Symmetrization of eigenfunctions of angular momentum in point groups

The eigenfunctions |jm〉 of angular momentum can combine linearly to make basis functions of irreducible representations of point groups. We surmount the projection operator and find a new method to calculate the combination coefficients. It is proven that these coefficients are components of eigenvectors of some hermitian matrices, and that for all pure rotation point groups, the coefficients can be made real numbers by properly choosing the azimuth angles of symmetry elements of point groups in the coordinate system. We apply the coupling theory of angular momentum to obtain the general formulas of the basis functions of point groups. By use of our formulas, we have calculated the basis functions with half‐integers j from 1/2 to 13/2 of double‐valued irreducible representations for the icosahedral group. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 286–302, 2001     

Laser-induced forward-transfer with light possessing orbital angular momentum

Helical light fields may carry both orbital angular and spin angular momentum which is respectively associated with their helical wavefronts (optical vortices) and rotating transverse electric fields. Interestingly, these helical light fields interact with materials and the orbital angular momentum of these fields can physically twist a range of materials, including metals, semiconductors, polymers, and liquids. With the aid of spin angular momentum, these fields can also form a range of helical structures. This light-matter interaction based on transfer of angular momentum has the potential to revolutionize industrial processes and enable technologies, such as advanced non-contact and nozzle-free printing. In this review paper, we focus on this printing technique, a process which we herein refer to as optical vortex laser induced forward transfer, and we show how it can be used for the production of next generation printed photonics/electronics/spintronics devices. Herein we review the interactions between the angular momentum of light and materials, and we discuss the ways in which optical vortices can be used to produce a variety of exotic structures. We also discuss the current state-of-the art of laser-induced forward-transfer technologies and detail some of the most novel devices, which have been fabricated using this optical vortex laser induced forward transfer, including hexagonal close-packed photonic-rings and plasmonic nanocores.     

Local angular momentum as ring strain descriptor

A new method is demonstrated to quantify local ring strain, which is based on the expectation value of orbital angular momentum along the internuclear axis. In contrast to energy based methods which provide overall ring strain, this method is able to identify the local strain in every part of the ring. The formalism is benchmarked on several cycloalkanes in which the presence of ring strain is well understood. The ring strain plays a decisive role in carbon nanotubes (CNTs) properties; for instance, the hydrogen storage capability of CNTs is related to their diameter, which in turn has a close relation to the ring strain in their C? C bonds. On this basis, the ring strain in five CNTs with different diameters is analyzed and the results reflected meaningful correlation between the CNTs diameter and ring strain. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Entanglement of angular momenta of atoms and molecules

In this article, we investigate the entanglement degrees of angular momenta of atoms and molecules. We demonstrate theoretically and numerically the guidelines, how to prepare maximally entangled states and how the entanglement of the angular momenta changes by changing the quantum numbers of atoms and molecules. We show that the entanglement degree reduces to the Clebsch–Gordan coefficients frequently encountered in angular momentum theory. The maximally entangled states found in this manner are useful for quantum computers and quantum information science. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Effects of angular momentum recoupling on depolarization spectra in alkali-rare gas optical half collisions

The depolarization mechanisms of excited alkali atoms interacting with ground state rare gas atoms are investigated using the method of far wing spectroscopy of collision pairs under single collision conditions. From a semiclassical theory explicite expressions for the spectra of alkali multipole components σ _k ^(?) (ω _L ) are derived assuming quasistatic excitation at a localized internuclear separationR _L(ω _L ) related to the laserfrequency ω _L as well as realistic models for the half collision following excitation. The collision models are characterized by different Hund's coupling regions, where excitation takes place and which are traversed on the collision path. Due to selection rules for excitation of populations and coherences and for support of multipoles the σ _k ^(?) (ω _L ) are shown to depend strongly on the collision model. From the spectra thus a labelling of the initial molecular states and mapping of the change in coupling case is possible. Estimations of the contributions of the various angular momentum recoupling effects are given.     

Combinatorics of angular momentum recoupling theory: spin networks,their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and mathematical physics. This paper starts with an account of the basics of such a theory, which represents the most exhaustive framework in dealing with interacting many-angular momenta quantum systems. We then outline the essential features of this algebra, that can be encoded, for each fixed number N = (n + 1) of angular momentum variables, into a combinatorial object, the spin network graph, where vertices are associated with finite-dimensional, binary coupled Hilbert spaces while edges correspond to either phase or Racah transforms (implemented by 6j symbols) acting on states in such a way that the quantum transition amplitude between any pair of vertices is provided by a suitable 3nj symbol. Applications of such a combinatorial setting—both in fully quantum and in semiclassical regimes—are briefly discussed providing evidence of a unifying background structure.     

Subshell-pair correlation coefficients of atoms in momentum space

Angular correlation coefficients τ ^{nl,n^′ l^′} [p] between linear momenta of an electron in a subshell nl and another electron in a subshell n′ l′ are studied for the 102 neutral atoms He through Lr in their ground states, where n and l are the principal and azimuthal quantum numbers, respectively. We theoretically find that electron momenta are negatively correlated or uncorrelated; τ ^{nl,n^′ l^′} [p] < 0 when |l − l′|=1, while τ ^{nl,n^′ l^′} [p]=0 when |l − l′| ≠ 1. Numerical examinations of the atoms show that except for the He–B atoms, negative correlations are largest between 1s and 2p subshells, which have the most diffuse electron distributions in momentum space.     

Recoupling coefficients of group SO(4) and their application to linear tetratomic molecules

The general expressions for the recoupling coefficients of group SO(4) are obtained by employing the subgroup chains (2) and (11). When considering a special case, we give results that can be used for tetratomic molecules. Using these recoupling coefficients, the matrix elements of C and M operators that are not diagonal can be given. Within the dynamic algebra framework, an effective algebraic Hamiltonian of linear tetratomic molecules is obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 293–301, 2003     

C-Cl bond fission dynamics and angular momentum recoupling in the 235 nm photodissociation of allyl chloride

The photodissociation dynamics of allyl chloride at 235 nm producing atomic Cl((2)P(J);J=1/2,3/2) fragments is investigated using a two-dimensional photofragment velocity ion imaging technique. Detection of the Cl((2)P(1/2)) and Cl((2)P(3/2)) products by [2+1] resonance enhanced multiphoton ionization shows that primary C-Cl bond fission of allyl chloride generates 66.8% Cl((2)P(3/2)) and 33.2% Cl((2)P(1/2)). The Cl((2)P(3/2)) fragments evidenced a bimodal translational energy distribution with a relative weight of low kinetic energy Cl((2)P(3/2))/high kinetic energy Cl((2)P(3/2)) of 0.097/0.903. The minor dissociation channel for C-Cl bond fission, producing low kinetic energy chlorine atoms, formed only chlorine atoms in the Cl((2)P(3/2)) spin-orbit state. The dominant C-Cl bond fission channel, attributed to an electronic predissociation that results in high kinetic energy Cl atoms, produced both Cl((2)P(1/2)) and Cl((2)P(3/2)) atomic fragments. The relative branching for this dissociation channel is Cl((2)P(1/2))/[Cl((2)P(1/2))+Cl((2)P(3/2))]=35.5%. The average fraction of available energy imparted into product recoil for the high kinetic energy products was found to be 59%, in qualitative agreement with that predicted by a rigid radical impulsive model. Both the spin-orbit ground and excited chlorine atom angular distributions were close to isotropic. We compare the observed Cl((2)P(1/2))/[Cl((2)P(1/2))+Cl((2)P(3/2))] ratio produced in the electronic predissociation channel of allyl chloride with a prior study of the chlorine atom spin-orbit states produced from HCl photodissociation, concluding that angular momentum recoupling in the exit channel at long interatomic distance determines the chlorine atom spin-orbit branching.     

Derivative of electron repulsion integral using accompanying coordinate expansion and transferred recurrence relation method for long contraction and high angular momentum

In this study, an early‐working algorithm is designed to evaluate derivatives of electron repulsion integrals (DERIs) for heavy‐element systems. The algorithm is constructed to extend the accompanying coordinate expansion and transferred recurrence relation (ACE‐TRR) method, which was developed for rapid evaluation of electron repulsion integrals (ERIs) in our previous article (M. Hayami, J. Seino, and H. Nakai, J. Chem. Phys. 2015, 142, 204110). The algorithm was formulated using the Gaussian derivative rule to decompose a DERI of two ERIs with the same sets of exponents, different sets of contraction coefficients, and different angular momenta. The algorithms designed for segmented and general contraction basis sets are presented as well. Numerical assessments of the central processing unit time of gradients for molecules were conducted to demonstrate the high efficiency of the ACE‐TRR method for systems containing heavy elements. These heavy elements may include a metal complex and metal clusters, whose basis sets contain functions with long contractions and high angular momenta.     

Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm

Orthogonal polynomials of a discrete variable have been widely investigated as fundamental tools of numerical analysis. This work aims to propose the extension of their use to quantum mechanical problems. By exploiting both their connection with coupling and recoupling coefficients of angular momentum theory and their asymptotic relationships (semiclassical limit) with spherical and hyperspherical harmonics, a discretization procedure, the hyperquantization algorithm, has been developed and applied to the study of anisotropic interactions and of reactive scattering. One of the most appealing features of this method turns out to be a drastic reduction of memory requirements and computing time for extensive dynamical calculations. Examples of the application of this technique to stereodirected dynamics via an exact representation for the S matrix as well as to the characterization of molecular beam polarization are also illustrated. Received: 17 September 1999 / Accepted: 3 February 2000 / Published online: 5 June 2000     

Expansion coefficients and moments of electron momentum densities for singly charged ions

Numerical Hartree–Fock calculations of the first three coefficients of the MacLaurin expansion and the leading coefficient of the large-p asymptotic expansion of the electron momentum densities Π(p) are reported for 54 singly charged atomic cations from He⁺ (atomic number Z = 2) to Cs⁺ (Z = 55) and 43 anions from H⁻ (Z = 1) to I⁻ (Z = 53) in their experimental ground states. We also report all the finite moments <p ^k > (−2≤k≤+4) of the momentum densities Π(p) for the above-mentioned 97 ionic species. The results are compared with the previous ones for neutral atoms [Koga and Thakkar (1996) J Phys B 29: 2973], and the dependence of the expansion coefficients and moments on nuclear charge is discussed among isoelectronic species. Received: 20 November 1998 / Accepted: 15 January 1999 / Published online: 7 June 1999     

Diatomic interactions in momentum space bond polarity

Some fundamental aspects of bond polarity embedded in diatomic molecular orbitals are studied from the viewpoint of the electron distribution in momentum space. Electron momentum density is expressible as a product of one-center and oscillation terms, and the effect of polarity appears mainly in the latter term. Since the oscillation is not spherically symmetrical, the bond polarity is then related to the anisotropy of momentum distribution. In order to investigate this relation, directional ratios of momentum moments are introduced and their behaviors are examined for a model heteronuclear diatomic system.     

Study of molecular orbitals in momentum space

Molecular orbitals are presented in configuration and momentum representations. We propose to minimize large oscillations present in calculated momentum wavefunctions by cancelling position factor phases. We illustrate the distortion introduced by different electron translation factors and show continuum states and dynamical wavefunctions in momentum coordinates. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

The role of angular momentum in collision-induced vibration-rotation relaxation in polyatomics

Vibrational relaxation of the 6(1) level of S(1)((1)B(2u)) benzene is analyzed using the angular momentum model of inelastic processes. Momentum-(rotational) angular momentum diagrams illustrate energetic and angular momentum constraints on the disposal of released energy and the effect of collision partner on resultant benzene rotational excitation. A kinematic "equivalent rotor" model is introduced that allows quantitative prediction of rotational distributions from inelastic collisions in polyatomic molecules. The method was tested by predicting K-state distributions in glyoxal-Ne as well as J-state distributions in rotationally inelastic acetylene-He collisions before being used to predict J and K distributions from vibrational relaxation of 6(1) benzene by H(2), D(2), and CH(4). Diagrammatic methods and calculations illustrate changes resulting from simultaneous collision partner excitation, a particularly effective mechanism in p-H(2) where some 70% of the available 6(1)-->0(0) energy may be disposed into 0-->2 rotation. These results support the explanation for branching ratios in 6(1)-->0(0) relaxation given by Waclawik and Lawrance and the absence of this pathway for monatomic partners. Collision-induced vibrational relaxation in molecules represents competition between the magnitude of the energy gap of a potential transition and the ability of the colliding species to generate the angular momentum (rotational and orbital) needed for the transition to proceed. Transition probability falls rapidly as DeltaJ increases and for a given molecule-collision partner pair will provide a limit to the gap that may be bridged. Energy constraints increase as collision partner mass increases, an effect that is amplified when J(i)>0. Large energy gaps are most effectively bridged using light collision partners. For efficient vibrational relaxation in polyatomics an additional requirement is that the molecular motion of the mode must be capable of generating molecular rotation on contact with the collision partner in order to meet the angular momentum requirements. We postulate that this may account for some of the striking propensities that characterize polyatomic energy transfer.     

Oil-water partition coefficients from solvent activities

The activity coefficients of 1-octanol and 2-octanol in benzene at 20°C are reported here. These have been combined with literature data and used to estimate the 1-octanol-water partition coefficient for benzene. The value of 1200±100 on the mole-fraction scale agrees well with the literature value of about 1200.     

Symplectic integrators and the conservation of angular momentum

In this article we observe that generally symplectic integrators conserve angular momentum exactly, whereas nonsymplectic integrators do not. We show that this observation extends to multiple timesteps and to constrained dynamics. Both of these devices are important for efficient molecular dynamics simulations. © 1995 by John Wiley & Sons, Inc.     

Extension of total angular momentum representation in energy space

A previously developed representation of total angular momentum in energy space for three atomic systems is extended to include reactions of the type CF₃CN → CF₃ + CN. The effects of internal CF₃ rotation on the angular momentum constraint are shown under different conditions. The representation is used for comparison with recent infrared multiphoton dissociation experiments on this system.     

Activity coefficients of adsorbed mixtures

Experimental and simulated data for adsorption of gas mixtures on energetically heterogeneous surfaces like activated carbon and zeolites exhibit negative deviations from ideality. The deviations are large in some cases, with activity coefficients at infinite dilution equal to 0.1 or less. Similar molecules form ideal mixtures, but molecules of different size or polarity are nonideal. Equations for bulk liquid mixtures (Wilson, Margules, etc.) do not apply to isobars for adsorbed mixtures. A two-constant equation for activity coefficients as a function of composition and spreading pressure is in good agreement with theory, simulation, and experiment.