Spreadsheet numerical approximations

The systematic treatment of equilibrium is a way to deal with all types of chemical equilibria, regardless of their complexity. After setting up general equations, we often introduce specific conditions or judicious approximations that allow simplification. Even simplified calculations are usually very tedious, so we make liberal use of spreadsheets for numerical... 

B2+ 1= 8 + 1 =9, hence we find the result 93 = 729 instead. In B4 the macro then computes 7303 = 3.87E + 08, and from there the size of the results quickly escalates to exceed the numerical capacity of the spreadsheet (approximately 9.9 X 10307) in B8, which is why we see the overflow sign. Therefore one should use For Each... Next loops only on cells containing values that are independent of all other data in the selected block. 

In the previous chapter, the normalization integrals for the exact hydrogenic Is atomic orbitals were determined as test cases for the various numerical integration procedures developed as spreadsheet operations and then the overlap integral between hydrogenic Is orbitals, as exact functions and sto-3g) approximations, was calculated for the case of H2. 

Figure 3.3 Demonstration of the good fit between the Slater 2s radial approximation, when rendered orthogonal to the Is function, and the numerical radial function as output of the Herman-Skillman program. The other details of the spreadsheet are as in Figures 3.1 and 3.2.

I started this manuscript with a quotation from Professor Hartree s famous book on numerical integration. With the development of the methodology of the Hartree-Fock-Slater approximation on a spreadsheet, perhaps to finish, I can be allowed the temerity to quote a well-known comment by one of the giants of quantum mechanics. Professor P.A.M. Dirac (69),... 

Using equation substitution, the fault tree based approximations can easily be solved. Using a spreadsheet tool, the Markov model solutions are obtained using linear algebra. For example, the numerical P matrix is ... 

In general, the accuracy of any numerical method depends on the value of the step size, h. If h is too large, Ay//i will not be an accurate approximation of the derivative, dy/dx. To test the accuracy of the above solution, the procedure was repeated for a smaller value oih,2 min, half of the original value. Now, the interval between t = 0 and t = 40 has been divided into 20 steps instead of the original 10. The results are shown in the following spreadsheet. 

This problem requires numerical differentiation. It is recommended to always apply the second-order correct finite difference formulas. The end-point formulas are applied at z = 0 and z = 2, while centered difference approximations are used at all other points. The results appear in the following spreadsheet, which includes a graph of percent solids volume versus axial position. 

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