Geometrical progression

Even-tempered basis sets (M. W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3961 (1979)) consist of GTOs in which the orbital exponents ak belonging to series of orbitals consist of geometrical progressions ak = a, where a and P characterize the particular set of GTOs. 

A geometric progression is a succession of terms such that each term, except the first, is derivable from the preceding by the multiph-cation of a quantity / called the common ratio. AU such progressions have the form a, ar, ar, . . . , ar" "h With a = first term, I = last term, / = ratio, n = number of terms, s = sum of the terms, the following relations hold ... 

Corollary 1.—If an ideal gas changes its volume reversibly without alteration of temperature, the quantities of heat absorbed or emitted form an arithmetical progression whilst the volumes form a geometrical progression (Sadi Carnot, 1824). 

This is a geometric progression that sums to lN)ia the growing chains with length / is... 

Thus, we see that this problem has a unique solution if the value y is given for some i. For the sake of simplicity let be known in advance for i = 0. With this, one can determine all the values y-, y, . . by the recurrence formula just established. In the case qi z= q = const and ipi = 0 this provides support for the view that the whole collection of yi constitutes a geometric progression. If qi = q and (pi qb 0 then... 

It is readily observed that the expression within brackets for the peak concentration C forms a geometric progression in which the first term equals one and with a common ratio of e . (The common ratio is the factor which results from dividing a term of the progression by its preceding one.) For the sum of the finite progression in eq. (39.41) we obtain after n cycles ... 

Figure 1.14 shows a typical distribution for the geochemically abundant elements in crustal rocks. It could be seen that the proportion of the volume of material available for exploitation increases in geometrical progression as grade falls in arithmetical progression. In a sense, therefore, there is no finite limit to the availability of such elements, however, dilution with host rock implies that revenue would be insufficient to cover the fixed cost of extraction. 

NMR has accomplished a growth in a geometrical progression since the early sixties and virtually developed into extremely potential analytical tools not only useful for elucidation of complex structural determinations but also equally beneficial in the assay of pharmaceutical substances. 

In selecting the stimulus continuum in threshold determination, the relation between just noticeable difference in relation to the intensity of stimuli is of interest. In accordance with Weber s law this quotient is assumed to be a constant. Therefore it would appear best to determine absolute thresholds on an intensity continuum in the form of a geometric progression. 

Now it has been observed that exponents which have been carefully optimized for atoms often form a good approximation to a geometric progression... 

Conversely, if it is assumed that the exponents do form a geometric progression and the parameters a and [3 are optimized for atoms then there is found to be little lost in accuracy. Basis sets developed in this way are termed even-tempered basis sets (for a discussion see [2]) and open up the possibility of constmcting the large and flexible basis sets that are inevitably required for calculations of high precision. 

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