The Structure of a Chemical System

Let us view a chemical system as a network of elementary chemical reactions linked to one another by common reactants. The broad question of what kinds of network configurations are possible for chemical systems can be answered mathematically if it is stated in the following way Given a hypothetical finite list of elementary chemical reactions, determine all the ways the reactions (or a subset of them) can be combined to form a specified overall reaction, or, more generally, to form any member of a specified family of overall reactions. The first stage in a mathematical solution of this problem is to define a chemical system formally. 

The mathematics we shall need is confined to the properties of vector spaces in which the scalar values are real numbers. From a mathematical viewpoint the whole discussion will take place in the context of two vector spaces, an S-dimensional space of chemical mechanisms and a Q-dimen-sional space of chemical reactions, which are related to each other by the fact that each mechanism m is associated with a unique reaction R(m) which it produces. The function R is a transformation of mechanisms to reactions which is linear by virtue of the fact that reactions are additive in a chemical system and that the reaction associated with combined mechanisms mt + m2 is R(m,) + R(m2). All mechanisms are combinations of a simplest kind of mechanism, called a step, which ideally consists of a one-step molecular interaction. Each step produces one of the elementary reactions which form a basis for the space of all reactions. 

For instance, let step s, be a collision process between a3 and a2 to form a3, and let step s2 be an isomerization of a3 to a4. Then R(s,) is a vector -a, - a2 + a3,R(s2)isavector — a3 + a4,andR(Sj + s2) is a vector which equals the sum -a - a2 + a4 of the first two vectors. This illustrates the linearity of R, which may be expressed in general by the linearity equation R(s, + s2) = R(Sj) + R(s2). In particular, if s is repeated a times, the linearity equation becomes R(ers) = erR(s). This principle can be extended to all real values of a, where a is regarded as the rate of reaction, including the possibility of a negative value for a to express the possibility of a reverse reaction. 

For simplicity we speak of a mechanism or a reaction, rather than a mechanism vector or reaction vector. The distinction lies in the fact that a reaction r (or mechanism) is essentially the same whether its rate of advancement is p or a, whereas pr and or are different vectors (for p a). Therefore, a reaction could properly be defined as a one-dimensional vector space which contains all the scalar multiples of a single reaction vector, but the mathematical development is simpler if a reaction is defined as a vector. This leaves open the question of when two reactions, or two mechanisms, are essentially different from a chemical viewpoint, which will be taken up 

Let us begin here with a formalization of some of the ideas which have already been introduced. A chemical system contains species which will be denoted by a, a2,. .., A and elementary reactions among these species which will be denoted by the S vectors in Eqs. (1), 

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The stmcture of a chemical system is the result of restrictions on the possible positions of its atomic nuclei and on the distributions of its electron density in space, due to potentials or other quantum mechanical restraints acting among the nuclei and the electrons the more restrictions there are, the more complex is the structure. The structure of a chemical system is described by a set of nuclear coordinates (jc, y, z), together... 

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