Quadratic constant

Triangular Crouzeix-Raviart Quadratic Constant Vertices and Centre... 

Although symmetry reduces the number of independent force constants, they still grow rapidly with the order of anharmonicity. For example, for a CH3X molecule of C.. symmetry there are 12 independent quadratic constants, 38 cubic constants, and 102 quartic constants. 

Equation [3] looks complicated at first sight, but is just Eq, [2] to which a cubic term has been added. The cubic constant has the value - 2.00 times the quadratic constant. The factor of 143.88 converts the units to kcal/mol. Judicious. selection of the force constant parameter for this cubic expression allowed for accurate treatments of bond length deformations in a wide variety of molecules. 

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means... 

Figure B3.2.11. Total energy versus lattice constant of gallium arsenide from a VMC calculation including 256 valence electrons [118] the curve is a quadratic fit. The error bars reflect the uncertainties of individual values. The experimental lattice constant is 10.68 au, the QMC result is 10.69 (+ 0.1) an (Figure by Professor W Schattke).

This approximation, known as Vegard s laM>, accurately describes the average lattice constant (but not the microscopic stmcture ) of most ternary compounds. However, the expression for the gap must be modified by the inclusion of a quadratic tenn... 

Determination of the paiameters entering the model Hamiltonian for handling the R-T effect (quadratic force constant for the mean potential and the Renner paiameters) was carried out by fitting special forms of the functions [Eqs. (75) and (77)], as described above, and using not more than 10 electronic energies for each of the X H component states, computed at cis- and toans-planai geometries. This procedure led to the above mentioned six parameters... 

Hach molecular mechanics method has its own functional form MM+. AMBER, OPL.S, and BIO+. The functional form describes the an alytic form of each of th e term s in th e poteri tial. For exam pie, MM+h as both a quadratic and a cubic stretch term in th e poten tial whereas AMBER, OPES, and BIO+ have only c nadratic stretch term s, I h e functional form is referred to here as the force field. For exam pie, th e fun ction al form of a qu adratic stretch with force constant K, and equilibrium distance i q is ... 

A larger value for the bending force constant K0 leads to a greater tendency for the angle to remain at its equilibrium value 0g. There may be cubic, quartic, etc. terms as with the corresponding bond stretch term in addition to the quadratic term shown here. 

The cubic stretch term is a factor CS times the quadratic stretch term. This constant CS can be set to an arbitrary value by an entry in the Registry or the chem. ini file. The default value for MM2 and MMh- is CS=-2.0. 

The functional form for bond stretching in AMBER is quadratic only and is identical to that shown in equation (11) on page 175. The bond stretching force constants are in units of kcal/mol per A and are in the file pointed to by the QuadraticStretch entry for the parameter set in the Registry or the chem.ini file, usually called =>istr.txt(dbf). 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

Figure 3.4.3 shows that the measured results fit the quadratic equation well for pressure generated. It also shows that the pressure generated is independent of flow since three different quantities of catalyst were used. Since the pressure drop remained constant, then flow must have been different over the three quantities of catalysts. The flow adjusted itself to match the constant pressure generated by the blower. 

On page 235-241 is the explicit solution used in Excel format to make studies, or mathematical experiments, of any desired and possible nature. The same organization is used here as in previous Excel applications. Column A is the name of the variable, the same as in the FORTRAN program. Column B is the corresponding notation and Column C is the calculation scheme. This holds until line 24. From line 27 the intermediate calculation steps are in coded form. This agrees with the notation used toward the end of the FORTRAN listing. An exception is at the A, B, and C constants for the final quadratic equation. The expression for B was too long that we had to cut it in two. Therefore, after the expression for A, another forD is included that is then included in B. 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

Therefore, with constant (steady-state) sound power, p decreases when r increases, nor linearly, hut quadratically. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

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