Tilde notation

We adopt the tilde-notation Km as a reminder that these equilibrium ratios are the ones appropriate to the AUa = 0 case considered. Notice then that the indicator function [1 - ba (A )] corresponds to the Boltzmann factor for the solute-solvent interactions of hard-core type, i.e., interactions that are infinitely unfavorable for molecular overlap, but otherwise zero. Thus [56]... 

Here and further on all calculations and results will be done in Hq representation, so that the tilde notation will be dropped. 

In Equation (47) we resume using the tilde notation when determining operators in the representation of the interaction. 

As for the spin-orbit interaction, we can transfer the derivatives from the electron coordinates to the nuclear coordinates and make the operators to the left of 1 /r y act to the left. With the tilde notation, the real spatial integrals are... 

Expanding (14.6.4), we obtain for the CCS right-hand side (in the tilde notation of Section 13.7.3 for T1-transformed fluetuation potential)... 

Exercise. Find from (2.7) - in matrix notation, the tilde indicates the transpose -... 

Configuration interaction and stationary wavefunctions In order to simplify the treatment, a correlated wavefunction P(r)corr built by Cl between two uncorrelated wavefunctions will be considered. (To shorten the notations, the tilde describing the antisymmetric character of the wavefunctions and the spin of the electron have been omitted, and the spatial vectors of all electrons are indicated by the symbol r only.) One has... 

The operator denotes the tensor product of the electric fields. The tildes above the electric field and polarization symbols portray the wave character of the fields and are used for better distinction with the field amplitudes E(r) and P(r) as defined below. For the notation of the polarization components follows... 

Again, in order to simplify the notation, we suppress the tilde, even though the k s can contain electrical fields and do definitely so for... 

Following the notation of the previous section, Eq. (4.2), p. 62, the tilde indicates distinguished solute-solvent interactions without electrostatic interactions. The contribution of electrostatic interactions is then isolated as... 

We now consider packing issues associated with the reference term of van der Waals approaches exemplified by Eq. (4.1), p. 61. As noted there, the simplest interaction model appropriate for those terms is a hard-core model. The distinguished molecule considered will perfectly repel solvent molecules from an overlap, or excluded, volume. The general issues we develop will apply to such molecules in general solvents, i.e., the solvent need not be simple in the same sense as the hard-core solute we treat. But we will exemplify our general conclusions with results on the hard-sphere solvent system. The notation here, however, continues to use the tilde, as /i (i "), to indicate that the distinguished solute might serve as a reference case for subsequent treatment of other interactions. 

Two properties of the spatial correlation may instantly be anticipated lattice likeness, to obtain a low energy of the considered valence electron gas, and com-mensurability to the crystal structure, to obtain a low interaction energy between electron gas and crystal structure. If (au, ai2, ai3) = a = aj are the elementary vectors of a structure and a c = aj. the elementary vectors of a valence electron correlation then ais = 2C ajc KcS is a commensurability where KcS = (K n, K. 2, K 13 K21.K22, K23 K31, K32,K33) is the commensurability matrix. The valence electron correlation and also the connected commensurability matrix are written with a prime in order to have a clear distinction from a lattice matrix of a crystal. In Fig. 3 a commensurability between crystal structures is written in the extended form and in shorter notations. The lattice matrix of Cu is given with tilde in order to indicate... 

Finally, the nvalue column of the vla4.property table can be populated when possible. This column stores the numerical value of the property. Since not all values are numerical, this column may have null entries. The purpose is to enable efficient use of numerical data when appropriate, for example, to select by value, sort, apply mathematical functions, etc. The following SQL will update the nvalue column when possible with a numeric value. The tilde operator in the where selects text values that match the regular expression. The expression shown here allows integers, decimal values, and scientific notation using E or e for the exponent, for example 6.023E23. 

In preparinj this table we have used bold notation to indicate data given in the problem statement (e.g., 100 C) or I rom information on an adjacent point in the cycle (also indicated by arrows connecting two points), italics to indicate numbers found in the steam tables (e.g.. 0.10135 MPa), and a tilde to indicate numbers obtained from the. steam tables by interpolation le.g.-. 2734.7 kJ/kg). 

Note.- From now on, we will work with perturbation (deviation) variables. We will use the normal notation (x, u, d, y, without tilde ). 

There are different ways to write down tensors and their components, whose usefulness depends on the context. If we talk about the tensor itself, independent of a coordinate system, we use the symbolic notation. Different typographical styles can be found in the literature. In this book, a first-order tensor is underscored once a, b,. ..), a, second-order tensor twice (and usually denoted with a capital letter. A, B,. ..). Higher-order tensors get a tilde... 

The vectors 8tOjp,) and (S/0/ appearing in Eqs. (559) and (560) correspond to the operators 8, Ojp(t) and 8tOk, where p(t) is the statistical density operator given by Eq. (341), and StOj and 8t0k are the operators defined by Eqs. (361) and (362), respectively. Unless stated otherwise, the tilde in the various expressions discussed here and in the remainder of our discussion of the maximum entropy approach to nonequilibrium processes should be interpreted as a generalized Kubo transform as defined by Eq. (363) or (450). As evident form Eq. (363), the generalized Kubo transform carries an additional time dependence. For the sake of notational simplicity, we have suppressed it. 

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