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FOR MORE than 60 years, chemists have tried to understand chemical reactions
by applying theories of quantum physics. Now, modern, powerful computers
are allowing theoreticians to make sophisticated quantum calculations of
the simplest chemical reaction. Parallel developments in high-power lasers
have allowed experimenters to study this reaction in the laboratory in unprecedented
detail. A new age of chemical physics is dawning.

Chemists try to understand how chemical reactions happen using a variety
of theories. The most sophisticated of these takes into account the quantum
nature of atoms and molecules. The quantum theory of reactions involving
more than just a few atoms, however, is horribly complicated. It requires
many simplifying approximations before it can be applied.

An exception is the simplest reaction you can imagine. Here, a hydrogen
atom replaces another hydrogen atom in a diatomic hydrogen molecule:

H + H₂→H₂ + H

This is called the hydrogen-exchange reaction. Unfortunately, the factors
that make the hydrogen-exchange reaction simple to the theoreticians also
make it an extremely difficult reaction to study experimentally. Until about
six years ago, theoreticians had little detailed experimental information
to compare with their calculations. Now, however, breakthroughs in applying
new laser techniques have produced just the kind of information the theoreticians
need. The experimenters have produced more data than the theoreticians can
cope with. The pressure is on theoreticians to come up with better calculations
– no less difficult to perform than the experiments.

When considering the theory of chemical reactions, chemists attach considerable
importance to the idea of the ‘transition state’ in a chemical reaction.
The transition state is an ‘in-between’ structure formed when atoms and/or
molecules come together in a new arrangement. Chemists had previously thought
that, because the transition state exists for no more than a few tens of
femtoseconds (million-billionths of a second), they would never be able
to study it in the laboratory.

It is not a separate molecule, so you cannot study it in isolation.
Nevertheless, chemists can assign definite properties to it that influence
the way in which the reaction happens. Measuring these properties in the
laboratory is one of the most exacting challenges facing experimental chemists
today (see ‘Molecules caught in the act’, ¿ìè¶ÌÊÓƵ, 17 June 1989).

In the hydrogen-exchange reaction, the two reacting chemical species,
the hydrogen atom (H) and the hydrogen molecule (H₂) come together
to form a transition state consisting of three hydrogen atoms (H₃). This
then rearranges its bonds to form a new hydrogen molecule and another hydrogen
atom.

H + H – H→H – H – H→H – H + H

It is relatively easy to calculate the properties of the H₃ transition
state because it contains so few particles. A hydrogen atom consists of
one proton and one electron. Thus the H₃ transition state contains only
three protons and three electrons. Modern computational quantum theory can
just about manage a system with this small number of particles. Even so,
an important approximation must be made.

A proton is nearly 2000 times heavier than an electron. This means that
the protons in the H3 system move much more slowly than the electrons. We
can suppose that when the protons change their positions, the electrons
readjust almost instantaneously to maintain the arrangement with the lowest
energy. This is the Born-Oppenheimer approximation. It allows us to draw
a ‘contour map’, or potential energy surface, showing how the energy in
the H₃ system changes as the protons move about. Theoreticians can calculate
the H₃ potential energy surface fairly easily using quantum theory without
resorting to further assumptions or approximations. In this sense, the hydrogen-exchange
reaction is unique.

The potential energy surface describes how energy relates to the electrons
binding a molecule together. The energy depends on the total number of electrons,
their distribution in space and their spin. Each possible arrangement of
electrons in a molecule will have a different potential energy surface.
But the same potential energy surface applies to molecules or reactions
involving isotopes – atoms with nuclei containing the same number of protons
but a different number of neutrons. For example, a molecule containing one
hydrogen atom and one deuterium atom (D), which has an extra neutron in
its nucleus, has the same potential energy surface as a molecule containing
two ordinary hydrogen atoms.

Substituting heavier deuterium for hydrogen allows chemists to distinguish
between the reacting molecules and products of the hydrogen-exchange reaction.
Studying all the possible permutations of the reaction (H + D₂, H + HD,
D + H₂, and so on) gives the chemists independent information
for each reaction, which all relate to the same potential energy surface.

Figure 1 shows two potential energy surfaces for the H3 system with
bonds between the hydrogen atoms at different angles. Although these contour
maps are mainly useful in computer calculations, they can help us to understand
how the energy of the H₃ system varies as a hydrogen atom approaches a hydrogen
molecule. If a hydrogen atom comes alongside a hydrogen molecule end-on,
so that there are three hydrogen atoms in a straight line, the energy of
the system increases to a maximum (see Figure 1a). The black line indicates
the path of minimum energy; it is the path that any conservative hill walker
would follow seeking to minimise the steepness of a climb in passing from
one valley to another. The maximum point of the minimum energy path is called
the ‘saddle’ point. The properties of the H3 system at the saddle point
define the transition state for the reaction. After crossing the saddle
point, the H₃ system slides ‘downhill’ to give the products.

You can imagine the progress of the reaction as a path traced out by
a marble rolling on a surface. The ‘trajectory’ traced out by the marble
can be calculated from the initial starting position and speed, using Newton’s
laws of motion. Similarly, chemists can work out the trajectory of a chemical
reaction on a potential energy surface. By averaging a large number of such
trajectories from a wide variety of starting conditions, they can obtain
enough information to predict what happens in the reaction.

As you can see, this procedure is based on classical physics. But, of
course, atoms and molecules obey the laws of quantum mechanics. This means
that their energies do not vary continuously; they can have only certain
values that go up in steps, or quanta. This applies not only to the energy
of the electrons in an atom or a molecule but also to all the types of motions
that they can undergo: translation, vibration and rotation. We can see the
effect that this has by looking at a two-dimensional slice of the H3 surface
when one of the hydrogen atoms is a long way from the other two bound hydrogen
atoms (see Figure 1e). What this curve actually shows is how the potential
energy changes with the distance between the two atoms of the hydrogen molecule.
The energy drops to a minimum as the atoms get close enough to form a bond,
but at even closer distances it rises due to repulsion between the positively
charged nuclei. At larger distances, the energy levels off, which corresponds
to the chemical bond breaking.

The chemical bond acts rather like a small spring holding the nuclei
together, allowing the molecule to vibrate. Quantum mechanics predicts that
the vibration gives rise to a set of discrete energy levels within the potential
energy curve (see Figure 1e). Each level corresponds to a different amount
of vibrational excitation, in which the nuclei move back and forth about
their lowest energy positions. Each level is associated with a number, called
the vibrational quantum number, v. Each vibrational level has a set of closely
spaced energy levels that result from the hydrogen molecule rotating in
space. Each rotational level has a rotational quantum number, j. The translational
movement of a molecule is also split into levels but they are so close together
as to be effectively continuous, so can be ignored. We can thus describe
the total energy of the hydrogen molecule by a particular set of the quantum
numbers for electronic, vibrational and rotational motions. These define
the molecule’s ‘quantum state’.

‘When calculating the trajectory of the hydrogen-exchange reaction,
then, theoreticians take into account the quantum nature of the particles
by choosing only those starting conditions that match the energies of the
quantum states of the reacting hydrogen molecule. They then calculate the
trajectory according to Newton’s laws, determine the energy of the product
hydrogen molecule and assign it to the nearest quantum state. Because the
calculations combine classical with quantum descriptions, they are called
quasi-classical trajectory calculations.

Chemists are very interested in how fast the reaction goes from a particular
quantum state of the reacting molecule to a particular quantum state of
the product – the ‘state-to-state’ rate of the reaction. They are also interested
in the numbers of molecules formed in the final product states (the product-state
distribution). For many years, theoreticians have been able to calculate
state-to-state rates and product-state distributions for the hydrogen-exchange
reaction. Until recently, however, they had no corresponding experimental
data against which to check how accurate their calculations were.

The reason was that the hydrogen-exchange reaction is almost too simple.
To measure the state-to-state rates and the product-state distribution,
experimenters must be able to monitor individual quantum states of hydrogen
molecules formed by the hydrogen-exchange reaction. In other reactions,
experimenters can often do this by studying how the product molecules absorb
and emit light of various wavelengths. If the energy of the light absorbed
or emitted matches the difference between the energy levels of the electrons
in the molecules, experimenters may be able to measure the resulting electronic
absorption or emission spectra in the ultraviolet and visible region. Differences
in vibrational and in rotational levels produce absorption spectra spanning
the infrared and microwave regions. These spectra can provide a more or
less complete picture of the possible energies of a product molecule.

In the case of the hydrogen molecule, there is a snag, however. For
a molecule to absorb radiation, so as to be ‘excited’ to a higher vibrational
or rotational level, it must have an uneven distribution of electrons –
an electric dipole – which can interact with the electric component of electromagnetic
radiation. The hydrogen molecule is symmetrical, with its electrons shared
equally between the nuclei, and so it has no permanent electric dipole.
It, therefore, does not have a vibrational or rotational absorption spectrum.

One way around this problem is to use light of much higher energy. This
excites an electron in the hydrogen molecule to a higher electronic state.
You can then obtain information about the vibrational and rotational states
by measuring how much light is absorbed or subsequently emitted by the molecule
as the electron falls back to different vibrational and rotational levels
in the lower electronic state. But the wavelengths of light needed for this
kind of study lie in the far ultraviolet, a region for which conventional
light sources are not adequate.

In 1983, researchers in two laboratories in California solved the problem
by using high powered lasers that produce bursts of ultraviolet light. James
Valentini and Dan Gerrity at the University of California at Irvine used
a technique known as coherent anti-Stokes Raman spectroscopy (or CARS) to
measure the distribution of quantum states in the hydrogen-deuterium molecule
produced by reacting hydrogen atoms with deuterium molecules . Jong-Chen
Nieh, David Phillips and Harold Levene have also worked with Valentini in
developing and applying these techniques. Richard Zare and his colleagues,
Ernesto Marinero and Charles Rettner at Stanford University, developed a
technique called multiphoton ionisation in which ultraviolet light was used
to detect the final hydrogen-deuterium molecule in individual quantum states
. Richard Blake, Klaus-Dieter Rinnen and Dahv Kliner have since worked with
Zare on the hydrogen-exchange reaction.

Both groups of researchers first created hydrogen atoms by using lasers
to break up hydrogen-containing molecules such as hydrogen iodide. Because
the energy they supplied was more than enough to break up the molecules,
the hydrogen atoms carried the excess away, consequently, moving very fast.
The energy in a fast-moving hydrogen atom is added to the energy contained
in the reacting hydrogen molecule to give the total ‘collision’ energy of
the H₃ system. You can vary the collision energy by choosing different hydrogen-containing
molecules and different laser wavelengths to break them up.

The two groups have continued to develop and refine their techniques,
and, until very recently, there has been excellent agreement in their results.
This is gratifying, because the two experimental techniques differ significantly
in their approach and in the ways they obtained the information.

Figure 2 gives an example of the kind of data obtained by Zare’s group
at Stanford. Valentini’s group had already obtained more detailed data,
that gave equivalent results. The figure shows the numbers of hydrogen-deuterium
molecules formed in specific rotational (j) and vibrational (v) levels plotted
against j for each v. The results show that the molecules usually form without
being excited into higher vibrational levels. The rotational excitation
in each vibrational level is also slight. This is because of the shape of
the H₃ potential energy surface which suggests that the energy at the saddle
point, representing the transition state of the reaction, H – H – H, is
lowest for a linear arrangement. A preference for this shape in the transition
state means that it is difficult to transfer energy into rotational motion
of the final molecule. It is much easier to set the product molecule rotating
if the departing hydrogen atom gives it a ‘kick’ at an oblique angle (see
Figure 3a).

The lack of vibrational excitation suggests that the H₃ system prefers
not to switch vibrational quantum states as the reacting molecules develop
into the product molecules. The amount of rotational excitation decreases
as the vibrational excitation increases from v = 0 to 1 to 2. This again
indicates that the atoms in the transition state prefer to be in a line.
To get any vibrational excitation in the product, the H₃ system must be
linear so that the departing hydrogen atom pushes against the central hydrogen
atom, compressing the newly formed hydrogen-hydrogen bond (see Figure 3b).

More recent experiments at Stanford have shown that the reaction between
a deuterium atom and a vibrationally excited hydrogen molecule gives a higher
rotational excitation in the hydrogen-deuterium molecule that forms, than
in the reaction with an unexcited hydrogen molecule. The researchers explain
this by suggesting that the hydrogen-hydrogen bond in the vibrationally
excited hydrogen molecule is on average longer than in the unexcited molecule,
so the reaction proceeds quite happily when the hydrogen atoms in the transition
state are at an angle.

The ultraviolet lasers used to create the initial hydrogen atoms from
hydrogen molecules tend to produce very fast moving atoms. This means that
in the exchange reaction, they collide with hydrogen molecules at high energies.
It is difficult to calculate reliably what happens during the hydrogen-exchange
reaction at such high collision energies, using quantum mechanics, although
theoreticians are rapidly developing the tools to do the job. Quasi-classical
trajectory calculations are relatively straightforward, however. Figure
2 shows the results of Normand Blais at Los Alamos National Laboratory and
Donald Truhlar at the University of Minnesota. Considering the approximations
and assumptions behind these calculations, the agreement between experiment
and theory is remarkable.

More recently, Nieh and Valentini obtained what they believe to be the
first evidence of the H₃ transition state, by studying the reaction between
a hydrogen atom and a para-hydrogen molecule (in which the spins of the
two protons point in opposite directions), Nieh and Valentini measured the
state-to-state rates for the formation of vibrationally excited hydrogen
molecules for many different collision energies. Some energies gave much
higher rates than others, as shown in Figure 4. At first, Nieh and Valentini
believed that these were ‘dynamic resonances’ – enhancements in the rate
that come about when vibrationally excited transition states form. Although
the transition state survives for only a few tens of femtoseconds, this
is long enough for it to vibrate a couple of times before falling apart.
Theoreticians had predicted such resonances for more than 20 years.

Zare and his group have strongly challenged Nieh and Valentini’s interpretation
of their results, however. Valentini now believes that, although his results
provide evidence for the transition state, they may be reflecting something
more subtle than the direct influence of dynamical resonances. Further research
should resolve the situation.

The experimental work has come a long way in six years. Powerful lasers
have laid bare the very heart of the hydrogen-exchange reaction. Although
the quasi-classical trajectory calculations agree with the experimental
results very well, theoreticians need to develop techniques that will allow
them to do full state-to-state quantum calculations at high collision energies.

Experimental chemists also need to do more work. They still need to
satisfy themselves that they are actually measuring the properties of the
H₃ transition state. The accumulation of further evidence will mean that
this most elusive of chemical species, which for more than 60 years has
been a device of the chemist’s imagination, will finally become a reality.
Further studies are in progress in both Zare’s and Valentini’s laboratories.
The simplest chemical reaction holds yet more secrets that the scientists
are eager to discover.

* * *

Good vibrations for scattered light

THE HYDROGEN molecule has no permanent electric dipole, so light cannot
excite its vibrations and rotations directly. These motions can, however,
be excited directly by light being scattered.

There are three types of light scattering. In Rayleigh scattering, a
photon of light ‘bounces’ off a molecule. There is no exchange of energy
between the molecule and the photon – its wavelength is unchanged. This
kind of scattering is strongest for light with a short wavelength and explains
why the sky is blue.

In Stokes Raman scattering, energy is transferred from the photon to
the molecule. The photon that bounces off the molecule has a longer wavelength
and the molecule jumps into a new state with excited vibrations and rotations.
This is different from absorption in that the molecule does not take up
all of the energy available in the photon (the photon is not ‘absorbed’).
In anti-Stokes Raman scattering, the molecule starts off in an excited vibration-rotational
state and energy is transferred to the photon. The photon that bounces off
then has a shorter wavelength.

You can enhance Stokes scattering by using two lasers, each with a different
wavelength. The wavelength of the first laser is fixed and is almost arbitrary:
it simply provides a source of photons that can be Stokes scattered by the
molecules. You can then vary the wavelength of the second laser to match
that of the light after it has been Stokes scattered. What happens is that
the first laser excites the molecule and the second laser then stimulates
the Stokes scattering process, enhancing the amount of Stokes scattered
light.

The high intensities of the lasers mean that many molecules are excited.
Light from the second laser further interacts with the excited molecules
to produce an intense output of light that is anti-Stokes scattered.

The anti-Stokes scattered light has a shorter wavelength than either
of the two lasers, and so is easy to detect separately. Furthermore, because
lasers are used to generate the anti-Stokes scattered light, it is coherent
– the peaks and troughs of the light waves are all in step much like in
laser light itself.

Finally, anti-Stokes scattering can happen only when the wavelength
of the second laser is properly matched to that of the original Stokes scattered
light. This, in turn, depends on the energy level of the initial vibration-rotational
state in the molecule. Thus, if you vary the wavelength of the second laser
continuously, you will see anti-Stokes scattering only when the combination
of light and vibration-rotational energies is right. This gives a spectrum
of the vibration-rotational levels without the light being directly absorbed.
This is coherent anti-Stokes Raman spectroscopy (CARS).

The intensity of the anti-Stokes scattered light depends on the number
of molecules produced in the excited vibration-rotational level which, in
turn, depends on the number of molecules in the initial vibration-rotational
state. The CARS spectra, therefore, reveal the vibration-rotational states
of the hydrogen molecule produced in the hydrogen-exchange reaction, and
how many molecules are formed in each state. This is the technique that
has been developed and used extensively by Valentini’s group. They have
used it to obtain data on the products from a wide variety of hydrogen-exchange
reactions including those between a hydrogen atom and deuterium molecule
and between a hydrogen atom and a para-hydrogen molecule (see main text).

* * *

Many photons make light work

WHEN molecules absorb light, they usually absorb one photon at a time.
Occasionally, a molecule can absorb two or more photons simultaneously but
this is unlikely to happen using conventional, sources of light of low intensity.
Lasers, however, produce high enough intensities of light to encourage a
molecule to absorb more than one photon simultaneously. This means that
the molecule can become highly excited.

The hydrogen molecule has to absorb a high energy ultraviolet photon
to excite one of its electrons. Such energetic photons are difficult to
generate in the laboratory, even with lasers. An alternative way to excite
the electron, therefore, is to bombard the hydrogen molecule with many photons
from a laser at a lower energy. The molecule can then absorb enough low
energy photons all at once to excite the electron. Under the right circumstances,
multiphoton absorption does not just excite the electron to a higher electronic
state, but it can rip the electron out of the molecule completely, leaving
a positively charged hydrogen molecular ion, H₂⁺.

Ions are easy to detect. ¿ìè¶ÌÊÓƵs can measure the time it takes for
the ions formed by the laser to travel through a vacuum into a chamber which
contains an ion detector in a so-called time-of-flight mass spectrometer.
The time taken is directly related to the mass of the ion, and this allows
scientists to detect different ions according to their mass.

In recent studies of two versions of the hydrogen-exchange reaction,
one with deuterium atoms and hydrogen molecules, and the other with hydrogen
atoms and deuterium molecules, Zare and his colleagues used a laser to excite
the resulting hydrogen-deuterium molecule to a high energy electronic state
by the simultaneous absorption of two photons with wavelengths around 210
nanometres. A third photon of the same wavelength then ionised the excited
molecule, and a time-of-flight mass spectrometer detected the molecular
ions that formed. This three-photon ionisation involves an intermediate
electronic state. The signal due to the ion is much bigger when the combined
energies of two photons correspond to that of the intermediate state. The
technique is called ‘resonance-enhanced’ multiphoton ionisation (see New
¿ìè¶ÌÊÓƵ, ‘How to count atoms’, 26 November 1987).

Matching the combined energies of the photons depends on the difference
between the energies of the initial vibration-rotational state of the excited
hydrogen-deuterium molecule and the lowest vibrational level of the intermediate
electronic state. This difference varies with the energy of the initial
state. The size of the signal due to the ion depends on how many hydrogen-deuterium
molecules there are in the initial state. When the researchers varied the
wavelength of the photons, they observed signals only when the combined
energies of the photons matched the energy differences for the different
initial vibration-rotation states. The spectra, therefore, revealed which
vibration-rotation states of hydrogen-deuterium were produced in the reactions,
and how many molecules were formed in each quantum state.

Jim Baggott is a freelance science writer based in Reading.