Second-order Taylor expansion

A further improvement can be seen for the situation depicted in Eigure lb. Let ( )i, (r) denote the potential due to the charges in the cell about point b, evaluated at the point r. Let a be the center of the subcell containing q. Then (j), (r) can be approximated by a second-order Taylor expansion about a ... 

The potential energy is approximated by a second-order Taylor expansion around the stationary geometry. 

There are two aspects in this. One is controlling the total length of the step, such that it does not exceed the region in which the second-order Taylor expansion is valid. The... 

This may again have multiple solutions, but by choosing the lowest A value the minimization step is selected. The maximum step size R may be taken as a fixed value, or allowed to change dynamically during the optimization. If for example the actual energy change between two steps agrees well witlr that predicted from the second-order Taylor expansion, the trust radius for the next step may be increased, and vice versa. 

For our second order Taylor expansions, n = 2. Moreover, Eq. (43) becomes exact as iVdata — oo, if59... 

Certainly, nonlinearities in real data can have several possible causes, both chemical (e.g., interactions that make the true concentrations of any given species different than expected or might be calculated solely from what was introduced into a sample, and interaction can change the underlying absorbance bands, to boot) and physical (such as the stray light, that we simulated). Approximating these nonlinearities with a Taylor expansion is a risky procedure unless you know a priori what the error bound of the approximation is, but in any case it remains an approximation, not an exact solution. In the case of our simulated data, the nonlinearity was logarithmic, thus even a second-order Taylor expansion would be of limited accuracy. 

Finally, the CECDC mechanism with more than one rate-determining step is considered. By a procedure similar to that described in the previous section, but now in terms of the second-order Taylor expansions of the individual rate equations, explicit expressions for the three second-order parameters in eqn. (140) have indeed been derived [7]. Although they are not as surveyable as their first-order counterparts, the expressions compiled in Table 7 are sufficiently simple to be suitable for fitting experimental data. 

The step size control parameter R, initially set to a value of order unity, is adaptive, in the sense that it is decreased (or increased) at each iteration depending on how well (or how badly) the energy change actually brought about by the corrections accords with the value predicted from a second-order Taylor expansion in the corrections themselves. In extreme cases, the corrections are rejected and recomputed with an increased value of R. Otherwise, any updates to R apply from the next iteration. The precise set of rules used to control R may affect efficiency but is not critical to the success of the minimization procedure, as long as the rules provide the correct qualitative behaviour (e.g. see Refs. [22] and [18]). 

In this approach, the external potential displacements that are responsible for a transition from stage (i) to stage (ii) create conditions for the subsequent CT effects, in the spirit of the Born-Oppenheimer approximation. Clearly, the consistent second-order Taylor expansion at M°(co) does not involve the coupling hardness t A B and the off-diagonal response quantities of Eqs. (168) and (170), which vanish identically for infinitely separated reactants. However, since the interaction at Q modifies both the chemical potential difference and the... 

Let 9m denote the mode (initially unknown) of the log-likelihood function L 9 y) for the single parameter 9. A second-order Taylor expansion of L around 9m, for a given model and pattern of experiments, gives... 

When data are obtained, we assume that the log-likelihood function will have a local maximum (mode) at some point 6 = 6m- A second-order Taylor expansion of L(6) around that point will give... 

A semiempirical model that has been particularly useful in zeolite studies is known as the electronegativity equalization method (EEM). EEM is based on the following assumptions (1) the electron density can be partitioned into spherical atomic contributions (2) T[p] and Ve [p] in Eq. [6] can be written as a second-order Taylor expansion of the effective charges and (3) each atom i carries an effective charge q,- = Z, - N,, where Z, and N, are the charge on the nucleus and the number of electrons in the atom, respectively. Then the total energy is given ... 

The solution of these equations may be obtained by use of the iterative Newton method. The Newton method can be derived from a second order Taylor expansion, which for the given Lagrangian (L) gives ... 

The relevant second-order Taylor expansion of the molecular electronic energy in powers of displacements of the canonical state parameters, [d/V, dl/(r)], is determined by the relevant principal derivatives of the energy representation ... 

Becke and Roussel [182] constructed a model exchange hole starting with the second-order Taylor expansion of the exact spherically averaged cr-spin hole [183]... 

By a near-quadratic surface, we mean a PES like that of Eq. (4.9) or (4.14), where the PES is taken to be a second-order Taylor expansion in directions orthogonal to the reaction path, although we will include cases where some higher-order terms are included. 

In order to relate CS of reactants A and B to the CT reaction rate an intersecting-state model (ISM) has recently been proposed [36], with the relevant potential energy curves defined in the electron population space. We again consider the reactive system = A—B and the associated potential energy surface Ejj(Na, Nb), given by the second-order Taylor expansion in the reactant population displacements from the initial configuration, = (A (B ), before CT (see Figs. 6 and 7) ... 

To ensure that the above result guarantees the minimum, we consider the second order Taylor expansion of / with respect to w at w, keeping the corresponding A fixed at v /2. Thus, for sufficiently small dw,... 

Sufficiently close to the root x, the second order Taylor expansion gives... 

The second-order Taylor expansion series of the function F(x) in the neighborhood of Xi is... 

If one is interested solely in the estimation of a characteristic decoherence time, an alternative to the generation and the analysis of a large set of diverging trajectories has been proposed by Schwartz et al Based on a second-order Taylor expansion of the positions and momentums of the nuclei at / = 0, the real part of the decoherence function can be approximated as ... 

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