Gradient tensor

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

Complex Harmonics and the Spherical Tensor Gradient Operator... 

The spherical tensor gradient operator (STGO) is a solid harmonic whose Cartesian coordinate arguments have been replaced by Cartesian derivatives. Hobson s theorem [25] is the result of acting a STGO upon any spherical function /(r )... 

By having written a point-multipole as the spherical tensor gradients passing through a point, one easily derives the particle mesh Ewald method for point multipoles. The main differences occur in the calculation of the structure factor, which requires spherical tensor gradients of the Cardinal B-spline weight, and the calculation of the short-range real-space correction (see Section 1.6.3). 

Giese, T. J., and York, D. M. (2008). Spherical tensor gradient operator method for integral rotation A simple, efficient, and extendable alternative to Slater-Koster tables,/. Chem. Phys. 129(1), 016102. 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Restored parameters for the evaluation of PDSM, may be different PMF of material tensor of stresses or its invariants, spatial gradients of elastic features (in particular. Young s modulus E and shear modulus G), strong, technological ( hardness HRC, plasticity ), physical (density) and others. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

The electric field gradient is again a tensor interaction that, in its principal axis system (PAS), is described by the tluee components F Kand V, where indicates that the axes are not necessarily coincident with the laboratory axes defined by the magnetic field. Although the tensor is completely defined by these components it is conventional to recast these into the electric field gradient eq = the largest component,... 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

The right Cauchy-Green strain tensor corresponding to this deformation gradient is thus expressed as... 

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

K set of referential internal state variables I velocity gradient tensor... 

The partial derivatives of x are the velocity vector y and the deformation gradient tensor f, respectively. 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

Comparing this with (A. 10), it is seen that is the velocity gradient when U = I, i.e., in a pure rotation. It is easily seen by differentiating the orthogonality condition (A.14i) that is antisymmetric. Analogously, the tensor / will be defined by... 

The dipole polarizability, the field gradient and the quadrupole moment are all examples of tensor properties. A detailed treatment of tensors is outside the scope of the text, but you should be aware of the existence of such entities. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

These equations, with six farther equations for the other components of a20 and au, when solved by the Burnett iteration procedure, yield the Navier-Stokes equations when solved simultaneously, however, there is no longer the simple dependence of the pressure tensor upon the velocity gradients, and of the heat flow upon the temperature gradient, but, rather, an interdependence of these relations. 

Fig. 21. Schematic drawing of the Taylor four-roll mill with the created flow field. The 2-dimensional velocity gradient can be represented by the tensor A ...

Fig. 4a-c. Sketch of simple motional mechanisms and resulting averaged field gradient tensors a Kink-3-bond motion b crankshaft-5-bond motion c 180° jump of phenyl ring... 

Electric field gradient tensor 24 Entanglements 124 Entropy model 200,201 Epoxy composites 192... 

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