Series Fourier

If we want to produce a series that will converge rapidly, so that we can approximate it fairly well with a partial sum containing only a few terms, it is good to choose basis functions that have as much as possible in common with the function to be represented. The basis functions in Fourier series are sine and cosine functions, which are periodic functions. Fourier series are used to represent periodic functions. A Fourier series that represents a periodic function of period 2L is 

Using trigonometric identities show that the basis functions in the series in Eq. (6.39) are periodic with period 2L. That is, show for arbitrary n that 

Fourier series occur in various physical theories involving waves, because waves often behave sinusoidally. For example, Fourier series can represent the constructive and destructive interference of standing waves in a vibrating string. This fact provides a useful way of thinking about Fourier series. A periodic fimc-tion of arbitrary shape is represented by adding up sine and cosine functions with 

There are some important mathematical questions about Fourier series, including the convergence of a Fourier series and the completeness of the basis functions. A set of basis functions is said to be complete for representation of a set of functions if a series in these functions can accurately represent any function from the set. We do not discuss the mathematics, but state the facts that were proved by Fourier (1) any Fourier series in x is uniformly convergent for all real values of x (2) the set of sine and cosine basis functions in Eq. (6.39) is a complete set for the representation of periodic functions of period 2L. In many cases of functional series, the completeness of the set of basis functions has not been proved, but most people assume completeness and proceed. 

In a power series, we found the coefficients by demanding that the function and the series have equal derivatives at the point about which we were expanding. In a Fourier series, we use a different procedure, utilizing a property of the basis functions that is called orthogonality. This property is expressed by the three equations  

An important mathematical tool for structure determination was developed by A. L. Patterson, allowing both the length and the direction of the lines between various of the atoms (interatomic vectors) to be evaluated from the intensities of the reflections (for which there may be no sign ambiguities). In ideal cases, this method should result in a series of interatomic distances to which the atoms known to exist in the unit cell must be fit. For complicated structures, such fitting is itself a difficult puzzle, and almost always the three-dimensional vector problem is broken down to a series of two-dimensional vector problems. Projections of the interatomic vectors on the faces of the unit cell are obtained, and from such projections, a three-dimensional picture may often be reconstructed. 

The computational labor associated with two-dimensional Fourier syntheses is not too formidable, and two-dimensional Fourier maps can be constructed without machine help. The labor associated with two-dimensional Patterson sysntheses is even less, and a two-dimensional vector map can often be obtained from measured intensities in a few hours. For Fourier and Patterson syntheses in three-dimensions, however, machine help is virtually indispensable. Before application of automatic computers to x-ray diffraction, the main obstacle standing in the way of a structure determination was generally the computational effort involved. In the 1950 8, the use of computers became commonplace, and the main obstacle became the conversion of measured intensities to amplitudes (the so-called phase problem ). There is still no general way of attacking this problem that is applicable in all situations, but enough methods have been developed so that by use of one, or a combination of them, all but very complicated structures may, with time and ingenuity, be determined. 

It is likely that in the near future the most important advances io 

Chemical Crystallography, Oxford University Press. London (1945). 

Physical Chemistry, 366-379, Prentice-Hall, Inc., New York (1950), 

As discussed in Section VI.A, any simple sine or cosine wave can therefore be described by three constants — the amplitude F, the frequency h, and the 

Each diffracted X ray that arrives at the film to produce a recorded reflection can also be described as the sum of the contributions of all scatterers in the unit cell. The sum that describes a diffracted ray is called a structure-factor equation. The computed sum for the reflection hkl is called the structure factor Fhkl. As / will show in Chapter 4, the structure—factor equation can be written in several different ways. For example, one useful form is a sum in which each term describes diffraction by one atom in the unit cell, and thus the series contains the same number of terms as the number of atoms. 

If diffraction by atom A in Fig. 2.15 is represented by fA, then one diffracted ray (producing one reflection) from the unit cell of Fig. 2.15 is described by a structure-factor equation of this form  

The structure—factor equation implies, and correctly so, that each reflection on the film is the result of diffractive contributions from all atoms in the unit cell. That is, every atom in the unit cell contributes to every reflection in the diffraction pattern. The structure factor is a wave created by the superposition of many individual waves, each resulting from diffraction by an individual atom. 

A widely used example of orthogonal functions is the set of sines and cosines. For example, given any real number a, and the function sin nx for integer values of n, 3) is equal to [a, a+ 2ji]. We can check that 

Similar orthogonality relationships can be shown for cosines. In addition, sines and cosines are mutually orthogonal, i.e., for any m and n 

Just as a vector is projected as components on orthogonal axes, a given function defined on a given domain can be projected onto an orthogonal set of functions. The Fourier series decomposition of a function /(x) defined over the interval [ —X, X] is a convenient example 

As in the case of regular vectors, the components a and b can be found by forming the inner products upon multiplication of the /(x) expansion by the adequate sine or cosine 

We note that, with the appropriate variable change, equation (2.6.11) reads 

The coefficient w(k) is the probability that the system exhibits the property k. For example, we could consider a mixture of polymers. Therein is a certain quantity of particles with the degree of polymerization of k. w(k) is then the probability, or the frequency, with which particles with the polymerization degree of k are present. We replace now x by exp(jx) and obtain the sum 

This is the discrete Fourier transform of the probability density function, also addressed as the characteristic function. We multiply nowEq. (12.21) by exp(—j/x), Z = 1, 2, 3. and integrate over an interval of In to get 

In this way, the probabilities w(k) can be obtained from Eq. (12.22). The definition of the Fourier transform is not unique, but it is historically grown from the different fields of science. Various conventions are in use. In Table 12.7 we show some pairs of Fourier transforms. The last definition in Table 12.7 involves two parameters a, b) and is the most versatile one. In the integral form, the characteristic function uses (1, 1). 

This is also consistent with the approximation that sinx rs x for x 0. 

Taylor series expansions, as described above, provide a very general method for representing a large class of mathematical functions. For the special case of periodic functions, a powerful alternative method is expansion in an infinite sum of sines and cosines, known as a trigonometric series or Fourier series. A periodic function is one that repeats in value when its argument is increased by multiples of a constant L, called the period or wavelength. For example. 

The functions sinn0 and cosn0, with n = integer, are also periodic in 2jt (as well as in Tjifn.) 

Writing the constant term as uq/I is convenient, as will be seen shortly. The coefficients % and hn can be determined by making use of the following definite integrals  

These integrals are expressed conpactly with use of the Kronecker delta, defined as follows  

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Patterson A L 1934 A Fourier series method for for the determination of the components of interatomic distances in crystals Phys. Rev. 46 372-6... 

In the smectic Aj (SmA ) phase, tlie molecules point up or down at random. Thus, tire density modulation can be described as a Fourier series of cosines ... 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

The Fourier Series, Fourier Transform and Fast Fourier Transform... 

Consider a periodic function x(t) that repeats between t = —r/2 and f = +r/2 (i.e. has period t). Even though x t) may not correspond to an analytical expression it can be written as the superposition of simple sine and cosine fimctions or Fourier series, Figure 1.13. 

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

Cooley J W and ] W Tukey 1965. An Algorithm for the Machine Calculation of Complex Fourier Series Aiathemalics of Computation 19 297-301. 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

The coefficients C are chosen to ensure that the function has a minimum at the appropriate reference bond angle. For linear, trigonal, square planar and octahedral coordination, Fourier series with just two terms are used with a Cq term and a term for n = 1, 2, 3 or 4, respectively ... 

XI. Complex Numbers, Fourier Series, Fourier Transforms, Basis Sets... 

Now let us eonsider a funetion that is periodie in time with period T. Fourier s theorem states that any periodie funetion ean be expressed in a Fourier series as a linear eombination (infinite series) of Sines and Cosines whose frequeneies are multiples of a... 

This form of the Fourier series is entirely equivalent to the first form and the an and bn ean be obtained from the Cn and viee versa. For example, the en amplitudes are obtained by... 

Hence we see that this simple periodic function has just two terms in its Fourier series. In terms of the Sine and Cosine expansion, one finds for this same f(t)=Sin3t that an = 0, bn =... 

Earlier, for Fourier series, we had the orthogonality relation among the Fourier functions ... 

If f is a function of several spatial coordinates and/or time, one can Fourier transform (or express as Fourier series) simultaneously in as many variables as one wishes. You can even Fourier transform in some variables, expand in Fourier series in others, and not transform in another set of variables. It all depends on whether the functions are periodic or not, and whether you can solve the problem more easily after you have transformed it. 

The only limitation on the function expressed is that it has to be a function that has the same boundary properties and depends on the same variables as the basis. You would not want to use Fourier series to express a function that is not periodic, nor would you want to express a three-dimensional vector using a two-dimensional or four-dimensional basis. 

Besides the intrinsic usefulness of Fourier series and Fourier transforms for chemists (e.g., in FTIR spectroscopy), we have developed these ideas to illustrate a point that is important in quantum chemistry. Much of quantum chemistry is involved with basis sets and expansions. This has nothing in particular to do with quantum mechanics. Any time one is dealing with linear differential equations like those that govern light (e.g. spectroscopy) or matter (e.g. molecules), the solution can be written as linear combinations of complete sets of solutions. 

In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. 

In case the curve y = fix) is symmetrical with respect to the origin, the a s are all zero, and the series is a sine series. In case the curve is symmetrical with respect to the y axis, the fc s are all zero, and a cosine series results. (In this case, the series will be valid not only for values of x between — c and c, but also for x = — c and x = c.) A Fourier series can always be integrated term by term but the result of differentiating term by term may not be a convergent series. 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

Hanna, R. Fourier Series and Integrals of Boundary Value Frohlems, Wiley, New York (1982). 

The original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

At temperatures above there is no instanton, and escape out of the initial well is accounted for by the static solution Q = Q with the action S ff = PVo (where Vq is the adiabatic barrier height here) which does not depend on friction. This follows from the fact that the zero Fourier component of K x) equals zero and hence the dissipative term in (5.38) vanishes if Q = constant. The dissipative effects come about only through the prefactor which arises from small fluctuations around the static solution. Decomposing the trajectory into Fourier series. 

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