Fixed valency

Figure 4 Time step dependence of the average total energy for two models of a partially hydrated dodecamer DNA duplex. Thinner traces show results for virtually harmonic conditions when temperature was lowered to 1 K. The DNA molecule has fixed bond lengths, rigid bases, and fixed valence angles except for the mtra- and extracyclic bond angles m sugars, (a) No modifications of inertia (b) inertia modified as explained m the text. (From Ref. 54.)...

Molecular Energetics. Molecular energies can be computed in a variety of ways including empirical fixed valence potentials, full force field potentials, and semi-empirical molecular orbital techniques (CNDO-2, INDO, MINDO-3, MNDO, PCILO). 

At low temperatures magnesium oxide, MgO, which adopts the sodium chloride structure, is virtually a stoichiometric phase, but at high temperatures in the MgO-A1203 system this is not so. At 1800°C the approximate composition range is from pure MgO to 5 mol % A1203 95 mol % MgO. The simplest way to account for this composition range is to assume that point defects are responsible. For this, because both Mg2+ and Al3+ cations in this system have a fixed valence, electronic compensation is unreasonable. There are then three ways to account for the composition range structurally ... 

A straightforward method is to incorporate ahovalent impurity ions into the crystal. These impurities can, in principle, be compensated structurally, by the incorporation of interstitials or vacancies, or by electronic defects, holes, or electrons. The possibility of electronic compensation can be excluded by working with insulating solids that contain ions with a fixed valence. 

The same analysis can be applied to compounds with a more complex formula. For example, the oxide LaCoCL, which adopts the cubic perovskite structure, usually shows a large positive Seebeck coefficient, of the order of +700 jjlV K-1, when prepared in air (Hebert et al., 2007). This indicates that there are holes present in the material. The La ions have a fixed valence, La3+, hence the presence of holes must be associated with the transition-metal ion present. Previous discussion suggests that LaCo03 has become slightly oxidized to LaCoCL+j, and contains a population of Co4+ ions (Co3+ + h or Coc0)- Each added oxygen ion will generate two holes, equivalent to two Co4+ ... 

Consistent with the proposed importance of Mn valence and LFSE, low fixed valence cations (e.g., AF+, Mg +, Li+) and electronegative multivalent... 

Y, and Z are connected by bonds of fixed length joined at fixed valence angles, that atoms W, X, and Y are confined to fixed positions in the plane of the paper, and that torsional rotation 0 occurs about the X-Y bond which allows Z to move on the circular path depicted. If the rotation 0 is "free such that the potential energy is constant for all values of 0, then all points on the circular locus are equally probable, and the mean position of Z, i.e., the terminus of , lies at point z. The mean vector would terminate at z for any potential function symmetric in 0 for any potential function at all, except one that allows absolutely no rotational motion, the vector will terminate at a point that is not on the circle. Thus, the mean position of Z as seen from W is not any one of the positions that Z can actually adopt, and, while the magnitude ll may correspond to some separation that W and Z can in fact achieve, it is incorrect to attribute the separation to any real conformation of the entity W-X-Y-Z. Mean conformations tiiat would place Z at a position z relative to the fixed positions of W, X, and Y have been called "virtual" conformations.i9,20it is clear that such conformations can never be identified with any conformation that the molecule can actually adopt... 

The term valence, of which ambivalence is not merely a variation, but a decidedly new and separate concept, derives from chemistry and atomic physics. Valence can refer to an extract or tincture, usually from an herb. In this connotation, it has obvious ties with the field of medical alchemy, or iatrochemistry. In the mid-i8oos, valence theory began to be used to signify the normal number of bonds that a given atom can form with other atoms—a register that links valence with philosophical materialism, matter, and Epicurianism. In recent scientific work, valence refers specifically to the number of electrons in the outermost shell of atoms. It is not provisional or occasional in its relation to the atom. Valence is atomicity. It defines a given chemical element, perhaps not in its essence, but in its capacity to combine with other elements—its potentiality. Valence is denoted by a simple number, and elements are said to be monovalent, bivalent, trivalent, quadrivalent, and so on. About one-fifth of all elements have a fixed valence (sodium is always i, or monovalent calcium is always 2, or bivalent and so on). Many elements have valences that are variable, depending on the other elements with which they are combined. 

These workers proceeded to carry out theoretical conformational analyses of a set of sweeteners, many of which have multiple A-H and/or B sites. The set of compounds analyzed are reported in Table I along with the sweet potency relative to sucrose. A fixed valence geometry molecular mechanics force field was used in the conformational analyses. The conformational search strategy was as follows ... 

Figure 2.8. Free rotation with fixed valency angle.

As an example, consider a chain of atoms— 1,/ ,/ + 1... with a fixed valence angle so that the angle between bonds / — 1, / and j>. [I + 1 is fixed at 6. Let dj. i = 1,2,3, be the unit base vectors of the fixed laboratory reference frame. In addition, we introduce a moving local Cartesian system with base vectors dj = 1,2,3, defined in Fig. 11. Then any vector r can be written in terms of components with respect to either frame as r = r, c, = fj dj, where we are adopting throughout the convention that superimposed bars denote components with respect to the local frame a,. The vector components with respect to the two frames are related in the usual way as... 

Calculation of the dimension FR/N of a chain with N segments with free rotation about valence bonds has not yet been carried out because of uncertainties in the dimensions of the six isomers of the repeat unit. The measured values of correspond to /IV on the order of 100 A.2, where a repeat unit is equated to a segment. It is expected that the freely rotating chain with fixed valence bonds will be a good model for BBB owing to the inappreciable steric interference to rotation about the single bonds connecting the repeat units. Such behavior has... 

Many different types of catalysts have been employed, including both metal oxides, such as PbO (5-7), which react in a redox cycle to Pb metal, and catalysts containing metals with fixed valence, such as Li promoted MgO (jg), which produce active Li 0" sites for methane dimerization. Such catalysts were discussed in four papers at the recent Ninth International Congress on Catalysis 8)- The most effective catalysts have several common traits. Low surface area has been found to be very important 2 9) in the conversion of methane. Also of importance is the basicity of the catalyst. (However, not every basic material causes C2 formation.)... 

In real pol3uners the bonds cannot assume arbitrary directions but there are fixed valence angles between them. In addition, rotation about bonds is not entirely free, because the potential energy shows minima and maxima as a function of the rotation angle. To account for these effects, the above equations may be modified with a rigidity constant which depends on the architecture of the chain. If we introduce a stiffness ["persistence ) parameter p, we may write... 

In 1891 Werner began to apply himself to the problem of valence with his Contributions to Theory of Affinity and Valence. At that time Kekul s principle of fixed valence, so useful in organic chemistry, was being applied by chemists such as Blomstrand and Jorgensen to complex metal salts resulting in formulations of, for example, hexammine cobaltic chloride as ... 

J ekul s theory of constant or fixed valency undoubtedly helped Werner and Hantzsch develop their theories of the stereoisomeric forms of oximes and diazo compounds. However, Werner soon felt that Kekul s ideas were too rigid and, being convinced that sufficient facts were now at hand to justify a new theory, outlined one for variable valency. He then spent the rest of his life producing experimental proofs. This is not the whole story, but it can serve to introduce Alfred Werner. Two of his pupils may be quoted Paul Pfeiffer (29) Contribution... 

Le Bel published his stereochemical ideas two months later, in November 1874, under the title, The relations that exist between the atomic formulas of organic compounds and the rotatory power of their solutions" (14). An English translation is presented in Le Bel (15). Le Bel approached the problem from a different direction from van t Hoff. His hypothesis was based on neither the tetrahedral model of the carbon atom nor the concept of fixed valences between the atoms. He proceeded purely from symmetry arguments he spoke of the asymmetry, not of individual atoms, but of the entire molecule, so that his views would nowadays be classed under the heading of molecular asymmetry. Only once does he mention the tetrahedral carbon atom, which he regarded as not a general principle but a special case. Today, substituted allenes, spiranes, and biphenyls are but a few examples of asymmetric molecules that do not contain any asymmetric carbons, thus confirming Le Bel s views on molecular asymmetry. The reason for the different approaches by van t Hoff and le Bel is easy to understand, van t Hoff came from the camp of structural chemists, and he... 

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