Taylor second-order

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Example Consider the differential equation for reaction and diffusion in a catalyst the reaction is second order c" — ac, c Qi) = 0, c(l) = 1. The solution is expanded in the following Taylor series in a. 

Sufficient conditions are that any local move away from the optimal point ti gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point ti up to second-order terms ... 

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 ... 

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression... 

Ebend is the energy required for bending an angle formed by three atoms A-B-C, where there is a bond between A and B, and between B and C. Similarly to Estr, Fbend is usually expanded as a Taylor series around a natural bond angle and terminated at second order, giving the harmonic approximation. 

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... 

The Rational Function Optimization (RFO) expands the function in terms of a rational approximation instead of a straight second-order Taylor series (eq. (14.3)). 

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. 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

That the reaction appeared to be second-order was confirmed by several groups (see compilation by Zollinger, 1961, p. 25) and all these arguments appear to be based essentially on the views originally developed by Hantzsch. However, Taylor s... 

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... 

Ruckenstein and Li proposed a relatively simple surface pressure-area equation of state for phospholipid monolayers at a water-oil interface [39]. The equation accounted for the clustering of the surfactant molecules, and led to second-order phase transitions. The monolayer was described as a 2D regular solution with three components singly dispersed phospholipid molecules, clusters of these molecules, and sites occupied by water and oil molecules. The effect of clusterng on the theoretical surface pressure-area isotherm was found to be crucial for the prediction of phase transitions. The model calculations fitted surprisingly well to the data of Taylor et al. [19] in the whole range of surface areas and the temperatures (Fig. 3). The number of molecules in a cluster was taken to be 150 due to an excellent agreement with an isotherm of DSPC when this... 

One of the fundamental assumptions of TST states that the reaction rate is determined by the dynamics in a small neighborhood of the saddle point, which we place at q = 0. It is then a reasonable approximation to expand the potential U(q) in a Taylor series around the saddle and retain only the lowest-order terms. Because VqU(q = 0) =0 at the saddle point itself, these are of second order and lead to the Hamiltonian... 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

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

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). 

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. 

Gordon, M., Taylor, J. S. Ideal copolymers and the second-order transitions of synthetic rubbers in non-crystalline copolymers. J. Appl. Chem, p. 493, 1952... 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in ... 

For a complex molecule with N internal degrees of freedom, the situation is a little more complicated. Let Rk (A = 1,. .., N) denote the components of R, and R j, their equilibrium values. A Taylor expansion up to second order gives now ... 

The second term in (5-4) is the second-order Doppler shift. This is the higher-order term of the Taylor expansion that we ignored in (5-3). Like , it can be calculated in the Debye model. Figure 5.6 shows plots of the second-order Doppler shift for the case of iron and for different values of the Debye temperature. Soft lattice vibrations are expected to decrease the isomer shift, although the effect becomes only significant at temperatures well above 80 K. 

Helfrich has shown [18] that the surface tension of a curved interface can be expressed as a Taylor series up to second order in the radius of curvature ... 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

Functional Taylor series expansion of the functional minimized in Eq. (87), in powers of noK ") = [nGs( ) - gs( )] has been employed first, and Eq. (88) used in the last step. So E " is close to KS correlation energy functional taken for the GS density of HF approximation, corrected by the (much smaller) HF correlation energy, and a small remainder of the second order in the density difference. The last quantity gives an estimate to the large parentheses term of Eq. (28) in [12]. 

We begin by defining a set of N atoms and writing their Cartesian coordinates as a single vector with 3N components, r = (r, ..., ro,N). If locating the atoms at ro is a local minimum in the energy of the atoms, then it is convenient to define new coordinates x = r ro. The Taylor expansion of the atom s energy about the minimum at ro is, to second order,... 

We can use exactly the same idea for the integral in the numerator of Eq. (6.10). The Taylor series expansion for the energy expanded around the transition state is, to second order,... 

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