly improved spectra for the D(4)HO(2)(+) and D(5)O(2)(+) systems. Band assignments are clarified in several cases using IR-IR hole-burning. We then investigate the physical origin of the anharmonic effects encoded in these spectra using a recently developed technique in which the anharmonic frequencies and intensities of transitions (involving up to two quanta of excitation) are evaluated using the ground state probability amplitudes [ J. Phys. Chem. A 2009 , 113 , 7346 ] obtained from diffusion Monte Carlo simulations. This approach has the advantage that it is applicable to low-symmetry systems [such as (HDO)H(+)(OH(2))] that are not readily addressed using highly accurate methods such as the multiconfigurational time-dependent Hartree (MCTDH) approach. Moreover, it naturally accommodates an intuitive evaluation of the types of motion that contribute oscillator strength in the various regions of the spectrum, even when the wave function is intrinsically not separable as a product of low-dimensional approximate solutions. Spectra for H(5)O(2)(+), D(5)O(2)(+), H(4)DO(2)(+), and D(4)HO(2)(+) that are calculated by this approach are shown to be in excellent agreement with the measured spectra for these species, leading to reassignments of two of the bands in the intramolecular bending region of D(4)HO(2)(+).     

Trivial, strongly minimal theories are model complete after naming constants

We prove that if is any model of a trivial, strongly minimal theory, then the elementary diagram is a model complete -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is .

     

Simple and immune relations on countable structures

 Let 𝒜 be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of 𝒜 in which the image of R (?R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on 𝒜 and R, a necessary and sufficient condition for the existence of a computable copy ℬ of 𝒜 in which the image of R (?R, resp.) is simple (immune, resp.). Received: 4 February 2001 Published online: 5 November 2002 RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899. RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899. RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899.