The virtual matter laboratory
M. J. G ILLAN
New types of computer simulation are being used to investigate matter on the atomic scale.
Unlike older simulation methods, the new techniques represent a solid or a liquid as a
collection of nuclei and electrons, and the laws of quantum mechanics are used to calculate
the energy and the forces from ® rst principles. The most successful method so far for doing
® rst-principles simulation is based on density functional theory and the pseudopotential
approximation. I describe the main ideas of these techniques and explain how they can be
used to perform simulations in which chemical bonds are made and broken. I give
illustrations of current work in two areas: the atomic-scale behaviour of water and other
liquids; and molecular processes at surfaces. I stress that the new methods are still evolving
rapidly and I point to likely developments in the next few years.
1. Introduction
In the last few years, more and more condensed-matter
scientists have become gripped by a powerful idea. The idea
is to use the fundamental laws of quantum mechanics to
produce highly realistic simulations of solids and liquids on
the atomic scale. In this article, I will explain how these new
simulations work and I will give some examples of the rapid
progress they are bringing to the understanding of
condensed matter.
We all have an image of the world in our mind. We
consult this mental image when we want to interpret the
world, or when making plans to change it. But our image is
too unreliable and approximate. We construct maps,
models and other artefacts to make it more precise. The
ideal would be to have a faithful working model of the
world which would tell us what would happen in any
situation Ð a simulation. Powerful computers are making
simulation a familiar concept. The aerodynamic perfor-
mance of a new aircraft design, the response of a building
to an earthquake, the behaviour of a car in a high-speed
collision: these and many other problems are now studied
by simulation, in the con® dence that what is seen on the
computer screen is a true picture of the real world. This
con® dence rests on one key thing Ð the laws of physics. A
knowledge of these laws makes simulation possible.
We know the laws governing atomic-scale matter. These
are the laws of quantum mechanics obeyed by the nuclei
and electrons of which matter is made. It should therefore
be possible to make a precise working model of any
material. It has taken many years to develop the techniques
to do this. We have to simulate systems of many atoms and
each atom may contain many electrons. Since we are doing
quantum mechanics, each electron must be represented by a
wave. The electrons interact with each other and the
motion of one aŒects the motion of all the others. There are
tremendous di� culties.
When people ® rst tried to simulate matter (Alder and
Wainwright 1957, 1959, Gibson et al. 1960), no-one
thought of modelling the behaviour of the electrons Ð it
just seemed too di� cult. Instead, the atoms were treated as
rigid objects interacting with each other. Their interaction
was usually modelled by some empirical potential function,
representing the energy of two atoms as a function of their
distance apart. A commonly used interaction model was the
Lennard-Jones potential shown in ® gure 1. With such an
empirical interaction, the computer could be used to ® nd
the equilibrium structure of a system of atoms; alterna-
tively, dynamical simulations could be performed, with all
the atoms moving according to Newton’ s equation of
motion. Since the interactions between the atoms were
represented by simple pair potentials, only very limited
kinds of system could be studied and much of this early
work was done on the rare-gas elements, particularly argon
(Rahman 1964, Verlet 1967). These elements have closed
electronic shells and the electrons are tightly bound to the
nuclei, so that a simple pair interaction is quite reasonable.
In spite of this limitation to simple systems, the early
simulations made a great impact. They represented the
properties of simple liquids and solids quite well. Later,
people found that many other kinds of materials, particu-
larly ionic and molecular materials (Woodcock and SingerAuthor’ s address : P hysics D epartm ent, K eele U niversity , K eele,
Staffordshire ST5 5BG, UK .
0010-7514/97 $12.00 Ó 1997 Taylor & Francis Ltd
Contemporary Physics, 1997, volume 38, number 2, pages 115 ± 130
1971, Cheung and Powles 1975) , could be tackled in the
same way, and simulation grew into one of the major ways
of investigating condensed matter (Allen and Tildesley
1987). But still, simulation was limited by its rather
empirical approach.
In the 1980s radical new ideas emerged and these have
completely changed the simulation of matter. The main
new idea was that the electrons should be included
explicitly in the simulations, so that the system would be
represented as a collection of nuclei and electrons. The
fundamental equations of quantum mechanics would be
used to calculate the energy of the system and the forces on
the nuclei, and the atoms would then move under the action
of these forces. The ® rst attempt to do this was reported by
Car and Parrinello (1985 ) and this paper has had a major
in¯ uence on simulation during the past ten years. The new
ideas changed the approach to simulation for two reasons.
First, they greatly expanded the range of materials and
problems that could be simulated. It was no longer
necessary to stick to closed-shell systems, because now all
kinds of ionic, covalent and metallic bonding could be
handled in the same way, and even the making and
breaking of chemical bonds could be simulated. Second,
instead of approaching every system in an empirical ad hoc
way, all of matter would now be treated using a uni® ed
method. M ost importantly, no adjustable parameters
would be needed. The whole of condensed matter would
be reconstructed from the ground up, using only the values
of the fundamental constants: the electronic charge e and
mass m and Planck’s constant h.
These new developments have brought us near to the
ideal of constructing a precise working model of atomic-
scale matter. I call this ideal the `virtual matter laboratory’ .
By observing virtual matter in this computer-generated
laboratory, we are observing a true image of real matter.
But more than this, by manipulating virtual matter and
observing the consequences, we are learning to make real
matter do what we want. What we have today is still an
approximation to the virtual matter laboratory Ð our
image of the real world is not perfect Ð but I hope to
convince you that the approximation is already good
enough to be extremely useful.
What about those tremendous di� culties that I men-
tioned Ð the large numbers of wave-like electrons and
their complicated interactions? This is an important part
of the story and I will explain in section 2 how an
extremely eŒective, but still incomplete, way of over-
coming the di� culties has been found. If you are not
interested in that part of the story, and you want to know
about the science that is coming out of the new methods,
the article will still make sense if you simply skip section 2.
In talking about the science, I have made arbitrary
choices. First-principles simulation is now contributing
to so many ® elds that it would be impossible to do justice
to them all. Guided by my own interests, I have chosen
illustrations concerning liquids and surfaces. In discussing
liquids, I will describe in section 3 some of the recent
progress in understanding liquid metals and semiconduc-
tors, and I will also say something about that most vital of
all liquids, water. The new methods are also bringing
rapid progress in surface science and are giving new
insights into the way molecules interact with surfaces. I
will talk in section 4 about recent work on the interaction
of the H 2 m olecule w ith metal surfaces, and the
interaction of the Cl2 molecule with the surface of silicon.
First-principles simulation is undergoing a tremendous
boom and I will end the paper by giving some predictions
of what the next ten years will bring.
2. First-principles simulation
We want to simulate atoms moving about in solids and
liquids. I will assume that as the nuclei move, the system of
electrons is always in its ground state: I am not interested in
electronic excitations. Th is assumption Ð technically
known as the Born ± Oppenheimer approximation Ð is
usually a good one. The electrons are so much lighter than
the nuclei that they can adjust themselves very rapidly to
what the nuclei do. From this point of view, the problem of
Figure 1. The Lennard-Jones form of interaction potential v(r)
between two rare-gas atoms, used in early simulations of liquid
argon. The curve shows v(r) divided by Boltzmann’s constant k,
in units of degrees K.
M. J. Gillan116
doing realistic simulations breaks into two questions: ® rst,
how do you ® nd the ground-state of the system of electrons
when the nuclei are sitting in certain positions and second,
how should the nuclei move? The ® rst question is by far the
most di� cult, and will occupy most of this section; the
second is rather straightforward, once we have answered
the ® rst. In fact, I will explain that the ® rst question consists
of three separate problems: electron correlation, the
representation of the electron waves, and the treatment of
large numbers of atoms. The reader who feels dissatis ® ed
with the brief details I will give here should consult the
original paper by Car and Parrinello (1985) and the more
recent reviews by, for example, Galli and Parrinello (1991)
and Payne et al. (1992).
2.1. Quantum mechanics
Let’ s start from basics by recalling the physics of the
hydrogen atom. The single electron bound to a proton is
very similar to a planet orbiting the sun, except that the
electron is attracted to the proton by the electrostatic
attraction between unlike charges, instead of gravitation.
The other great diŒerence is that we must use quantum
mechanics instead of Newton’ s laws of motion, so that the
electron is described by a wavefunction w (r). The wave-
function itself is not directly observable, but its square
| w (r)|2
gives the probability of ® nding the electron is at any
point r.
To ® nd the allowed energies E n of the hydrogen atom
and the corresponding wavefunctions w n(r) of the electron,
you have to solve the SchroÈ dinger equation, which says
that
2±h
2
2m,
2w n (r) 1 V( r) w n (r) 5 En w n (r) (1)
Here, ±h is Planck’ s constant h divided by 2 p and V(r)
represents the potential energy of the electron in the
electrostatic ® eld of the proton:
V( r) 5 2 e2/ 4 p ²0r, (2)
where r is the distance between the electron and the proton.
The problem of the hydrogen atom can be solved
exactly, and there is a well-known formula for its allowed
energies:
En 5 2 me4/ (4 p ²0 )
22 ±h
2n
2. (3)
The state of lowest possible energy Ð the ground-state Ð is
obtained by putting n = 1, and in this case the wavefunc-
tion is given by the simple formula:
w 1 (r) 51
( p a30 )
1/ 2exp ( 2 r/ a0 ) (4)
where a0 is the Bohr radius, which has a value of about
0 × 529 AÊ .
Unfortunately, as soon as you consider atoms with more
than one electron, it becomes impossible to do the quantum
mechanics exactly, and it is even worse for many atoms. To
see what the problem is, let us look at the helium atom. We
now have two electrons, acted on by the electrostatic
attraction of the nucleus and their own electrostatic
repulsion. It is this repulsion that causes all the trouble.
If there was no repulsion, each electron would behave as if
the other was not there and we would be back to a single-
electron problem like equation (1), which is easy to solve.
In this situation, the two electrons would have their own
wavefunctions w a(r1) and w b(r2) and it turns out that the
wavefunction w (r1 ,r2) representing the two-electron system
is just the product of the two:
w (r1, r2 ) 5 w a (r1 ) w b ( r2 ) (5)
The repulsion between the electrons completely spoils
this beautiful simplicity. The true wavefunction, instead of
being expressible as the product form shown in equation
(5), is some very complicated function of the electron
positions, which we can never hope to ® nd exactly. This
re¯ ects the fact that the electrons do not move indepen-
dently: their motion is correlated. We are now face-to-face
with the ® rst and most profound problem of doing ® rst-
principles calculations.
2.2. Ignoring correlation: Hartree theory
We cannot simply ignore the repulsion between electrons.
When the distance between two electrons is 1 AÊ , their
electrostatic interaction energy is roughly 14 eV, and if we
ignored an energy as large as this we would get completely
wrong results. However, instead of ignoring their interac-
tion, we can ignore their correlation. This idea leads to a
method called Hartree ± Fock theory, which is a step in the
right direction.
What does it mean to ignore correlation? Suppose I am
an observer sitting at some point r inside an atom. I observe
an electron orbiting around the nucleus. At my observation
point I measure the potential V(r) due to the charge on the
nucleus and the charge on the orbiting electron. Because
the electron is moving, the potential V(r) ¯ uctuates. Let me
take the average value of the potential at my observation
point r, which I will call Vm (r). At every point in space,
there is an average potential. Now instead of being an
observer, I will be an electron. I will orbit the nucleus under
the action of this average potential Vm (r). The other
electrons will do the same. We will all move, not acted on
by the true ¯ uctuating potential, but acted on by the
average potential due to the nucleus and the electrons.
Since we are all moving in a static potential, we are
behaving like independent electrons, but at the same time
we feel each other’ s repulsion, though only in an average
sense.
The virtual matter laboratory 117
What I have just described is the essence of Hartree
theory (the Fock part comes later). It replaces the
(insoluble) many-electron problem by the (soluble) problem
of single electrons moving in the average potential Vm(r):
we are back to equation (1). But we have paid a price: by
ignoring correlation, we are making a deliberate error,
which may have serious consequences.
2.3. The lonely electron: Hartree ± Fock theory
Up to now, I have avoided mentioning two very important
facts about electrons. Indeed, some of the things I said were
not quite correct. Now I must put things right. The ® rst fact
is that electrons are spinning about their own axis.
Quantum mechanics says that the component of the spin
angular momentum along a given direction can only have
the two values 6 12
±h . The spin can only be `up’ or `down’ .
This means that the wavefunction w r (r) for an electron
should depend on a spin variable r , which has the values
`up’ or `down’ , which I will write and ¯ . For example,
| w (r)|2
gives the probability for ® nding the electron at
point r with its spin pointing up.
The second important fact is that electrons are indis-
tinguishable: there is absolutely no way of telling which is
which. According to quantum mechanics, this has extra-
ordinary consequences. The main point for us is that the
wavefunction of a system of electrons must change its sign
when any two electrons are interchanged. This requirement,
called `exchange symmetry’ , means that for two electrons
the simple product wavefunction shown in equation (5) is
not correct. Instead, if the two electrons both have their
spins up, the two-electron wavefunction must be:
w a (r1 ) w b (r2 ) 2 w a (r2 ) w b (r1 ) . (6)
This exchange symmetry is the origin of the Pauli exclusion
principle, which says that two electrons cannot be in the
same quantum state. It also says that two electrons with the
same spin cannot be found at the same place: if you put
r1 = r2 into equation (6), the wavefunction vanishes, so that
there is zero probability of ® nding them at the same place.
Electrons are unsociable: they keep away from each other.
The unsociability of electrons is not included in Hartree
theory, because the theory does not respect exchange
symmetry. But all we have to do is to replace the simple
product of wavefunctions like equation (5) by the correct
`antisymmetrized’ product like equation (6). The resulting
scheme is Hartree ± Fock theory. The inclusion of exchange
symmetry lowers the energy. This is because it keeps the
electrons away from each other so that they feel less
positive repulsive energy. The energy reduction due to
exchange symmetry is called exchange energy.
Hartree ± Fock theory has been widely used, but it is not
very accurate and would not be good enough for the virtual
matter laboratory. It is not enough simply to neglect the
correlation due to electronic repulsion and this is why we
need density functional theory.
2.4. Coping with correlation: density functional theory
We have seen how the Hartree and Hartree ± Fock theories
include the repulsive interaction between electrons by
ignoring correlations. All the electrons move independently
under the action of a static potential, and the average
repulsive interaction is included as part of this potential.
Amazingly, it turns out to be possible to include correlation
simply by modifying the static potential Ð and the miracle
is that this can in principle be done exactly. This is the idea
behind density functional theory (DFT) (Hohenberg and
Kohn 1964, Kohn and Sham 1965, Jones and Gunnarsson
1989).
To understand what DFT does, remember how exchange
symmetry results in the lowering of the energy by keeping
electrons apart. Correlation has exactly the same eŒect. The
electrostatic repulsion between electrons also tends to keep
them apart and this is the essence of the eŒect I am calling
correlation. Electrons avoid each other for two reasons:
® rst, because of exchange symmetry; and second, because
of correlation. Both mechanisms lower the energy. Because
exchange and correlation have such similar eŒects, they are
often lumped together. The reduction of energy caused by
exchange and correlation is called the exchange ± correla-
tion energy and denoted by E xc. The key statement of DFT
for us is that Exc can be expressed solely in terms of the
electron density distribution.
Now there is one system for which we know all about the
exchange ± correlation energy. This is the uniform gas of
interacting electrons, sometimes called jellium. Theorists
have spent a lot of eŒort on jellium, and the exchange ±
correlation energy per electron is accurately known for a
wide range of electron densities (Ceperley and Alder 1980 ,
Perdew and Zunger 1981). Let us call e xc (n) the exchange ±
correlation energy per electron in jellium when the electron
density is n. The great breakthrough in dealing with
correlation was the discovery that the exchange ± correla-
tion energy of electrons in a system of atoms is very similar
to that in jellium. In collections of atoms, the electron
density varies from place to place Ð let’ s call it n(r). Now
assume that the exchange ± correlation energy per electron
at point r is given by e xc (n(r)), the quantity appropriate to
jellium. Then the amount of exchange ± correlation energy
per unit volume is n(r) e xc (n(r)), and the total exchange ±
correlation energy is given by:
Exc 5 n(r)²xc (n(r))dr. (7)
This remarkably simple expression, called the local density
approximation (LDA), has proved astonishingly successful
for many condensed-matter systems. Recently even better
M. J. Gillan118
expressions have been found, which allow for the gradients
of n(r). These new expressions are called `generalized
gradient approximations’ (GGA) (Perdew 1986, Becke
1988).
If you have followed me this far, you can permit yourself
a large sigh of relief because our discussion of correlation is
now done. I called this question of electron correlation the
® rst and most profound problem, but the approximate
solution has turned out to be amazingly simple. The LDA
and GGA form the basis for all the simulations I will
discuss later.
2.5. Representing the wavefunctions Ð pseudopotentials
and plane waves
We now come to the second major problem: how do you
represent wavefunctions on a computer? Computers are
discrete, ® nite machines designed to store and manipulate
lists of numbers, so this means that wavefunctions must be
represented as lists of numbers. There are two opposing
schools of thought about how to do this.
One school says that the wavefunctions w i(r) of electrons
in an assembly of atoms are like the wavefunctions / a (r) of
electrons in isolated atoms. The idea is that the wavefunc-
tion w i(r) can be represented by adding together the atomic
wavefunctions / a (r):
w i(r) 5a
ci a u a (r) . (8)
We know about the atomic wavefunctions; they are like the
wavefunctions of the hydrogen atom given in equation (4).
The l̀ist of numbers’ manipulated by the computer is the
set of coe� cients ci a . The computer has to vary these
coe� cients until the electrons are in the ground-state as
near as possible. The / a (r) functions are called `basis
functions’ , because they provide a basis for representing the
wavefunctions w i(r).
The other school of thought also uses basis functions,
but starts from a very diŒerent viewpoint. It says that the
electrons in condensed matter can run about rather freely
so that they are like free particles. Now the wavefunction
for a free electron is exp (ik × r), where k is the momentum of
the electron divided by ±h ; in fact, k is just the wavevector of
the de Broglie wave. So the idea is to use these plane-waves
exp (ik × r) as basis functions:
w i (r) 5k
cik exp ( ik . r) , (9)
and now the coe� cients c ik will be the l̀ist of numbers’ to
be varied.
The two schools of thought have both produced strong
arguments why their method is best. But in practice, for
large numbers of atoms in condensed matter, the plane-
wave method has been much more successful. This is
surprising, because the electrons in matter are clearly
nothing like free particles. In most atoms, there are tightly-
bound core electrons con® ned to small regions around the
nucleus, so that the plane-wave method seems to contradict
common sense. In fact, the method only makes sense when
combined with another idea: the pseudopotential concept.
What is a pseudopotential? It is basically a modi® ed
form of the true potential experienced by the electrons
(Heine 1970, Cohen and Heine 1970, Heine and Weaire
1970). When they are near the nucleus, the electrons feel a
strong attractive potential and this gives them a high kinetic
energy. But this means that their de Broglie wavelength is
very small, and their wavevector k is very large. Because of
this, a plane-wave basis would have to contain so many
wavevectors k in equation (9) that the calculations would
become impossible. A remarkable way of eliminating this
problem was discovered about 35 years ago by Heine,
Cohen and others, who showed that you can represent the
interaction of the valence electrons with the atomic cores by
a weak eŒective `pseudopotential’ and still end up with a
correct description of the electron states and the energy of
the system. In this way of doing it, the core electrons are
assumed to be in exactly the same states that they occupy in
the isolated atom, which is usually valid.
The discovery that the true potential can be replaced by a
much weaker pseudopotential is an extraordinary one and
it has had a deep in¯ uence on condensed-matter physics.
There is now a highly developed theory which speci® es how
pseudopotentials should be constructed for all the elements
in the periodic table so that they reproduce the properties
of the real potentials as exactly as possible (Bachelet et al.
1982). With these pseudopotentials, plane-wave basis sets
can be used for any element.
Plane waves have proved to be very successful for many
reasons. The wavefunctions can be made as accurate as
necessary by increasing the number of plane waves, so that
the method is systematically improvable. Plane waves are
simple so that the computer’ s job is easy. It also turns out
that the forces on the ions are straightforward to calculate,
so that it is easy to move them. Finally, plane waves are
unbiased. The calculations are unaŒected by the prejudices
of the user Ð an important advantage for any method that
is going to be widely used.
2.6. Supercells
I have now ® nished with the di� cult ideas, but there is still
one more important problem. In condensed matter, we are
dealing with enormous numbers of atoms Ð numbers like
1023
. But realistically we can never do ® rst-principles
simulations on numbers like that. Fortunately, this does
not matter in most situations. Like short-sighted people in
a crowd, atoms are aware only of their neighbours. More
scienti ® cally, the electrons arrange themselves so that the
The virtual matter laboratory 119
physical interactions between atoms have a range of only a
few atomic diameters. This means that the properties of
matter containing many atoms can be understood by
calculations on quite small numbers of atoms.
There are two ways of doing this: the cluster method and
the supercell method. In the cluster method, we carve out a
small piece of material and do the simulations on that. The
small piece is a cluster of atoms, which in the crudest
approximation is simply in free space, but more commonly
is embedded in a simple representation of the surrounding
material. In principle, one should try to increase the size of
the cluster until the properties of interest cease to vary. The
cluster method is not usually a good idea, for a simple
reason: its properties are dominated by its surface, unless
the cluster is very large. (Of course, real physical clusters
are interesting in their own right Ð but that is another
matter.)
In the supercell method, the calculations are also done
on a limited set of atoms, but there is a big diŒerence. The
set of atoms is surrounded on all sides by images of itself,
which are periodically repeated as shown in ® gure 2. This
device has the excellent eŒect of eliminating unwanted
boundaries and surfaces. The set of atoms that are
actually simulated are fooled into thinking that they are
part of an in® nite system. The price we pay is that the
system is made arti ® cially periodic, but the eŒects of this
generally disappear rapidly as the size of the system is
increased.
There is another great advantage of supercells: they ® t
very well with plane-wave basis sets. In a periodic system, a
plane-wave representation is the same as a Fourier series.
Since Fourier series are expressly designed to represent
periodic functions there is something extremely natural
about the combination of the supercell method and plane-
wave basis sets.
2.7. Move the atoms!
Now the ideas are all in place. The di� cult problem of
electron correlation is handled by the local density
approximation for exchange and correlation or by one of
the improved approximations. The wave functions of the
electrons are represented in terms of plane waves, using the
pseudopotentials scheme. The supercell device makes the
atoms behave as they would in bulk matter.
By doing things this way, we can ® nd the ground state of
a system of many atoms, by varying the plane-wave
coe� cients until the energy is a minimum. We can also
calculate the forces F i on all the atomic cores. These forces
come in two parts: the electrostatic forces between the
charges of the cores and the forces exerted on the cores by
the electrons. This second part can be calculated once we
know the wavefunctions of the electrons in the ground
state.
We are now ready to move the atoms. We let our system
evolve in time by making the atomic positions R i follow
Newton’s equation of motion:
d2R i / dt
2 5 F i / M i , (10)
where M i is the mass of atom i. Our working model of
matter moves into action! In fact, this way of moving the
atoms was already being used many years ago in the old-
sty le simulations based on empirical models for the
interaction between atoms (Gibson et al. 1960, Rahman
1964, Verlet 1967 , Allen and Tildesley 1987) . But now there
is no model. Everything is calculated from the quantum
mechanics of the electrons. As the nuclei move, the
electrons follow, and for each new set of atomic positions
R i the electronic ground state and hence the forces on the
atoms are recalculated.
I must explain a striking feature of these simulations.
We are using quantum mechanics for the electrons but
classical mechanics for the nuclei. Clearly the electrons
demand quantum mechanics, because they are in the
ground state, but what justi® es the use of classical
mechanics for the nuclei? The answer is that quantum
mechanics goes over to classical mechanics for highly
excited states and for most situations the nuclei are in
highly excited states. Put another way, the de Broglie
wavelength of the nuclei is very short compared with the
distance between the atoms so that quantum eŒects are
negligible. But this is not always justi® ed. For very light
atoms like hydrogen, we cannot use Newton’ s laws and I
will discuss an example of this later.
Figure 2. An illustration of the supercell method. The
simulation cell, in this case a cube containing 6 atoms, is
repeated to form an in® nite periodic system.
M. J. Gillan120
3. Liquids
The physical concepts that go into the simulation of liquids
are fairly simple. We take a collection of electrons and
nuclei in our periodically repeating cell, with the volume of
the box chosen to give the density we want. The electrons
are brought to the ground state and we determine the
energy of the system and the forces on the atoms. We then
give the nuclei some kinetic energy and let the system
evolve in time. In the language of statistical mechanics, the
system explores the microstates (atomic positions and
momenta) associated with the thermodynamic state having
the given density and energy. In other words, our
simulation generates the microcanonical ensemble (Mandl
1988). The temperature T of the system can be found using
the equipartition principle: the average kinetic energy per
atom is 32 kBT , where kB is Boltzmann’s constant. If we want
to adjust the temperature, we add or remove kinetic energy
by rescaling the velocities.
3.1. Liquid silicon
The ® rst liquid studied with the new methods was liquid
silicon (SÏ tich et al. 1989 , 1991), and this is an excellent
illustration because its covalent bonding makes it very
di� cult to model realistically with empirical interaction
potentials.
Silicon is in the same column of the periodic table as
carbon and crystallizes in the diamond structure shown in
® gure 3. In this structure, each Si atom is connected by
covalent bonds to four neighbouring atoms. Crystalline Si
is a semiconductor with a band gap of 1 × 1 eV. (Its
semiconducting properties form the basis of the computer
industry, so that with delightful circularity we are using Si
to study Si!) When Si melts, its structure changes
completely, as we shall see, and it becomes a metal. As
the atoms move around in the liquid, there must be a
constantly shifting pattern of bonding to their neighbours.
This means that the electrons in the system must be
continually rearranging themselves in response to the
motion of the atoms. This is exactly the kind of problem
that the new simulation methods are designed to address.
SÆtich et al. performed a completely ® rst-principles simula-
tion of liquid silicon (I will use the abbreviation l-Si),
using the molecular dynamics methods sketched in section
2.7. The repeating simulation cell they used contained only
64 atoms and the duration of the simulation (after
equilibration) was only 1 × 2 ps, but subsequent work has
shown that this is just about enough to represent the bulk
liquid.
The most direct way of testing that such a simulated
system is faithfully mimicking its counterpart in the real
world is by examining the radial distribution function, and
I need to say a few words about this. Suppose you are
riding on a Si atom as the simulation evolves. You observe
the average density of atoms at some distance r away from
you. If the atoms were arranged completely randomly (as in
a perfect gas, for example), this density would just be the
bulk number density q . But the interactions between atoms
make the arrangement far from random. To describe this,
we say that the average density at a distance r away from
your atom, instead of being q , is q g(r), where g(r) is the
radial distribution function. At distances r where g(r) is
greater than unity, there is more than the usual probability
of ® nding atoms and where it is less than unity the
probability is less.
The radial distribution function is easy to monitor in the
simulation, basically by constructing a histogram of the
interparticle distances. It is also directly measurable in
either X-ray or neutron diŒraction experiments. The
comparison of the radial distribution function of simulated
and real l-Si reported by SÏ tich et al. just above the melting
point is reproduced in ® gure 4. Notice how, in both the
simulated and the real systems, g(r) is zero for short
distances Ð there is no probability of ® nding atoms less
than a certain distance apart, because of the strong
repulsion between them. There is a pronounced peak at a
distance of 2 × 46 AÊ , representing the ® rst shell of neighbours
surrounding any given atom. In the crystal, this ® rst peak
would contain four neighbours, but in the liquid the area
under the peak corresponds to 6 × 5 neighbours, according to
both simulation and experiment. This is the change of
structure that I mentioned before, which is spontaneously
reproduced by the simulated system. At larger distances,
there are weaker peaks, indicating the presence of rather ill-
de® ned further shells of neighbours. The good agreement
between the radial distribution functions of the simulated
and real systems is even more remarkable when one recallsFigure 3. The diamond crystal structure.
The virtual matter laboratory 121
that there are no adjustable parameters whatever involved:
the only experimental input to the simulations is the values
of the fundamental constants e, m and h. The combination
of SchroÈ dinger’ s equation for the electrons and Newton’s
equation of motion for the ions provides an almost perfect
working model of the real liquid. (It is only `almost perfect’
because there is an approximation Ð the local density
approximation mentioned in section 2.4.)
3.2. More complicated liquids
The success of these simulations on l ± Si stimulated
investigations of many other liquids (see e.g. Zhang et al.
1990, Gong et al. 1993) . As an illustration of current work,
I will describe some results from the work of my own
group, on liquid alloys of silver and selenium (KirchoŒet
al. 1996). Silver is a typical metal and in the solid state its
atoms pack together like billiard balls in the face-centred
cubic structure. Selenium is completely diŒerent, because it
is a covalent material, with its atoms held together by
strong directional chemical bonds. Since it has a valency of
two, it likes to form chains and the normal form of the Se
crystal consists of spiral chains stacked parallel to each
other (® gure 5). Liquid alloys of very diŒerent elements,
like Ag and Se, are fascinating because the type of bonding
in the liquid varies continuously as the composition
changes. Many experiments have been done on such liquid
alloys, to ® nd out how their structure changes with
composition and how this aŒects their electrical properties.
(For a review, see Enderby and Barnes (1990). )
Our ® rst-principles molecular dynamics simulations were
done in a repeating cell containing 69 atoms and they lasted
for about 3 ps. Hence they are rather similar to the l-Si
simulations of SÏ tich et al. The diŒerence is that we now
have two kinds of atoms and we made simulations at three
diŒerent compositions Ag1 ± xSex for which x = 0 × 33, 0 × 42
and 0 × 65. As with l-Si, a direct check against the real world
can be made through the radial distribution functions.
Since there are two kinds of atoms, there are now three
diŒerent radial distribution functions. There is gA gA g(r)
describing the distribution of Ag atoms around an Ag
atom; gSeSe(r) describing the distribution of Se around Se;
and gA gS e(r) for Se around Ag Ð or Ag around Se, which is
the same thing. Recently, all three radial distribution
functions have been measured for real liquid Ag2Se by
neutron diŒraction (Lague et al. 1996), so we can verify
completely that simulation is mimicking reality. The
comparison shown in ® gure 6 leaves no doubt that this is
being achieved and I stress again that the values of e, m and
h are the only data entering the simulation.
Now let us ® nd out what happens when we change the
composition. Experimentally, the structure has been
measured only for the composition Ag2Se. But if the
simulations mimic reality, we can con® dently use them to
® nd out about the structure at diŒerent compositions.
Figure 5. The crystal structure of Se, with bonds drawn
between atoms in spiral chains. The outline indicates one unit cell
of the crystal.
Figure 4. The radial distribution function g(r) of simulated
liquid Si (solid lines) compared with the experimental results
obtained by neutron diŒraction (dotted line) and X-ray diŒrac-
tion (dash-dotted line). Simulation was performed by SÏ tich et al.
(1989), neutron diŒraction by Gabathuler and Steeb (1979) and
X-ray diŒraction by Waseda and Suzuki (1975). Reproduced
from SÏ tich et al. (1989), with permission.
M. J. Gillan122
Figure 7 shows what our simulations predict for the radial
distribution functions at the three compositions we have
studied. Nothing much changes in gA gA g(r) and gA gSe(r),
but there is clearly something rather dramatic happening in
gSeS e(r): as the amount of Se increases, a completely new
peak grows up at the distance 2 × 35 AÊ . On going more
deeply into the simulations, we found that this new peak is
caused by Se atoms linking together by covalent bonds.
This process became completely clear when we took
snapshots of the atom arrangements, like the ones shown
in ® gure 8. The fascinating thing here is that the formation
of chemical bonds between Se atoms happens completely
spontaneously in the simulations. The laws of quantum
mechanics for the electrons and Newton’s law for the nuclei
make it happen. Although this bonding between Se atoms
has not yet been observed experimentally in liquid Ag ± Se
alloys, diŒraction measurements by Barnes and Enderby
(1988 ) on liquid CuSe display the same short-distance peak
in the radial distribution function that we see in our
simulations. This leaves little doubt that the Se bonding
eŒects shown in ® gure 8 occur in the real world.
3.3. Water
If you were asked to name the most important liquid, you
would probably say water. The medium of all biology, a
major force in geology, a key agent in the physics and
chemistry of the atmosphere, a universal solvent: it is
important for many reasons. But it has proved one of the
most di� cult liquids to understand on the atomic scale.
The work of the virtual matter laboratory is starting to give
new insights.
Figure 6. The three radial distribution functions g a b (r) of liquid
Ag2Se obtained from ab initio simulations of KirchhoŒ et al.
(1996) (solid lines), compared with the neutron diŒraction results
of Lague et al. (1996).
Figure 7. The radial distribution functions g a b (r) of liquid
Ag1 ± xSex at three values of the Se fraction x obtained from
ab initio simulations by KirchhoŒet al. (1996).
Figure 8. Snapshots of typical atomic arrangements observed in ab initio simulations of liquid Ag1 ± xSex by KirchhoŒet al. (1996).
Dark and light spheres represent Ag and Se atoms, respectively, and panels (a), (b) and (c) show snapshots for Se fractions of x= 0 × 33,
0 × 42 and 0 × 65 respectively. Bonds are drawn between pairs of Se atoms separated by less than 2 × 9 AÊ .
The virtual matter laboratory 123
The water molecule, shown in ® gure 9, has a bent form,
with an angle of 105 8 between the O ± H bonds. The oxygen
atom attracts electrons to itself (it is electronegative) and
this means that it acquires a negative charge, leaving the H
atoms positively charged. The molecule is thus an electric
dipole and interacts electrostatically with other water
molecules and with dissolved ions. It is this electrostatic
property of the H2O molecule that makes water such a
good solvent and a lot of its action in biology also depends
on this.
Because of their shape and their dipole moment, H2O
molecules like to arrange themselves as shown in ® gure 10,
with the positive H atoms pointing towards the negative O
atoms. But there is more to it than just electrostatics. The
positive charge on a H atom attracts e lectrons on
surrounding molecules towards itself. The electron clouds
on the surrounding molecules get pulled out of shape as the
H atom tries to form a weak chemical bond with an O atom
on a neighbouring molecule. Chemists refer to this as a
hydrogen bond. There have been many models of water
and many attempts to make simulations based on empirical
interactions, none of them fully satisfactory. The new
simulation methods promise to change this situation. The
electrons are explicitly included in the simulation and the
distortion of the electron clouds as the molecules move is all
represented in the ® rst-principle calculations.
But does it work in practice? Recent simulations by
Parrinello, Car and their co-workers are encouraging
(Laasonen et al. 1993, Fois et al. 1994). They reported
simulations on a collection of 32 water molecules in a
periodically repeated box at the experimental density and
room temperature (300 K). The simulations ran for 1 × 5 ps,
which is just long enough to represent thermal equilibrium.
As usual, the crucial test of realism is the radial distribution
functions, and the comparison with experiment is shown in
® gure 11. The agreement is impressive, given that no
adjustable parameters are involved. Under the action of
® rst-principles quantum mechanics, the hydrogen and
oxygen atoms spontaneously arrange themselves almost
exactly as in real water.
Building on this success, Parrinello’ s group have now
moved on to the next challenge: the structure of hydrogen
Figure 9. The geometry of the H2O molecule.
Figure 10. Illustration of the general way in which H2O
molecules arrange themselves in liquid water so as to lower their
energy. Dashed lines indicate hydrogen bonds between H and O
atoms.
Figure 11. The three radial distribution functions g a b (r)
between H and O nuclei in liquid water from ab initio simulation
of Laasonen et al. (1993) (solid lines), compared with neutron-
diŒraction results of Soper and Philips (1986) (dotted lines).
Reproduced from Laasonen et al. (1993), with permission.
M. J. Gillan124
and hydroxyl ions in water (Tuckerman et al. 1995). Even
in pure water, the molecules have a very slight tendency to
dissociate into hydrogen and hydroxyl ions:
H 2O j H1 1 OH 2 . (11)
In thermal equilibrium, there is a certain concentration of
these ions, which is aŒected by whatever is dissolved in the
water. (Chemists describe the concentration of H+
ions by
the pH of the solution.) However, life is not so simple. No
one seriously believes that H+
could be present in the form
of isolated protons, because electrostatic attraction will
surely make the protons attach themselves to surrounding
water molecules. Sometimes, people talk of hydronium ions
(H3O+
). But are these stable objects? Experiments show
that hydrogen ions diŒuse very rapidly and this suggests
that H3O+
ions (if they exist) must be continually breaking
apart and reforming. The true nature of the OHÐ
ions has
been equally obscure.
The ® rst glimpses of the true situation are now emerging
from the virtual matter laboratory. By simulating the
hydrogen ion in water, Parrinello’ s group have indeed
observed the H 3O+
ion, linked by hydrogen bonds to
surrounding water molecules. But it turns out to be a
transient and ever-changing object. It is easy for one of the
protons in the H 3O+
to move to another water molecule.
The simulations show that for about 40% of the time the
extra proton is attached more or less equally to two water
molecules. This picture of hydrogen diŒusion occurring by
the continual swapping of partners has long been the
favoured model among physical chemists Ð it is called the
Grotthus mechanism (Atkins 1994) Ð but this is the ® rst
time it has been revealed by simulation. The simulations are
also giving insights into the structure and dynamics of the
OHÐ
ion.
Clearly, this story has a long way to go. Bigger systems
and longer simulations are needed. An interesting challenge
for the future will be to study whether the simulations
reproduce the celebrated density maximum of water at 4 8 C.
But the long-term impact on our understanding of water,
and of all kinds of aqueous solutions, will certainly be
immense.
4. Surfaces
Surfaces make the world what it is. When we observe
physical objects, it is usually their surfaces that we are
seeing. Materials placed in contact deform each other or
stick together at their surfaces. They rub on each other and
are worn away, or are corroded by their environment
through surface processes. The growth of crystals from
liquids or vapours depends on the dynamics of atoms at
surfaces. Many chemical processes would be virtually
impossible if they were not catalysed by surfaces. The
desire to understand adhesion, wear, corrosion, growth,
catalysis and many other surface phenomena has stimu-
lated an enormous experimental eŒort over the past thirty
years and a vast panoply of techniques has been deployed
to probe surfaces on the atomic scale.
In spite of all this eŒort, experiments are still unable to
answer many basic questions and the virtual matter
laboratory is playing an increasingly important role. To
give a glimpse of what is happening, I will present some
examples of recent work on the interaction of molecules
with surfaces. The break-up of molecules when they land
on a surface, known as dissociative chemisorption, is
important for many reasons. For example, it is involved in
corrosion and it underlies the operation of many important
catalysts, such as the catalytic converter used in cars.
However, these cases are too complex to describe here and I
will use simpler examples to illustrate the ideas, starting
with the H2 molecule on metal surfaces.
4.1. Hydrogen on metal surfaces
The case of H2 is a good place to start, because it is the
simplest molecule. However, the small mass of the H atom
means that quantum eŒects are important for the motion of
the nuclei and the use of Newton’s laws of motion would be
questionable. Instead, the quantum methods described in
section 2 are being used to map out how the energy varies
as the molecule goes from place to place. This energy map
can then be used to work out what happens when the
molecule hits the surface.
One of the ® rst cases studied was copper surfaces (White et
al. 1994 , Hammer et al. 1994). I will talk about the work of
White et al., which was concerned with the (100) surface.
(The notation (100) just means that the surface is made by
cutting the crystal along a plane perpendicular to one of the
cubic axes of the face-centred cubic Cu lattice.) As I
explained in section 2.6, periodic boundary conditions are
used, and to make this possible, the metal is represented as a
slab containing a certain number of layers of atoms. Since the
H atom is so light, it moves very rapidly and it can then be
assumed that the Cu atoms do not move much when the
molecule hits the surface, so the calculations are done with
the Cu atoms in the slab frozen in their positions. The
supercell approach means that the calculations are done on a
periodic array of H2 molecules on the surface. How does one
ensure that the arti® cial periodicity does not aŒect the
results? The system one really wants to study is a large piece
of metal with a single molecule on the surface, but the
supercell method does not allow this. However, it does allow
a good approximation to it, provided the slab is thick
enough, the vacuum between neighbouring slabs is wide
enough and the H 2 molecules are far enough apart. It turns
out that this is quite easy to achieve. The calculations of
White et al. were done with ® ve layers in the slab, a vacuum
width of 7 AÊ and a distance of 3 × 6 AÊ between H2 molecules; a
The virtual matter laboratory 125
plan view of their periodically repeated system, looking down
onto the surface, is shown in ® gure 12. Their tests showed
that increasing the distance between H 2 molecules and
changing the number of layers made almost no diŒerence to
the energetics. For this size of system, a single calculation is
very quick Ð and speed is of the essence, because the
calculations have to be repeated hundreds of times.
The reason so many calculations are needed is that the H
atoms can be in many diŒerent positions. This is the energy
map that I just mentioned. In making a map of a country,
the height of the land is systematically measured at a large
number of places. This large set of data can then be used to
construct a contour map, allowing us to understand the
shape of the mountains and valleys. In the same way, for a
molecule on a surface, one needs a map of the total energy
as a function of the atomic positions. There is a big
diŒerence though. In the map of a country, the height
depends on a two-dimensional position. For a molecule on
a surface there are far more variables. Even if the atoms in
the surface are frozen, each atom in a molecule has a three-
dimensional position, so that six variables are needed.
What is needed is a map of the energy as a function of
these six variables. That is why many calculations are
needed.
A six-dimensional map is rather hard to display, and it is
easier to look at chosen two-dimensional sections. A
convenient choice is to ® x the orientation of the molecule
and the surface site directly below the molecule, and to plot
the energy as a function of the distance d between the atoms
in the molecule (the molecular bond length) and the height
z of the molecule above the surface. An example of this
contour plot is shown in ® gure 13, which is taken from the
work of White et al. In looking at this plot, notice that in
the top left-hand corner the H atoms are far from the
surface and we are in a deep valley in which the minimum
energy corresponds to the bond length of the free molecule
(0 × 74 AÊ ). In the bottom right-hand corner, we are again in
a valley, but now with the H atoms bound to the surface
and far apart, so that the molecule has dissociated. The
contours make it clear that in order for the molecule to
approach the surface from the gas phase and then
dissociate and stick to the surface, an energy barrier has
to be overcome. The path marked by dots in the ® gure
shows a favourable way for this to happen. The search for
the most favourable path is rather similar to crossing a
range of mountains by the lowest pass.
What does this energy barrier mean in terms of real
experiments? Well, clearly it means that the molecule
cannot stick to the surface and dissociate unless it arrives
with at least enough energy to overcome the barrier. The
sticking of molecules to surfaces has been intensively
studied in experiments and is characterized by the `sticking
coe� cient’ S . This gives the fraction of incoming molecules
that succeed in sticking; the others bounce back into the
vacuum. If there is an energy barrier, then S will generally
increase as the energy of the incoming molecule increases.
Figure 12. The periodically repeated system used in the ab
initio calculations of White et al. (1994) on the dissociative
adsorption of H2 on the Cu (100) surface. Large and small
spheres represent Cu and H atoms.
Figure 13. Contour plot of the total energy of the H2 molecule
on the Cu (100) surface as a function of the height z of the centre
of the molecule above the surface and the separation d between
the H nuclei. In the case shown, the H ± H bond is parallel to the
surface, and the mid-point of the bond is sited as in ® gure 12.
Contour spacing is 0 × 05 eV, with dashed contours at 0 × 5 and
1 × 0 eV above energy of free molecule. Dotted line shows a
favourable path for dissociative adsorption of the molecule.
Reproduced from White et al. (1994) with permission.
M. J. Gillan126
Once the energy surface has been mapped out, it is possible
to calculate S as a function of the incident energy, and this
has been done with some success for H 2 on one of the
copper surfaces by Gross et al. (1994).
Real laboratory experiments show that the sticking
coe� cient for H2 on metals does not always increase with
incident energy. On many transition metals, S increases
when the incident energy decreases to low values. This has
given rise to much controversy, and suggestions have been
made that perhaps the molecule becomes trapped in a so-
called precursor state on the surface. One well-known case
is the (100) surface of tungsten and this has very recently
been investigated by ® rst-principles simulation (White et al.
1996). It turns out that there is no energy barrier at all in
this case: there are paths that take the molecule from the
gas phase to the dissociated absorbed state, which go
downhill in energy all the way. The detailed calculations
have given a very clear explanation for the energy
dependence of S . In a recent paper, Kay et al. (1995 ) have
taken the energy surface calculated for H2 on W (100) and
have used it to predict the dynamics of the H 2 molecule
when it hits the surface, allowing for the quantum
behaviour of the H nuclei. The results agree with
experiment in showing a mimimum sticking coe� cient at
a certain energy and the calculations show that the eŒect
arises from `steering’ . Even though the most favourable
dissociation path does not involve a barrier, it can require
the system to follow a rather tortuous path. Just as a
motorist may fail to negotiate a sharp corner if he arrives
too fast, so a molecule with too much energy may strike a
hill in the energy plot and be re¯ ected back into the gas
phase. A slow molecule will have time to adjust its path to
the twists and turns and will succeed in following the valley.
The increase of sticking probability at very high energies is
simply because the molecule is then able to ride over
obstacles in the contour plot.
At the same time as the work of White et al. (1996) and
Kay et al. (1995) , similar calculations for H2 dissociation on
the Pd (100) surface were reported by Gross et al. (1995).
Here again, S shows a minimum at a certain energy, as shown
in ® gure 14, and the calculations give strong evidence that
steering is the reason. The distinctive feature of these
calculations for H 2 on Pd (100) is that the quantum dynamics
of the H nuclei were included in a very complete way.
Taken together, all these ab initio studies on H 2
dissociation represent a really decisive step forward. The
reader who wants to ® nd out more about this area is
strongly recommended to look at the very recent review by
Darling and Holloway (1995) .
4.2. The dynamics of molecular break-up
Hydrogen absorption on metals is a very special case in one
respect: it is one of the few cases where the response of the
surface can be ignored. For many problems, the surface
itself may be strongly distorted by its interaction with the
molecule. This means that the energy will depend on the
positions of many atoms and any method based on the
study of energy surfaces seems doomed to failure. But there
is one saving feature: in many cases, quantum eŒects in the
dynamics of the atoms will be small, so that dynamical ab
initio simulations based on Newton’ s equation of motion
can be used, as discussed in section 2.7.
The ® rst attempt to use direct dynamical simulation to
study molecular adsorption was made by my group at
Keele in collaboration with the group of Payne in
Cambridge (De Vita et al. 1993). The example chosen was
the adsorption of the Cl2 molecule at the (111) surface of Si.
The geometry of this surface is very interesting in its own
right, because it is very diŒerent from what you get if you
simply cut the bulk crystal. Instead, the atoms at the
surface rearrange themselves, so as to satisfy the `dangling’
chemical bonds at the surface and lower the energy. The
equilibrium structure of the Si (111) surface was itself
greatly clari® ed by dynamical simulations done some years
ago (Ancilotto et al. 1990).
The main aim in our simulations on Cl2 adsorption at the
Si (111) surface was to ® nd out whether the molecule would
dissociate spontaneously, whether this depended on where
the molecule hit the surface and whether the response of the
surface played an important part. To study this question, we
performed ® ve simulations, in which the Cl2 molecule was
sent from the gas phase towards diŒerent sites on the surface
and in diŒerent orientations. At the rather high incident
energies we used, we found that the molecule dissociated on
Figure 14. Comparison of calculated and experimental sticking
coe� cients as a function of incident kinetic energy for H2 on the
Pd (100) surface, from Gross et al. (1995). Dashed line: H2
molecules initially in the rotational ground state; solid line: H2
molecules with an initial rotational and energy distribution
appropriate for molecular beam experiments; circles: experiment
(Rendulic et al. 1989). Reproduced with permission.
The virtual matter laboratory 127
hitting the surface in all cases and the Cl atoms formed new
chemical bonds with the surface atoms. An example of this
process is displayed in ® gure 15, which shows how the
electron density evolves as the molecule is adsorbed. The
breaking of the Cl ± Cl bond and the formation of new Cl ±
Si bonds are very clear. One clear conclusion from these
simulations was that surface distortions are very large for
this system, so that no approach based on contour plots
would stand much chance of success.
Many other simulations of molecular absorption have
been reported in the last few years, including some on oxide
surfaces. A good recent example of dynamical simulations
used to study molecular break-up was the work of Langel
and Parrinello (1994 ) on the adsorption of water on
stepped MgO surfaces.
5. Where are we going?
All I have been able to do here is to point to a few random
examples of work done recently in the virtual matter
laboratory. It is important to understand that these are
only a very small part of a large worldwide eŒort now going
into the new simulation methods. Some of the develop-
ments of the next few years are easy to predict by simple
extrapolation from what is happening now.
For example, the new methods will certainly make a
tremendous impact on our understanding of surfaces.
Many of the important catalytic processes underlying
industrial chemistry are still very poorly understood and
the ability to observe these processes directly in simulation
models will bring a completely new level of understanding.
It is also highly likely that it will bring the insights needed
to modify and improve these processes in a rational way.
The examples of surface simulation that I have given
concerned mainly metals and semiconductors, but ® rst-
principles simulations will be increasingly used for oxides
(Pugh and Gillan 1994, Manassidis et al. 1995 , Kantor-
ovich et al. 1995 , Goniakowski and Gillan 1996) and for
molecular processes in microporous materials such as
zeolites (Shah et al. 1996a,b, Nusterer et al. 1996a,b ).
There are also many other reasons apart from catalysis for
wanting to understand surfaces. The growth of crystals by
deposition of atoms from the vapour phase is important in
applications as diŒerent as fabricating computer chips and
growing diamond ® lms (Ashfold et al. 1994). Here too, the
virtual matter laboratory will be important.
There will also be enormous progress in the under-
standing of real-world liquids. The work on the liquid Ag-
Se mixture that I mentioned shows how one can observe a
kind of primitive chemical reaction involving the sponta-
neous formation of covalently bonded clusters. The general
idea of using the quantum simulation methods to observe
chemical reactions in liquids is an immensely powerful one.
Given more space, I could have talked about recent
Figure 15. Evolution of the electron density during the
dissociative chemisorption of a Cl2 molecule on the Si (111)
surface, according to the ab initio dynamical simulations of De
Vita et al. (1993). From top to bottom, the three panels show the
molecule above the surface, its ® rst contact as the bond between
Cl atoms is broken and the full formation of chemical bonds with
surface atoms. Each panel shows an isovalue surface of the
electron density.
M. J. Gillan128
simulations by Chiarotti et al. (1995) , in which they have
observed the spontaneous polymerization of acetylene
under high pressures. Parrinello’ s work on water that I
described in section 3.3 will surely form the starting point
for investigations of all kinds of problems in aqueous
solutions. I believe that the examples that we have seen so
far are only a taste of the tremendous advances that will be
made in the next few years.
Before I conclude, I want to point to one area that I
have not had space to mention at all: materials under high
pressures. This area is extremely important for the
understanding of planetary interiors, but also for practical
applications such as the behaviour of explosives. There
has been a strong interest among earth scientists for many
years in the use of simulation to help understand the
materials of the earth’ s core and mantle under the extreme
conditions that exist at great depths. The reason for this
interest is easy to understand. The fact is that it is still
experimentally di� cult to study materials under the
conditions of the earth’ s deep interior. The wonderful
thing about simulation is that extreme conditions present
no problem. If you want to increase the pressure two-fold
(or a thousand-fold), it is only a question of resetting a
parameter. The last few years have seen the beginning of
an eŒort to investigate the mantle material magnesium
silicate at very high pressures using ® rst-principles
simulation (Wentzcovitch et al. 1993 , 1995). The same
methods will certainly be enlisted in the study of liquid
iron, which is believed to make up much of the earth’ s
core.
I want to end with this thought: the ® rst-principles
simulation of matter is still young. The techniques are not
in their ® nal state. The plane-wave pseudopotential method
has been spectacularly successful, but still better methods
may yet be found. Ever more complex problems are being
tackled. First-principles simulations on systems containing
hundreds of atoms are becoming common; the next few
years will bring simulations on thousands, or even tens of
thousands. The last ten years have been a time of
extraordinary progress. The next ten will be equally
exciting.
Acknowledgements
I am grateful to Professor A. M. Stoneham FRS for his
comments on the manuscript. I also thank Professors D.
Bird, M. Parrinello and M. Sche‚ er for permission to
reproduce ® gures from their publications.
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Michael Gillan obtained a D. Phil. from Oxford
University, and worked at the University of
Minnesota and at Harwell Laboratory before
being appointed Professor of Theoretical Physics
at Keele University in 1988. Since moving to
Keele, his main research interest has been in the
ab initio simulation of solids and liquids and
particularly the use of simulation to study liquid
metals and molecular processe s at surfaces. He is
currently coordinator of the UK Car ± Parrinello
consort ium, a collaboration of 10 research
groups which is using the Edinburgh Cray T3D
supercom puter to study a range of problems in
condensed-matter physics and chemistry.
The virtual matter laboratory130