# Mach number basis

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. [c.101]

Lm. The coarseness results from the relatively low power dissipation per mass on distillation trays. This means that it is relatively easy to remove by a device such as a wire mesh pad. Over 50 percent is typically captured by the underside of the next higher tray or by a turn in the piping leaving an evaporator. Conversely, though small on a mass basis, the smaller drops are extremely numerous. On a number basis, more than one-half of the drops in the lower curve are under 5 [Lm. These can sei ve as nuclei for fog condensation in downstream equipment. [c.1413]

Thus, the mean size varies from 65 to 186 im for the same sample. It is thus most important to carefully define which mean is intended. Clearly, the mean size on a number basis is much smaller than that a weight basis (since mass oL ). These calculations show that in this example most mass lies between 125-250 pm but most crystals lie between 0-90 pm. [c.25]

One of the approximations inherent in essentially all ah initio methods is the introduction of a basis set. Expanding an unknown function, such as a molecular orbital, in a set of known functions is not an approximation, if the basis is complete. However, a complete basis means diat an infinite number of functions must be used, which is impossible in actual calculations. An unknown MO can be thought of as a function in the infinite coordinate system spanned by the complete basis set. When a finite basis is used, only the components of the MO along those coordinate axes corresponding to the selected basis can be represented. The smaller die basis, the poorer the representation. The type of basis functions used also influence the accuracy. The better a single basis function is able to reproduce the unknown function, the fewer are basis functions necessary for achieving a given level of accuracy. Knowing that the computational effort of ah initio methods scales formally as at least it is of course of prime importance to make the basis set as small as possible widiout compromising the accuracy. [c.150]

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than [c.379]

Split valence basis sets generally give much better results than minimal ones, but at a cost. Remember that the number of two-electron integrals is proportional to kf , where W is the number of basis functions. Whereas STO-3G has only live ba.sis functions for carbon, 6-31G has nine, resulting in more than a tenfold increase in the size of the calculation, [c.385]

There is no definitive method for generating basis sets, and the construction of a new basis set is very much an art. Nevertheless, there are a number of well-established approaches that have resulted in widely used basis sets. We have already seen how linear combinations nf Gaussian functions can be fitted to Slater type orbitals by minimising the overlap (see Figure 2.6 and Table 2.3). The Gaussian exponents and coefficients are derived by lea.st-squares fitting to the desired functions, such as Slater type orbitals. When using basis sets that have been fitted to Slater orbitals it is often advantageous to use Slater exponents that are different to those obtained from Slater s rules. In general, better results for molecular calculations are obtained if larger Slater exponents are used for the valence electrons I hi has the effect of giving a smaller, less diffuse orbital. For example, a value of 1.24 is widely used for the Slater exponent of hydrogen rather than the 1.0 that would be suggested by Slater s rules. It is straightforward to derive a basis set for a different Slater exponent if the Gaussian expansion has been fitted to a Slater type orbital with ( = 1.0. If the Slater exponent is replaced by a new value, then the respective Gaussian exponents a and a are related by [c.92]

A much-quoted fact is that ab initio calculations scale as the fourth power of the number of basis functions for ground-state, closed-shell systems. This scaling factor arises because each two-electron integral (pr Arr) involves four basis functions, so the number of two-electron integrals would be expected to increase in proportion to the fourth power of the number of basis functions. In fact, the number of such integrals is not exactly equal to the fourth power of the number of basis functions because many of the integrals are related by symmetry. We can calculate exactly the number of two-electron integrals that are required in a Hartree-Fock ab initio calculation as follows. There are seven different types of two-electron integral [c.139]

Tn this approach the domain of the solution is first divided into a number of large sub-domains without leaving any gaps or overlapping. I liis division provides a very coarse unstruetured mesh which is used as the basis tor the generation of structured grids in each of its zones. The union of these local grids gives a computational grid for the entire domain, called a block-structured (or a multi-block) mesh. The flexibility gained by this approach can be used to handle complicated domains having multiply connected boundaries, problems involving heterogeneous physical phenomena and mathematical non-uniformity. Figure 6.1 shows representative examples of block-structured grids with different forms of linking or communication interface between adjacent sub-regions. [c.193]

The disadvantage of ah initio methods is that they are expensive. These methods often take enormous amounts of computer CPU time, memory, and disk space. The HF method scales as N, where N is the number of basis functions. This means that a calculation twice as big takes 16 times as long (2" ) to complete. Correlated calculations often scale much worse than this. In practice, extremely accurate solutions are only obtainable when the molecule contains a dozen electrons or less. However, results with an accuracy rivaling that of many experimental techniques can be obtained for moderate-size organic molecules. The minimally correlated methods, such as MP2 and GVB, are often used when correlation is important to the description of large molecules. [c.28]

When we say cycloheptatriene is not aromatic but cycloheptatrienyl cation is we are not comparing the stability of the two to each other Cycloheptatriene is a stable hydrocarbon but does not possess the special stability required to be called aromatic Cycloheptatrienyl cation although aromatic is still a carbocation and reasonably reac tive toward nucleophiles Its special stability does not imply a rock like passivity but rather a much greater ease of formation than expected on the basis of the Lewis struc ture drawn for it A number of observations indicate that cycloheptatrienyl cation is far more stable than most other carbocations To emphasize its aromatic nature chemists often write the structure of cycloheptatrienyl cation m the Robinson circle m a ring style [c.457]

The total number of two-electron integrals is proportional to for a molecular system with m basis functions. Some of these integrals may be zero because of the symmetry, and some may be very small in magnitude. Using the regular integral format or the Raffenetti integral format, each integral value and its indices take 16 bytes and all are saved to the computer main memory or disk. All saved two-electron integrals are then used in forming the Fock matrix in every iteration. Those integrals with zero value or with a very small magnitude do not make much contribution to the Fock matrix and to the total energy. Neglecting these integrals may not affect the accuracy of ab initio calculations. Thus, in order to save computer main memory or disk space and speed up the calculation of the SCFprocedure, a two-electron integral cutoff is introduced. HyperChem uses the two-electron integral cutoff to determine which of the two-electron integrals need to be saved. The value of 10" ° (Hartree) generally is good enough for most calculations. A small value is recommended for tight calculations and a large value for loose calculations. [c.265]

A number of chain-growth polymers are commercially produced on a high tonnage basis, and the technology of polymerization deserves some comment even though it is not our main emphasis in this volume. Many important monomers polymerize with the evolution of large amounts of heat. This can result in temperature increases, increased kinetic constants, and accelerated reaction rates-in short, to runaway reactions-unless the heat is dissipated. Many important monomers are toxic, carcinogenic, or both and hence must be processed carefully for the safety of workers in the industry and users of the products. These facts, plus the diversity of monomers employed and the assortment of end uses for polymer products, make the choice of a polymerization technique a highly specific matter. In this section we shall discuss some of the possibilities. [c.396]

The overall form of each of these equations is fairly simple, ie, energy = a constant times a displacement. In most cases the focus is on differences in energy, because these are the quantities which help discriminate reactivity among similar stmctures. The computational requirement for molecular mechanics calculations grows as where n is the number of atoms, not the number of electrons or basis functions. Immediately it can be seen that these calculations will be much faster than an equivalent quantum mechanical study. The size of the systems which can be studied can also substantially ecHpse those studied by quantum mechanics. [c.164]

The countercurrent arrangement (Fig. 5c) represents the best compromise between the objectives of high extract concentration and a high degree of extraction of the solute, for a given solvent-to-feed ratio. The feed entering stage 1 is brought into contact with a B-rich stream which has already passed through the other stages, while the raffinate leaving the last stage has been in contact with fresh solvent. Because of the economic advantages, continuous countercurrent extraction is normally preferred for commercial-scale operations. For the case of a partially miscible ternary system, the number of ideal stages in a countercurrent cascade can be estimated graphically on a triangular diagram, using the Hunter-Nash method (53). The feed and solvent compositions and the resulting mixture point M are first located on the diagram as in Figure 2a. If in addition one of the exit stream (extract or raffinate) compositions is given, a point representing the composition of the net flow in the countercurrent cascade can be located. This point, called the delta point, provides the basis for constmction of material balance lines and tie-lines representing a sequence of ideal stages for the countercurrent extractor. The Hunter-Nash procedure is well known and useflil (5,28). For dilute systems, it is often more convenient to use the delta point constmction on a diagram with solvent-free coordinates (5,28). In this case a rectangular diagram is plotted in which the horizontal axis is the mass fraction of the solute C on a B-free basis, and the vertical axis is the mass ratio of B to A + C. [c.65]

The geothermal drilling industry is much smaller than that of oil and gas drilling and the active geothermal rig count is generally less than 10. Thus, there is not a commercial basis for the development of specialized materials and equipment for geothermal drilling. For a number of years, the U.S. Department of Energy has sponsored the development of high temperature drilling fluids and cements especially designed for geothermal operations (9). Efforts have been concentrated on lightweight, carbon dioxide-resistant cements, thermally conductive and scale-resistant protective liners, improved materials to control lost circulation, and bonding agents. [c.265]

Buttress-Shaped Grooves. The cones of the Vickers-Anderson coupling may be replaced by a series of paraUel grooves of buttress section machined on the outside surface of the vessel and its end cover, which mate with similar ones counterbored on the inside of a spUt coUar as shown in Figure 22b. This arrangement is only possible for a ring such as the wave ring which requires no initial end load to make a pressure-tight seal. Instead of the three component spUt coUar shown in Figure 25a, two half laps may be used as in Figure 25b. This closure has formed the basis of the commonly adopted design for the largest autoclaves used for LDPE production, although there are variations in the number and size of the buttress grooves employed. Some vessels use a spUt coUar to engage with a buttress-shaped thread and so overcome the problem of ensuring that the buttress grooves are of uniform pitch. Seals for vessels commonly used in the German chemical industry at pressures of 30 to 400 MPa (4350—58,000 psi) have been reviewed (140). [c.94]

The vapor-phase catalytic oxidation of anthracene to anthraquinone was first described in 1916 and was the basis for a plant operated at Ludwigshafen (27). In 1940, a smaller plant at Milan, Italy incorporated the latest information available at the time and was beUeved to be far in advance, technically, of the Ludwigshafen plant. The entire plant was designed exclusively for the manufacture of anthraquinone from anthracene (28). The main feature of the plant was the so-called "contact oven" where the oxidation of the anthracene occurred. The oven was a cylindrical iron vessel about 5.8 m high and 2.4 m in diameter. Within the oven were nine perforated sheet-iron shelves placed one on top of the other. On each shelf was a layer of iron vanadate catalyst (29) embedded in high pressure coils which heated and cooled the reaction mass during the oxidation. In conjunction with the contact oven were a number of auxiUary pieces of equipment a preheater for heating an air-stream mixture, a vaporizer containing molten anthracene, various metering and temperature control devices, several heat exchangers, and a Dowtherm heating system. Briefly, 93% anthracene was vaporized at 318 °C by means of the preheated air—steam mixture. The steam was incorporated to minimize the possibiUty of an explosion during the oxidation several had [c.421]

Cumulative frequency data, whethei on a number, surface area, or mass basis, allows for easy estimates of the total number, surface area, or mass of particles less than a given size. For the data in Table 1, 12 particles out of 1000 (1.2%) had a diameter less than 2 p.m, 7.4% less than 4 p.m, etc. [c.128]

It was not until the eighteenth century that carbon was recognized as a chemical element, and it is quite certain that no early metallurgist was aware of the basis of the unique properties of steel as compared to those of wrought iron. Carbon can be alloyed with iron in a number of ways to make steel, and all methods described herein have been used at various times in many locaUties for perhaps 3000 or more years. [c.373]

When the United States entered World War II, access to the source of natural mbber was eliminated and the federal government set up a consortium of American mbber manufacturers under the auspices of the Office of Rubber Reserve. A total of 15 styrene—butadiene mbber (SBR) plants were constmcted between 1941 and 1942. These plants, devoted to the production of a natural mbber substitute, were mn by the four principal mbber companies in the United States Goodyear, U.S. Rubber, B.F. Goodrich, and Firestone. To increase the number of mbber companies participating in the program, two additional companies were formed from smaller mbber producers Copolymer Corporation and National Synthetic Rubber Corporation (2). The product agreed upon was called GR-S, an abbreviation for Government Rubber—Styrene, and was based on Buna S. It was the standard general-purpose SBR manufactured until right after World War II, when Germany s work on a low temperature system to generate free radicals for cold emulsion polymerization of SBR became known. The properties of this product were so much better than the hot GR-S that the Office of Rubber Reserve ordered all of the consortium s plants to install refrigeration equipment and begin phasing in the cold-polymerized GR-S. This product has been the basis of the emulsion polymer industry in the United States ever since. [c.493]

Surface Energy of Gas—Liquid Interfaces. Each unit of a Hquid surface is associated with a quantity of free energy called the free-surface energy (mj/m (= erg/cm )). Work equivalent to the iacrease ia free-surface eaergy must be added to a Hquid to iacrease its surface area. A pure Hquid teads toward the coaditioa of minimum eaergy, which is the shape of minimum area. Thus, the shape of a free-falling droplet would be spherical under idealized conditions. The tendency of a pure Hquid to minimize its surface can be explained on the basis of intermolecular attractive forces. Even though a Hquid is a freely flowing fluid mass, the individual molecules are so close together that van der Waals-type forces are the dominant attraction between molecules. Thus, a molecule in the interior of a Hquid is not subject to a net attractive force in any direction because the number of molecules in the sphere surrounding it is, on the average, the same in all directions. However, a molecule at the surface of the Hquid is subject to a strong inward attraction because the number of molecules in the hemisphere of the vapor close enough to exert an attractive force pulling out of the Hquid is small as compared to the number in the hemisphere of Hquid pulling inward from the surface. A reasonable ratio of vapor-to-Hquid molecules is 1 1000. This unbalanced attraction causes as many molecules as possible to move out of the surface into the bulk Hquid, reducing its area to a minimum. [c.234]

Over 30% of the chlorine produced on a global basis goes to make PVC. Not only is chlorine essential to the chemical composition of PVC, it provides a number of unique properties that give this versatile plastic a distinct advantage in product appHcations and the marketplace. It makes PVC inherently flame-retardant. PVC is the world s leading electrical material, with over 250 x 10 t (500 x 10 lb) used aimually for wire and cable insulation and sheathing, electrical conduit, boxes, and components. PVC is over 50% chlorine and, as a result, one of the most energy-efficient polymers. Chlorine makes PVC far more environmentally acceptable than other materials that are totally dependent on petrochemical feedstocks. In addition, recycling PVC is easier because the chlorine in PVC acts as a marker, enabling automated equipment to sort PVC containers from other plastics in the waste stream (168). [c.508]

Rule 1. Priority is assigned to atoms on the basis of atomic number. Higher priority is assigned to atoms of higher atomic number. If two atoms are isotopes of the same element, the atom of higher mass number has the higher priority. For example, in 2-butene, the carbon atom of each methyl group receives first priority over the hydrogen atom connected to the same carbon atom. Around the asymmetric carbon atom in chloroiodomethanesulfonic acid, the priority sequence is I, Cl, S, H. In 1-bromo-l-deuteroethane, the priority sequence is Cl, C, D, H. [c.45]

As these methods are explored, it is quickly realized that the numerical effort in the theoretical description grows prohibitively large with the number of atoms in a molecule. The difficulty lies in precisely what makes molecular motion fiindamentally qiiasi-classical, i.e. the large molecular masses (relative to the mass of the electron). Consequently, a molecular wavefiinction has many oscillations and is difficult to model numerically. There have been many attempts at developing alternate approaches for representing quantum wavefiinctions and observables without the use of large grids or basis sets, ranging from approximations to patli-mtegral descriptions. The basics of these approaches are described in Sections B3.4.8. Later, Sections B3.4.9. describes the issues involved in the study of non-adiabatic phenomena. [c.2291]

Within the time-dependent framework, the Time-Dependent Self-Consistent Field (TDSCF) approximation is known, in the context of electronic structure theory, from the early years of quantum theory, but its applications to problems of molecular dynamics are much more recent. Methods of much improved accuracy were developed on the basis of the TDSCF approximation. " Also, some technical hurdles in applying the method for larger systems were overcome in the last few years at least for some types of interaction potentials, so recently calculations have been reported for problems of significantly increased number of degrees of freedom. Different methods for time-dependent quantum simulations of large systems were pursued vigorously by several research groups, and much progress was made. Some of the novel methods proposed are still confined to models or special systems, while for a few of the others realistic applications are already at [c.366]

VThile a large number of measurements of this type can be made, there are certain experiments whose results are of such general importance that all flux models would be expected to conform with them. These experiments, which form the empirical basis for an understanding of transport in porous media, are described in Chapter 6. Just as the theory of transport in porous media owes much to ideas of Clerk Maxwell, the experimental basis is dominated by the work of another 19th century scientist, Thomas Graham. Graham s work has been extensively misquoted, with the consequence that his results were largely ignored until their eventual rediscovery about a century later. However, E. A. Mason, who has been instrumental in the modern development of the dusty gas model, has actively drawn attention to the true nature and importance of Graham s contributions, which are here summarized briefly in Appendix II. [c.4]

Each eigenvalue has an eigenveetor assoeiated with it. The eigenveetors tell us as much about the atomie or moleeular system as the eigenvalues do. Like eigenvalues, eigenveetors ean be exaet for simple systems but in general we know them only approximately. Eigenveetors define a vector space that has a number of dimensions equal to the number of basis funetions. Unfamiliar and perhaps daunting as you may find spaees with dimensions beyond the three dimensions of Euelidean spaee, we shall soon be working in many-dimensional spaees so frequently you will find them quite ordinary. [c.201]

For many projects, a basis set cannot be chosen based purely on the general rules of thumb listed above. There are a number of places to obtain a much more quantitative comparison of basis sets. The paper in which a basis set is published often contains the results of test calculations that give an indication of the accuracy of results. Several books, listed in the references below, contain extensive tabulations of results for various methods and basis sets. Every year, a bibliography of all computational chemistry papers published in the previous [c.89]

Most of the computational techniques discussed in this text have been included in a number of software packages. The general techniques are uniquely defined, meaning that a HF calculation with a given basis set on a particular molecule will give the exact same results regardless of which program is used. However, the choice of software is still important. Software packages dilfer in cost, functionality, elficiency, ease of use, automation, and robustness. These concerns make an enormous difference in determining what computational projects can be completed successfully and how much work will be involved. [c.322]

Other correlations based partially on theoretical considerations but made to fit existing data also exist (71—75). A number of researchers have also attempted to separate from a by measuring the latter, sometimes in terms of the wetted area (76—78). Finally, a number of correlations for the mass transfer coefficient itself exist. These ate based on a mote fundamental theory of mass transfer in packed columns (79—82). Although certain predictions were verified by experimental evidence, these models often cannot serve as design basis because the equations contain the interfacial area as an independent variable. [c.37]

Each of the elements has a number of isotopes (2,4), all radioactive and some of which can be obtained in isotopicaHy pure form. More than 200 in number and mosdy synthetic in origin, they are produced by neutron or charged-particle induced transmutations (2,4). The known radioactive isotopes are distributed among the 15 elements approximately as follows actinium and thorium, 25 each protactinium, 20 uranium, neptunium, plutonium, americium, curium, californium, einsteinium, and fermium, 15 each herkelium, mendelevium, nobehum, and lawrencium, 10 each. There is frequently a need for values to be assigned for the atomic weights of the actinide elements. Any precise experimental work would require a value for the isotope or isotopic mixture being used, but where there is a purely formal demand for atomic weights, mass numbers that are chosen on the basis of half-life and availabiUty have customarily been used. A Hst of these is provided in Table 1. [c.212]

Agricultural Chemicals. Plasticizer range alcohols are used as intermediates in the manufacture of a number of herbicides (qv) and insecticides, the largest use being that of 2-ethylhexanol and isooctyl alcohol to make the octyl ester of 2,4-dichlorophenoxyacetic acid (2,4-D) [94-75-7] for control of broadleaf weeds. Surfactants made from these alcohols are used as emulsifiers and wetting agents for agricultural chemicals. A mixture of octanol and decanol and the proper surfactants is able to kiH the young meristemic tissue of some plants without harming more mature tissue. This is the basis for formulations that kiH unwanted buds (suckers) in tobacco (59) and other plants and serve as a selective herbicide. Both decanol and 4-methyl-2-pentanol can be used as fungicides (qv) (60). [c.450]

Two diverse technical approaches to fusion power, magnetic confinement fusion, also known as magnetic fusion energy (MFE) and inertial confinement fusion, also known as inertial fusion energy (lEE) are being pursued worldwide. These form the basis of a large number of fusion research programs. Magnetic confinement techniques, studied since the 1950s, ate based on the principle that charged particles such as electrons and ions, ie, deuterons and tritons, tend to be bound to magnetic lines of force. Thus the essence of the magnetic confinement approach is to trap a hot plasma in a suitably chosen magnetic field configuration for a long enough time to achieve a net energy release, which typically requires an energy confinement time of about one second. In the alternative lEE approach, fusion conditions ate achieved by heating and compressing small amounts of fuel ions, contained in capsules, to the ignition condition by means of tightly focused energetic beams of charged particles or photons. In this case the confinement time can be much shorter, typically less than a millionth of a second. [c.151]

Combinatorial Hbraries are limited by the number of sequences that can be synthesized. For example, a Hbrary consisting of one molecule each of a 60-nucleotide sequence randomized at each position, would have a mass of >10 g, weU beyond the capacity for synthesis and manipulation. Thus, even if nucleotide addition is random at all the steps during synthesis of the oligonucleotide only a minority of the sequences can be present in the output from a laboratory-scale chemical DNA synthesis reaction. In analyzing these random but incomplete Hbraries, the protocol is efficient enough to allow selection of aptamers of lowest dissociation constants (K ) from the mixture after a small number of repetitive selection and amplification cycles. Once a smaller population of oligonucleotides is amplified, the aptamer sequences can be used as the basis for constmcting a less complex Hbrary for further selection. [c.236]

Glasses are metastable and, under the appropriate conditions, revert to a thermodynamically more stable state. Glasses particularly susceptible to uncontrolled crystal growth or phase separation have traditionally created problems for their manufacturer. However, glass is also a good medium for controlled crystalli2a tion (50), and has become the basis for a number of unique crystalline materials known as glass-ceramics (qv). The separation of a single glass iato multiple glassy phases can make the article cloudy and adversely affect its chemical durabiUty. Controlled phase separation, however, can produce opaque, white opal glass or, after a leaching step, lead to new materials such as porous glass or 96% siUca glass. The colloidal suspension of multiple phases ia transparent glass produces precise colors for products such as optical filters (qv). Furthermore, photosensitive and photochromic glasses change their optical transmission and color, sometimes reversibly, upon stimulus by a combination of light and heat treatment (see Chromogenic MATERIALS, photochromic). ah these transformations generally depend on the phenomena of diffusion, nucleation, and growth. [c.289]

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. [c.379]

See pages that mention the term

**Mach number basis**:

**[c.159] [c.52] [c.33] [c.2174] [c.2184] [c.2482] [c.410] [c.265] [c.90] [c.136] [c.174] [c.214] [c.1279] [c.29]**

Pressure safety design practices for refinery and chemical operations (1998) -- [ c.312 ]