Cancellation approximation

The particular example of cyclodextrin complexes led to the identification of another special case as the partial cancellation approximation-, in this case we assume = Kj < Ki, and the result is, approximately. ... 

The important difference between (46) and the CI(D) equations is that the right-hand side of (42) cancels (approximately) the correlation energy on the left-hand side of (42). 

As already noted, we cannot use eq. (III-6) unless we have a complete vibrational analysis of all four molecules. This is particularly true if some of the vibrations are degenerate, since degenerate frequencies must be counted more than once and it may be critically important which of two similar frequencies in AD corresponds to a degenerate frequency in AH and which to a nondegenerate frequency. If all of the frequencies are high, (m< and Uj 1 for all i and j) the first two factors cancel approximately in the numerator and in the denominator separately, and we have ... 

The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

The most important classes of functionalized [60]fullerene derivatives, e.g. methanofullerenes [341, pyrrolidinofullerenes [35], Diels-Alder adducts [34i] and aziridinofullerene [36], all give rise to a cancellation of the fivefold degeneration of their HOMO and tlireefold degeneration of their LUMO levels (figure Cl.2.5). This stems in a first order approximation from a perturbation of the fullerene s 7i-electron system in combination with a partial loss of the delocalization. 

The number of steps is always much larger than the displacement x, since there is a good deal of back-and-forth cancellation. Hence the ratio x/nl is less than unity and the logarithms may be approximated by the leading terms of a series expansion... 

The two effects almost cancel one another to yield an approximation for the minimum work potential used in a distillation (3,4). 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

LW) interactions refer to the purely physical London s (dispersion), the Keesom s (polar) and Debye s (induced polar) interactions and correspond to magnitudes ranging from approximately 0.1 to 10 kJ/mol (but in rare cases may be higher). The polar forces in the bulk of condensed phases are believed to be small due to the self-cancellation occurring in the Boltzmann-averaging of the multi-body... 

The proliferation of acidity functions is a consequence of the activity coefficient cancellation assumption. According to Eq. (8-89), a plot of log(cB/cBH+) against Hq should be linear with unit slope. Such plots are usually linear (for bases of closely related structure), but the slopes often differ from unity. - This behavior is an indication that the cancellation assumption (also called the zero-order approximation) is not valid, and several groups have devised alternatives. We will use the symbolism of Cox and Yates. ... 

The first term is referred to as the diamagnetic contribution, while the latter is the paramagnetic part of the magnetizability. Each of the two components depend on the selected gauge origin however, for exact wave functions these cancel exactly. For approximate wave functions this is not guaranteed, and as a result the total property may depend on where the origin for the vector potential (eq. (10.61)) has been chosen. 

It can be seen that both of the CPA s reproduce the concentration dependence of Umix quite well, even though it is not a simple parabola. The fact that a cancellation of errors causes the SCF-KKR-CPA results to be quite near the LSMS ones may be one of the reasons that results obtained over the years with that approach have been quite acceptable. After the inclusion of the Coulomb bonding, the CPA-LSMS gives a Umix that is surprisingly good considering other aspects of the approximation that Will be illustrated below. 

It is possible to explain these trends in terms of the electron configurations of the corresponding atoms. Consider first the increase in radius observed as we move down the table, let us say among the alkali metals (Group 1). All these elements have a single s electron outside a filled level or filled p sublevel. Electrons in these inner levels are much closer to the nucleus than the outer s electron and hence effectively shield it from the positive charge of the nucleus. To a first approximation, each inner electron cancels the charge of one pro-... 

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