Linear notations definition

Another linear notation is the ROSDAL coding. It has very elaborate stereochemical syntax elements, which make it useful for the definition of organic as well as inorganic compounds. It covers even conformation representations. ROSDAL strings are extensively used in the Beilstein XFIRE system. 

In the following, a general treatment of arbitrary binary excitation sequences will be given. Since the proper definition of the excitation and the response function is not unambiguously possible, a problem-independent notation will first be given, which will later be mapped to the actual experiment. For the moment, it is sufficient to picture a linear system with an input x(t), an output y(t) and a linear response function h(t), as sketched in Fig. 22. The input x(t) may be a pulse of finite duration, as discussed in the previous sections, or a pseudostochastic random binary sequence as in Fig. 22. 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

The specification super-operator is common in quantum chemical emd physical literature for linear mappings of Fock-space operators. It is very helpful to transfer this concept to the extended states A, B) and define the application of super-operators by the action on the operators A and B. We will see later how this definition helps for a compeict notation of iterated equations of motion and perturbation expansions. In certain cases, however, the action of a super-operator is fully equivalent to the action of an operator in the Hilbert space Y. The alternative concept of Y-space operators allows to introduce approximations by finite basis set representations of operators in a well-defined and lucid way. 

Finally, definitions of and the customary notation for some special matrices used in special linear systems [Eq. (1)] are given in Table I. 

Here S t) is an independent source vector and M i) a matrix describing the properties of the system. Although the describing or state equation is apparently of first order only, this is no real restriction. For example, a single second-order equation could always be represented as two simultaneous first-order equations, and so on. Although the notation is convenient for linear problems where M is independent of the density N, the more general case where Af is a function of the density is not ruled out by this notation. (By definition, S is not a function of N.)... 

The primitive body-fixed basis functions as described above do not have definite parity, except for K = 0. However, since parity is known to be a rigorously good quantum number, it is usually advantageous to choose basis functions which do have definite parity, and it is straightforward to define linear combinations of the primitive functions for which this is the case. Adopting the notation 0 = K, these are... 

Each of the L states in molecules belonging to C ov or 1 exhibits a definite behavior (+ or —) under the reflection operation, and this is always indicated in the superscript that accompanies the IR notation. For the doubly degenerate states (II, A, O,...) it is always possible to choose linear combinations analogous to those in Eq. 4.10 in order that each of the two combinations is either unaffected or changes sign with respect to a particular reflection plane. The reflection symmetries of the cos0 and sincf) combinations in Eq. 4.10, for example, are respectively (+) and (-) with respect to a reflection plane containing the x and z axes. 

Electromechanical and mechano-electiical transduction is often based on inverse and direct piezoelectricity, respectively. According to the definition of the piezoelectric coefficient (cf Eq. 1 and Fig. 1 above), direct piezoelectricity is the linear change of the electrical displacement D caused by mechanical stress T, while inverse piezoelectricity is the linear change of the mechanical strain S due to electrical stress E. Here, either the electric field E or the mechanical stress T has to be kept eonstant, respectively. Under this condition, the electric displacement D may be replaced by the electric polarization P in the definition of the direct d coefficient d = dP/dT, which may also be written in tensor notation as Pj = djkiTki (Newnham 2005). 

Viscosity coefficients measured in these geometries when n is immobilised by boMiesowicz viscosities. (Note, that in the literature a variety of alternative notations are common in particular the definitions of r i and r 2 are frequently interchanged.) If the orientation of n is fixed in an arbitrary direction with respect to v and Vv, then the effective viscosity coefficient is given by a linear combination of the Miesowicz viscosities, and another viscosity constant Tju, which cannot be visualised in a pure shear-flow ... 

---