The Current Approximation

As mentioned above in Sect. 6.2, use of the 5- or 6-point current approximation will eliminate the error from this source for all practical purposes that is, use Eq. 4.86 to compute G and Eq. 4.93 to compute Cq from a known concentration profile. The expressions for n = 5 or 6 are not too unwieldy and, in any case, if they are used routinely, it makes sense to write function procedures to evaluate them. In our laboratory we use library functions GOFUNC and COFUNC, with variable n (see Chapt. 9 for the FORTRAN code). 

In this way, the coefficients for any y - ri) can be calculated. Table A.l in Appendix A shows a number of these, as whole numbers mPu where m is the multiplier mentioned above. For each n, the table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. In case the reader wonders why all this is of interest the forms ( ) will be used to approximate the current or, in general, the concentration gradient, in simulations (see the next section) the backward forms y (n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the table will be used for the Kimble and White (high-order) start of the BDF method, aiso described in these chapters. The coefficients have a long history. Collatz [1] derived some of them in 1935 and presents more of them in [2]. Bickley tabulated a number of them in 1941 [3]. The three-point current approximation, essentiaily 34(3) in the present notation, was first used in electrochemistry by Randles [4] (preempted by 2 years by Eyres et al. [5] for heat flow simulations), then by Heinze et al. [6] Newman [7, p. 554] used a five-point current approximation, and schemes of up to seven-point were provided in [8]. 

As shown in Chap. 2, Eq.(2.26), the current in its dimensionless form G is the dimensionless gradient of C with respect to X at X = 0. This implies that a forward difference must be used, as we normally have C-values starting at X = 0. There are algorithms with points at negative X values, but they are not generally very successful or popular. The approximation can therefore be expressed as the n-point approximation 

The symbol H is the interval along the normalised spatial axis. The symbol G will sometimes be used, to mark n, the number of points used. The simplest formula is the two-point form. 

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At each iteration k, the new positions are obtained from the current positions x, the gradient gj. and the current approximation to the inverse Hessian matrix... 

Size-consistency is a desirable feature of any approximate theory. Since we truncate in the operator space (as opposed to the Hilbert space of wavefunc-tions), the current approximation is naturally size-consistent. Consider two widely separated systems X and Y. Then we can construct two bases of... 

While in the first step the deviation from the exact solution stems only from approximating the solution curve by its tangent line, in further steps we calculate the slope at the current approximation y1 instead of the unknown true value (), thereby introducing additional errors. The solution of (5.2) is given by (5.3), and the total error Ei = y(t ) - y for this simple equation is... 

For small departures from E°, the exponential term in Eq. (1.27) may be linearized, with the current approximately proportional to iy ... 

First we consider the current approximation presented in the above two sections. A question left untouched, for example, the equation for the current approximation (3.25) above, is just what terms were dropped when generating a particular form. The order of what was dropped is given in Sect. 3.3, but not extended to actual higher terms. This must be done now. Bieniasz [108] presents a table of these and we can write the first few of these. For this, it is convenient to use a more compact notation for the higher derivatives let... 

The above treatment includes the current approximation on an unequal grid, and the subroutine U DERIV can compute it. It is, however, a little unwieldy, and a simpler interface to it is also mentioned in the same Appendix, function GU, which only requires the three arguments (C, x, n). Similarly, the function CU computes Co from a given concentration profile and a known current,... 

For a small number n of points, it may be worthwhile using the algebraic solutions for the coefficients. The procedure is as described above, but instead of inserting actual hk values into the matrix in (3.48), that matrix is inverted algebraically and the coefficients expressed as a genera] formula. These are given, for a few approximations, both for first and second derivatives (restricted to those that are deemed to be of practical interest) in Appendix A. All the current approximations up to n = 4 are provided there, as well as... 

In this chapter, the current approximation function Q, defined in Chap. 3, (3.25), will be used extensively. Note also that since this function is a linear combination of the array argument (for example, C as in Q(C,n. H)), the function of a weighted sum of two arrays, such as the arrays u and v (to be met later), the following holds (a being some scalar factor) ... 

There are the usual boundary conditions depending on the experiment performed on this system. One possible way to handle all this is simply to write out the whole system as a large linear system, expand that to include the boundary conditions, and solve. This, brute force approach (see below), has in fact been used [138] and can even be reasonably efficient if the number of equations is kept low, by use, for example, of imequal intervals, described in Chap. 7. If the equations in such a system are arranged in the order as above (6.55), it will be found that it is tightly banded, except for the first two rows for the boundary conditions, which may have a number of entries up to the number n used for the current approximation. 

As already described in some detail in Chap. 3, a one-sided first derivative such as the current approximation G can be raised to higher-order by a Hermitian scheme, as introduced by Bieniasz [108], This can then be used both to obtain better current approximations, and also in those cases where G enters a boundary condition. For the simpler case of the current approximation on a concentration grid already calculated, see the relevant Sect. 3.6 in Chap. 3. Here we need to go into some detail on the boundary conditions application. 

Once we have the subroutine U DERIV shown above, it is simple to construct a more convenient function to calculate the current approximation, if that is all we want (that is, if we do not want the coefficients that make it up). The function GU does the job, calling the more complicated U DERIV to do the hard work. 

U DERIV can be used to compute Co, given the current (as in chronopoten-tiometry) and the concentration profile. As for equal intervals, the current approximation formula on n points is adapted, by the function CU. 

For the current approximate configuration part of the solution vector, find the exact solution of the orbital part. This reduces the number of configuration gradients < K MC> [Eq. (96)] that must be calculated. 

As with bacteria, fungi are ubiquitous and have been isolated from every conceivable organic substrate examined regardless of where in the world it was collected. The number of known fungal species is approximately 80000. In 2001, it was estimated that 74000 species had been described in the literature with a further disclosure rate of one thousand per year, this provides the current approximation—which may also be an underestimation.25,26... 

The basic idea in these methods is building up curvature information progressively. At each step of the algorithm, the current approximation to the Hessian (or inverse Hessian, as we shall see) is updated by using new gradient information. The updated matrix itself is not necessarily stored explicitly, as the updating procedure may be defined compactly in terms of a small set of stored vectors. This economizes memory requirements considerably and increases the appeal to large-scale applications. 

At each iteration, the current approximations to the eigenvalues of H are given by the eigenvalues of the small matrix G, which is the Hamiltonian in the subspace spanned by the expansion vectors bj, with matrix elements Gij = (b,Hbj). Likewise, the current approximate eigenvectors are linear combinations of the subspace vectors with coefficients given by the eigenvectors... 

In our implementation we have taken a different approach, for which it will be seen that the above sorts of terms can be treated exactly without posing computational problems. Our approach fits naturally within the iterative method of solution of the CPKS equations, in which it is not the individual elements of A which are calculated, but rather the contraction of these with the current approximate solution vectors. The XC contribution to these quantities may be expressed as... 

C4. Using the quantities IF bJ - IF i l as coefficients and the calculated phases, a density difference map is constructed. This map indicates where adjustments of the current approximation Kj of the theoretical nuclear configuration is required. One potential problem involves the effects of cut-off errors related to limitations of with regard to scattering angles. Another potential problem is that the phases may need adjustments. Nevertheless, the difference density map so obtained is expected to indicate the likely components of the nuclear geometry correction AK- for step D. 

During the iterative solution of the LCAO equations, at each iteration, the diagonalisation of the current approximation to the self-consistent Hartree-Fock matrix generates such a partition of the total function space, i.e. a current (non-self-consistent) set of occupied orbitals and a set of current victuals a current occupied space and a current virtual space. These current spaces share some of the properties of the final self-consistent spaces in particular the current single-determinant is invariant against linear transformations within the current occupied space. 

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