Approximating Complicated Functions

Another common application of derivatives is to generate a simple approximation to some complicated function /(x). We can rearrange Equation 2.1 and substitute y = 

If Ax is not infinitesimal, but is sufficiently small, it must be true that 

Equation 2.18 is actually the first term in what is known as a Taylor series, which can be extended to include higher derivatives as well for a better approximation. The more general expression is  

The ratio of the nth term in Equation 2.21 to the immediately preceding term is x/(n — 1). So if x 1, each term is much smaller than the one before it, and the series converges rapidly. For example, if x = 0.01, e0 01 = 1.010050167, which differs only slightly from the value of 1.01 predicted by Equation 2.20. 

Ifx 1 the series starts out with growing terms, but no matter how large a number we choose for x, x/n C 1 for large enough n. Thus eventually the terms start getting progressively smaller. In fact this series converges for all values of x. 

---

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei< . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

Extension of this method for correcting the energies of approximate wave functions to systems containing more electrons and orbitals would be very useful. But difficulties quickly arise. The interelectronic effects become complicated because of exchange and correlation. More importantly, in DFT, it is only the highest occupied orbital whose energy is equal to the electronic chemical potential. This potential is valid for the total electron density. 

Medium-range interactions can be defined as those which dominate the dynamics when atoms interact with energies within a few eV of their molecular binding energies. These forces determine a majority of the physical and chemical properties of surface reactions which are of interest, and so their incorporation in computer simulations can be very important. Unfortunately, they are usually many-body in nature, and can require complicated functional forms to be adequately represented. This means that severe approximations are often required when one is interested in performing molecular dynamics simulations. Recently, several potentials have been semi-empirically developed which have proven to be sufficiently simple to be useful in computer simulations while still capturing the essentials of chemical bonding. 

In the intermediate situation the measured parameters are complicated functions of the full effects experienced in the paramagnetic site (Acom, Rim Rim) and the exchange rate r 1. The r 1 dependence of such functions has been summarized in Figs. 4.2 and 4.3. There are, however, particular regions in which the measured parameters are simpler functions of r 1. These regions correspond to cases in which Eqs. (4.8), (4.10) and (4.11) can be approximated by... 

The Gj(t) functions of Eq. (15) have been calculated by Lin [60] when summing over Franck-Condon factors obtained from all possible (infinite) wavefunctions in the harmonic oscillator approximation. These Gj(t) are rather complicated functions of the frequencies arf, co and reduced masses M j, M which are attributed to the corresponding normal coordinates Qf and Q j. They are collected in parameters describing the frequency relation ft2 and the potential minimum shift Aj of the excited state with respect to the ground state... 

The polarization P is generally a complicated function of an external electric field E. The real function P(E) can be approximated with a Taylor expansion around zero electric field strength E = 0. 

Although Qext is a complicated function of both a and m, in its extremes (e.g., very small d or large d) it can be roughly approximated by the form Qext = u + za or Qext = 2, where z is a constant slope and u is a constant. These approximations are illustrated in Fig. 16.5 for a nonabsorbing and an absorbing particle. 

With regard to the second point, it is important to note that an approximate wave function which is more general than those of Eqs. (2)-(5) cannot be described in terms of either bent bonds or wave function (general MCSCF, GVB-CI or Cl) will be a complicated combination of these two descriptions (as well as others, e.g., the atoms-in-molecule picture (10)) or in certain approximate wave functions the descriptions are related by a transformation and are thus in some sense equivalent (10). Hence the best one can do is decide on a criterion to measure the extent to which a particular picture is contained in the general wave function. One possible measure would be the overlap of a unique or unique bent bond description with the general wave function. 

It is common to call tvp the beam-waist radius. In the literature one often finds the phrase at the beam waist, which refers to that value of z for which the function u has its minimum radial extent. For u defined by (8), this occurs at z = 0. The distance Zp is called the confocal distance. When z < Zp we say that the Gaussian beam is in the near field. When z > Zp, the Gaussian beam is in the far field. The majority of this chapter is concerned with the behavior of u in the range 0 < z < Zp, the near-field region. The phase and amplitude of m is a complicated function of position in the near field. When z Zp and p z or when we are in the far field and the paraxial approximation is valid, it is straightforward to show that the asymptotic behavior of u approaches a diverging spherical wave from a point source at z = 0. 

If the Laplace transform of a function /(/) is f s), then f(t) is the inverse Laplace transform of f(s). Although an integral inversion formula can be used to obtain the inverse Laplace transform, in most cases it proves to be too complicated. Instead, a transform table (1), is used to find the image function f f). For more complicated functions, approximate methods are available. In many cases the inverse of a ratio of two polynomials must be... 

In the generalized gradient approximation (GG A) to DFT, the XC potential depends on the electron density p and its gradient Vp and is a complicated function in three-dimensional space. This makes an analytical solution of the XC integrals impossible and numerical quadrature is used to compute the XC matrix elements,... 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

Reference-qnality EOS typically have a complicated functional form, with many parameters. However, software is available [9, 10] that implements these EOS for many flnids for example, the database from NIST [9] incorporates formulations for approximately 80 pnre flnids. A subset of this information is available in the NIST Chemistry Webbook [5]. 

As mentioned in the last section, the XC integrals are in practice evaluated numerically, because of the complicated algebraic forms used for the approximate XC functionals. As with fitting procedures, there have been a great many quadrature schemes which have been proposed and are in use. The areas in which advances are needed in order to treat the quadrature properly may be succinctly categorized as standardization, orientation and differentiation. 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

---