Smallest number

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

The four steric energy sums in Eq. (4-13) corresponding to sketching, bending, and torsional modes of motion and van der Waals inhamolecultu interaction appear to be about the smallest number one can use in an accurate MM geomehy minimization. 

J Chem. Phys., 52, 431 (1970)] is a relatively inexpensive one and can be used for calculations on quite large molecules. It is minimal in the sense of having the smallest number of functions per atom required to describe the occupied atomic orbitals of that atom. This is not exactly true, since one usually considers Is, 2s, and 2p, i.e., five functions, to construct a minimal basis set for Li and Be, for example, even though the 2p orbital is not occupied in these atoms. The 2sp (2s and 2p), 3sp, 4sp, 3d,. .., etc. orbitals are always lumped together as a shell , however. The minimal basis set thus consists of 1 function for H and He, 5 functions for Li to Ne, 9 functions for Na to Ar, 13 functions for Kand Ca, 18 functions for Sc to Kr,. .., etc. Because the minimal basis set is so small, it generally can not lead to quantitatively accurate results. It does, however, contain the essentials of chemical bonding and many useful qualitative results can be obtained. 

Equation 7.7 only provides an estimate for the smallest number of samples expected to produce the desired sampling error. The actual sampling error may be substantially higher if the standard deviation for the samples that are collected is signihcantly greater than the standard deviation due to sampling used to calculate n. ... 

Diketones contain two carbonyl groups and are named by adding the suffix -dione to the parent hydrocarbon, and by indicating the position of the carbonyl groups using the smallest numbers possible. Diketones are generally used as specialty chemical intermediates in the pharmaceutical, flavor, fragrance, and dye industries. 

All stage-to-stage methods that work from both ends of the column toward the middle suffer from two other disadvantages. First, the top-down and the bottom-up calculations must me somewhere in the column. Usually the mesh is made at a feed stage, and if more than one feed stage exists, a choice of mesh point must be made for each component. When the components vary widely in volatility, the same mesh point cannot be used for all components if serious numerical difficulties are to be avoided. Second, arbitrary procedures must be set up to handle nondlstrihuted components. (A nondistributed component is one whose concentration in one of the end-product streams is smaller than the smallest number carried by the computer.) In the LM and TG equations, the concentrations for these components do not natur ly take on nonzero values at the proper point as the calculations proceed through the column. 

This is a highly recommended method of capacitor switching for installations that are large and require very fine monitoring and correction of p.f. with the smallest number of banks. The entire reactive requirement is arranged in only a few steps yet a small correction up to the smallest capacitor unit is possible. The relay is sequenced so that through its binary counter the required switching is achieved in small steps, with just four or six sets of capacitor units or banks. The operation of the entire sequence can be illustrated as follows ... 

The state variables are the smallest number of states that are required to describe the dynamic nature of the system, and it is not a necessary constraint that they are measurable. The manner in which the state variables change as a function of time may be thought of as a trajectory in n dimensional space, called the state-space. Two-dimensional state-space is sometimes referred to as the phase-plane when one state is the derivative of the other. 

Use process flow modeling. Using removable sticky notes, you can arrange the various requirements identified during the fishboning activity into a process flow model. This model needs to be carefully reviewed to make sure that it consists of the smallest number of steps necessary to deliver the required process. 

Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

Most measured quantities are not end results in themselves. Instead, they are used to calculate other quantities, often by multiplication or division. The precision of any such derived result is limited by that of the measurements on which it is based. When measured quantities are multiplied or divided, the number of significant figures in the result is the same as that in the quantity with the smallest number of significant figures. 

When measured quantities are added or subtracted, the uncertainty in the result is found in a quite different way than when they are multiplied and divided. It is determined by counting the number of decimal places, that is, the number of digits to the right of the decimal point for each measured quantity. When measured quantities are added or subtracted, the number of decimal places in the result is the same as that in the quantity with the greatest uncertainty and hence the smallest number of decimal places. 

To find the mole ratios, divide by the smallest number, 0.170 mol K ... 

The polynomials finally chosen as having the smallest number of terms needed to represent the high-velocity regime data are as follows ... 

Some elements—particularly the halogens—form more than two kinds of oxoanions. The name of the oxoanion with the smallest number of oxygen atoms is formed by adding the prefix hypo- to the -ite form of the name, as in the hypochlorite ion, CIO-. The oxoanion with the most oxygen atoms is named with the prefix per- added to the -ate form of the name. An example is the perchlorate ion, C104-. The rules for naming polyatomic ions are summarized in Appendix 3A and common examples are listed in Table D.l. 

STRATEGY Divide each mass percentage by the molar mass of the corresponding element to obtain the number of moles of that element found in exactly 100 g of the compound. Divide the number of moles of each element by the smallest number of moles. If fractional numbers result, then multiply by the factor that gives the smallest whole numbers of moles. 

SOLUTION Name the compounds by counting around the ring in the direction that gives the smallest numbers to the substituents. 

Addition and subtraction When adding or subtracting, make sure that the number of decimal places in the result is the same as the smallest number of decimal places in the data. For example, 0.10 g + 0.024 g = 0.12 g. 

The simplest acid-base relationships should involve the smallest number of molecular species and the smallest number of phases. Adding... 

How then are these ions/decompositions chosen Before considering this we must define, very carefully, the requirements of the analysis to be carried out. Is a single compound to be determined or are a number of compounds of interest If a single compound is involved, its mass spectrum and MS-MS spectra can be obtained and scrutinized for any appropriate ions or decompositions. If the requirement is to determine a number of analytes, their chromatographic properties need to be considered. If they are well separated, different ions/decompositions can be monitored for discrete time-periods as each compound elutes, thus obtaining the maximum sensitivity for each analyte. If the analytes are not well separated, this approach may not be possible and it may then be necessary to monitor a number of ions/decompositions for the complete duration of the analysis. If this is the case, the analyst should attempt to find the smallest number of ions/decompositions that give adequate performance for all of the analytes (remember the more ions/decompositions monitored, then the lower the overall sensitivity will be). 

The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms... 

The Australian TGA is by far the largest in terms of the number of staff Venezuela and the Netherlands rank second and third, respectively. Cypms has the smallest number of registration staff—two pharmacists and three administrative staff—who are also responsible for licensing and GMP inspection. Of all the dmg regulatory authorities, only those of Australia, Cypms and Tunisia have more administrative staff than professional staff The dmg regulatory authorities of Estonia, Uganda and Zimbabwe do not employ administrative staff for the registration unit, and it is not clear whether administrative matters are handled by the professionals. 

There is one report on the use of micro reactors for cosmetics [323] and none for foods. Of aU the topics mentioned, these have the smallest number of reports. 

The consequence of all these (conscious and unconscious) simplifications and eliminations might be that some information not present in the process will be included in the model. Conversely, some phenomena occurring in reality are not accounted for in the model. The adjustable parameters in such simplified models will compensate for inadequacy of the model and will not be the true physical coefficients. Accordingly, the usefulness of the model will be limited and risk at scale-up will not be completely eliminated. In general, in mathematical modelling of chemical processes two principles should always be kept in mind. The first was formulated by G.E.P. Box of Wisconsin All models are wrong, some of them are useful . As far as the choice of the best of wrong models is concerned, words of S.M. Wheeler of New York are worthwhile to keep in mind The best model is the simplest one that works . This is usually the model that fits the experimental data well in the statistical sense and contains the smallest number of parameters. The problem at scale-up, however, is that we do not know which of the models works in a full-scale unit until a plant is on stream. 

Fig. 29.7. Illustration of a pattern of points with rank of 2. The pattern is represented by a matrix X with dimensions 5x4 and a linear dependence between the three columns of X is assumed. The rank is shown to be the smallest number of dimensions required to represent the pattern in column-space 5 and in row-space S".

B Na2C03 produces the smallest number of moles in solution. 

When it is employed to specify an ensemble of random structures, in the sense mentioned above, the MaxEnt distribution of scatterers is the one which rules out the smallest number of structures, while at the same time reproducing the experimental observations for the structure factor amplitudes as expectation values over the ensemble. Thus, provided that the random scatterer model is adequate, deviations from the prior prejudice (see below) are enforced by the fit to the experimental data, while the MaxEnt principle ensures that no unwarranted detail is introduced. 

The empirical formula (or any formula) must be in the ratio of small integers. Thus, we attempt to get the ratio of moles of carbon to moles of hydrogen into an integer ratio we divide all the numbers of moles by the smallest number of moles ... 

What is the smallest number of carbon atoms possible in a molecule of (a) a ketone (b) an aldehyde (c) an aromatic hydrocarbon ... 

With the knowledge of /x(AT) we may derive all the bundle statistics. As an example, assuming that m is the smallest number of chain atoms in a loop, the average loop length is given by (see Fig. 1, m is a number of chain atoms)... 

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