Symbolic Mathematics Programs

As an example of a symbolic calculation. Fig. 5 displays a Mathcad solution of one 4X4 block of the secular determinant of a Hiickel molecular orbital calculation done in Exp. 41 for orf/ro-benzosemiquinone (compare with Table 41-2). Mathcad is a software package for numerical analysis but also makes use of a subset of the symbohc routines of Maple. The algebraic expansion of the determinant is generated and solved with two 

if orJy numerical values are needed, get energies and waveiiinction coefTicients from 

As one other example illustrating the power of symbolic mathematics programs. Fig. 6 shows a Mathematica calculation of Franck-Condon factors for the I2 B — X electronic absorption spectrum studied in Exp. 39. These factors are the squares of the overlap integrals of the vibrational wavefunctions for the lower (v ) and upper (v ) vibrational levels involved in a transition  

The absorption spectrum consists of sequences of transitions from v = 0, 1, 2 to various v levels in the upper state, and the relative intensities of the vibration-rotation bands are given primarily by the product of the FCF value and a Boltzmann term, which can be taken to be exp — hcv v /kT). Common choices for the i/r s are harmonic oscillator and Morse wavefunctions, whose mathematical form can be found in Refs. 7 and 9 and in other books on quantum mechanics. The harmonic oscillator wavefunctions are defined in terms of the Hermite functions, while the Morse counterparts are usually written in terms of hypergeometric or Laguerre functions. All three types of functions are polynomial series defined with a single statement in Mathematica, and they can be easily manipulated even though they become quite complicated for higher v values. 

In Fig. 6, section 1 defines the constants for the I2 molecule and calculates parameters to make the arguments of the wavefunctions unitless. One need only change these numbers to perform the calculations for some other molecule, such as N2 or Br2. Section 

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More complex enzymatic reactions usually display Michaelis-Menten kinetics and can be described by Eq. (2). However, the forms of constants Km and Vm can be very complicated, consisting of many individual rate constants. King and Altman (7) have provided a method to readily derive the steady-state equations for enzymatic reactions, including the forms that describe Km and Vm. The advent of symbolic mathematics programs makes the implementation of these methods routine, even for very complex reaction schemes. The P450 catalytic cycle (Fig. 2) is an example of a very complicated reaction scheme. However, most P450-mediated reactions display standard hyperbolic saturation kinetics. Therefore, although the rate constants that determine Km and Vm are... 

Descriptions of symbolic mathematics programs and some of their applications can be found, for example, in W. H. Cropper, Mathematica Computer Programs for Physical Chemistry, Springer-Verlag, New York (1998) and in J. F. Ogilvie, Mathematics for Chemistry with Symbolic Computation, an electronic text containing Maple worksheets, available at http //www.maplesoft.com. 

As we saw in Sect. 6.1, the input/output operators are the special operators. The mechanisms for working with symbol mathematics programming instruments are called operators too (buttons — and — on the toolbar Evaluation in Fig. 6.39). 

Mathematica is a complete mathematics package that can carry out both numerical and symbolic mathematics. Before we discuss the solution of equations using Mathematica, we provide an elementary introduction to the program. When you open Mathematica, a blank untitled window appears on the video screen. This window is called a notebook. Mathematica is now ready to accept instructions. 

Complicated algebraic expressions are best handled nowadays using symbolic math programs such as Mathematical, Maple , or Mathcad . Cancellation is a wonderful way to simplify formulas. Consider... 

ABSTRACT. This paper describes a methodology for the sensitivity analysis and optimization of planar constrained mechanical systems. Direct differentiation methods and finite difference techniques have been used in the design sensitivity calculations. The optimization process is developed within the framewoik of mathematical programming techniques. The sensitivity equations were constructed symbolically and subsequently integrated in the dynamic analysis equations of motion and solved simultaneously. 

In standard high level language programming the dimension of the NSS n, signals the number of nested do loops which are necessary to reproduce the structure in a computational environment. But the mathematical usefulness of this entity can be easily recognized when the particular characteristic of this symbolic unit is analyzed the involved vector parameters could be chosen with arbitrary and variable dimensions. There are many scientific and mathematical formulae which will benefit of this property, when written in a paper or computationally implemented. 

The systematic application of artificial intelhgence methods for the programmed manipulation of symbols in accordance with mathematical rules. Symbohc computing allows one to obtain symbolic solutions to equilibrium... 

KEY is written in the "C" programming language because of all of the mathematical and pointer manipulations that need to take place. An excellent symbolic debugger for PC-based "C" is available. However, the finished system is easily ported back to the VAX environment. 

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