Quadratic eigenvalues

I - FH) r on a quadratic surface. Here I is the unit matrix and denotes the nth cycle. The ultimate convergence rate is governed by the magnitude of the largest eigenvalue of the matrix (I - FH). This will be... 

HyperChein has two synch ron ons transit meth ods im piemen ted. The linear synchronous transit method (LST) searches for a maximum along a linear path between reactants and products. It may happen that this method will end up with a structure having two or more negative eigenvalues. The quadratic synchronous transit method (QSTlisan improvement of LST approach and searches for a maximum along a parabola connecting reactants and products, and for a minimum in all directions perpendicular to the parabola. 

It uses a linear or quadratic synchronous transit approach to get closer to the quadratic region of the transition state and then uses a quasi-Newton or eigenvalue-following algorithm to complete the optimization. 

The roots of Eq. (11-26) are the eigenvalues of the secular equation they are the reciprocal of the relaxation times (of which there are two since Eq. (11-26) is a quadratic). 

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

Figure 11. Quantum monodromy in the spectrum of the quadratic Hamiltonian of Eq. (38). The solid lines indicate relative equilibria. Filled circles mark the eigenvalues of the most stable isomer and those above the relevant effective potential barrier in Fig. 8. Open circles indicate interpenetrating eigenvalues of the secondary isomer. The transported unit cell moves over the hlled circle lattice, around the curved fold line connecting the two spectra.

The reciprocals of the time constants, x, and x2, are the rate constants k, and k2. The weights of the exponentials (ii and w2) are complicated functions of the transition rates in Eq. (6.25). Flowever, the rate constants are eigenvalues found by solving the system of differential equations that describe the above mechanism. A, and k2 are the two solutions of the quadratic equation ... 

The numerical values were obtained from a simple Hartree-Fock variational treatment, as described in Section 1.3 below. Note that, in this simple case, we could obtain the exact eigenvalues of the 2x2 matrix H by solving a quadratic equation. The present use of perturbation theory to approximate these eigenvalues is for illustrative purposes only. 

Note that the eigenvalue pfx) does not depend on time, which is a consequence of Eq. (17). In particular, for a quadratic Hamiltonian, the operator satisfying Eq. (17) can be obtained explicitly. This canonical method has an advantage that quantum statistical information can... 

If a function of two variables is quadratic or approximated by a quadratic function /(x) = b0 + bxxx + 62 2 + 11 1 + 22 2 + 12 1 2 then the eigenvalues of H(x) can be calculated and used to interpret the nature of fix) at x. Table 4.2 lists some conclusions that can be reached by examining the eigenvalues of H(x) for a function of two variables, and Figures 4.12 through 4.15 illustrate the different types of surfaces corresponding to each case that arises for quadratic function. By... 

Geometry of a quadratic objective function of two independent variables—elliptical contours. If the eigenvalues are equal, then the contours are circles. 

Figure 2.5 The ellipse 2x2 — 2xy + 2y2 = 1. //, and u2 are the eigenvectors associated with the quadratic form, 3 and 1 the corresponding eigenvalues.

For the Kratzer potential, which is quadratic in the variable (r - re)/r, the eigenvalues for / 0 can be obtained in closed form. This is also the case for the potential quadratic in (r-r2/r) (Gol dman et al., 1960). This potential does not, however, tend to a finite value as r —> oo. 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

Our generalized eigenvalue problem thus depends upon three parameters, a, b, and s. Denoting the eigenvalue by W and solving the quadratic equation, we obtain... 

Therefore, the two eigenvalues A,i and X2 may be solved from the following quadratic equation ... 

To solve a diffusion equation, one needs to diagonalize the D matrix. This is best done with a computer program. For a ternary system, one can find the two eigenvalues by solving the quadratic Equation 3-lOOe. The two vectors of matrix T can then be found by solving... 

Case ii). C) is automatically satisfied if B), B0) hold and, moreover, the spectrum of IIt) at least belotu Aq consists of finite discrete eigenvalues with finite multiplicities. To show this we may assume U0 and i cl) to be positive definite (see the remark of 7.1). Then the quadratic form ((i70 + /) is an increas-... 

In textbooks on linear algebra and dynamical modeling, it is shown that the eigenvalues X of such matrixes are negative or zero (see, e. g., Arrowsmith and Place, 1992). Eigenvalues and eigenfunctions of quadratic matrixes are defined in Box 21.8. 

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