The ortho-para conversion of H(3) (+) and H(2) in the reaction H(3) (+)+H(2)-->(H(5) (+))(*)-->H(3) (+)+H(2) in interstellar space is possible by scrambling the five protons via (H(5) (+))(*) complex formation. The product distribution of the ortho-para conversion reaction can be given by ratios of cumulative reaction probabilities (CRP) calculated by microcanonical statistical theory with conservation of energy, motional angular momentum, nuclear spin, and parity. A statistical method to calculate the state-to-state reaction probabilities for given initial nuclear spin species, rotational states, and collision energies is developed using a simple semiclassical approximation of tunneling and above-barrier reflection. A new calculation method of branching ratios for given total nuclear spins and scrambling mechanisms is also developed. The anisotropic long-range electrostatic interaction potential of H(2) in the Coulomb field of H(3) (+) is taken into account using the first-order perturbation theory in forming the complex. The CRPs and the product distribution of the ortho-para conversion reaction at very low energies with reactants in their ground vibronic and lowest rotational states for given initial nuclear spin species are presented as a function of collision energy assuming complete proton scrambling or incomplete proton scrambling. The authors show that the product distribution at very low energies (or very low temperatures) differs substantially from the high energy (or high temperature) limit branching ratios.

The spin-modification probability (SMP) method, which provides fundamental and detailed quantitative information on the nuclear spin selection rules, is discussed more systematically and generalized for reactive collision systems involving more than one configuration of reactant and product molecules, explicitly taking account of the conservation of the overall nuclear spin symmetry as well as the conservation of the total nuclear spin angular momentum, under the assumption of no nuclear hyperfine interaction. The values of SMP once calculated can be used for any system of identical nuclei of any spin as long as the system has the corresponding nuclear spin symmetry. The values of SMP calculated for simple systems can also be used for more complex systems containing several kinds of identical nuclei or various isotopomers. The generalized formulation of statistical scattering theory which can easily represent various rearrangement mechanisms is also presented.

The use of energy selected bases (ESB) with iterative diagonalization of the Hamiltonian matrix is described for vibrations of tetra-atomic systems. The performance of the method is tested by computing vibrational states of HOOH below 10,000 cm(-1) (1296 A+ symmetry states) and H(2)CO below 13,500 cm(-1) (729 A(1) symmetry states). For iterative solutions, we tested both the implicitly restarted Lanczos method (IRLM) and the standard (nonreorthogonalizing) Lanczos approach. Comparison with other contracted basis approach as well as direct product grid representation shows superior performance of the ESB/IRLM approach. Of the two systems, H(2)CO is found to be more challenging than HOOH since it has much stronger couplings among vibrational modes, which leads to a drastically larger primitive basis set. For H(2)CO we also discuss some interesting behavior of the molecule in the high internal energy regime.

Two recent papers presented calculations of the highly excited vibrational states of ozone. The nature and energies of these states may hold the key to the anomalous isotopic distribution of ozone in the atmosphere. Even though the same potential energy surface of Babikov et al. was used in both calculations, the number of bound van der Waals states reported below dissociation differed significantly. In order to resolve the issue we present here the results of an independent computation of all the bound vibrational states of (16)O(16)O(16)O and (16)O(16)O(18)O up to dissociation. Our methods differ from both earlier calculations since we use hyperspherical coordinates and a direct product discrete variable representation of the Hamiltonian. The results of present work support the existence of several van der Waals states for J=0 on this potential energy surface.

An efficient and accurate quantum method for the calculations of many large amplitude vibrational states of polyatomic molecules is proposed and tested on three triatomic molecules; H2O, SO2, and HCN. In this approach we define zero-order reduced dimensional Hamiltonians ĥk using minimum energy reduced dimensional potentials. The eigenfunctions and eigenvalues of ĥk, φn(k), and εn(k), are used to form an energy selected basis (ESB) for the full system including all the product functions Πkφn(k) for which ∑ε(k)⩽Ecut. We show that ESB can be used efficiently in an iterative solution of the Schrödinger equation by the transformation between the ESB and the direct product quadrature grid. Application of the ESB of one-dimensional basis functions is shown to be very efficient for vibrational states of H2O and SO2 up to 30 000 and 23 000 cm−1, respectively. A combined two-dimensional/one-dimensional basis is used very effectively for HCN above the isomerization energy to HNC. The present approach is shown to be substantially more efficient than either the direct product discrete variable representation (DVR) bases or compact bases from the DVR with the sequential diagonalization/truncation method.

The theoretical (quantum) description of large amplitude vibrations of systems containing four or more atoms using orthogonal internal coordinates requires three or more angular coordinates. The basis commonly used to represent these coordinates is the coupled angular momentum basis. We show that a direct product angular discrete variable representation (DVR) can be used advantageously, particularly for systems with high permutation-inversion symmetry and nonlinear equilibrium geometry. The DVR permits full symmetry projection and solution by the sequential diagonalization and truncation method. Application to the dimer of rigid CO2 demonstrates the accuracy and efficiency of the approach.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTApplication of the WKB Approximation in the Solution of the Schrödinger EquationZbigniew L. Gasyna and John C. Light View Author Information Department of Chemistry, The University of Chicago, Chicago, IL 60637-1403Cite this: J. Chem. Educ. 2002, 79, 1, 133Publication Date (Web):January 1, 2002Publication History Received3 August 2009Published online1 January 2002Published inissue 1 January 2002https://pubs.acs.org/doi/10.1021/ed079p133https://doi.org/10.1021/ed079p133research-articleACS PublicationsRequest reuse permissionsArticle Views635Altmetric-Citations2LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose SUBJECTS:Approximation,Energy levels,Oscillation,Potential energy,Schrodinger equation Get e-Alerts

We establish a quantum kinetic equation describing the transport properties of the vibrons in a molecular monolayer adsorbed on a dielectric substrate. A renormalization procedure is applied to the Hamiltonian of the system which is then separated in a vibron Hamiltonian, a bath Hamiltonian connected the external motions and a coupling Hamiltonian between the vibrons and the external modes. A perturbative analysis based on the projector method allows us to eliminate the irrelevant information related to the bath dynamics. The use of conventional approximations (Markov limit and Wick theorem) leads us to write the kinetic equation in a form exhibiting linear and nonlinear contributions. The linear term characterizes irreversible processes connected to the bath fluctuations whereas the nonlinear term represents a self-modulation of the dynamical matrix with respect to the vibron distribution. An application of the transport of CO vibrons on NaCl(100) illustrates the method.

We introduce quasirandom distributed Gaussian bases (QDGB) that are well suited for bound problems. The positions of the basis functions are chosen quasirandomly while their widths and density are functions of the potential. The basis function overlap and kinetic energy matrix elements are analytical. The potential energy matrix elements are accurately evaluated using few-point quadratures, since the Gaussian basis functions are localized. The resulting QDGB can be easily constructed and is shown to be accurate and efficient for eigenvalue calculation for several multidimensional model vibrational problems. As more demanding examples, we used a 2D QDGB-DVR basis to calculate the lowest 400 or so energy levels of the water molecule for zero total angular momentum to sub-wave-number precision. Finally, the lower levels of Ar3 and Ne3 were calculated using a symmetrized QDGB. The QDGB was shown to be accurate with a small basis.

Mo/ller operators in the formulation of reaction probabilities in terms of wave packet correlation functions allow us to define the wave packets in the interaction region rather than in the asymptotic region of the potential surface. We combine Mo/ller operators with the semiclassical propagator of Herman and Kluk. This does not involve further approximations and can be used with any initial value representation (IVR) semiclassical propagator. Time propagation in asymptotic regions of the potential due to Mo/ller operators reduces the oscillations of the propagator integrand and improves convergence of the results with respect to the number of trajectories. The effectiveness of Mo/ller operators for semiclassical reaction probability calculation is demonstrated for the collinear hydrogen exchange reaction. Full convergence is achieved and the number of classical trajectories is reduced by a factor of 10 compared to the calculation without Mo/ller operators.

We examine the effect of the diagonal Born–Oppenheimer correction on dynamics in two simple systems – the Hooke's atom in an external harmonic potential and the collinear hydrogen exchange reaction. The transmission probability for the Hooke's atom, calculated within the Born–Oppenheimer approximation, is simply shifted in energy with respect to the exact result, and this is corrected by the diagonal adiabatic contribution. The reaction probability for the H3 system reflects the fact, that the diagonal Born–Oppenheimer correction raises the barrier to the reaction by approximately 70 cm−1.

In an earlier paper [J. Chem. Phys. 111, 4869 (1999)] we introduced a quasiclassical phase space approach for generating a nearly optimal direct-product basis for representing an arbitrary quantum Hamiltonian within a given energy range of interest. From a few reduced-dimensional integrals, the method determines the optimal one-dimensional marginal Hamiltonians, whose eigenstates comprise the direct-product basis. In this paper the method is applied to three-body vibrational systems expressed in radial and angular coordinates. Numerical results are obtained for the bound state eigenenergies of the nonrotating HCO molecule, determined to ∼0.01 cm−1 accuracy using a phase space optimized direct-product basis of 1972 functions. This represents a computational reduction of several orders of magnitude, in comparison with previous calculations.

We present a simple method of calculation of the stability (monodromy) matrix that enters the widely used semiclassical propagator of Herman and Kluk and almost all other semiclassical propagators. The method is based on the unitarity of classical propagation and does not involve any approximations. The number of auxiliary differential equations per trajectory scales linearly rather than quadratically with the system size. Just the first derivatives of the potential surface are needed. The method is illustrated on the collinear H3 system.

The self-diffusion of hydrogen on the (100) copper surface is investigated using a quantum kinetic equation approach. The dynamics of the adatom is described with a multiple-band model and the surface phonons represent the thermal bath responsible for the diffusion mechanism. Using the Wigner distribution formalism, the diffusive motion of the adatom is characterized in terms of the correlation functions of the adatom–phonon interaction. The diffusion coefficient exhibits two terms related to phonon mediated tunneling (incoherent part) and to dephasing limited coherent motion (coherent part). The competition between these two contributions induced a transition from a thermally activated regime to an almost temperature independent regime at a crossover temperature T*. A numerical analysis is performed using a well-established semiempirical potential to describe the adatom–surface interaction and a slab calculation to characterize the surface phonons. These calculations show that two-phonon processes represent the relevant contribution involved in the adatom–phonon coupling. The temperature dependence of the diffusion constant is thus presented and the relative contribution of the incoherent versus the coherent part is analyzed. Both contributions exhibit a change of behavior around 100 K from an exponential to a power law temperature dependence as the temperature decreases. This change is due to the confinement of the motion of the adatom in the ground energy band at low temperature. The incoherent part is shown to be the dominant contribution at high temperature and is characterized by an activation energy and a prefactor equal to ΔE=0.49±0.01 eV and D0≈2.44×10−3 cm2/s, respectively. At low temperature, the power law dependence of the two contributions is different since the coherent part increases slowly as the temperature decreases whereas the incoherent part decreases. The crossover temperature is estimated to be equal to T*=125 K. Below T*, the coherent part becomes the main contribution and the diffusion constant exhibits an almost temperature independent behavior.

This chapter contains sections titled: Introduction and history “Pointwise“ representations in one dimension Multidimensional DVRs and applications Caveats Summary and conclusions