Quartic equation

It can be shown [ ] that the expansion of the exponential operators truncates exactly at the fourth power in T. As a result, the exact CC equations are quartic equations for the t y, etc amplitudes. The matrix elements... 

These quartic equations are solved in an iterative maimer and, as such, are susceptible to convergence difficulties. In any such iterative process, it is important to start with an approximation reasonably close to the final result. In CC theory, this is often achieved by neglecting all of tlie temis tliat are nonlinear in the t amplitudes (because the ts are assumed to be less than unity in magnitude) and ignoring factors that couple different doubly-excited CSFs (i.e. the sum over i, f, m and n ). This gives t amplitudes that are equal to the... 

For polyatomics, ordinarily only the last two tenns of equation (C3.5.6), the cubic and quartic anlrannonic tenns, need be considered [34]. In a cubic anlrannonic process, excited vibration D relaxes by interacting with two other states, say airother vibration cr aird one phonon (or alternatively two phonons). In the quartic process, D relaxes by interacting with tlrree other states, say two vibrations aird one phonon. The total rate constairt for energy loss from Q for cubic... 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-and-error enters, however, because (GSi)r and are mild functions of Tg and related Te, respectively, and aprehminary guess of Tg is necessaiy. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. 

By eliminating the concentrations in turn from eqns.(2), (6) and (8), we obtain a quartic equation which is most simply obtained numerically. However, we can obtain good approximate solutions in limiting cases. Firstly, we can set the Caa and Cbb equal to unity to a... 

As a first example we consider a system bounded periodically in two coordinates and by thermal walls in the other coordinate. The two thermal walls are at rest and maintained at the same temperature, T. The system is subjected to an acceleration field which gives rise to a net flow in the direction of one of the periodic coordinates. For this system, the hydrodynamic equations yield solutions of quadratic form for the velocity and quartic for the temperature. 

KACSYKA can also solve quadratic, cubic and quartic equations as well as some classes of higher degree equations. However, it obviously cannot solve equations analytically in closed form when methods are not known, e.g. a general fifth degree (or higher) equation. 

When v > 1 the equation equivalent to equation 15 is of order higher than quadratic, a cubic when v = 2 and a quartic when v = 3. The solution of these equations, though possible, does not lead to a value of X of acceptable precision. 

One may require additional solution points for purposes of presentation or analysis. In many cases, interpolating polynomials prove to be useful. Thus, one might approach our example problem using products of quartic equations of the general foxmX f (x) = + bjX + CjX + djX + ej and similarly for An appropri-... 

This equation leads to a quartic in n, but for certain cases (e.g., SI GaAs), we can neglect n and nf/n with respect to JVAS and iVDs. That is, the carrier concentrations are typically only 106-107 cm-3 in SI GaAs, whereas the impurity concentrations are typically 1016 cm 3. In this approximation, Eq. (B48) becomes... 

Force fields up to quartic anharmonic terms are now known with reasonably high accuracy for several triatomic molecules and the results shown in Table 3 for H2O are typical. However, even for these there has had to be an assumption that some of the quartic interaction terms are zero in order that the equations from which the constants are derived shall have unique solutions. It can be seen moreover that some of the cubic and quartic terms have uncertainties which are larger than the values of the constants themselves. 

The quartic expression for tr(J) conveniently only involves even powers of fi. The condition for the change of local stability, tr(J) = 0, therefore is a quadratic equation in fi2 with roots given by... 

Equation (11) is a quartic equation for the steady-state value of 0b. Because the only values of 0b that make physical sense are those between zero and one, Sturm s method (Takoudis et al. 1981b) can be used to show that there can be either one or three real roots within this interval depending upon the values of the parameters. The region of triplicity is the finger-shaped domain in the ax, a2-plane of figure 1. Its boundary is labelled turning points because its equation may be found by eliminating 0A and 0b between = /2 = 0 and det[0(/1,/2)/0(0A, 0b)] = 0. 

Equations (19)-(22) offer a method for generating level curves of the rs-catastrophe surface by solving a quadratic equation, rather than the quartic g(0a) = 0. This is done by choosing a fixed rate and then calculating S for several values of a. The t i and a2 coordinates are then calculated from (22). 

Second-order optical nonlinearities result from introduction of a cubic term in the potential function for the electron, and third-order optical nonlinearities result from introduction of a quartic term (Figure 18). Two important points relate to the symmetry of this perturbation. First, while negative and positive p both give rise to the same potential and therefore the same physical effects (the only difference being the orientation of the coordinate system), a negative y will lead to a different electron potential than will a positive y. Second, the quartic perturbation has mirror symmetry with respect to a distortion coordinate as a result, both centrosymmetric and noncentrosymmetric materials will exhibit third-order optical nonlinearities. If we reconsider equation 23 for the expansion of polarization of a molecule as a function of electric field and assume that the even-order terms are zero (i.e., that the molecule is centrosymmetric), we see that polarization at a given point in space is ... 

For small displacements, of the order of vibrational amplitudes at room temperature, the terms in the power series expansion (1) converge fairly rapidly, and higher-order terms are related to successively smaller-order effects in the spectrum, so that they become more and more difficult to determine. Almost all calculations to this date have been restricted to determining quadratic, cubic, and quartic force constants only [the first three terms in equation (1)], and in this Report we shall not consider higher-than-quartic terms in the force field. The paper by Cihla and Chedin11 is one of the few exceptions in which force constants involving up to the sixth power have been determined for a polyatomic molecule, namely COa. 

The conclusion is that if the spectrum can be analysed in terms of equations (3)—(7), then the force constants can be determined. The bond length re can be determined from the equilibrium rotational constant Bc then the quadratic force constant /3 can be determined either from the harmonic wavenumber centrifugal distortion constant De then the cubic force constant /3 can be determined from aB and finally the quartic force constant /4 can be determined from x. It is necessary to determine the force constants in this order since in each case we depend upon already knowing the preceding constants of lower order. The values of re,f2,f3, and /4 calculated in this way for a number of diatomic molecules are shown in Table 2. 

In solving the function for Rp it was necessary to use a numerical scheme to get roots of the quartic equation and then decide which root was the meaningful one. Since this would have been too time consuming, we decided to use Newtons method, with the observed value of Rp as a first guess, and tentatively accept the value on which the method converged as the meaningful root. This proved to be a workable approach. 

Upon convergence of the iterative procedure and using the least-squares estimates of the parameters herein reported, the other roots of the quartic were investigated at each of these data points. In all cases, there was one other positive root (Rp) and either two negative or two complex roots. The second positive root was in all cases further from the observed value of Rp than the root which was selected by Newtons method. Furthermore, it was found that in all cases for which we had data this second positive root when substituted in the original equations gave a negative value of R. Thus, our numerical procedure led to the proper least squares solution. 

Therefore, one and only one root of the quartic equation has physical meaning. If it is desired to estimate fis for other polymerization systems, a better way of writing the function is in its most elementary form. That is, merely equating Equations 9a and 10a [eliminating the negative square root in Equation 9a since it gives negative R s]. 

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