Essentials of Computational Chemistry

3
Simulations of Molecular Ensembles
3.1 Relationship Between MM Optima and Real Systems
As noted in the last chapter within the context of comparing theory to experiment, a
minimum-energy structure, i.e., a local minimum on a PES, is sometimes afforded more
importance than it deserves. Zero-point vibrational effects dictate that, even at 0 K, the
molecule probabilistically samples a range of different structures. If the molecule is quite
small and is characterized by fairly ‘stiff’ molecular coordinates, then its ‘well’ on the PES
will be ‘narrow’ and ‘deep’ and the range of structures it samples will all be fairly close
to the minimum-energy structure; in such an instance it is not unreasonable to adopt the
simple approach of thinking about the ‘structure’ of the molecule as being the minimum
energy geometry. However, consider the case of a large molecule characterized by many
‘loose’ molecular coordinates, say polyethyleneglycol, (PEG,–(OCH2CH2)n –), which has
‘soft’ torsional modes: What is the structure of a PEG molecule having n = 50? Such a
query is, in some sense, ill defined. Because the probability distribution of possible structures
is not compactly localized, as is the case for stiff molecules, the very concept of
structure as a time-independent property is called into question. Instead, we have to accept
the flexibility of PEG as an intrinsic characteristic of the molecule, and any attempt to understand
its other properties must account for its structureless nature. Note that polypeptides,
polynucleotides, and polysaccharides all are also large molecules characterized by having
many loose degrees of freedom. While nature has tended to select for particular examples
of these molecules that are less flexible than PEG, nevertheless their utility as biomolecules
sometimes derives from their ability to sample a wide range of structures under physiological
conditions, and attempts to understand their chemical behavior must address this issue.
Just as zero-point vibration introduces probabilistic weightings to single-molecule structures,
so too thermodynamics dictates that, given a large collection of molecules, probabilistic
distributions of structures will be found about different local minima on the PES at non-zero
absolute temperatures. The relative probability of clustering about any given minimum is a
function of the temperature and some particular thermodynamic variable characterizing the
system (e.g., Helmholtz free energy), that variable depending on what experimental conditions
are being held constant (e.g., temperature and volume). Those variables being held
constant define the ‘ensemble’.
Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer
© 2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)