Representation equivalent

Aside from the infinite number of representations equivalent to (9.25), there are other representations of C3 >. The matrices (9.25) are each in block-diagonal form. If we partition them into submatrices of the form (2.30), then the submatrices S12 and S2i will be column and row matrices whose elements are all zero. The product of any two of these partitioned... 

Let Rao be the matrix representing the symmetry operation R in the representation rAG. Equation (9.64) means that there is a similarity transformation that transforms RAO to RAO, where RAO is in block-diagonal form, the blocks being the matrices R],R2,...,Rfc of the irreducible representations ri,r2,...,ri. Let rAO denote the block-diagonal representation equivalent to TAO and let A be the matrix of the similarity transformation that converts the matrices of I AO to TAO Rao = A, RaoA. We form the following linear combinations of the AOs ... 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

Furthermore, since CWT is the space-scale representation equivalent to the zero scale asymptotic relations in Eq.(1.5), the same conditions should also apply within the wavelet representation. We refer to this procedure as Turning Point Quantization (TPQ). 

Thus, (x, y) produce a reducible, two-dimensional representation equivalent to the irreducible representation given by exp( /< ). The reason for listing (x, y) as bases... 

If we have found one representation of i-dimensional matrices Mj, M2, M, ... of the group A, then, at least for 1, we can define infinitely many other equivalent representations consisting of the matrices... 

The traces of the representation matrices are called the characters of the representation, and (equation Al.4.36) shows that all equivalent representations have the same characters. Thus, the characters serve to distingiush inequivalent representations. 

If we start with an t -dimensional representation of A consisting of the matrices M, M2, M3,. .., it may be that we can find a matrix V such that when it is used with ( equation A1.4.34) it produces an equivalent representation M, M 2, M 3,. .. each of whose matrices is in the same block diagonal form. For example, the nonvanishing elements of each of the matrices could fonn an upper-left-comer ... 

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3.8.13 (and/or the equivalent time integral of the flux-flux correlation fiinction). While this approach has to date only been applied to problems with simplified hannonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a fiirther classical or quasiclassical approximation. 

Benzene was probably the fust compound in chemical history where the valence bond concept proved to be insufficient. Localizing the nr-systems, one comes up with two equivalent but different representations. The true bonding in benzene was described as resulting from a resonance between these two representations (Figure 2-46). 

Figure 7-18. Schematic representation of the LCAO scheme in a, T-only calculation for ethylene, The AOs

Non-covalent interactions between molecules often occur at separations where the van der Waals radii of the atoms are just touching and so it is often most useful to examine the electrostatic potential in this region. For this reason, the electrostatic potential is often calculated at the molecular surface (defined in Section 1.5) or the equivalent isodensity surface as shown in Figure 2.18 (colour plate section). Such pictorial representations... 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

Remember that ai is the representation of g(x) in the fi basis. So the operator eigenvalue equation is equivalent to the matrix eigenvalue problem if the functions fi form a complete set. 

We could take any set of functions as a basis for a group representation. Commonly used sets include coordinates (x,y,z) located on the atoms of a polyatomic molecule (their symmetry treatment is equivalent to that involved in treating a set of p... 

In this section we resume our examination of the equivalency of time and temperature in the determination of the mechanical properties of polymers. In the last chapter we had several occasions to mention this equivalency, but never developed it in detail. In examining this, we shall not only acquire some practical knowledge for the collection and representation of experimental data, but also shall gain additional insight into the free-volume aspect of the glass transition. 

Fig. 2. Schematic representation of relevant electrolyte transport through the renal tubule, depicting the osmolar gradient ia medullary iaterstitial fluid ia ywOj yW where represents active transport, —passive transport, hoth active and passive transport, and passive transport of H2O ia the presence of ADH, ia A, the cortex, and B, the medulla. An osmole equals a mole of solute divided by the number of ions formed per molecule of the solute. Thus one mole of sodium chloride is equivalent to two osmoles, ie, lAfNaCl = 2 Osm NaCl. ADH = antidiuretic hormone.

Variations in measurable properties existing in the bulk material being sampled are the underlying basis for samphng theory. For samples that correctly lead to valid analysis results (of chemical composition, ash, or moisture as examples), a fundamental theoiy of sampling is applied. The fundamental theoiy as developed by Gy (see references) employs descriptive terms reflecting material properties to calculate a minimum quantity to achieve specified sampling error. Estimates of minimum quantity assumes completely mixed material. Each quantity of equal mass withdrawn provides equivalent representation of the bulk. 

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