The Error Function

This function occurs often in probability theory, and diffusion of heat and mass, and is given the symbolism  

Dummy variables within the integrand are used to forestall possible errors in differentiating erf(jc). Thus, the Leibnitz formula for differentiating under the integral sign is written 

Fundamentally, erf(jc) is simply the area under the curve of exp(- ) between values of = 0 and = x thus it depends only on the value of x selected. The normalizing factor 2/ yfir is introduced to ensure the function takes a value of unity as - 00 (see Problem 4.10) 

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Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffiisivity is constant, the solution is the error function, a tabulated function. 

The only remaining question is the nature of the error function. Pulay suggested the difference FDS — SDF (S is the overlap matrix), which is related to the gradient of the SCF energy with respect to the MO coefficients. This has been found to work well in practice. 

The error function erfx has a power series expansion for small x and an asymptotic expansion for large x... 

By performing the derivative of the error function, one finds easily that the optimal position ZG of the FSGO necessarily satisfies the following equation ... 

Here erfc( ) is the error function complement, a mathematical function of the argument u given by 1 - erf( ), where erf( ) is the error or Euler-Laplace integral. This integral in turn is defined by the expression... 

The variable y in the expression under the integral sign is an auxiliary variable the value of the integral depends only on the limits of integration (i.e., on the value of u). The numerical values of the error function vary from zero for m = 0 to an upper limit of unity for m —(this value is practically attained already for u 2). Plots of functions erf(n) and erfc(n) are shown in Fig. 11.2. 

The error function E explicitly depends on the set of coefficients aj, j 6 / that parameterize the vector field and has the major advantage of having a unique minimum of zero in the ideal Porod-Kratky case. 

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

The error function complement erfcy is defined by the relationship erfcy = 1 — erf y. The concentration at the reference plane c is... 

There are different ways to define the error function to be minimized. A few possibilities are as follows ... 

The integral in Eq. (10) is the usual definition of the error function. A closely related function is the complementary error function... 

The error function cannot be evaluated analytically, although it is readily available in the form of tables and evaluated in many computer programs. 

Central to the development of new semi-empirical parameter sets is the construction and subsequent minimisation of an appropriate error function, S (Eq. 5-2). The error function contains the molecular quantities calculated at the semi-empirical level... 

Choice of Reference Data and Construction of the Error Function... 

In view of the fact that recent parameterisations make use of reference data from high-level calculations, the corresponding error functions used to develop these methods can in principle involve any given property that can be calculated. Thus, in addition to structural information, the error function can involve atomic charges and spin densities, the value for the wavefunction, ionisation potentials and the relative energies of different structures within the reference database [26, 32], Detailed information concerning the actual wavefunction can be extremely useful for... 

The error function is tabulated in most handbooks of mathematical tables. It is useful to note that erf( —x) = — erf(x) and that... 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

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