2.7 Feinberg, Horiuti and Wegscheider Criterion
2.7.1 The Feinberg Deficiency Theorems and Stability
In the physical sciences, many natural and man-made systems exhibit instabilities,
in particular, dampened, indefinitely periodic and undampened oscillations. Such
oscillations are found in mechanical systems, electrical systems, biological systems
and indeed chemical systems. Perhaps the most well-known chemical oscillations
are those associated with the liquid phase redox systems, the Belousov–
Zhabotinsky system which is cerium catalysed and the Briggs–Rauscher system
which is manganese catalysed and their related chemistries [81], which, in addition
to showing temporal oscillations, may show spatio-temporal oscillations as well in
both 3D and 2D environments. Most of the conceptual foundations for this area can
be traced back to the mathematician and computer scientist Alan Turing and his
seminal paper in which he proposed the existence of all these classes of chemical
oscillations [82].
Given the non-linear structure of the CBER mechanisms [M] CBER and
[M] CBER+UNI and given the pronounced rate enhancements that can take place,
their stability should be questioned. Said another way, is it possible that such
mechanisms contribute to product formation from repeated fast and slow rate
intervals, or is it possible that such mechanisms contribute to product formation
for a brief time but then decay?
Over a three-decade period, Feinberg [83] developed the criteria for answering
chemical network stability questions, given any reaction with any given number of
reactants. The basis for the so-called deficiency theorems is rooted in topology
(structure), but in short summary, one takes each reaction with each set of inputs
and outputs and defines new quantities called complexes. After summing over all
individual reactions, all reactants and all complexes, one obtains the criterion.
Our group has applied the Feinberg deficiency criteria for quadratic, bilinear,
linear–quadratic and linear–bilinear CBER systems for both realized and hypothesized hydroformylations and hydrogenations with varying degrees of complexity/
selectivity. As far as we can tell, there is no fundamental reason that CBER systems
should be unstable. The important converse conclusion is that CBER systems are
synthetically useful in the generalized case since they are kinetically stable.
2.7.2 Enumeration of Synthetic Pathways: The Horiuti Criteria
For symmetric substrates in hydroformylation, the non-linear structure of the CBER
mechanisms [M] CBER and [M] CBER+UNI are visually, diagrammatically and conceptually rather straightforward to follow. When considerations of selectivity are
included, be these considerations regio-, chemo- and stereoselective or a combination thereof, issues rapidly become very complex. A question that one may ask is
this: given a CBER mechanism with simultaneous regio-, chemo- and stereoThe Catalytic Binuclear Elimination Reaction: Importance of Non-linear. . .
211
2.7.1 The Feinberg Deficiency Theorems and Stability
In the physical sciences, many natural and man-made systems exhibit instabilities,
in particular, dampened, indefinitely periodic and undampened oscillations. Such
oscillations are found in mechanical systems, electrical systems, biological systems
and indeed chemical systems. Perhaps the most well-known chemical oscillations
are those associated with the liquid phase redox systems, the Belousov–
Zhabotinsky system which is cerium catalysed and the Briggs–Rauscher system
which is manganese catalysed and their related chemistries [81], which, in addition
to showing temporal oscillations, may show spatio-temporal oscillations as well in
both 3D and 2D environments. Most of the conceptual foundations for this area can
be traced back to the mathematician and computer scientist Alan Turing and his
seminal paper in which he proposed the existence of all these classes of chemical
oscillations [82].
Given the non-linear structure of the CBER mechanisms [M] CBER and
[M] CBER+UNI and given the pronounced rate enhancements that can take place,
their stability should be questioned. Said another way, is it possible that such
mechanisms contribute to product formation from repeated fast and slow rate
intervals, or is it possible that such mechanisms contribute to product formation
for a brief time but then decay?
Over a three-decade period, Feinberg [83] developed the criteria for answering
chemical network stability questions, given any reaction with any given number of
reactants. The basis for the so-called deficiency theorems is rooted in topology
(structure), but in short summary, one takes each reaction with each set of inputs
and outputs and defines new quantities called complexes. After summing over all
individual reactions, all reactants and all complexes, one obtains the criterion.
Our group has applied the Feinberg deficiency criteria for quadratic, bilinear,
linear–quadratic and linear–bilinear CBER systems for both realized and hypothesized hydroformylations and hydrogenations with varying degrees of complexity/
selectivity. As far as we can tell, there is no fundamental reason that CBER systems
should be unstable. The important converse conclusion is that CBER systems are
synthetically useful in the generalized case since they are kinetically stable.
2.7.2 Enumeration of Synthetic Pathways: The Horiuti Criteria
For symmetric substrates in hydroformylation, the non-linear structure of the CBER
mechanisms [M] CBER and [M] CBER+UNI are visually, diagrammatically and conceptually rather straightforward to follow. When considerations of selectivity are
included, be these considerations regio-, chemo- and stereoselective or a combination thereof, issues rapidly become very complex. A question that one may ask is
this: given a CBER mechanism with simultaneous regio-, chemo- and stereoThe Catalytic Binuclear Elimination Reaction: Importance of Non-linear. . .
211