An Alternative Basis

Aqueous geochemists, however, are interested not only in the composition of all parts of the system , but also in the concentrations of all aqueous species in the system, including charged ions. But we cannot describe the concentration of Na+ or H+ ions in a salt solution using the formulas NaCl and H2O no combination of the formulas NaCl and H2O will result in the formula Na+. Furthermore, modelers want not only to describe the composition of systems, but to control, or constrain, their evolution during some process, such as maintaining equilibrium with some solid or gas phase. 

To do these things, we need to use building blocks different from those we would choose to merely describe bulk compositions as illustrated above. We must use not only a different basis, but a different kind of basis. We could use the elements themselves (Al, B, N,. .., etc.), plus the electronic charge, because this would certainly allow us to describe the composition of any species or phase. However, it has proved to be convenient to use as building blocks , or descriptive composition terms, entities which do exist - ordinary charged ions such as IICO and Na+. These are called basis species, component species, or master species, and they make up a new kind of basis, which is the minimum number of chemical formulas needed to describe the composition of all phases and all species, charged and uncharged in the system. If mineral or gas phases are present, their compositions must also be included in the basis, as described below. 

Suppose a system contains 1 mole of NaCl and 1 kg of water. The components as defined in 3.6.1 are NaCl and H2O, but the basis species needed to describe all the ions present are (in most programs other choices are always possible) Na+, Cl-, H+, and H2O. A speciation calculation gives the following results  

Note that the composition of all seven actual species can be described by some combination of the four basis species (e.g., NaOH = Na+ + H2O — H+), and that each basis species (other than H+ and H2O) represents the total amount of some element. Thus 

Most modeling programs have a selection of 30 to 80 or more basis species, plus a collection of minerals and gases, from which the modeler chooses those required to describe the composition of all aqueous species, gases, and minerals in a particular system. If an element, say rubidium (Rb), does not occur as a basis species (as say, Rb+) in the database, the program is of course then unable to calculate the amounts of various rubidium species or minerals, even if we have an analysis for the rubidium content of our system. 

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Moreover, a constant section can only contain integer n and I values separated by Al, An) = m+ —m - - 1), which means that the horizontal and vertical lattice vectors in Fig. 22 are given by x = 0,m + 1) and (1,0), respectively. Hence an alternative basis may be dehned with a unit cell having sides Av,APj = (l,m+ 1), in which case the argument in Section IIC3 shows that the two-dimensional monodromy matrix for the fixed km section transforms to... 

For each irreducible representation JW there are different projection operators, so in all there are 2 n — g such operators. As we shall see presently, they are linearly independent, and thus form an alternative basis for the group algebra any element of U can be represented as a linear combination of the... 

The customer required isothiazoline and DBNPA to be used on an alternating basis, but was agreeable to the supply of a biodispersant. However, the practice was for the biodispersant to be added 4 hours after each application of biocide, rather than before or at the same time, which tended to defeat the purpose. 

The use of quarks in atomic shell theory provides an alternative basis to the traditional one. The transformations between these bases can be complicated, but there are many special cases where our quarks can account for unusual selection rules and proportionalities between sets of matrix elements that, when calculated by traditional methods, go beyond what would be predicted from the Wigner-Eckart theorem [4,5], This is particularly true of the atomic f shell. An additional advantage is that fewer phase choices have to be made if the quarks are coupled by the standard methods of angular-momentum theory, for which the phase convention is well established. This is a strong point in favor of quark models when icosahedral systems are considered. A number of different sets of icosahedral Clebsch-Gordan (CG) coefficients have been introduced [6,7], and the implications of the different phases have to be assessed when the CG coefficients are put to use. 

Thus, vhile the value of executing phase studies using svelling methods has been established beyond doubt, the analysis of composition using index-of-refraction data has been disappointing. A different kind of data vhich is useful in the context of the DIT experiment and vhich could serve as an alternative basis for analysis is needed. 

Recent attention has been directed towards circular magnetization modes (i.e., clockwise and counterclockwise), rather than linear modes as an alternative basis for storage logic [22],... 

An alternative basis combining moderate size with the accuracy of the... 

If an alternative basis set is used in which the basis functions are just linear combinations of the original basis functions, then the same wavefunction can be written as a linear... 

V ariants of this model may be derived by assuming an alternative basis model (such as the implicit or explicit penultimate models) or by making further assumptions as to nature of the complexation reaction or the behavior of the complexed radical. For instance, in the special case that the complexed radicals do not propagate (that is, Sd = 0 for all i), the reactivity ratios are not affected (that is, r = rj for all i) and the complex formation serves only removal of radicals (and monomer, if monomer is the complexing agent) from the reaction, resulting in a solvent effect that is analogous to a Bootstrap effect (see Section 13.2.3.4). 13.2.3.2.3 Experimentai evidence... 

In addition to the use of paths as molecular descriptors, the connectivity indices X " X X offer an alternative basis. As mentioned before, in the case of... 

Kekule valence structures continue to offer an alternative basis for description of benzenoid and non-benzenoid conjugated hydrocarbons, which complements characterization of molecules by quantum chemistry, which tends almost exclusively to consider molecular orbital approaches as the frame for discussion of molecular properties. We have already seen applications of conjugated circuits (which are extracted from individual Kekule valence structures) for clarification of the notion of aromaticity in polycyclic conjugated hydrocarbons and characterization of the relative magnitudes of molecular resonance energies and will also later consider the use of conjugated... 

An alternative approach to solving the coupled equations is to use a basis set expansion for the R coordinate as well as for the angular variables. The angular basis sets used in such calculations are generally the same as in coupled channel calculations. This approach was pioneered by Le Roy and Van Kranendonk, who used numerical basis sets for the radial (R) functions. Such basis sets are adequate for the rare gas-H2 systems, but converge very poorly for more strongly anisotropic systems. An alternative basis set, based on Morse-oscillator-like functions, has been used extensively by Tennyson and coworker... 

The alternative approach to determining requirements is to estimate the total body content of vitamin C, then measure the rate at which it is metabolized, by giving a test dose of radioactive vitamin. This is the basis of both the 1989 US RDA of 60 mg/ day for adults and the Netherlands RDA of 80 mg/day. Indeed, it also provides an alternative basis for the RNI of 40 mg/day adopted in Britain in 1991. 

Despite tireless and somewhat tiresome efforts to find an alternate basis for citizenship, the status has no logic, power, or moral force outside a nation state (Hansen 2009). A citizen is one who enjoys the firll panoply of rights—civil, social, economic, and political—accorded by a nation state a citizen can call on his or her nation state, and onty that rration state, to claim diplomatic protection and the nation state can in turn demand the ultimate loyalty of its citizens, including the obligation to fight and die. 

The enveloping nature of the design basis event relative to other events will be confirmed. The design basis event shall be a double-ended guillotine break (DEGB) loss-of-coolant accident (LOCA), unless an alternative basis is demonstrated to be... 

The basis set requirements for F12 calculations differ strongly from those of conventional calculations. In particular the rather steep functions with high angular momentum are not necessary. These mainly enter into the correlation-consistent series as they describe short-range correlation, that now is taken care of by the correlation factor. Therefore, Peterson and coworkers have developed an alternative basis set series, specially optimized for F12 calculations, the cc-pVXZ-F12 series. For these basis sets, taylor-made auxiliary basis sets have been provided, too. The above basis set series is optimized for valence correlation, a corresponding series cc-pCVXZ-F12 that is suitable for all-electron calculations has been developed as well. It is also possible to apply extrapolation schemes for these basis sets. ... 

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