Tensor formalism

A general introduction to tensor analysis and its relation to Dirac s notation may be found in the book of Schouten.  

Every vector x in an -dimensional linear vector space may be expressed as a linear combination of basis vectors e,-  

The basis vectors are nonorthogonal in general. That is, the scalar product of every pair gives a number S  

So far we have restricted ourselves to vectors so as to simplify the discussion. Now we turn to tensors. A tensor T of rank k may be seen as an entity whose components are described by 

Consider the following examples for illustration A vector a in n-dimensional space is described completely by its n components a,. It may therefore be seen as a one-index quantity or a tensor of rank one. A matrix A has components A,/ (two indices) and is a rank two tensor. A tensor of rank three has n components, and its components have three indices, T, and so on. As a special case, scalars have only = 1 component and are tensors of rank zero. 

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The Hamiltonian describing the quadrupolar interaction in the laboratory frame (L), in the units of radians/s, can be written using the spherical tensor formalism as [1,6, 24]... 

Tensor formalism can be used to generate all aspects of the electromagnetic field. The inhomogeneous Maxwell equation (2) in tensor form... 

Wayman describes in detail how the tensor formalism can be used to solve the crystallographic problem [5]. A simple graphical demonstration, in two dimensions, of how an invariant line (habit plane) can be produced by the deformations B, S, and R is given in Exercise 24.6. 

In many cases in this book, particularly when /I-doubling is involved, we also consider the q 0 components of the dipolar interaction, which follow naturally from the irreducible tensor formalism. The matrix elements of such components are off-diagonal in A, and their angular dependence is described by the spherical harmonics Y2) n(0, < />) or Y2, 2(0,0). In such cases, we shall use the constants t and t 2, defined by analogy with to in equation (8.515). To be specific, the parameter d of Frosch and Foley is related to our t2 by the expression... 

SYMMETRY-ADAPTED STRAIN, SYMMETRY-BREAKING STRAIN, NON-SYMMETRY-BREAKING STRAIN AND SOME TENSOR FORMALITIES... 

The development of DFT computations of electronic g-tensors has mainly focused on improving the accuracy and applicability for isolated systems, while only little attention has been devoted to account for environmental effects. Most studies of solvent or matrix effects on electronic g-tensors have adopted the supermolecular approach, in which the solvent molecules are explicitly introduced into the model used in the calculations. Recently, we developed an electronic g-tensor formalism in which solvent effects are accounted for by the polarizable continuum model [154]. We applied this approach to investigate solvent effects on electronic g-tensors of di-r-butyl nitric oxide (N-I) and diphenyl nitric oxide (N-II). Calculations were... 

Which agrees with the result obtained using the irreducible tensor formalism.34 The triple-quantum filtration pulse sequence, applied to a spin- quadrupolar nucleus, produces almost exactly the same result as does the double-quantum filtration sequence, except for two differences. The double-quantum relaxation rate is faster than the triple-quantum relaxation rate, and the triple-quantum FID is a factor of 1.5 larger than the double-quantum FID.34 The triplequantum filter will thus require less than half the number of acquisitions to equal the signal-to-noise of the double-quantum sequence. 

In tensor formalism the difference between electric and magnetic fields practically disappears. What one observer interprets as an electric process another may regard as magnetic, although the actual particle motions that they predict will be identical. It can be shown (Schwarz, 1972) that the two tensors are related through a completely antisymmetrical tensor of fourth rank... 

Returning to the phenomenological approach, beyond the matrix and tensor formalism just sketched, several fundamental conclusions can be expressed, seen as consequences of the postulates of special relativity (Einstein), namely ... 

Let us consider a molecule placed in a cavity surrounded by a dielectric continuum (fig. 1). The relative dielectric permittivity of the continuum is assumed to be e and in the cavity it is taken as equal to the permittivity of a vacuum. In the following we shall assume that the charge distribution of the solute is represented by a single center multipole expansion. An equivalent distributed multipole 2,3] representation may be used without further difficulty. We shall use the spherical tensors formalism [4,5] for the multipoles in which the 2/4 1 components of the multipole of rank / at the origin are defined from unnormalized spherical harmonics [6] by the equation ... 

It has recently become clear that classical electrostatics is much more useful in the description of intermolecular interactions than was previously thought. The key is the use of distributed multipoles, which provide a compact and accurate picture of the charge distribution but do not suffer from the convergence problems associated with the conventional one-centre multipole expansion. The article describes how the electrostatic interaction can be formulated efficiently and simply, by using the best features of both the Cartesian tensor and the spherical tensor formalisms, without the need for inconvenient transformations between molecular and space-fixed coordinate systems, and how related phenomena such as induction and dispersion interactions can be incorporated within the same framework. The formalism also provides a very simple route for the evaluation of electric fields and field gradients. The article shows how the forces and torques needed for molecular dynamics calculations can be evaluated efficiently. The formulae needed for these applications are tabulated. 

A more practical objection to the spherical tensor formulation has been that it is inefficient to use in computationally-intensive applications such as molecular dynamics, because of the supposed need to describe the molecular orientations in terms of Euler angles and to evaluate trigonometrical functions of these angles. This objection is completely unfounded. The Euler angles have traditionally been used to describe orientation, but they are by no means necessary, and the formulae to be presented below will not mention them at all. It is possible, within the spherical tensor formalism, to describe orientation in terms of the same direction cosines /. ... 

Sometimes it is useful to represent the multipole electrical moments in a spherical form. The spherical form of these moments allows us to apply effectively the theory of irreducible spherical tensor formalism. For this aim these 2 -pole moments may be written in terms of the regular spherical harmonics using their both complex Rlm r) and real (Rimc r) and Rims r)) forms defined, for m > 0, as... 

The power of the tensor formalism and the concept of effective slippage have then been demonstrated by exact solutions for two other potential applications optimization of transverse flow and anal3Aical results for hydrodynamic resistance to the approach of two surfaces. These examples demonstrate that properly designed superhydrophobic surfaces could generate a very strong transverse flow and significantly reduce the so-called "viscous adhesion." Finally, we have discussed how superhydrophobic surfaces could amplify electro kinetic pumping in microfluidic devices. 

Standard tensor formalism is employed throughout this paper, for example, the Einstein convention of implicit summation over two repeated Greek subscripts is in force. The notation adopted in previous references [15, 39,40] is used. The SI system of units has been adopted. 

Terminology and notation adopted in previous papers and reviews [2-5] are used, allowing for standard tensor formalism, e.g., smnmation over repeated Greek indices is implied according to the Einstein convention. 

In this Sect. 1 focus on two topics the assignment of chemical bonds within clusters and the electrostatic interaction, a paramount component of intermolecular forces. The latter topic involves the well-known multipolar expansion [4,5], performed in the spherical tensor formalism [6], which An-... 

Due to the spherical-tensor formalism, the molecular multipole moments can be easily converted between two coordinate frames and E2 according to... 

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