Contracted basis

The contracted basis set created from the procedure above is listed in Figure 28.3. Note that the contraction coefficients are not normalized. This is not usually a problem since nearly all software packages will renormalize the coefficients automatically. The atom calculation rerun with contracted orbitals is expected to run much faster and have a slightly higher energy. 

The contracted basis in Figure 28.3 is called a minimal basis set because there is one contraction per occupied orbital. The valence region, and thus chemical bonding, could be described better if an additional primitive were added to each of the valence orbitals. This is almost always done using the even-tempered method. This method comes from the observation that energy-optimized exponents tend to nearly follow an exponential pattern given by... 

HyperChem quantum mechanical calculations are ab initio and semi-empirical. Ab initio calculations use parameters (contracted basis functions) associated with shells, such as an s shell, sp shell, etc., or atomic numbers (atoms). Semi-empirical calculations use parameters associated with specific atomic numbers. The concept of atom types is not used in the conventional quantum mechanics methods. Semi-empirical quantum mechanics methods use a rigorous quantum mechanical formulation combined with the use of empirical parameters obtained from comparison with experiment. If parameters are available for the atoms of a given molecule, the ab initio and semi-empirical calculations have an a priori aspect when compared with a molecular mechanics calculation, letting... 

Home desalinators are possible only for industrialized countries with a central service organization. They will eventually become available on a rental/service contract basis, as is standard practice for water softeners in many communities. Although rental of water softeners is common in the United States, home membrane-system rental is not estabUshed. 

A useful approximation of B for a conical hopper is B = 22f/a, where a is the bulk density of the stored product. The apparatus for determining the properties of solids has been developed and is offered for sale by the consulting firm of Jenike and Johansen, Winchester, Massachusetts, which also performs these tests on a contract basis. The flow-factor FF tester, a constant-rate-of-strain, direct-shear-type machine, gives the locus of points for the FF cui ve as well as ( ), the... 

Take note of Dunning s notation. He writes the primitives (10s6p) and the contracted basis functions in square brackets [5s3p]. To give a detailed example, consider the oxygen atom set in Table 9.7. 

There are many different contracted basis sets available in the literature or built into programs, and the average user usually only needs to select a suitable quality basis for the calculation. Below is a short description of some basis sets which often are used in routine calculations. 

McLean and Chandler developed a similar set of contracted basis sets from Huzinaga s primitive optimized set for second row elements. A DZ type basis is derived by contracting (12s8p) —> [5s3p], and a TZ type is derived by contracting (13s9p) —> [6s4p]. The latter contraction is 6,3,1,1,1,1 for tlie s-functions and 4,2,1,1,1 for the p-functions, and is often used in connection with the Pople 6-3IG when second row elements are present. 

All of the above basis sets are of the segmented contraction type. Modem contracted basis sets aimed at producing very accurate wave functions often employ a general contraction scheme. The ANO and cc basis sets below are of die general contraction type. 

Meters are accurate within close limits as legislation demands. However, gas is metered on a volume basis rather than a mass basis and is thus subject to variation with temperature and pressure. The Imperial Standard Conditions are 60°F, 30inHg, saturated (15.56°C, 1913.7405 mbar, saturated). Gas Tariff sales are not normally corrected, but sales on a contract basis are. Correction may be for pressure only on a fixed factor basis based on Boyle s Law or, for larger loads, over 190,000 therms per annum for both temperature and pressure using electronic (formerly mechanical) correctors. For high pressures, the compressibility factor Z may also be relevant. The current generation of correctors corrects for pressure on an absolute basis taking into account barometric pressure. 

Industry in general requires user-friendly equipment for industrial monitoring without the need for expertise for operation and data interpretation. Equipment and techniques that do require corrosion expertise will therefore, be limited to a service , i.e. specialist, company providing both equipment and personnel on a contract basis. 

The services of an outside maintenance contractor are usually on an annual contract basis and should clearly define the work which... 

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Triatomic Vibrational Eigenstates Product or Contracted Basis Sets, Lanczos or Conventional Eigensolvers What Is the Most Efficient Combination ... 

Contracted Basis Functions with a Lanczos Eigensolver for Computing Vibrational Spectra of Molecules with Four or More Atoms. 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

It was discovered that the optimum combination of speed and accuracy (when comparing to calculations using STOs) was achieved for M = 3. Figure 6.2 compares a Is function using die STO-3G formalism to the corresponding STO and shows also the 3 primitives from which the contracted basis function is constructed. STO-3G basis functions have been defined for most of die atoms in the periodic table. 

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