Harmonic oscillator tensors. I. The nondegenerate case

It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J₊, J₀, J_?} of LASU (2). The Hamiltonian of the system is proportional to J₀, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.     

On quantum groups and their potential use in mathematical chemistry

The quantum algebrasu _q (2) is introduced as a deformation of the ordinary Lie algebrasu(2). This is achieved in a simple way by making use ofq-bosons. In connection with the quantum algebrasu _q (2) we discuss theq-analogues of the harmonic oscillator and the angular momentum. We also introduceq-analogues of the hydrogen atom by means of aq-deformation of the Pauli equations and of the so-called Kustaanheimo Stiefel transformation.     

Minimum uncertainty wavelets in non-relativistic super-symmetric quantum mechanics

We consider the connection to the harmonic oscillator, super-symmetric quantum mechanics (SUSY-QM) and coherent states of the recently derived constrained Heisenberg “minimum uncertainty” (μ-) wavelets [Phys Rev Lett 85:5263 (2000); Phys Rev A65: 052106-1 (2002); J Phys Chem A107:7318 (2003)]. We analyze several new types of raising and lowering operators,which also can be viewed as arising from a (non-unitary) similarity transformation of the Harmonic Oscillator Hamiltonian and ladder operators. We show that these new ladder operators lead to a new SUSY formalism for harmonic oscillation, so that the μ-wavelets naturally manifest SUSY properties. Using these new ladder operators, we construct coherent and supercoherent states for the oscillator. In the discussion, we consider possible implications of similarity transformations for quantum mechanics. In an appendix we consider the classical limit of the μ-wavelet oscillator.     

On the time‐dependent solutions of the Schrödinger's equation. II. The one‐mode field perturbed harmonic oscillator

The solution of the time‐dependent Schrödinger's equation for a perturbed harmonic oscillator is obtained using a solvable Lie algebra. We choose a harmonic oscillator interacting with a one‐mode field, where the perturbation happens to be periodic in time. This leads to one of the simplest Floquet problems. Using the Wei–Norman theorem, the Floquet wave function is obtained as well as the semiclassical Floquet shift in the energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Symmetry functions adapted to the subgroup chain U(7) ⊃ SO(7) ⊃ G2 ⊃ SO(3) ⊃ G

Summary In this paper the Lie algebra technique is used to construct symmetry functions adapted to the subgroup chain U(7) SO(7) G ₂ SO(3) G, which is one of symmetry group chains appearing in the weak ligand field scheme for f ^N ions. The functions are expressed in terms of the Gelfand states.     

The Wigner function of the rotating Kratzer oscillator mapped onto the Morse oscillator

In this article, the rotating Kratzer oscillator in quantum phase space is studied. The Langer transformation is used to map the Kratzer oscillator with centrifugal term onto a one‐dimensional Morse oscillator. As a result, the Wigner distribution functions for the Morse oscillator are obtained. The quantum states of the system are visualized in the phase space for a few vibrational and rotational quantum numbers. The results obtained in the phase space correspond to those derived in the standard quantum theory. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Application of the Lie algebraic approach to diffractionally and rotationally inelastic molecule–surface scattering

The Lie algebraic approach of Alhassid and Levine [Phys. Rev. A 18 , 89 (1978)] is applied to the molecule–surface scattering. Specially, the diffractionally and rotationally inelastic scattering of a diatomic molecule from a solid surface is dealt with. Within the framework of the close-coupling method, we construct a Hamiltonian for the scattering system and use it to generate a dynamical algebra h₆. By solving equations of motion for the group parameters, the scattering wave functions near the surface are obtained. Computed transition probabilities of diffractively and rotationally inelastic scattering of H₂ from LiF(001) surface with the use of Lie algebraic method are seen to agree well with the coupled-channel calculations. The Lie algebraic method thus appears to have a wide range of validity for describing the dynamics of gas–surface scattering. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 981–989, 1997     

Allowed transitions in silicon isoelectronic ions

Dipole‐allowed transitions have been studied for the first few members of the Si isoelectronic sequence. Transition energies, oscillator strengths, transition probabilities and quantum defect values have been estimated for the low‐ and high‐lying excited states of s and d symmetries up to the principal quantum number n=7 for these 3p open shell ions from P⁺ to Cr¹⁰⁺. Time‐dependent coupled Hartree–Fock (TDCHF) theory has been utilized to calculate such transition properties. Most of the results for transition energies, oscillator strengths, and transition probabilities for higher excited states are new. The transition energies for low‐lying excited states agree well with experimental data wherever available. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Calculations of D‐wave bound states and resonance states of the screened helium atom using correlated exponential wave functions

We have investigated the effects of screened Coulomb (Yukawa) potentials on the bound ^1,3D states and the doubly excited ^1,3 D^e resonance states of helium atom using highly correlated exponential basis functions. The Density of resonance states are calculated using stabilization method. Highly correlated exponential basis functions are used to consider the correlation effect between the charged particles. A total of 18 resonances (nine each for ¹ D^e and ³ D^e states) below the n = 2 He ⁺ threshold has been calculated. For each spin states, this includes four members in the 2pnp series, three members in the 2snd series, and two members in 2pnf series. The resonance energies and widths for various screening parameters ranging from infinity to a small value for these ^1,3 D^e resonance states are reported along with the bound‐excited 1s3d ^1,3 D state energies. Overall behavior of the spectral profile of 1s3d ¹D state of helium atom due to electron‐electron and electron‐nucleus screening are also presented. Accurate resonance energies and widths are also reported for He in vacuum. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Multiple‐State Emissions from Neat,Single‐Component Molecular Solids: Suppression of Kasha's Rule

Three rigid and structurally simple heterocyclic stilbene derivatives, (E)‐3H,3′H‐[1,1′‐biisobenzofuranylidene]‐3,3′‐dione, (E)‐3‐(3‐oxobenzo[c] thiophen‐1(3H)‐ylidene)isobenzofuran‐1(3H)‐one, and (E)‐3H,3′H‐[1,1′‐bibenzo[c] thiophenylidene]‐3,3′‐dione, are found to fluoresce in their neat solid phases, from upper (S₂) and lowest (S₁) singlet excited states, even at room temperature in air. Photophysical studies, single‐crystal structures, and theoretical calculations indicate that large energy gaps between S₂ and S₁ states (T₂ and T₁ states) as well as an abundance of intra and intermolecular hydrogen bonds suppress internal conversions of the upper excited states in the solids and make possible the fluorescence from S₂ excited states (phosphorescence from T₂ excited states). These results, including unprecedented fluorescence quantum yields (2.3–9.6 %) from the S₂ states in the neat solids, establish a unique molecular skeleton for achieving multi‐colored emissions from upper excited states by “suppressing” Kasha's rule.     

MS Xα calculations of the TiC20−6 cluster in titanium carbide

The Lie algebra, and the group governing configuration interaction in many of these quasi-bound states are a compact form of D₂, and the compact O(4), respectively. An isomorphic time-dependent algebra and group governs their ionization.     

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     

Usage de l'Algèbre de Lie su (n) dans l'Etude des Systèmes Quantiques à n Etats. V. Application aux Systèmes composés

We adapt the use of the Lie algebra su (n) we proposed in former papers to a system made up of two subsystems (respectively, n_A and n_B states). We show how reduced density operators, correlation problems, and overall and reduced evolution equations, may be made precise with this formalism.     

Kinetics of the hydrogen abstraction reactions of 1,1‐ and 1,2‐difluoroethane with hydroxyl radical: an ab initio study

The hydrogen abstraction reactions of 1,1‐ and 1,2‐difluoroethane with the OH radical have been investigated by the ab initio molecular orbital theory. The geometries of the reactants, products, and transition states have been optimized at the (U)MP2=full level of theory in conjunction with 6‐311G(d,p) basis functions. Single‐point (U)MP2=full with larger basis set, such as 6‐311G(3d,2p), and QCISD(T)=full/6‐311G(d,p) calculations have also been carried out to observe the effects of basis sets utilized and higher order electron correlation. Three and four reaction channels have been identified for 1,1‐ and 1,2‐difluoroethane, respectively. In the case of 1,1‐difluoroethane, hydrogen abstraction from the α‐carbon has been found to be easier than that from the β‐carbon. The barriers of the four reaction channels for 1,2‐difluoroethane are close to each other. Weak hydrogen bonding interactions have been observed between hydroxyl hydrogen and a fluorine atom in the transition states. Rate constants for the reactions of 1,1‐ and 1,2‐difluoroethane with the OH radical have been calculated using the standard transition state theory and found to be in good agreement with the experimental results. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1305–1318, 2000     

The construction of the Gilmore–Perelomov coherent states for the Kratzer–Fues anharmonic oscillator with the use of the algebraic approach

By applying the algebraic approach and the displacement operator to the ground state, the unknown Gilmore–Perelomov coherent states for the rotating anharmonic Kratzer–Fues oscillator are constructed. In order to obtain the displacement operator the ladder operators have been applied. The deduced SU(1, 1) dynamical symmetry group associated with these operators enables us to construct this important class of the coherent states. Several important properties of these states are discussed. It is shown that the coherent states introduced are not orthogonal and form complete basis set in the Hilbert space. We have found that any vector of Hilbert space of the oscillator studied can be expressed in the coherent states basis set. It has been established that the coherent states satisfy the completeness relation. Also, we have proved that these coherent states do not possess temporal stability. The approach presented can be used to construct the coherent states for other anharmonic oscillators. The coherent states proposed can find applications in laser-matter interactions, in particular with regards to laser chemical processing, laser techniques, in micro-machinning and the patterning, coating and modification of chemical material surfaces.     

High‐lying core‐excited quartet states in Li‐like N4+ and F6+ ions

The variational method with mutilconfiguration interaction wave function is used to obtain the energies, fine structures, and hyperfine structures of high‐lying core‐excited quartet states 1s2lnl' ⁴P^o(m) (m = 1–5) and 1s2pnp ⁴P(m) (m = 1–5) in Li‐like N⁴⁺ and F⁶⁺ ions, including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The oscillator strengths, lifetime, wavelengths, fine structure, and hyperfine structure for this system are also investigated to compare with other theoretical and experimental data in the literature. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Degeneracy in one‐dimensional quantum mechanics: A case study

In this work we study the isotonic oscillator, V(x) = Ax² + Bx^?2, on the whole line ?∞ < x < + ∞ as an example of a one‐dimensional quantum system with energy level degeneracy. A symmetric double‐well potential with a finite barrier is introduced to study the behavior of energy pattern between both limit: the harmonic oscillator (i.e., a system without degeneracy) and the isotonic oscillator (i.e., a system with degeneracy). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

A Laplace transform approach to find the exact solution of the $$$$-dimensional Schrödinger equation with Mie-type potentials and construction of Ladder operators

The second order \(N\)-dimensional Schrödinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution theorem. The Ladder operators are also constructed for the Mie-type potentials in \(N\)-dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.     

1,4‐Dilithio‐1,1,4,4‐tetraphenylbutane in diethyl ether: An effective initiator for the living anionic polymerization of dienes

The anionic polymerization of butadiene initiated with 1,4‐dilithio‐1,1,4,4‐tetraphenylbutane (LiTPB) in diethyl ether (DEE) gives polybutadiene (PBD) with high 1,2 content (>70%), narrow polydispersities (1.04 < M_w/M_n < 1.20), and predicted molecular weights. In THF, this polymerization does not work very well. After removal of DEE and addition of THF, the PBD dianion is end capped quantitatively by addition of 1,1‐diphenylethylene (DPE) to give the diphenylalkyl end capped PBD dianion. Subsequent addition of methyl methacrylate at low temperatures results in the formation of well‐defined PMMA‐b‐PBD‐b‐PMMA triblock copolymers. The results are accounted for by taking into account the effects of Li ion solvation on the BD initiation and end capping of the PBD anion by DPE. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2198–2206, 2009