Taylors series

There is a systematic procedure for deriving power-series expansions for all well-behaved functions. Assuming that a function /(x) can be represented in a power series about xq = 0, we write 

We have, therefore, determined the coefficients oq, a, a2. .. in terms of derivatives of f(x) evaluated at x = 0, and the expansion 7.32 can he given more explicitly by 

If the expansion is carried out around xo = a, rather than 0, the result generalizes to 

This result is known as Taylor s theorem, and the expansion is a Taylor series. The case xq = 0, given by Eq. (7.40), is sometimes called a Maclaurin series. 

In order for a Taylor series around x = a to be valid, it is necessary for all derivatives f a) to exist. The function is then said to be analytic at x = a. A function that is not analytic at one point can still be analytic at other points. For example, Inx is not analytic at x = 0 but is at x = 1. The series (7.20) is equivalent to an expansion of hix around x = 1. 

If fix) is analytic within and on a circle C centered at Xq of radius R, then the Taylor series representation of fix) is given by 

It is particularly useful for approximating some functions in the vicinity of x = 0  

Isotopic fractionation provides illustrative examples of first-order expansions of unknown functions. In general, the mass spectrometric measurement r/ of the ratio between two isotopes of mass m( and m, of the same element, differs from the natural value R/. Only a very small fraction of the original sample produces ions and different processes taking place in different parts of the mass spectrometer act differently on the sensitivity of each isotope. We assume that instrumental isotopic fractionation is mass-dependent. 

Equilibrium fractionation. A simple fractionation law, called the linear law (e.g., Hofmann, 1971), relates the measured and natural isotopic ratios through a function f (Am/) of the mass difference Am/ = m,—m, between the isotopes defining the ratios 

Mass discrimination with distillation effects. Let us assume that the isotope composition of an element is being measured by thermal ionization. This method consists in ionizing the sample atoms by evaporation on a metal filament. Statistical thermodynamics (e.g., Denbigh, 1968) tells us that, while vapor pressure is a function 

Expanding the function g in a Taylor series to the first order with respect to m, we get the approximation 

Finally, we briefly mention the concept of an integrating factor, a multiplicative factor (L) that converts an inexact differential (ctf) to an exact differential (dg), namely, 

A common situation in thermodynamics is that some property z(x) and its lower derivatives (z , z , z , . ..) have been measured at a certain point x0, and one wishes to use this information to approximate the behavior of the function z(x0 + Ax) in the Ax-neighborhood of x0. For this purpose, the fundamental Taylor series (or MacLaurin series, the special case for x0 = 0) yields approximations that are useful for sufficiently small Ax  

The student of thermodynamics should be able to generate such Taylor series expansions for common algebraic and trigonometric functions. 

Exercises Use the first few terms of the Taylor series expansion (1.23) to develop small-x approximations for the functions 

The techniques that we will use are iterative. We start with some initial guess of the solution, and apply an algorithm to refine this guess to generate a sequence P. .. tiiat 

We first consider Newton s method, an iterative technique that is based on the use of Taylor series expansions. As Taylor series are used extensively in numerical mathematics, we briefly review their use. 

Let us say that we have some function f(x) that we wish to represent as a polynomial in the vicinity of some point xo. 

Given only information of the function at xo the value of the function itself plus the values of all of its derivatives (we assnme derivatives of /(x) to all orders exist at xo), what coefficients ao, a, ... should we use to match the polynomial to /(x) First, we see that 

Continuing this process, we find that if all derivatives to infinite order of /(x) exist at xo, we may represent the function as the infinite series 

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This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

Alternatively, expansion of equation (A2.5.1). equation (A2.5.2) or equation (A2.5.3) into Taylor series leads ultimately to series expressions for the densities of liquid and gas, / and p, in temis of their sum (called the diameter ) and their difference ... 

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. 

Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

The molecular dipole moment (not the transition dipole moment) is given as a Taylor series expansion about the equilibrium position... 

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

Under steady-state flow conditions (coherent motion), a Taylor series can be applied to describe the time-dependent position of the fluid molecules ... 

The Taylor series by itself is not numerically stable, since the individual temis can be very large even if the result is small, but other polynomials which are highly convergent can be found, e.g. Chebyshev [M, M and M] or Lancosz polynomials [, 68]. 

The high-field output of laser devices allows for a wide variety of nonlinear interactions [17] between tire radiation field and tire matter. Many of tire initial relationships can be derived using engineering principles by simply expanding tire media polarizability in a Taylor series in powers of tire electric field ... 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

The basic idea of NMA is to expand the potential energy function U(x) in a Taylor series expansion around a point Xq where the gradient of the potential vanishes ([Case 1996]). If third and higher-order derivatives are ignored, the dynamics of the system can be described in terms of the normal mode directions and frequencies Qj and Ui which satisfy ... 

Theexact eigenfunctions and eigen values can now be expanded in a Taylor series in A. 

Th e vibrational potential in ay be ex pan ded in a Taylor series abon t the etinilibrmni positions of the atoms. 

A Maclaunn series is a specific form of the Taylor series for which Xq = 0. Some standard expansions in Taylor series form are ... 

When discussing derivative methods it is useful to write the function as a Taylor series expansion about the point jc. ... 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

Truncating this series after the first derivative and integrating provides the basis for the hermodynamic integration approach. Moreover, if the Taylor series expansion is continued intil it converges then Equation (11.45) is equivalent to the thermodynamic perturbation brmula, so providing a link between the two approaches. In practice, it is always necessary... 

III- slow growth expression can be derived from the thermodynamic perturbation expres sion (Equation (11.7)) if it is written as a Taylor series ... 

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