Actual 360 basis

The actual basis functions are formed as linear combinations of such primitive gaussians ... 

Linear combinations of primitive gaussians like these are used to form the actual basis functions the latter are called contracted gaussians and have the form ... 

These primitive Gaussians form the actual basis functions which are called contracted Gaussians. The atomic orbitals are then expressed as ... 

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

The actual basis for the belief in transmutation consisted in just such operations as we have seen illustrated in the two Theban papyri. That these arts in Egypt were... 

From the results an important conclusion follows viz, that for large basis sets the differences in actual basis sets are not reflected significantly in the computed molecular properties. It is just this feature which is typical for ab initio calculations. 

Tlie exact definition is slightly more complicated, since the wave function has to be properly antisymmetrized and projected onto the actual basis, but for illustration the bove form is sufficient. Such R12 wave functions may then be used in connection with the Cl, MBPT or CC methods described above. Consider for example a TT calcnlation... 

In generating a Mulliken population from a ZDO calculation, however, an assumption such as that given in Eq. [14a] must be made that is, a model that is characterized principally through integrals must be related to an actual basis. The symmetrically orthogonalized basis is not an inherent feature in any of the ZDO methods discussed above, with the exception of the SINDOl method. 

The observations just made with respect to b and are necessarily qualitative, but they can serve as useful guidelines in equation-of-state building, and as criteria for comparison and assessment of cubic equations of state. Most of the cubic equations proposed to date, whatever the actual basis for their development, conform more or less to these guidelines. 

If a UHF single determinant is used where each 2p electron can have its own radial distribution the dilemma is removed and a solution is generated which, within the confines of the actual basis used, is an oxygen atom and a diffuse electron. 

As mentioned in the introduction, the calculations were initially carried out by using the well-known clamped nuclei approximation for the nuclear dynamics which is the basis for the Bom-Oppenheimer (BO) decoupling approach. The actual basis set chosen was the one corresponding to a triple-zeta expansion plus polarisation usually labelled by the acronym cc-pVTZ in the current literature [29]. Thus, our clamped nuclei calculations will be labelled throughout the present work as B3LYP/cc-pVTZ calculations. 

The coupon arrived at, now in money terms, is truncated to two decimal places for the two oldest existing linkers (2006 and 2011), and truncated to four decimal places for all others bar one. The exception is the new 2035 issue, which uses natural rounding to six decimal places. Accrued interest is calculated on the money value, not real value, of the coupon to be paid on an actual/actual basis. 

The actual basis of any chromatographic separation is that the various analytes have differing distribution constants between the stationary and mobile phases. This causes the analytes to move through the column at different rates and to appear as time-separated peaks on the final chromatogram provided there is a reasonable difference in migration rates and the column is of sufficient length. 

The actual basis for separation can be the differential solubility in the membrane material or the membrane hole size. When separating gases hollow membrane fibres are popular, as the small diameter of the fibres allows than to withstand the substantial driving pressures involved. Separation of the components in liquid systems usually relies on some form of spirally wound membrane sheet and spacer. Once again, substantial differential pressures are involved across the membrane surface and this conventionally requires the use of some form of spacer to prevent closure of the channels. Not only do these spacers (e.g. coarse woven fabric)... 

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