Confined systems and the modified virial theorem from semiclassical considerations

We show that two simple semiclassical strategies, one based on the Wilson–Sommerfeld rule and the other on the uncertainty principle, yield the exact modified form of the virial theorem for confined systems. An alternative, easier quantum mechanical route to arrive at this result is also sketched. Pilot calculations on confined oscillators reveal decisive trends. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Semiclassical calculation of the vibrational echo

The infrared echo measurement probes the time scales of the molecular motions that couple to a vibrational transition. Computation of the echo observable within rigorous quantum mechanics is problematic for systems with many degrees of freedom, motivating the development of semiclassical approximations to the nonlinear optical response. We present a semiclassical approximation to the echo observable, based on the Herman-Kluk propagator. This calculation requires averaging over a quantity generated by two pairs of classical trajectories and associated stability matrices, connected by a pair of phase-space jumps. Quantum, classical, and semiclassical echo calculations are compared for a thermal ensemble of noninteracting anharmonic oscillators. The semiclassical approach uses input from classical mechanics to reproduce the significant features of a complete, quantum mechanical calculation of the nonlinear response.     

WKB and MAF quantization rules for spatially confined quantum mechanical systems

A formalism is developed to obtain the energy eigenvalues of spatially confined quantum mechanical systems in the framework of the usual Wentzel–Kramers–Brillouin (WKB) and modified airy function (MAF) methods. To illustrate the working rule, the techniques are applied to three different cases, viz. the confined one‐dimensional harmonic and quartic oscillators and a boxed‐in charged particle subject to an external electric field. The energies thus obtained are compared with those from shifted 1/N expansion, variational, and other methods, as well as the available exact numerical results. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 497–504, 1999     

A Hierarchical-Equation-of-Motion Based Semiclassical Approach to Quantum Dissipation?

We present a semiclassical (SC) approach for quantum dissipative dynamics, constructed on basis of the hierarchical-equation-of-motion (HEOM) formalism. The dynamical components considered in the developed SC-HEOM are wavepackets' phase-space moments of not only the primary reduced system density operator but also the auxiliary density operators (ADOs) of HEOM. It is a highly numerically efficient method, meanwhile taking into account the high-order effects of system-bath couplings. The SC-HEOM methodology is exemplified in this work on the hierarchical quantum master equation[J. Chem. Phys. 131 , 214111 (2009)] and numerically demonstrated on linear spectra of anharmonic oscillators.     

Transition state for intramolecular C–H bond cleavage in [(LCu)2(μ-O)2]2+ (L = 1,4,7-tribenzyl-1,4,7-triazacyclononane)

Hybrid quantum mechanical/molecular mechanical electronic structure calculations reveal the transition state for C–H bond cleavage in [(LCu)₂ (μ-O)₂]²⁺ (L=1,4,7-tribenzyl-1,4,7-triazacyclononane) to be consistent with a hydrogen-atom-transfer mechanism from carbon to oxygen. At the MPW1K/double-zeta effective core potential(+)|univeral force field level, 0 K activation enthalpies for the parent, p-CF3, and p-OH substituted benzyl systems are predicted to be 8.8, 9.5, and 7.8 kcal/mol. Using a one-dimensional Eckart potential to estimate quantum effects on the reaction coordinate, reaction in the unsubstituted system is predicted to proceed with a primary kinetic isotope effect of 22 at 233 K. Structural parameters associated with the hydrogen-atom transfer are consistent with the Hammond postulate. Received: 10 October 2000 / Accepted: 3 November 2000 / Published online: 3 April 2001     

Exact formulas for overlap integrals of Slater-type orbitals with equal screening constants

Exact formulas for 147 overlap integrals between Slater-type orbitals with equal screening constants are presented in the most simplified form. This represents all combinations of orbitals with quantum numbers: 1 ≤ N ≤ 5, 0 ≤ L ≤ 3, and M ≤ L. The formulas are automatically generated by computer using the “C-matrix” single-center expansion method. There are no limitations to the applicability of this method to orbitals of higher quantum numbers.     

A semiclassical formulation of the time dependent perturbation theory: The N coupled harmonic oscillator model

A new semiclassical formulation of the time dependent perturbation theory is proposed to describe the evolution ofN discrete quantum levels perturbed by time dependent couplings. A formal identity exists between the evolution operator matrix elements of the two following systems: (i) the real molecular system composed of N coupled quantum levels, (ii) a system composed of N harmonic oscillators, specifically prepared to account for a given transition, interacting via a specific potential. By working on the second system rather than the first one a formal expression of the time displacement operator results and an original interaction representation technique with respect to the diagonal and non-diagonal couplings can be derived. This allows one to “dress each transition channel” with slowly varying phase shifts generated by the coupling terms involving the transition itself. An application to the collision system N₂ + CO (OC) illustrates a generalized Mies effect due to a coupling term between quasi-resonant levels.     

Application of the strong-coupling correspondence principle to atom–triatom collinear collisions

The strong-coupling correspondence principle has been used to calculate the T–V transition probabilities in the collinear collision of He and Kr with CO₂. For a harmonic CO₂ potential the results for He agree well with published quantum mechanical probabilities. For Kr the agreement is less satisfactory but at worst the ratio of quantum to semiclassical transition probability is approximately 0.2. Introduction of anharmonicity in the CO₂ potential was found to increase the semiclassical transition probabilities but this may just be due to the lowering of the vibration frequencies.     

Experimental and Theoretical Study of Helium Broadening and Shift of HCO+ Rotational Lines

An experimental and theoretical study of pressure broadening and pressure shift of HCO⁺ rotational lines perturbed by collisions with He is presented. Results are reported from measurements at 88 K for the lines j=4←3, 5←4 and 6←5 with frequencies ranging from 0.35 to 0.54 THz. Using a new CCSD(T)/aug‐cc‐pVQZ potential energy surface for the He–HCO⁺ interaction, the collisional line shape parameters are studied from fully quantum and semiclassical calculations. Results from the quantum treatment are in satisfactory agreement with experiments whereas the semiclassical approach can lead to appreciable differences. A study of the dependence of line width Γ and shift s as a function of the translational energy shows the presence of quantum oscillations. Calculations on a previous Hartree–Fock‐based potential energy surface lead to quite similar results for the collisional line shape parameters. Using a simplified version of the potential morphing method it is found that the line width Γ is particularly sensitive to the long‐range part of the potential energy surface. This also explains the success of the first line‐broadening calculations which date back to the 1950s.     

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

In this work, we study the position and momentum information entropies of multiple quantum well systems in fractional Schrödinger equations, which, to the best of our knowledge, have not so far been studied. Through a confining potential, their shape and number of wells (NOW) can be controlled by using a few tuning parameters; we present some interesting quantum effects that only appear in the fractional Schrödinger equation systems. One of the parameters denoted by the L_d can affect the position and momentum probability densities if the system is fractional (1 < α < 2). We find that the position (momentum) probability density tends to be more severely localized (delocalized) in more fractional systems (ie, in smaller values of α). Affecting the L_d on the position and momentum probability densities is a quantum effect that only appears in the fractional Schrödinger equations. Finally, we show that the Beckner Bialynicki-Birula-Mycieslki (BBM) inequality in the fractional Schrödinger equation is still satisfied by changing the confining potential amplitude V_conf, the NOW, the fractional parameter α, and the confining potential parameter L_d .     

Semiclassical Trajectory Studies of Reactive and Nonreactive Scattering of OH(A 2Σ+) by H2 Based on an Improved Full-Dimensional Ab Initio Diabatic Potential Energy Matrix

We present a new full-dimensional diabatic potential energy matrix (DPEM) for electronically nonadiabatic collisions of OH(A ²Σ⁺) with H₂, and we calculate the probabilities of electronically adiabatic inelastic collisions, nonreactive quenching, and reactive quenching to form H₂O+H. The DPEM was fitted using a many-body expansion with permutationally invariant polynomials in bond-order functions to represent the many-body part. The dynamics calculations were carried out with the fewest-switches with time uncertainty and stochastic decoherence (FSTU/SD) semiclassical trajectory method. We present results both for head-on collisions (impact parameter b equal to zero) and for a full range of impact parameters. The results are compared to experiment and to earlier FSTU/SD and quantum dynamics calculations with a previously published DPEM. The various theoretical results all agree that nonreactive quenching dominates reactive quenching, but there are quantitative differences between the two DPEMs and between the b=0 results and the all-b results, especially for the probability of reactive quenching.     

Quantum generalized Langevin equation approach to gas/solid collisions

The classical generalized Langevin equation (GLE) approach to gas/solid collisions is generalized to quantum scattering. Using Feynman's method of partial path integration, the full gas/solid propagator is reduced to a form in which only the dynamics of the incident atom and the surface oscillator(s) directly struck appear explicitly. Solving this effective dynamical problem in the semiclassical limit yields a stationary phase equation of motion identical in form to the classical GLE. The noise, however, is distributed according to quantum rather than classical statistics. From the GLE a quantum phase can be constructed and an S-matrix computed. The resulting theory is capable of describing inelastic-diffractive scattering which has been seen experimentally by Williams.     

Micellar structure of an oligopeptide surfactant “trimeric N -dodecanoyl-L-proline potassium salt” in aqueous solution – small-angle neutron scattering study

Structures of the micelles which are formed by the chiral oligopeptide surfactant N-dodecanoyl-L-proline tripeptide anions have been examined using small-angle neutron scattering spectral analysis. Results show that the chiral N-dodecanoyl-L-proline trimeric anions may form a spherical micelle with an aggregation number of 36 and that the oligopeptide portions with a poly-L-proline I-type helical structure are saturated with water. Received: 21 March 2001 Accepted: 5 April 2001     

Classical theory of vibration energy transfer in collinear collisions

Classical theory of collisions is cast in a form which also includes the uncertainty principle. This theory is used for analyzing the vibration energy transfer in the collinear collision which approximates the He-H₂ system. The results are compared with the quantum calculations and several classical and semiclassical approaches. Very good agreement with quantum theory is found, for all the parameters investigated.     

Adiabatic coordinate separation and large N-dimensional limit in two-electron ions

The 1/N expansion is used to generate semiclassical effective potentials for two-electron ions. These potentials can be considered an improvement over the conventional ones computed by Lin and Fano for the full two-electron coulomb potential insofar as they incorporate some of the quantum mechanical effects. No falling into the nucleus is allowed. Furthermore, the minima of these effective potentials are in surprising agreement with those computed by Fano, Macek, and Lin. This would indicate that the collective variable R, associated with N = 6, leads to a much smaller expansion parameter N, making the adiabatic separation more justified. The role and significance of these broken symmetry solutions with geometric localization are discussed as well as generalizations. The procedure is semiclassical in the sense that the kinetic energy becomes negligible by having a large effective mass.     

SWKB Approach to Confined Isospectral Potentials

Recently we had formulated the supersymmetric Wentzel–Kramers–Brillouin (SWKB) quantization rule for one-dimensional confined quantum systems and applied the same to two trigonometric potentials, tangentially limited by infinite walls at x=0 and x=L, viz., V(x)=V ₀cot²(x/L) and the Pöschl–Teller potential, V(x)=V ₀₁cosec²({x/(2L))}+V ₀₂sec²(x/(2L)). Both the potentials have received quite a lot of attention by various authors because of their importance in molecular physics. Though these potentials have been studied in the framework of WKB, BS (Bohr–Sommerfeld), mBS (matrix formulation of BS) formalisms, it was observed that the supersymmetric approach not only rendered the calculations simpler and more transparent, it also reproduced the exact analytical energies in both the cases.In this study, we shall generate isospectral Hamiltonians of the above potentials with the help of a modified form of Darboux's theorem. We shall show that though the new potentials look different from the original ones, and have different eigenfunctions, they too, are confined in the same region of space, and share the same energy spectrum as their original counterparts. This may be of substantial importance in determining the energy spectrum of highly non-trivial systems.     

On exact calculation of response properties of oscillators in static electric field: A Fourier grid Hamiltonian approach. I. One-dimensional systems

The Fourier grid Hamiltonian method is used to calculate the response properties of different types of 1-d (one-dimensional) quantum oscillators in a uniform static electric field. The calculations are potentially exact. Excepting the harmonic oscillator, the other model oscillators studied are seen to possess nonlinear polarizabilities. In general, the polarizabilities are not monotonic functions of appropriate vibrational quantum numbers. The exact nature of this vibrational-state dependence of polarizabilities is shown to depend on the type of mechanical anharmonicity in which the nuclei move and the nature of electrical anharmonicity characterizing the field–oscillator coupling. The large vibrational contribution to nonlinear polarizabilities often predicted for real diatomics could therefore originate from the mechanical and electrical anharmonicities of the potential in which the nuclei move when placed in a static electric field. © 1994 John Wiley & Sons, Inc.     

Organogels from ABA triblock copolymers

ABA triblock copolymers with two polystyrene endblocks connected by a poly(ethylene/butylene) midblock form highly elastic gels in a solvent which is incompatible for the endblocks but a good solvent for the midblock, for example, paraffin oil. In this situation the polystyrene endblocks aggregate into micelles. The midblocks can either form loops or build up bridges between different micelles; thus, domains and networks of interconnected micelles are produced. We have studied organogels of this kind consisting of a polymer with a molar mass of 90,000 and a styrene content of 31% per weight (Kraton G 1650) in paraffin oil. Rheological, calorimetric (differential scanning calorimetry) and small-angle X-ray scattering experiments were performed on these systems. An interesting result of our work which was not described previously is that the size (r˜ 6.8 nm) and the separation (d˜ 36 nm) of the micellar aggregates does not seem to be influenced by the block copolymer content in the concentration range investigated. Received: 12 March 2001 Accepted: 5 April 2001     

One-pot Synthesis of a Truncated Cone-shaped Porphyrin Macrocycle and Its Self-assembly into Permanent Porous Material

Here, we report the synthesis of a truncated cone-shaped triangular porphyrinic macrocycle, P₃L₃ , via a single step imine condensation of a cis-diaminophenylporphyrin and a bent dialdehyde-based linker as building units. X-ray diffraction analysis reveals that the truncated cone-shaped P₃L₃ molecules are stacked on top of each other by π⋯π and CH⋯π interactions, to form 1.7 nm wide hollow columns in the solid state. The formation of the triangular macrocycle is corroborated by quantum chemical calculations. The permanent porosity of the P₃L₃ crystals is demonstrated by several gas sorption experiments and powder X-ray diffraction analysis.     

What happens when translational energy levels are quantized?

The conventional translational partition function (CTPF) that is widely used in textbooks is essentially semiclassical, whose form is found by using the particle‐in‐a‐box model to represent the particles motion of an ideal gas. This form assumes continuum translational energy levels, thereby replacing the sum over the energy levels with an integral at high temperature. This is only valid if de Broglie wavelength is much shorter than the container dimension in which the particle is placed. Additionally, de Broglie wavelength must also be much smaller than the mean separation of the constituent particles of the gas. To ensure this, one will have to assume large mass and container size and high temperature (T). This assumption is a restriction in itself. Should de Broglie wavelength be larger than the microscopic size of interest, the CTPF will be invalid to use which in turn will lead to erroneous conclusions, and a pure quantum mechanical treatment must then be employed for accurate results. A perfect example of the failure of using the CTPF would be the conduction electrons in a metal or in conjugated dienes, whereby the translational energy levels are significantly quantized. This quantization manifests itself in the quantum translational partition function (TPF) as its curve starts at zero and remains zero at low T till further increase in T stimulates accessible translational thermal states to be populated, at which point a rise in quantum TPF is observed, whereas the CTPF is only zero at T = 0, and then starts rising as T increases since energy levels are continuum and their discretization is insensitive to the CTPF. Therefore this article explores quantum mechanical treatment of the TPF and its effects on thermodynamic functions using our closed‐form expression of quantum TPF developed in [M. Toutounji, Int J Quantum Chem Early View (unpublished)] (IJQC) which is valid at all Ts, sizes, and masses. Although this TPF appeared in IJQC in an abstract form before, there was not any discussion of its applicability and physical aspects therein. Also, some model calculations are presented to show the failure of the CTPF in the current literature in case of a purely quantum conditioned environment. Thermodynamic functions such as Helmholtz free energy, entropy, and heat capacity are explored using the herein closed form quantum partition function. A closer look at the quantum translational heat capacity shows a curve starts at zero at low T, and then starts sharply rising as T increases going through a maximum after which it levels off at its classical value k/2, where k is Boltzmann constant. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011