Descriptions of operators in quantum mechanics

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.

Wigner distribution for angle coordinates in quantum mechanics

The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase‐space language is much clarified by this extension.

Molecular Mechanics Studies of Homopolymeric and Mixed Sequence Py.Pu*Pu Triple Helices,

DNA triple helices containing two purine strands and one pyrimidine strand (C.G*G and T.A*A) have been studied, using model building followed by energy minimisation, for different orientations of the third strand resulting from variation in the hydrogen bonding between the Watson-Crick duplex and the third strand and the glycosidic torsion angle in the third strand. Our results show that in the C.G*G case the structure with a parallel orientation of the third strand, resulting from Hoogsteen hydrogen bonds between the third strand and the Watson-Crick duplex, is energetically the most favourable while in the T.A*A case the antiparallel orientation of the third strand, resulting from reverse Hoogsteen hydrogen bonds, is energetically the most favourable. These studies when extended to the mixed sequence triplexes, in which the second strand is a mixture of G and A, correspondingly the third strand is a mixture of G and APT, show that though the parallel orientation is still energetically more favourable, the antiparallel orientation becomes energetically comparable with an increasing number of thymines in the third strand. Structurally, for the mixed triplexes containing G and T in the third strand, it is seen that the basepair non-isomorphism between the C.G*G and the T.A*T triplets can be overcome with some changes in the base pair parameters without much distortion of either the backbone or the hydrogen bonds.

Ab initio molecular orbital calculations on ion-molecule and ion pair-molecule complexes of the water-lithium cyanide system

Ab initio molecular orbital (MO) calculations with the 3-21G and 6-31G basis sets were performed on a series of ion-molecule and ion pair-molecule complexes for the H2O + LiCN system. Stabilisation energies (with counter-poise corrections), geometrical parameters, internal force constants and harmonic vibrational frequencies were evaluated for 16 structures of interest. Although the interaction energies are smaller, the geometries and relative stabilities of the monohydrated contact ion pair are reminiscent of those computed for the complexes of the individual ions. Thus, interaction of the oxygen lone pair with lithium leads to a highly stabilised C2v structure, while the coordination of water to the cyanide ion involves a slightly non-linear hydrogen bond. Symmetrical bifurcated structures are computed to be saddle points on the potential energy surface, and to have an imaginary frequency for the rocking mode of the water molecule. On optimisation the geometries of the solvent shared ion pair structures (e.g. Li+cdots, three dots, centered OH2cdots, three dots, centered CN−) revealed a proton transfer from the water molecule leading to hydrogen bonded forms such as Li-O-Hcdots, three dots, centered HCN. The variation in the force constants and harmonic frequencies in the various structures considered are discussed in terms of ion-molecular and ion pair-molecule interactions.

A simple computational model for predicting .pi.-facial selectivity in reductions of sterically unbiased ketones. Relative importance of electrostatic and orbital interactions

Various factore controlling the preferred facial selectivity in the reductions of a number of sterically unbiased ketones have been evaluated using a semiempirical MO procedure. MNDO optimized geometries do not reveal any significant ground-state distortions which can be correlated with the observed face selectivities. Electrostatic effecta due to an approaching reagent were modeled by placing a test negative charge at a fixed distance from the carbonyl carbon on each of the two faces. A second series of calculations was carried out using the hydride ion as a test nucleophile. The latter calculations effectively include orbital interactions involving the u and u* orbitals of the newly formed bond in the reaction. The computed energy differences with the charge model are generally much larger compared to those with the hydride ion. However, both models lead to predictions which are qualitatively consistent with the experimentally determined facial preferences for most of the systems. Thus, electrostatic interactions between the nucleophile and the substrate seem to effectively determine the face selectivities in these molecules. However, there are a few exceptions in which orbital interactions are found to contribute significantly and occasionally reverse the preference dictated by electrostatic effecta. The remarkable succew of the hydride model calculations, in spite of retaining the unperturbed geometries of the substrates, points to the unimportance of torsional effeds and orbital distortions associated with the pyramidalized carbonyl unit in the transition state in most of the substrates considered. Additional experimental results are reported which provide useful calibration for the present computational approach.

Chiral polyhydroxylated tetrahydrothiophene derivatives: novel synthesis and structural elucidation by X-ray crystallography, NMR spectroscopy and molecular mechanics calculations

The dideoxygenation reaction of 1,3;4,6-di-O-alkylidene-2,5-di-S-methylthiocarbonyl-D-mannitol derivatives under Barton-McCombie reaction conditions gave the hexahydrodipyranothiophenes 4 and 7 instead of the expected 2,5-dideoxy products. Structural and conformational information on these novel derivatives has been obtained by NMR spectroscopy, single-crystal X-ray crystallography and molecular mechanics calculations.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.

Ab initio molecular orbital calculations on ion pair-water complexes of metal halides and oxides

Ab initio MO calculations are performed on a series of ion-molecular and ion pair-molecular complexes of H2O + MX (MX = LiF, LiCl, NaCl, BeO and MgO) systems. BSSE-corrected stabilization energies, optimized geometrical parameters, internal force constants and harmonic vibrational frequencies have been evaluated for all the structures of interest. The trends observed in the geometrical parameters and other properties calculated for the mono-hydrated contact ion pair complexes parallel those computed for the complexes of the individual ions. The bifurcated structures are found to be saddle points with an imaginary frequency corresponding to the rocking mode of water molecules. The solvent-shared ion pair complexes have high interaction energies. Trends in the internal force constant and harmonic frequency values are discussed in terms of ion-molecular and ion-pair molecular interactions.

Diastereospecific Coupling of Imines by Low-Valent Titanium: An Experimental and Computational Study

The reaction of phenylsilane (PhSiH3) and titanium(IV) isopropoxide Ti(OiPr)(4)] generates low-valent titanium alkoxides that reduce and reductively couple imines. The C-C coupling reaction is diastereospecific, with exclusive formation of the (+/-)-isomer. The yield is dependent on the concentration of titanium(IV) isopropoxide used. Using imines with varying electronic demand revealed that the coupling is most efficient for unsubstituted imines. The involvement of a trimeric titanium biradical species is invoked to explain the observed concentration dependence and diastereospecificity in the reaction. Multilayer ONIOM (''our own n-layered integrated molecular orbital and molecular mechanics'') calculations were carried out on the plausible intermediates involved by using the Guassian 09 suite of programs. Formation of the trimeric biradical species leading to the formation of the (+/-)-isomer is more favored than formation of the trimeric biradical species, which leads to the meso-isomer. Similar calculations with dimeric intermediates do not predict the (+/-)-selectivity. Mass spectral analysis of the reaction mixture indicates the presence of a trimeric titanium alkoxide species.

E. C. G. Sudarshan and Symmetry in Classical Dynamics, Optics and Quantum Mechanics

We review work initiated and inspired by Sudarshan in relativistic dynamics, beam optics, partial coherence theory, Wigner distribution methods, multimode quantum optical squeezing, and geometric phases. The 1963 No Interaction Theorem using Dirac's instant form and particle World Line Conditions is recalled. Later attempts to overcome this result exploiting constrained Hamiltonian theory, reformulation of the World Line Conditions and extending Dirac's formalism, are reviewed. Dirac's front form leads to a formulation of Fourier Optics for the Maxwell field, determining the actions of First Order Systems (corresponding to matrices of Sp(2,R) and Sp(4,R)) on polarization in a consistent manner. These groups also help characterize properties and propagation of partially coherent Gaussian Schell Model beams, leading to invariant quality parameters and the new Twist phase. The higher dimensional groups Sp(2n,R) appear in the theory of Wigner distributions and in quantum optics. Elegant criteria for a Gaussian phase space function to be a Wigner distribution, expressions for multimode uncertainty principles and squeezing are described. In geometric phase theory we highlight the use of invariance properties that lead to a kinematical formulation and the important role of Bargmann invariants. Special features of these phases arising from unitary Lie group representations, and a new formulation based on the idea of Null Phase Curves, are presented.

Investigation on the Magnetic Anomaly and the Role of Orbital Moment on the Magnetic Properties of LaMn0.5Co0.5O3

Two distinct ferromagnetic phases are present in LaMn0.5Co0.5O3 for which the spin-only magnetic moment calculated from the high temperature dc susceptibility is found to be unusually high. Such a high moment can only be accounted for by assigning the valence state of the cations to Mn2+-Co4+. This is unrealistic as the earlier report based on X-ray absorption spectroscopy (XAS) has suggested the valence state to be mainly Mn4+-Co2+ with traces of Co3+. Also from our studies using XAS, it is found that the valence state is mainly Mn4+-Co2+. In addition, no notable difference is observed in the minor Co3+ present in both phases. Our results based on X-ray magnetic circular dichroism studies (XMCD) reveal the presence of ``distinct'' high orbital moment associated with Co2+ for both phases. Thus it is found that the distinctness of the orbital moment also plays a vital role in determining the magnetic moment and T-c of both phases of LaMn0.5Co0.5O3. By considering the orbital moment obtained from XMCD, the anomaly in the paramagnetic susceptibility is resolved and thus we are able to assign the valence state to Mn4+-Co2+ configuration. The difference in the magnitude of orbital moment in both phases is believed to be due to the crystal field effects.

Electron donor-acceptor complex of ICl with diethyl ether - He I photoelectron spectroscopy and ab initio molecular orbital study

The He I photoelectron spectrum of the diethyl ether-ICl complex has been obtained. The oxygen orbitals are shifted to higher binding energies and that of ICl to lower binding energies owing to complex formation. Ab initio molecular orbital (MO) calculations of the complex molecule showed that the bonding is between the sigma-type lone pair of oxygen and the I atom and that the complex has C-2v symmetry. The binding energy of the complex is computed to be 8.06 kcal mol(-1) at the MP2/3-21G* level. The orbital energies obtained from the photoelectron spectra of the complex are compared and assigned with orbital energies obtained by MO calculations. Natural bond orbital analysis (NBO) shows that charge transfer is from the sigma-type oxygen lone pair to the iodine atom and the magnitude of charge transfer is 0.0744 e.

Extended Huckel molecular orbital calculations on hydroxobridged heterotrinuclear cobalt(III) complexes

EHT calculations on heterotrinuclear cobalt(III) complexes of the type [Cu{(OH)(2)Co(L(4))}(2)](4+) where L(4) denotes (en)(2) or (NH3)(4), en = ethylenediamine and their component species have been carried out. The results regarding bonding and structure for the trinuclear complexes are compared with those for the monomer components such as [Co(en)(2)(OH)(2)](+), [Co(NH3)(4)(OH)(2)](+) and [Cu(OH)(4)](2-) are discussed.

Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation

Atomic vibration in the Carbon Nanotubes (CNTs) gives rise to non-local interactions. In this paper, an expression for the non-local scaling parameter is derived as a function of the geometric and electronic properties of the rolled graphene sheet in single-walled CNTs. A self-consistent method is developed for the linearization of the problem of ultrasonic wave propagation in CNTs. We show that (i) the general three-dimensional elastic problem leads to a single non-local scaling parameter (e(0)), (ii) e(0) is almost constant irrespective of chirality of CNT in the case of longitudinal wave propagation, (iii) e(0) is a linear function of diameter of CNT for the case of torsional mode of wave propagation, (iv) e(0) in the case of coupled longitudinal-torsional modes of wave propagation, is a function which exponentially converges to that of axial mode at large diameters and to torsional mode at smaller diameters. These results are valid in the long-wavelength limit. (C) 2011 Elsevier Ltd. All rights reserved.

Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics

In this paper, the critical budding temperature of single-walled carbon nanotubes (SWCNTs), which are embedded in one-parameter elastic medium (Winkler foundation) is estimated under the umbrella of continuum mechanics theory. Nonlocal continuum theory is incorporated into Timoshenko beam model and the governing differential equations of motion are derived. An explicit expression for the non-dimensional critical buckling temperature is also derived in this work. The effect of the nonlocal small scale coefficient, the Winkler foundation parameter and the ratio of the length to the diameter on the critical buckling temperature is investigated in detail. It can be observed that the effects of nonlocal small scale parameter and the Winkler foundation parameter are significant and should be considered for thermal analysis of SWCNTs. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the thermal buckling properties of embedded single-walled carbon nanotubes. (C) 2011 Elsevier B.V. All rights reserved.