Total pairing energy

These two pairing terms add to produce the total pairing energy, 11 ... 

Relative to the total pairing energy II, A is strongly dependent on the ligands and the metal. Table 10.6 presents values for aqueous ions, in which water is a relatively weak-field ligand (small A ). The number of unpaired electrons in the complex depends... 

These are all valid ways of deploying one 2s and three 2p atomic orbitals—in the case of sp2 hybridisation there will be one unhybridised p orbital also available (p. 8), and in the case of sp1 hybridisation there will be two (p. 10). Other, equally valid, modes of hybridisation are also possible in which the hybrid orbitals are not necessarily identical with each other, e.g. those used in CH2C12 compared with the ones used in CC14 and CH4. Hybridisation takes place so that the atom concerned can form as strong bonds as possible, and so that the other atoms thus bonded (and the electron pairs constituting the bonds) are as far apart from each other as possible, i.e. so that the total intrinsic energy of the resultant compound is at a minimum. 

The series inside the parentheses converges to a sum that is 2 ln2 or 1.38629. This value is the Madelung constant for a hypothetical chain consisting of Na+ and Cl- ions. Thus, the total interaction energy for the chain of ions is —1.38629N0e2/r, and the chain is more stable than ion pairs by a factor of 1.38629, the Madelung constant. Of course NaCl does not exist in a chain, so there must be an even more stable way of arranging the ions. 

Note that, due to their infinite-range character, pure Coulombic potentials can actually lead to significant bond non-additivity for any proposed separation into bonded and nonbonded units. This reflects the fact that classical electrostatics is oblivious to any perceived separation into chemical units, because all Coulombic pairings (whether in the same or separate units) make long-range contributions to the total interaction energy. 

The total potential energy of a given configuration may be written as the sum of the pair potentials. Each pair potential is... 

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

The potassium salt of the 2,2 -dipyridyl acetylene anion-radical represents another important example. In this case, the spin and charge are localized in the framework of N-C-C=C-C-N fragment. The atomic charge on each nitrogen atom is -0.447, that is, close to unity in total. The energy of this ion pair is minimal when the potassium counterion is located midway between the two rather close nitrogen lone pairs. Such a structure is consistent with the fact that the ESR spectrum of this species is almost insensitive to temperature. It means that the counterion does not hop between two remote sites of the anion-radical (Choua et al. 1999). 

Depending on the solvent polarity and redox potentials of a donor and an acceptor, the ions resulting from electron transfer may remain associated either as a contact IRP or as a solvent-separated IRP. In the contact pair, back electron transfer can take place. For such electron back-transfers, the solvent reorganization energy is less than 5% of the total reorganization energy (Serpa and Arnaut 2000). 

Here the notation Vi( 8s(i) ) is meant to imply that whereas the contribution Vi of residue i to the total conformational energy of the carbohydrate may depend on all of the variables in the set 85, it need not do so and may indeed depend only on some limited number of them. The partitioning in eqn. (10) yields the greatest simplification if the residue contribution Vi depends upon a set of variables (8s(i)) upon which no other residue contribution Vj depends. Stated more directly, there is in this case no intersection of the coordinate sets 8s(i) and 8s(j) for any pair of residues i and j, 8s(i) n 8s(j) =< ). 

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