The Taylor Expansion

It is useful at this point if we examine the Taylor expansion for a general diatomic potential U R) about the equilibrium bond length R.  

We often choose the zero of potential such that t/(J e) = 0 and the term 

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According to the Porod law [28], the intensity in the tail of a scattering curve from an isotropic two-phase structure havmg sharp phase boundaries can be given by eqnation (B 1.9.81). In fact, this equation can also be derived from the deneral xpression of scattering (61.9.56). The derivation is as follows. If we assume qr= u and use the Taylor expansion at large q, we can rewrite (61.9.56) as... 

A good starting point for understanding finite-difference methods is the Taylor expansion about time t of the position at time t + At,... 

If we start from the Taylor expansion about r(t) then... 

In the completely confluent case where each = x0, the result is the Taylor expansion up to the power. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The much better performance of the diagonal Fade approximants compared to the Taylor expansion is not unexpected since similar convergence trends were found previously for the expansion of the first hyperpolarizabilities and u>) of ammonia. The good... 

The different curves obtained by increasing the number of terms in the Taylor expansion are represented in Figure 3.3 on top of the Gompertz curve itself. The exponential growth model can thus be now justified not only because it fits well the data but also because it can be seen as a first approximation to the Gompertz growth model, which is endowed with a mechanistic interpretation, namely, competition between the catabolic and anabolic processes. 

Equation (E.4) may be obtained by considering the Taylor expansion of the laser field up to first order... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

Although there are some general textbook approaches to equation (8), see reference [11] for example, we have not found the expression of the Taylor expansion in full as simple as it has been presented here. Moreover, many potential Taylor expansions are used in various physical and chemical applications for instance in theoretical studies of molecular vibrational spectra [12] and other quantum chemical topics, see for example reference [13]. Then, the possibility to dispose of a compact and complete potential expression may appear useful. 

From the position of any one data point, Z(i), the remaining data points appear scattered around it in the 3N — 6 dimensional space. The error in the Taylor expansion T), E[Z(j) — Ti[Z(j), can be measured at each neighboring data point, j i. If the data density is sufficiently high, there are a set of M points (typically M ss 20-100) which are close enough to Z(i) that the error in T) is dominated by the first term neglected in the Taylor expansion. Then... 

To derive this result we only have to rely on Eq. (54) and the idea that the distribution of errors is a normal distribution. Of course, the error in the Taylor expansions is not just a function of distance, it is also a function of direction. Hence, a better model would assign each Taylor expansion confidence lengths for each direction in space. For various reasons, it is much simpler to associate a confidence length with each element of Z, and to define the weight function as... 

This method is based on the Taylor expansion of f(x) about the kth iterate, x, ... 

The errors in (6.20)-(6.23) are given for the exponential form of the free energy difference, and the inaccuracy in ZL4fwd and ZL4rvs can be obtained from them easily. Note that when 5e is small, (6.22) and (6.23) give the absolute systematic error in (3AA itself (through the Taylor expansion of 6e to the second order). 

We develop y into a Taylor-series around a set of initial values for the k- which must not deviate too much from the optimised final values. With this condition we may truncate the Taylor-expansion after the linear terms and obtain the following system of linear relations ... 

Because the concept of minimum energy path is not well-defined when multiple electronic states are involved, the initial data set is simply taken as the union of points which one considers important on each of the electronic states—for example, local minima on each electronic state. The weights of each data point, Wi in Eq. (2.34), were taken to be the same on all electronic states because they only depend on the location of the data points. Hence, the difference between electronic states (V he[)ard(R)) is manifested only in the parameters of each of the Taylor expansions ... 

Dynamical variables are estimated from the gradient information generated during energy minimization. In this way velocities are derived (in one dimension) from the Taylor expansion... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

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