A JWKB analysis of the sextic anharmonic oscillator in d dimensions

On the basis of a radial generalization of the JWKB quantization rule, which incorporates higher orders of the approximation, an explicit analytical formula is derived for the energy levels of the three-dimensional sextic anharmonic oscillator. The formula exhibits the scaling property of the exact eigenvalues, and is readily generalized to any dimension. The predicted results are in good agreement with known numerical values.     

Multidimensional tunneling dynamics of proton transfer in malonaldehyde molecule and its isotopomers

The twenty-one-dimensional Hamiltonian of malonaldehyde molecule and a number of its isotopomers (H/D, ¹³C/¹²C) was reconstructed in the low-energy region (<3000 cm^–1). Parameters of the Hamiltonian were obtained from quantum-chemical calculations of the energies, equilibrium geometries, and eigenvectors and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and transition state. Despite substantial variation of the barrier height calculated using different quantum-chemical methods (from 2.8 to 10.3 kcal mol^–1), the corresponding potential energy surfaces can be matched with high accuracy by scaling only one parameter (the semiclassical parameter , which defines the scales of potential, energy, and action). Scaling invariance allows optimization of the Hamiltonian in such a way that the calculated ground-state tunneling splitting coincides with the experimental value. The corresponding potential barrier height is estimated at 4.34±0.4 kcal mol^–1. The quantum dynamics problem was solved using the perturbative instanton approach without reducing the number of degrees of freedom. The role of all transverse vibrations in proton tunneling is characterized. Vibration-tunneling spectrum is calculated for the ground state and low-lying excited states and mode-specific isotope effects are predicted.     

Optimized Tensor-Product Approximation Spaces

This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard Sobolev spaces. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finite element subspaces with a lower dimension than the standard full-grid spaces. These new approximation spaces preserve the approximation order of the standard full-grid spaces, provided that certain additional regularity assumptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functions. We show in which cases the so-called curse of dimensionality can be broken. The theory covers elliptic boundary value problems as well as boundary integral equations. September 17, 1998. Date revised: March 5, 1999. Date accepted: September 20, 1999.     

Charmonium and upsilon systems with fine-hyperfine splittings

We show that both charmonium and upsilon spectra can be simultaneously fitted by an effective non-coulombic power-law potential with lowest order relativistic corrections. The fine-hyperfine splittings of these spectra can be well understood without considering the short distance coulombic part of the potential as suggested byqcd, if the Lorentz structure of the central potential is taken as an almost equal admixture of vector and scalar gluon exchanges. With this model we make several specific predictions for the upsilon spectra which could hopefully be tested in the near future.     

Calculated properties of the active site of oxidized rubredoxins

For a molecular model of the Fe-S active site complex in oxidized rubredoxin, we have calculated the spin-orbit coupling between the ground sextet state and excited quartet and doublet states which gives rise to the observed zero field splitting of the sextet ground state into three spin-mixed Kramers doublets. Additionally, we have used the six spin-mixed sextet state components to calculate effective magnetic moments, magnetic field energies and nine g values corresponding to transitions between the three pairs of Kramers doublets in applied magnetic fields along three perpendicular axes. We have calculated these properties for eight conformational variations of the ligands around the Fe at the active site. The results of these calculations clearly show the origin of the observed g=4.3 signal previously described only in terms of the phenomenological spin-Hamiltonian formalism. For the eight conformations considered, five have this characteristic signal. Zero field splitting comparable to the observed values could be obtained for all symmetries studied. In addition, the calculated values of magnetic moment in all symmetries correspond to that of high spin ferric ion and do not vary appreciably with temperature above 77° K, in agreement with experimental results. From comparison of all our calculated results with experiment, it appears that the active site in oxidized rubredoxins could have small conformational variations in different rubredoxins and under the various experimental conditions used.     

Six-dimensional tunneling dynamics of internal rotation in hydrogen peroxide molecule and its isotopomers

The six-dimensional torsion-vibration Hamiltonian of the H₂O₂ molecule and its H/D- and ¹⁸O/¹⁶O-isotopomers is derived. The Hamiltonian includes the kinetic energy operator, which depends on the tunneling coordinate, and the potential energy surface represented as a quartic polynomial with respect to the small-amplitude transverse coordinates. Parameters of the Hamiltonian were obtained from DFT calculations of the equilibrium geometries, eigenvectors, and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and both the cis- and trans-transition states, carried out with the B3LYP density functional and 6-311+G(2d,p) basis set. The quantum dynamics problem is solved using the perturbative instanton approach generalized for the excited states situated above the barrier top. Vibration-tunneling spectra are calculated for the ground state and low-lying excited states with energies below 2000 cm^–1. Strong kinematic and squeezed potential couplings between the large-amplitude torsional motion and bending modes are shown to be responsible for the vibration-assisted tunneling and for the dependence of tunneling splittings on the quantum numbers of small-amplitude transverse vibrations. Mode-specific isotope effects are predicted.     

Extended study on a quasi-exact solution of the Bohr Hamiltonian

A unified presentation of the quasi-exact solutions of the Bohr Hamiltonian with a sextic oscillator potential is offered by evidencing the similarities and the differences between them. An extended study of the Bohr Hamiltonian with sextic potential for γ$\gamma$-rigid triaxial nuclei is conducted, focusing on the evolution of the energy spectra and B(E2)$B\left(E2\right)$ transition probabilities when the order of the quasi-exactly solvable system is increased. The effect is analyzed throughout the shape phase transition from a γ$\gamma$-rigid triaxial vibrator to an anharmonic one and especially in its critical point. The new results are tested by several applications for experimental data, improving the previous calculations and providing new possible triaxial nuclei.     

Tunneling dynamics of internal rotation in the nitric acid molecule

The Hamiltonian of internal rotation about theC ₂ axis in the HNO₃ molecule and its H/D-, O¹⁸/O¹⁶-, and N¹⁵/N¹⁴-isotopomers was reconstructed using the results of quantumchemical calculations. The Fermi resonance between the torsional (2v₉) and ONO bending (v₅) vibrations is a characteristic feature of the molecule. Tunneling splittings in the ground and excited states were calculated using the perturbative instanton approach. Abnormally large changes in the splittings upon isotope substitution of heavy atoms are predicted. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2052–2060, November, 1999.     

An ab initio study of the structures and energetics of the planar ground and 90°‐twisted excited states of substituted ethylenes

Ab initio molecular orbital calculations are performed on the planar ground states (S₀), the 90°‐twisted triplet (T₁), and pyramidalized singlet (S₁) excited states of ethylene, methaniminium cation (MC), monocyano‐ (MCE), 1,1‐dicyano‐ (DCE), 1,1‐dihydroxy‐ (DHE), and 1,1‐dicyano‐2,2‐dihydroxy (DCHE) ethylenes. Equilibrium geometries are optimized at the Hartree–Fock (HF) level with the 6‐31G* basis set. Electron correlation corrections are estimated by optimizing the HF/6‐31G* geometries at the (U)MP2/6‐31G* level and then by carrying out single‐point calculations at the fourth‐order Møller–Plesset perturbation theory ((U)MP4/6‐311G**//MP2/6‐31G*). The effects of various types of perturbations on the structures, energetics, dipole moments, and state ordering of S₀, S₁, and T₁ are carefully investigated. “Positive” S₁–T₁ splittings are estimated at the HF level for all studied molecules, while “negative” S₁–T₁ splittings are obtained at the MP2 level for MC, DHE, and DCHE. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 242–254, 2001     

High distance Heegaard splittings via fat train tracks

We define fat train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in S³.