Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Starting from one-range addition theorems for Slater-type functions, which are expansion in terms of complete and orthonormal functions based on the generalized Laguerre polynomials, Guseinov constructed addition theorems that are expansions in terms of Slater-type functions with a common scaling parameter and integral principal quantum numbers. This was accomplished by expressing the complete and orthonormal Laguerre-type functions as finite linear combinations of Slater-type functions and by rearranging the order of the nested summations. Essentially, this corresponds to the transformation of a Laguerre expansion, which in general only converges in the mean, to a power series, which converges pointwise. Such a transformation is not necessarily legitimate, and this contribution discusses in detail the difference between truncated expansions and the infinite series that result in the absence of truncation.

The Full Rotation Group and Irreducible Spherical Tensors

The anisotropic character of the chemical shift can be measured by experiments that provide shift tensor values and comparing these experimental components, obtained from microcrystalline powders, with 3D nuclear shielding tensor components, calculated with quantum chemistry, yields structural models of the analyzed molecules. The use of a metric tensor for evaluating the mean squared deviations, d2, between two or more tensors provides a statistical way to verify the molecular structure governing the theoretical shielding components. The sensitivity of the method is comparable with diffraction methods for the heavier organic atoms (i.e., C, O, N, etc.) but considerably better for the positions of H atoms. Thus, the method is especially powerful for H‐bond structure, the position of water molecules in biomolecular species, and other proton important structural features, etc. Unfortunately, the traditional Cartesian tensor components appear as reducible metric representations and lack the orthogonality of irreducible icosahedral and irreducible spherical tensors, both of which are also easy to normalize. Metrics give weighting factors that carry important statistical significance in a structure determination. Details of the mathematical analysis are presented and examples given to illustrate the reason nuclear magnetic resonance are rapidly assuming an important synergistic relationship with diffraction methods (X‐ray, neutron scattering, and high energy synchrotron irradiation). © 2009 Wiley Periodicals, Inc.Concepts Magn Reson Part A 34A: 217–237, 2009.

All Hamiltonians of spin interactions, which are based on second-rank tensors (see Tables 46.1 to 46.5), can be expressed in the general form $$ \mathcal{H}_{i}=a_{i}\sum_{l=0,2} \sum_{m=-2}^{l} \left ( -1 \right)^{m}A_{l},-mT_{l,m} $$ (49.1) where ai is a constant specific for the interaction. The quantities A l ,_ m are functions of the spatial orientation and distances (usually given in polar coordinates (r, ϕ, ϑ) as defined in Chap. 48), whereas T l,m represents functions of spin operators. The analytical form of these expressions are chosen in such a way that the Hamiltonians transform under rotations like spherical harmonics of rank l. They are therefore referred to as irreducible spherical tensor (IST) operators of the spatial and the spin coordinates, respectively. “Irreducible” means that no decomposition in terms transforming under rotations other than spherical harmonics of the same rank l is possible.1 The subscript m indicates the order of multiple-quantum transition induced by T l,m .

Simulations are essential in NMR spectroscopy for understanding and quantitatively predicting the outcomes of experiments. Dynamic NMR experiments are influenced by relaxation and chemical exchange phenomena, which complicate their interpretation. The quantum mechanical theory behind NMR simulations is well established and enables simulations under various conditions or states of the system. In particular, the representation of the spin system in terms of the density matrix, energy and pulse operators, and relaxation and chemical exchange in Liouville space is straightforward and allows for the routine simulation of dynamic NMR experiments. In fact, several advanced open-source software packages are currently available-such as Spinach, GAMMA (pyGAMMA), Simpson, ChemEx, and NMR TITAN-which users can choose from based on their simulation needs. Many of these software packages are optimized for computational speed and make use of sophisticated bases, such as the irreducible spherical tensor basis. Here, we present simulations using the direct method (spin product basis), as implemented in GAMMA, which we believe will help users better understand the underlying theory and thereby explore the other available software resources more effectively. We chose two experiments-CPMG-RD and CEST-as a case study to understand the conformational changes in proteins or macromolecules occurring in the fast and slow exchange regimes, which are frequently investigated by structural biologists.

We consider the exterior domain problems of Dirichlet and Neumann type of the two-dimensional Stokes equations. For the solution of this boundary value problem we choose a potential ansatz and show that for the reduction of the computational costs, the fast multipole method of Greengard and Rokhlin can be used. Therefore, we find a complex representation of the hydrodynamical potentials and provide statements about the corresponding multipole and Taylor expansions, as well as the appropriate translation, rotation and conversion operators. The theoretical statements are illustrated by numerical experiments. Bibliography: 15 titles.

We present an efficient solution of the initial-value problem of the torque-free rotation of a triaxial rigid body in a complete manner. The solution transforms, by means of a transition matrix, the coordinate triad attached to the rotating body as well as the body-fixed components of the rotational angular momentum vector from their initial values to the values at an arbitrary time. The transition matrix is expressed as one rotation matrix sandwiched by two wobble matrices. Their computation is significantly accelerated by (1) the rewriting of wobble matrices in terms of the body-fixed components of rotational angular momentum, (2) the evaluation of elliptic functions by addition theorems, and (3) the utilization of Maclaurin series expansion in computing various kinds of transcendental functions including the incomplete elliptic integral of the third kind for small arguments. The resulting formulation is a counterpart, in rotational dynamics, of the method of the so-called f- and g-functions to solve the initial-value problem of a Keplerian orbit efficiently.

To develop a framework for 3D sodium (23 Na) MR fingerprinting (MRF), based on irreducible spherical tensor operators with tailored flip angle (FA) pattern and time-efficient data acquisition for simultaneous quantification of T1 , , , and in addition to ΔB0 . 23 Na-MRF was implemented in a 3D sequence and irreducible spherical tensor operators were exploited in the simulations. Furthermore, the Cramér Rao lower bound was used to optimize the flip angle pattern. A combination of single and double echo readouts was implemented to increase the readout efficiency. A study was conducted to compare results in a multicompartment phantom acquired with MRF and reference methods. Finally, the relaxation times in the human brain were measured in four healthy volunteers. Phantom experiments revealed a mean difference of 1.0% between relaxation times acquired with MRF and results determined with the reference methods. Simultaneous quantification of the longitudinal and transverse relaxation times in the human brain was possible within 32 min using 3D 23 Na-MRF with a nominal resolution of (5 mm)3 . In vivo measurements in four volunteers yielded average relaxation times of: T1,brain = (35.0 ± 3.2) ms, = (29.3 ± 3.8) ms and = (5.5 ± 1.3) ms in brain tissue, whereas T1,CSF = (61.9 ± 2.8) ms and = (46.3 ± 4.5) ms was found in cerebrospinal fluid. The feasibility of in vivo 3D relaxometric sodium mapping within roughly ½ h is demonstrated using MRF in the human brain, moving sodium relaxometric mapping toward clinically relevant measurement times.

Irreducible spherical tensor analysis of quadrupolar nuclei

A low frequency stable plane wave addition theorem

The bisection method to localize the solution of a nonlinear equation [Fukushima (1996, M, 112, 2858)] was extended to handle a long sequence of bisections in a concise manner. This was done by means of the addition theorem of transcendental functions appearing in the equation. The localizer extended was combined with a variation of Newton's method where the functions are evaluated by their Taylor series expansions. As its application, we developed a procedure solving an extended form of Kepler's equation for the hyperbolic case. Our procedure is robust and fast. It finds sufficiently precise solutions even when Danby's starter [1988, Fundamentals of Celestial Mechanics, 2nd Ed. (Willmann-Bell, Richmond, VA), Section 6.9] fails. For typical cases, it requires less CPU time than that for evaluating the equation itself for an arbitrary argument.

We present the full quantum mechanical theory of the angular momentum distributions of photofragments produced in photolysis of oriented/aligned parent polyatomic molecules beyond the axial recoil limit. This paper generalizes the results of Underwood and Powis(28,29) to the case of non-axial recoil photodissociation of an arbitrary polyatomic molecule. The spherical tensor approach is used throughout this paper. We show that the recoil angular distribution of the angular momentum polarization of each of the photofragments can be presented in a universal spherical tensor form valid for photolysis in diatomic or polyatomic molecules, irrespective of the reaction mechanism. The angular distribution can be written as an expansion over the Wigner D-functions in terms of the set of the anisotropy-transforming coefficients c(K(i)q(i))(K) (k(d), K(0)) which contain all of the information about the photodissociation dynamics and can be either determined from experiment, or computed from quantum mechanical theory. An important new conservation rule is revealed through the analysis, namely that the component q(i) of the initial reagent polarization rank K(i) and the photofragment polarization rank K onto the photofragment recoil direction k is preserved in any photolysis reaction. Both laboratory and body frame expressions for the recoil angle dependence of the photofragment angular momentum polarization are presented. The parent molecule polarization is shown to lead to new terms in the obtained photofragment angular distributions compared with the isotropic case. In particular, the terms with |q(i)| > 2 can appear which are shown to manifest angular momentum helicity non-conservation in the reaction. The expressions for the coefficients c(K(i)q(i))(K) (k(d), K(0)) have been simplified using the quasiclassical approximation in the high-J limit which allows for introducing the dynamical functions and the rotation factors which describe the decreasing of the photofragment angular momentum orientation and alignment due to the rotation of the molecular axis during photodissociation. In this case, the resultant recoil angle dependence is also presented in a form where the anisotropy of the parent molecular ensemble is expressed in terms of the molecular axis distribution, rather than in terms of the molecular density matrix.

A cogent theory of collective multipole-like quantum correlations in symmetric multiqubit states is presented by employing SO(3) irreducible spherical tensor representation. An arbitrary bipartite division of this system leads to a family of inequalities to detect entanglement involving averages of these tensors expressed in terms of the total system angular momentum operator. Implications of this theory to the quantum nature of multipole-like correlations of all orders in the Dicke states are deduced. A selected set of examples illustrate these collective tests. Such tests detect entanglement in macroscopic atomic ensembles, where individual atoms are not accessible.

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

Rotational line structure in three-photon scattering by symmetric top molecules