We have emphasized
the unique model building power of contemporary computers. How can we exploit
this power to learn about the behavior of atoms and molecules?
Two broad strategies are available to contemporary scientists. In the first,
we attempt to solve the Schrödinger equation  to learn in
detail about the distribution and energies of the electrons in the molecule;
and by calculating the energy of the molecule for a variety of geometries,
we can try to predict the lowest energy structure. Remember
that the Schrödinger equation can be solved exactly only for the hydrogen
(and other one electron) atoms. And there exist a whole range of approximate
methods for solving the equation for more complex systems. The best
of these can now achieve highly
accurate descriptions for molecules (and solids) of increasing size and
complexity.
Here we see a molecule produced in nature, caffeine, the image below
shows the calculated electron density for the important mineral
MgSiO_{3}. In both cases the computer
provides a detailed
understanding of the distribution of electrons which govern the
molecule's or material's properties.
The electron density of a caffeine molecule.
Electron density in MgSiO_{3}.
Indeed, calculations on molecules such as caffeine and
crystals such as MgSiO_{3} are
playing an increasingly
important role in understanding the behavior of the
nature's molecules and the materials from which the earth is constructed.
These methods yield an enormous
wealth of information. They allow us to understand how electron density
is redistributed when atoms form molecules  the process which, as we have
seen, is at the very heart of chemical bonding. In addition, from our
knowledge of the density of electrons we can calculate the electrostatic
potentials around molecules  a crucial quantity which controls the way
in which the molecule interacts with other surrounding molecules. The image
on the left shows the electrostatic potential on a plane through
a water molecule (left) and methanol molecules (right).
The electrostatic potential about H_{2}O and
CH_{3}OH.
With the continuing growth
of computer power, calculations of the distribution of electrons in molecules
and hence of molecular structures and properties, will extend to increasingly
large and complex molecules and solids. However, even with the largest,
most sophisticated computers available today (and in the foreseeable future),
such methods will be limited as the amount of computer time and memory
increases rapidly with the size of the system studied. But for many
of the problems which are posed by highly complex molecules or materials,
we can use alternative simpler procedures, based on an old concept in chemistry
and physics known as the interatomic potential. We can understand
this simple idea by reference to the diagram on the left showing schematically
how the energies of a pair of atoms vary with their distance apart (more
accurately, their internuclear spacing). When the two atoms are close to
each other,
their nuclei will repel strongly, as will the 'core' electrons. The energy
will therefore begin to increase very rapidly as seen in the diagram. Conversely,
when they are a long way apart they will attract each other, due to chemical
bonding or the weak 'nonbonding' interactions. The variation of energy
with distance will therefore be expected to have a minimum as shown in
the diagram.
Interatomic potential schematic. The separation of two atoms is
shown on the x axis, the potential energy of the atom pair is
plotted on the y axis.
Over the last fifty or so
years, scientists have built up detailed information on such interatomic
potentials for a wide variety of atom pairs. Moreover, the concept can
be extended to larger numbers of atoms. We can learn how the energy of
not just two, but three or four or larger numbers of atoms depend on their
relative positions (although it may be more difficult to represent these
diagramatically). The source of this information is first experiment: the
properties of molecules and larger aggregates of atoms depend on the ways
in which the atoms interact; and the experimental data may therefore be
inverted to yield interatomic potentials. But of course direct calculations
of the type described above are, as we have argued, to a growing extent
able to give accurate information on the ways in which atoms interact.
They are therefore an increasingly reliable source of high quality interatomic
potentials.
A data base of good accurate
interatomic potentials is a rich repository of information on matter at
the atomic level. And the computer is well suited to the processing
of this information content into detailed models of the behavior of molecules
and materials. A variety of computational approaches may be used. The simplest
and most economical (but very powerful) procedure is to exploit a fundamental
principle of nature, that is that systems tend to run down hill. We will
discuss in more detail the factors which control the
direction in which systems evolve. But if we want to predict the structure
of a molecule or a solid, it is normally a valid procedure to search for
the lowest energy arrangement of atoms.
This simple approach  often
referred to as energy minimization (or molecular mechanics when
used specifically for molecules) is illustrated diagramatically on the left.
We must, of course, start by guessing a structure for our molecule
 the starting point, S, and we will consider later
how these points are defined. But once the initial configuration of atoms
has been defined, the procedure is simple. The atoms are systematically
moved around in such away that the molecule is driven 'down hill' in energy.
In practice, this 'minimization' proceeds by a succession of moves or 'iterations',
each of which involves a small displacement of some or all the atoms; the
energy of the system is calculated from the data base of potentials, as
are normally the forces on each atom which guide the direction and magnitudes
of the atomic displacements. The calculation ends when a minimum is located;
that is a point is reached (like M on the left) which represents the
lowest energy, and any displacement from this point will raise the energy.
This simple but powerful
method can work remarkably well. On the left we show trajectories of
in the first case a simple peptide and in the other a crystal structure
that have been successfully energy minimized. In both cases, the minimized
structure is an accurate model of reality; and in each the 'starting point'
is remote from the final minimum.
Energy minimization and its effect on the conformation of a peptide molecule.
Adjusting the calculated structure of a benzene crystal structure to match a powder diffraction pattern.
Energy minimization is now
a standard tool of the computational chemist. But the technique is, however,
limited. There are fundamental limitations arising from the whole
basis of the concept: the energy minimum is often a good guide to the structure
of a molecule or crystal; but the science of thermodynamics
tells us that other factors must be considered
in a detailed theory of the stability of matter at the atomic level. A
second (and related) limitation arises from the use of purely 'static'
models. No account is taken of the fact that
atoms are in constant motion; and the dynamical behavior of molecules
influence their structures and stabilities. Then there may be serious practical
limitations, of which possibly the most important is illustrated in above
in the energy surface schematic.
Here we illustrate the course of a minimization which starts from
'S' and ends at the point 'ML'. This is indeed a minimum, but it is not
the lowest one; for if we run 'up hill' a little and pass over barrier
B, we will move down into the lower minimum, 'M'. The minimum 'ML' is known
as a 'local minimum'. And calculations starting from any point in the valley
around this minimum will end up at this point. In complicated molecules
(like those in biological systems) or solids, there may be many hundreds
of local minima, and the task of finding the lowest energy or 'global'
minimum becomes increasingly difficult. many different starting points
must be sampled, and even then there is no guarantee that the global minimum
will have been located. Minimization, therefore, although widely used by
the computational scientist, is often only the first stage of a computational
study of matter at the atomic level.
The next level of sophistication
remedies one of the major deficiencies of minimization calculations. It
represents the dynamic nature of matter at the atomic level.
The conceptual basis of the
molecular dynamics technique is again simple. In the system to be
simulated  for example, the molecules, cluster of molecules or crystal
 we assign all atoms not just positions but velocities; and the latter
are chosen with a target temperature in mind. We then allow the system
to evolve in time subject to the forces acting on all the atoms (which
can be calculated using the interatomic potentials). Normally, 'classical'
mechanics is used; that is, the equations of motion first formulated by
Newton are applied to the dynamics
of atoms. This classical approach works surprisingly well for all but the
lightest of atoms; and more sophisticated 'quantum' methods are available
when, for example, the dynamics of hydrogen atoms are modeled.
Let us look, however, in
a little more detail at the way in which simulations work. In many ways
they work like a movie: they consist of a succession of snapshots closely
spaced (in time) which when run together create a representation of continuous
motion, as shown schematically on the left, where we consider four atoms:
each has a position and a velocity at the 'first frame'.
Schematic illustration showing molecular dynamics time steps. (Click
on the image to see the four stages illustrated in greater detail). The
last illustration on this page shows snapshots from a molecular
dynamics simulation of liquid water. This animation is quite large and
may take a little while to download to your browser.
We now allow time
to move forward a small amount (Dt)
to the second frame. Since we know
how fast the atoms are moving, it is straightforward to work out their
new positions in the new frame; they will simply have moved by their velocity
times Dt.
But we also know the forces acting on the atoms, which we can
work out from our knowledge of the interatomic potentials. And if we know
the force, Newton's celebrated Second Law of motion
tells us we can work out accelerations  that is the extent to which the
velocities are changing with time  so we can calculate the new velocities
in the new frame. We then move onto the next frame and the next; and in
a real simulation tens, if not hundreds, of thousands of frames will be
used in creating a dynamical record of the system.
The choice of the time step
(Dt) is, of course, crucial. If it is too long, it
will give a simulation
which is like an old style 'jumpy' movie. And indeed the criterion for
the choice of 'time step' (Dt) is very similar
to that used in choosing
the time lapse between frames in movies. It must be short compared with
the time of any important process in the system: in movies this may be
the time to kick a ball or to pull and fire a gun. In the molecular world, it
is the time taken by a molecule to vibrate or rotate. The time taken to kick
a ball is roughly a second or so; and the time between frames in movies
is a hundredth to a thousandth of a second. Molecules have different time
scales: they vibrate and rotate in periods of a million millionths
(10=2>^{12})
seconds. The time steps must be a hundred or a thousand times smaller than
this (i.e. 10^{15} seconds). So if we
collect a million time frames (roughly
the limit of current calculations), we can watch how our simulated system
evolves in a thousand millionths (10^{9})
seconds. But a lot can happen in molecules and materials during this period!
We show
three typical dynamics trajectories below:
the first relating to the biological molecule
oxytocin.
The second shows the trajectories of ions moving in a solid electrolyte
at high temperatures. The migrating ions carry electric charge and hence
this material is a good conductor of electricity  a fact which has stimulated
interest in its use in advanced batteries. Finally we show an
animation of successive frames from a simulation of water which shows
that the position of water molecules rapidly change in liquid water.
A molecular dynamics trajectory calculated for the hormone oxytocin.
A molecular dynamics trajectory showing the tracks of lithium
ions in a conducting polymer.
Snapshots from a simulation of liquid water. One of the molecules is
shown in yellow to highlight its trajectory through the course of
the simulation.
Molecular dynamics has proved
to be an enormously productive technique for the computational chemist,
physicist and biochemist. The method has been applied to an extraordinary
variety of systems  proteins, pharmaceuticals, industrial polymers, complex
crystals, solids, glassy materials and a huge range of liquids. Simulations
using this method yield a rich range of information on both structures
and dynamics of the system simulated. Indeed, molecular dynamics has often
been referred to as a 'molecular microscope' which yields details on the
behavior of matter at the atomic level which are inaccessible from experiment.
Again, however, the method is limited. We have already discussed the time
scale of the simulations  at most 10^{9}
seconds even with the biggest, most powerful modern computers. So the processes
we are interested in must take place within this time. And although this
is long on the molecular time scale, important events may often not be
sampled in this period. Two typical examples relate firstly to the 'relaxation'
and reorientation processes in polymers, which exert a crucial influence
on the dynamics of these systems and which may have time scales in the
range 10^{12} to
10^{1} seconds; and secondly to diffusion
in solids (which although
in some cases relatively fast as shown on the left) is
normally slow, taking place by infrequent 'hops' of atoms between different
sites in the solid. For the size of system simulated (which is discussed
in greater detail below) few, if any, such jumps will take place over the
time scale of a molecular dynamics simulation. So phenomena like corrosion,
which depend on slow diffusion processes in solids, cannot be usefully
investigated by this technique; although it turns out that simpler methods,
akin to the energy minimization approach, but in which we chart the change
in energy of a migrating atom as it moves between two sites, are
effective for probing atomic dynamics in these systems.
A related but distinct point
concerns the size of the simulated system. The simplest of statistical
considerations suggests that the larger the number of atoms in the simulation,
the greater will be the probability of observing events (such as the hops
of atoms between sites as discussed above). Modern simulations are performed
typically on thousands of atoms (although calculations on millions of atoms
are becoming increasingly common). In simulating solids, which extend indefinitely
in three dimensions, an ingenious strategy is used. The group of atoms
being simulated (often referred to as the simulation box or cell) is surrounded
by images of itself that extend to infinity as illustrated schematically
(in two dimensions) on the left. A finite group of particles can therefore
be made to represent an infinite system. In the case of crystals
periodicity that this procedure imposes  the content
of all the boxes are identical  may correspond to reality since, as we
will see, the defining feature of crystals at the atomic level is that
the arrangement of atoms is periodic. For liquids and glasses this
periodicity is artificial; but the procedure is nevertheless
useful.
Other computational methods
will be discussed at appropriate points later on. We hope,
however, that the discussion above has shown how the computer is a uniquely
powerful tool in constructing models for matter at the atomic level. The
horizons of the field of computational studies of matter have expanded
rapidly in the last ten years. Developments in both hardware and software
have played their role in this spectacular growth of the subject.
We have concentrated
on the model building power of computers at the microscopic level.
Their unique capabilities are also being exploited
to model engineering, global and cosmic systems.
We can extend our understanding of matter at the atomic level by exploring
the properties and behavior of gigantic numbers of atoms and molecules.
