Quadratic Representation

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

For radial concentration profiles, a quadratic representation may not be adequate since application of the zero flux boundary conditions at r, = cp0 and r, = 1.0 leads to d2 = d3 = 0. Thus a quadratic representation for the concentration profiles reduces to the assumption of uniform radial concentrations, which for a highly exothermic system may be significantly inaccurate. 

However, a quadratic representation of the radial concentration profile may not be adequate since application of zero flux conditions at the inner thermal well and outer cooling wall with a quadratic profile reduces to an assumption of uniform radial concentrations. Although additional radial... 

In this chapter, we have derived the two-dimensional finite element penalty formulation for creeping flows where the pressure was eliminated by assuming a compressible flow. Here, we will use a mixed formulation, where the pressure is included among the unknown variables. In the mixed formulation, we use different order of approximation for the pressure as we will for the velocity. For instance, if tetrahedral elements are used, we can use a quadratic representation for the velocity (10 nodes) and a linear representation for the pressure (4 nodes). Hence, we must use different shape functions for the velocity and pressure. For such a formulation we can write... 

Combination of Mulliken s formalism with the Marcus quadratic representation [10] of the initial and final (diabatic) states allows the energy profile of the ET reaction coordinate to be constructed. As illustrated in Figure 8, an increase in Hab results in (i) a lowering of the ET barrier, (ii) a stabilization of the precursor and successor (CT) complexes, and (iii) a shift of their positions along the reaction coordinate (i.e. the charge is partially transferred from the... 

We can also consider cases in which the intrinsic barrier is altered. Two such effects are steric hindrance and contribution of charge-separated structures to the transition state. Steric hindrance raises the energy of the transition state compared to that of a similarly exothermic unhindered model. This can be accomodated by considering an increase in the intrinsic barrier, which therefore makes the isotope effect rise. In ref.11 this is alternatively interpreted in a quadratic representation of the surface as an increase in the interaction force constant, and thus also correlated with an increase in the tunnel correction. An example of such an enhancement is the large value of the isotope effect in the trityl radical mesitylenethiol reaction in Table 1. 

The simple Quadratic Representation 4 of f(p T) is definitely applicable in a large neighborhood of the coexistence boundary, and evolves to a form equivalent to a virial expansion with second and third virial coefficients at supercritical temperatures. 

The feature of this quadratic representation which is of interest in our present discussion is the dependence of the surface on the values of ki, /cj and ki2- First, if /cj = /c2, the slope of the line representing the reaction coordinate is exactly — 1, irrespective of the value of... 

The original explanation of this effect [34] can be restated in terms of the considerations of the quadratic representation, namely that steric hindrance is a repulsive interaction present in the transition state but absent in the initial and final states. It can therefore be... 

Figure B3.5.1. Contour line representation of a quadratic surface and part of a steepest descent path zigzagging toward the minimum.

In this Fourier representation the Hamiltonian is quadratic and the equipartition theorem yields for the thennal... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Representation of the trust region to select the step length. Solid lines are contours of fix). Dashed lines are contours of the convex quadratic approximation of fix) at x. The dotted circle is the trust region boundary in which 8 is the step length. x0 is the minimum of the quadratic model for which H(x) is positive-definite. 

This appendix provides a summary of the functional form of the algebraic Hamiltonian used in the text for tri- and tetratomic molecules. Values of the parameters are reported both for a low-order realistic representation of the spectrum and for accurate fits using terms quadratic in the Casimir operators. 

Levine, R. D. (1983), Representation of One-Dimensional Motion in a Morse Potential by A Quadratic Hamiltonian, Chem. Phys. Lett. 95, 87. 

One may now proceed to write out y+y and w+w and, to make the connection to the present work, retain only the terms that are particle conserving. The result are representability conditions, and they include terms quadratic in/I and terms quadratic in/2, but no mixed terms. It will be clear then that the extreme conditions—and they are all that matter—involve either /I or /2, but not both moreover, one will observe that 0 < y+y and 0 < w+w lead to the same conditions, which are the real cases of the T1 and the strengthened T2 conditions. 

These density matrices are themselves quadratic functions of the Cl coefficients and they reflect all of the permutational symmetry of the determinental functions used in constructing T they are a compact representation of all of the Slater-Condon rules as applied to the particular CSFs which appear in Tk They contain all information about the spin-orbital occupancy of the CSFs in Tk The one- and two- electron integrals < (f>i I f I (f>j > and < (f>i(f>j I g I ( >k4>i > contain all of the information about the magnitudes of the kinetic and Coulombic interaction energies. 

Using the fact that the quadratic operators transform according to the irreducible representations ... 

Fig. 11.5. Phase plane representation of a travelling wavefront for quadratic autocatalysis. The trajectory emerges from the initial singularity at p = 1, g = 0 and tends to the final state fi = 0,...

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