Mathematical Challenges from Theoretical/Computational Chemistry
BOX 4.2 Tutorial on Statistical Mechanics and the Importance of Minima and Saddlepoints in Condensed Matter Systems
A large part of computational chemistry is concerned with the properties of systems at or near thermal equilibrium. The statistics of configurations at thermal equilibrium therefore dominate many of the que
stions studied by chemists. The principles of the statistical mechanics of equilibrium systems are quite simple to state but are profound and sometimes surprising in their results.
A fundamental postulate of equilibrium statistical mechanics is that in the long run, all states of an isolated system that are consistent with conservation of energy will be observed with equal probability. The thermodynamic quantit
y S, the entropy, is simply a measure of the number 
of these equally probable states that the system might access, S = kB log . Here kB is Boltzmann's constant. Thus, combinatorial and various counting problems play a special role in our thinking about the thermodynamics of chemical systems. Although each of the states of an isolated system is equally pro
bable, this is not the case when we consider only a part of a system. When only part of a system is being examined, we must ask the question, How many states of the entire system are accessible when a subsystem is in a given configurat
ion? The answer is given by
where X refers to the specified subsystem and the entropy refers to the subsystem's environment.
A very interesting and powerful special case of this formula is used constantly in equilibrium statistical mechanics. If the subsystem considered is only weakly coupled to the rest of a much larger system, we can decompose the tota
l energy of the entire system into parts:
The energetic coupling is small and can be neglected if we are considering a system that is itself fairly large and therefore has a relatively small surface of interaction with the remainder of the system. In this case the counting problem can
be solved since we know that the entropy of the environment is a smooth function of its total energy. This then gives a count of states expressed by
The probability then of an exactly specified state of a subsystem that is part of a larger one is proportional to this number of states. It is given by the Boltzmann distribution law
The temperature entering here is the thermodynamic derivative of the entropy and is proportional to the average kinetic energy of each particle in the system. This distribution law contains within it many of the great pheno
mena of chemistry and physics. First we see that the most important states are those that have the lowest energy. If the energy then is a continuous function of the coordinates of part of the system, the most probable configurations are
those that give the minima of this potential. Indeed, the coefficient 1/kBT in the Boltzmann distribution law ensures that at the lowest temperatures only the deepest or global minima are important.
Chemistry is usually a low-temperature phenomenon-most chemical reactions are studied around room temperature, although, of course, many do occur under greater extremes of conditions-and room temperature corresponds to only one-fo
rtieth of the typical energy scale of chemical bonds. Thus, the Boltzmann distribution law tells us that chemistry will mostly concern itself with the specific configurations that minimize the energy.
Of course, if molecular systems remained entirely at their energy minima, little would go on. Occasionally, a molecular system must make a transition between one minimum on the energy surface and another. To do this, the system must occ
asionally find itself in an intermediate high-energy configuration, which the Boltzmann distribution law tells us is rather improbable. If we ask which of the relatively improbable intermediate states between two minima are the most
probable, it is clear that these should correspond to saddlepoints of the energy. These saddlepoint configurations are known as transition states to chemists. The probability of a system being found at a transition state determines the
rate of a chemical transformation. We see, therefore, that the geometry of minima and saddlepoints of potential energy surfaces is extremely important in determining the chemical properties of a molecular system.
Sometimes only certain aspects of a system are considered explicitly. For example, when we study the shapes, structures, and motions of a biological molecule (e.g., a protein immersed in water), we are interested only secondarily i
n the configurations of the water molecules around this macromolecule.
A special case of these geometrical problems arises when the subsystem being considered is itself rather large and involves strong interactions between its molecular subunits. In this case, it sometimes happens that the minimum-energy
saddlepoint actually possesses an extremely high energy. We then say that the transformation between two minima has a large barrier and the transformation will be extremely slow. Sometimes as the subsystem studied grows larger and la
rger, the transformation barrier itself also grows larger and larger. Thus, for a macroscopic system, certain transformations may actually take place effectively only on infinite time scales. We can then treat each part of the co
nfiguration space very nearly as separate regions. This situation arises when a phase transition occurs. The theory of phase transitions is then concerned with the problem of how a many-dimensional configurational space gets f
ragmented into parts that are separated by very high energy barriers.
The Boltzmann distribution law applies only to a completely specified subsystem that is interacting weakly with its environment. The biological macromolecule is interacting strongly with its solvent environment, and so the Boltzmann
distribution law using the energy alone is inappropriate for describing its configurations. On the other hand, for different configurations of the biomolecule, we can in principle compute the number of configurations of the surrounding
solvent that are compatible with that configuration of the biomolecule. Thus, the probability of a particular configuration of the biomolecule would have the form
where the probability has been rewritten in a Boltzmann-like form in which the energy of the molecular system is combined with the entropy in its environment to form a free energy F(x) = E(X) - TS(X), which gives the probability of the subsy
stem's configuration. For this reason, the geometry of free energy surfaces is often also of great interest to chemists and physicists.
Occasionally the distinction between energies and free energies is blurred in offhand writing by chemists and physicists, and the uninitiated reader must be careful about these distinctions when applications are made.
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