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This latter result is perhaps represented more clearly in Fig. 5.9. The binding energy is a non-monotonic function of the well width and peaks at a relatively small width value. This is a very similar result to that of the more complex case of exciton binding energies which will appear again in 6; in the latter case, such non-monotonic behaviour has been observed experimentally. The non-monotonic behaviour in both cases is merely due to the probability of the electron being within the well. For very narrow wells, the increasing confinement energy, as seen in Fig. 5.8, pushes the electron closer to the top of the well, eventually forcing it to 'spill over the top', thus leading to decreases in the binding energy. Returning to the earlier point, as the well width increases then the binding energy of the donor tends towards its bulk value, as previously hoped. These results show that the theoretical approach is sound and, when implemented computationally, the model assumes simple limits according to sensible physical thinking. Therefore, confidence can be placed in this method. However, this form

[Salomon and Van Hemmen 1996] R Salomon, JL Van Hemmen, Accelerating Backpropagation through Dynamic Self-Adaptation, Neural Networks, Vol 9, No 4, 1996, pp 589 601 [Sanger 1989] T Sanger, Optimal Unsupervised Learning in a Single-Layer Linear Feedforward Neural Network, Neural Networks, Vol 2, 1989, pp 459 473 [Sartori and Antsaklis 1991] MA Sartori, PJ Antsaklis, A Simple Method to Derive Bounds on the Size and to Train Multilayer Neural Networks, IEEE Transactions on Neural Networks, Vol 2, No 4, 1991, pp 467 471 [Schwefel 1974] H-P Schwefel, Adaptive Mechanismen in der biologischen Evolution und ihr Einflufi auf die Evolutionsgeschwindigkeit, Technical Report, Technical University of Berlin, 1974 [Schwefel 1975] H-P Schwefel, Evolutionsstrategie und numerische Optimierung, PhD Thesis, Technical University Berlin, 1975 [Schwefel 1981] H-P Schwefel, Numerical Optimization of Computer Models, Wiley, 1981 [Schwefel 1995] H-P Schwefel, Evolution and Optimum Seeking, Wiley, 1995 [Schittenkopf et al.

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The notions of 'free' and 'bound' occurrences of variables were discussed in Section 11.2.2 in the context of summation. The definitions apply to all quantifications, as do the side conditions on the use of rules. (Briefly, capture or release is forbidden, as is repetition of a variable in a list of dummies.) We do not repeat the side conditions below but trust in the reader's understanding of the concepts.

for the wave function would probably never be used in real calculations. The reason being, that while it appears to be the simplest choice of trial wave function, and although it does appear to follow the expected limits, a three-dimensional trial wave function, as discussed in the next section, supersedes it. Calculations will show that a spherically symmetric wave function gives lower variational energies, which are therefore considered to be better approximations, and in addition, the computational implementation of the mathematics is more efficient than in this, the two-dimensional case.

The dummies (or bound variables) in a quantification serve to relate the range and term. There are three rules governing their use. [Dummy Renaming! ( j:R:T) = (<&k:R[j := k]:T[j := k]) (11.44) (11.45) [Nesting] ( jJs:R*S:T) = ( j:R: $js:S:T)) . (11.46) [Rearranging! ( j,k:R:T) = (@k,j:R:T) .

Figure 5.9 The neutral donor binding energy EDo for a donor at the centre of a well, as a function of the well width

1997] C Schittenkopf, G Deco, W Brauer, Two Strategies to Avoid Overfitting in Feedforward Neural Networks, Neural Networks, Vol 10, No 30, 1997, pp 505 516 [Schoonderwoerd et al 1996] R Schoonderwoerd, O Holland, J Bruten, L Rothkrantz, Ant-Based Load Balancing in Telecommunications Networks, Adaptive Behaviour, Vol 5, 1996, pp 169 207..

The range part is a boolean-valued expression that determines the set of values over which the dummy ranges. There are four rules, two simplification rules and two splitting rules. [EmptyRangel < fc:false:T> = 1 . (11.47) [One-Point! ( k:k = e : T ) = T[k:=e] . (11.48) [Splitting! ( k : P : r > e ( 0 k : Q : r > = < k : P v Q : r > { f c : P A Q : r > .

5.4 THREE-DIMENSIONAL TRIAL WAVE FUNCTION Driven by knowledge of bulk material and the spherical symmetry of the hydrogenic wave function, the next obvious choice of the spatial coordinate r" is given by:

(11.49)

which is just the electron donor separation, defined earlier as r'. Again, the investment in the general formalism developed in Section 5.2 pays dividends in that the full requirements for the calculation merely require an evaluation of the integrals Ij, (j = 1,2,3,4) but with the new form for r". Consider first I1 as originally defined in equation (5.19). Using the new form for r" = r' and switching to plane polar coordinates gives:

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