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It possible to view the periodic table as a portrayal of various aspects of symmetry and its breaking. This article does not add anything new to our understanding of the table, but explores its foundations in a different way from the conventional approach.

I consider the challenges of sharing thermodynamic concepts with both the general public and with students, whom I regard mostly as students of chemistry. I deal with the following challenges: (1) The order of presentation: what are the issues relating to presenting thermodynamics before or after quantum theory? (2) What should we identify as the foundations of our subject? (3) How do we convey to the general public (and the starting student) the insights that we get from the Second Law? (4) How do we bridge the gap from the qualitative to the quantitative? (5) Does visualization of calculations and concepts always help or can it make matters more complicated? (6) How do we make the transition to discussions in terms of Helmholtz and Gibbs energies, and start doing chemically useful calculations? (7) How do we keep track of the seemingly overwhelmingly large number of equations that a systematic treatment of thermodynamics inevitably generates? (8) How should we introduce statistical thermodynamics and enrich our understanding of classical thermodynamics? (9) How do we extend calculations to show the fascinating and broad scope of elementary thermodynamics?

En su presentación, el Dr. Peter Atkins nos habló de los retos y las oportunidades alrededor de la química (ver la portada de este número, tomada con la autorización del autor). Se refirió a los retos con el color rojo y los describió como La Abstracción, La complejidad y Las matemáticas, mientras que las oportunidades llevaron el color verde y fueron Los conceptos, El currículo y Los Gráficos.

Martyn Christian Raymond Symons was born on 12 November 1925 in Ipswich, Suffolk. The talents he was later to develop were a reflection of his genetic and cultural environment. Thus, his grandfather was William Christian Symons, who achieved contemporary minor fame as a painter in water colour and oils even though his work is now largely forgotten. His grandmother, Cecilia Davenport, was a concert pianist before her marriage. Symons was to display both artistic attributes, for he was a skilful self–taught pianist and an accomplished water colourist. The environment was richer than that, though, for the three sons of the grandparent's marriage were Mark, a painter, Phillip, who became a Benedictine monk and served as organist at Downside, and Stephen, Martyn's father. Painting, as already remarked, was one of Symons's great relaxations, and in early life (but not in middle age and after) the Catholic version of the Christian religion gave him guidance and solace. Indeed, there was a stage when he was poised to become a priest, but the passion passed and after the suffering and death of his first wife, who had become a Catholic in order to marry him, he rejected religion.The technical contribution to Symonss environment came from his father, Stephen White Symons, a consultant mechanical engineer, ably supported in the female manner of the day by his wife Marjorie. Here, though, the environment temporarily withdrew its support, for the young Symons entered the John Fisher School in Purley (1933–40), and hated every minute of it. Ill taught (he claimed) and bullied (he interpreted), the teachers—with the freedom of the age–almost literally hammered knowledge of a sort into him, not realizing the sensitivity of the child in their care and presumably contributing at least a little to his unusual psychology.

Science is the only path to understanding. It would be contaminated rather than enriched by any alliance with religion. Such should be the attitude of a scientifically alert atheist. This article elaborates and justifies this core attitude. There are those who consider that the domain of science is restricted to some kind of ‘physical world’, whereas religion deals with the ‘spiritual’. A scientific atheist holds that the domain of science is the physical world, but considers there is no other variety of world, and that the ‘spiritual’ is an illusion generated by a physical brain. The discussion considers the nature of this belief and distinguishes it from religious belief.

What are the central ideas of chemistry that we should ensure that our students carry with them as they travel through an educational system and out into the world? Indeed, what are the great ideas that give a chemist's vision of the world such a distinctive character, and which we would wish the general public to comprehend? Chemistry is such a central science for both our students and our communities that we should ensure that we do not dissuade our students and our public from discovering the insights it provides. In this talk I shall endeavour to identify ®rst the dif®culties of teaching our subject, then the dozen or so great ideas that, in my view, should be the spine of our courses. I will examine the general principles that make chemistry such a central part of any scienti®c education and in particular a component of the physical sciences. Then I shall identify the individual topics that I regard as the foundations of chemistry. The latter I shall do at three levels: at freshman level, for those who need to be aware of chemistry but not in great depth, to those in physical chemistry, who do need a deep understanding of our remarkable subject, and to the general public, who should be aware of a subset, at least, of our ideas. What are the principal concepts of chemistry that we should teach to our students? I shall concentrate on introductory chemistry in this talk, and try to identify the concepts that we chemists should hope our students will carry away from our courses into their careers, whatever these careers may be. At the same time I will be directing my remarks at instructors who are devising courses to introduce chemistry to students. I shall also have in mind one of the most important of our tasks: the communication of our remarkable subject to that most suspicious and unwelcoming audience, the general public. First, I would like to share my general attitude to education in chemistry. The principal target of our education should be to ®nd a way to bridge the imagined to the perceived. By that, I mean, we should show people how to look at a lump of matter, and in their mind's eye, see it as a collection of atoms and molecules. Then we should teach them to judge between con icting in uences. That is the essence of our subject, for it is rare that a single property governs the outcome of a reaction. We need to train our students to judge the likely outcome of con ict. Third, we need to show how to express qualitative ideas quantitatively. That ability brings chemistry into the domain of the physical sciences and puts the enormous power of mathematics into our hands. But what is it that makes our subject so dif®cult? One feature, which I have already touched on, is that chemistry is the science of con ict. The problem of knowing whether it is electronegativity, hardness, or some other property that is governing a physical or chemical property undermines con®dence and makes our subject seem dif®cult to penetrate, let alone master. Second, ours is an intricate subject, and it is very dif®cult for people to master suf®cient detail to give themselves con®dence to make rationalizations, let alone predictions. The third dif®culty is that, despite our subject being the most tangible of all it is also highly abstract. Our currency of discourse seems to the general public at least, highly abstract. As soon a chemists start to speak, out tumble all manner of abstractions, such as atoms, molecules, energy, entropy, *Lecture presented at the 7th International Chemistry Conference in Africa & 34th Convention of the South African Chemical Institute, Durban, South Africa, 6±10 July 1998, pp. 919±1024. 2Correspondence: E-mail: peter.atkins@chem.ox.ac.uk and so on. We know that our principal concepts are not abstractions, and a part of the battle is to render our concepts so that they seem real. So, with those thoughts behind us, let us turn to our principal concepts: what are the great ideas of chemistry? at this point, I do not want to promise too much. These are fundamental ideas, and as such are necessarily simple ideas that everyone already knows. These are the ideas that, in my view underpin and mark out our subject. 1 Matter consists of about 100 elements. It is a wonderful achievement of chemistry that it has shown that the world is a composition of so few entities. Of course. physicists have gone further, and have reduced the world to far fewer fundamental entities, but at 100 fundamental entities the entities retain their personalities and hence give chemistry its richness. 2 Elements are composed of atoms. The atom is the fundamental unit of our currency of discourse. and is the foundation not only of our understanding of chemical properties but also of the manipulations of chemistry and stoichiometry. Here Lavoisier is the particular god from whom this concept ultimately springs, for he ®rst brought the chemical balance to bear on chemistry and achieved the extraordinary feat of attaching numbers to matter. 3 The orbital structure of atoms accounts for their periodicity. The periodic table is, of course, the icon of our subject, and it is extraordinary that it can be explained in terms of a few simple ideas about orbitals, their energies, and the Pauli principle. It is remarkable that so much that can be rationalized by so little. 4 Chemical bonds form when electrons pair. Another god of chemistry is G. N. Lewis, who among his many achievements had the insight to identify the electron pair as the central content of the chemical bond. It is all the more remarkable that he did so before the ideas of quantum mechanics had been fully formulated, We know that there are exceptions to Lewis's approach, but his simple, central concept carries us far into chemistry. 5 Shape is central to function. We all know that it is simple ideas, such as the atomic radii of the element and related properties that determines, to a large extent, the bonding characteristics of the element, and how the shapes of molecules, particularly of enzymes, determine their properties. Incidentally, once the general public understands that chemistry's abstractions are in fact tangible, they should feel more comfortable with our so-called abstractions. 6 Molecules attract and repel each other. At this point we encounter the all-important bridge that takes us from the world of atoms (in a sense, the word of the imagined) to the world of bulk matter (in a sense, the world of the perceived). To make the connection, we need to be able to translate the properties of individual entities into emergent properties, and for that we need intermolecular forces. 7 Energy is blind to its mode of storage. Related to the bridge from atoms to the bulk, we have the foundations of statistical thermodynamics. The Boltzmann distribution is probably the most profound concept in chemistry. It is ostensibly a highly determined structure (the exponential decline of population with increasing energy) but its derivation shows that it is based on complete randomness; the distribution of populations over the available states with equal a priori probabilities. Then, with the structure established, we have an expression of enormous power, particularly for establishing the properties of bulk matter. Moreover, the distribution neatly captures the spirit of chemistry. Most of the population lies at low energy levels: and hence we have an understanding of why most matter survives for long periods. Yet the distribution also has a tail at high energies. So it also allows for the possibility of change from one structure into another. Indeed, if I were participating in a `balloon debate' (in which each participant as argue that he should not be thrown out of a sinking balloon), then I would feel very comfortable arguing on behalf of Boltzmann and his distribution. 8 Reactions fall into a small number of types. We are all familiar with the classi®cation of reactions into proton transfer (Brunsted acid±base), electron transfer (redox), and electron-pair sharing (Lewis acid± base). Here we encounter the second great simpli®cation, in which we see not only that all matter can be reduced to 100 or so elements, but that the transformations of matter can be expressed in terms of about three types of process. Here we see part of the nobility of the scienti®c attitude, the reduction of the complex into concatenations of simple entities. 928 P. ATKINS q 1999 IUPAC, Pure Appl. Chem. 71, 927±929 9 Reaction rates are summarized by rate laws. One of the techniques that physicists are very good at is building differential equations to summarize phenomena, and then extracting the juice of their solutions. We chemists are not nearly as good at this technique, but we are becoming good at it in one very special region, that of rate laws. Rate laws provide a window on to mechanism. However, until recently, all chemists have been able to do is to trivialize their rate laws and have found solutions that have very little content. Now, though, with the aid of computers, we can solve the highly nonlinear differential equations that result in periodicity. At last, the patterns of nature have come within chemistry's grasp. At this point I would like to indulge in the luxury of talking more explicitly about physical chemistry. Of course, almost all I have said already can rightfully be considered as lying within this domain, but I would like to focus here on the equations of physical chemistry. What are the most important equations? I can identify a small handful of absolutely crucial equations. One obvious one is the SchroEdinger equation

Though hard going for the general reader and highly personal in its selectivity, Elegant Solutions: Ten Beautiful Experiments in Chemistry provides reflections of a thoughtful author that will delight chemists

The World of Physical Chemistry. By Keith J. Laidler. Oxford University Press: 1993. Pp. 476. £55, $85.

The Chemical Bond: Structure and Dynamics. Edited by Ahmed Zewail. Academic: 1992. Pp. 313. $49.95, £33.

The charge and current densities for variational molecular wavefunctions in finite basis spaces are examined, and the consequences of their failure to satisfy a continuity equation, and the local charge generation and annihilation, are discussed. The current can be used to compute observables even though it might not satisfy the continuity equation. When approximate wave-functions are used charge is not conserved locally, but is conserved globally, and the details of the structure of the charge sources and sinks are discussed in the case of benzene. A valence bond technique for the calculation of current density is described and applied to benzene. Valence bond and molecular orbital methods are compared.

The decay of a classical hindered rotor over the whole domain of motion from almost free overall rotation to harmonic vibration in a well is examined. The decay is described by a modification of a projection operator formalism involving the separation of dynamical and lattice time scales, and depends on a canonical transformation into a hindered rotating frame that rotates and oscillates in accord with the unperturbed equation of motion of two coaxial dipoles.

Chemically induced electron polarization (CIDEP) has been observed for the durosemiquinone radical anion generated in the flash photolysis of solutions of duroquinone in the presence of various amines. The initial polarization has been measured directly by using a fast response time-resolved E.S.R. spectrometer. The magnitude of polarization is shown to depend on amine concentration and identity, and the solvent medium. Conventional nanosecond flash photolysis has been used to measure duroquinone triplet lifetimes under various conditions. The results are discussed in terms of the triplet mechanism and the radical pair mechanism.

The density matrix equations of motion arising in the triplet mechanism of chemically induced electron spin polarization are solved exactly without the imposition of the Redfield approximation. It is shown that the triplet spin relaxation time occurring in the final expression is not the true relaxation time because the spectral density involved depends both on the rotational correlation time and on the quenching rate. The effective spin relaxation time differs only slightly from the true time. The equations are extended to the case where the initial triplet passes on its polarization to the secondary triplet and exact solutions for the polarizations of the latter's doublets are obtained in the form ΠB = cΠA; an explicit expression for c is presented. The consequences of the secondary triplet being able to pass back its polarization to the initial triplet are explored and a ‘coherence effect’ on the polarization on the first triplet's doublets is analysed.

The two-dimensional Coulomb problem is examined from the point of view of a mapping onto a harmonic oscillator basis which is suitable for the introduction of a species of coherent state. These multispace coherent states are especially suitable for the discussion of the time-evolution of the system under the influence of a stochastic perturbation, and the method permits the deduction of explicit forms for the matrix elements of a general evolution operator in terms of parameters describing the perturbation.

Schwinger's theory of sources is examined in connection with the electromagnetic interactions of molecules. Using his constructive principles of causality and space-time uniformity it is shown how to arrive at the starting-points of conventional calculations of photon-molecule scattering and of molecule-molecule interactions through the electromagnetic field.

On the basis that every molecule acquires chirality by virtue of its rotation, it can be anticipated that the dispersion energy between two spherical tops depends on the magnitude and relative orientation of their rotations. An explicit calculation for two spherical tops is reported.

The effect of a magnetic field on the intensity of fluorescence from chemiluminescent reactions occurring in fluid solutions is calculated. The calculation first considers pairs of triplet molecules rotating in each other's vicinity and calculates the effect of a magnetic field on the rate at which the overall singlet state is populated. The field diminishes this rate, as in the solid-state situation. The triplets are then allowed to diffuse apart, and the process of populating the overall singlet state of the pair is treated as a relaxation process occurring during the diffusive trajectory. In this case too at high fields the intensity diminishes. The calculations are repeated for triplet-doublet quenching, and, in accord with the solid-state results, the intensity is enhanced at high fields because the quenching rate for doublet-triplet collisions is diminished, and the resultant fluorescence arises via triplets which escape abortive D-T encounters. Explicit expressions involving the dynamical parameter...

The effect of an electric field on the relaxation of a spin- 1 2 nucleus by a paramagnetic ion in fluid solution is calculated. The model relates to a spin—echo experiment, and depends on inter-ionic magnetic dipole—dipole relaxation. It is shown that under optimum conditions, using electric fields of strength ⪅ 10 MV m ±1 , the value of the nuclear T 2 may be changed by about 50%.

The triplet model of electron spin polarization in fluid media is evaluated. The model consists of an initial singlet molecule rotating in a static, externally applied magnetic field. Intersystem crossing into different zero-field states is represented by a rate matrix diagonal in the molecular frame, and this matrix is expressed as an effective spin operator. The triplet rotates, and the motion affects the polarization in the laboratory frame, and also causes spin relaxation in the triplet manifold. The triplet is chemically quenched, and the polarization appears in the doublet fragments. The model is treated in a density matrix formalism and on the basis of anisotropic rotational diffusion of the triplet molecule. Explicit expressions are obtained in terms of the molecular parameters, the various rate constants, and the rotational correlation time.

The proportion of singlets generated during a recombination of atoms or ions in a radiolytic spur reaction is examined from the viewpoint of a combinatorial analysis. The results obtained differ from other, already conflicting, published work. The problem is formulated in a way that enables spin relaxation effects to be incorporated.

The electron spin resonance line-shape of a tumbling radical is treated by a projection operator formalism that enables analytical expressions to be obtained. These are valid from the fast-motion limit to well into the slowmotion region so long as non-secular terms may be ignored.

A quantum-mechanical approach based on a projection-operator formalism is presented for the ion motion and the instantaneous power absorption in ion cyclotron resonance. The role of non-linearities in the power absorption is investigated by incorporating the formalism in a non-linear response theory, and their effect on the line-shape is discussed.

The rate of interconversion of ortho and para hydrogen in a solution of paramagnetic ions is calculated. The rate depends on the correlation of the magnetic fields at the two protons, and this is calculated on the basis of a dipolar interaction modulated by rotational and translational motion of the molecule and the ion. The theory is applied to experimental data on conversion caused by lanthanon and transition metal ions, and is used to estimate the radii of first hydration spheres.

Analytical expressions for population transfer and spin polarization are deduced for a two-level system in which the levels converge linearly or exponentially from a finite separation. The asymptotic populations and polarizations are discussed for rapid and slow approach trajectories and, in the exponential case, a simple closed form exact to second order in the perturbation for all rates of approach is discussed.

A general description of linear optical birefringence effects is presented in terms of the effect of forward scattering on the photon polarization density matrix. The approach enables explicit statements to be given for the polarization changes involved in any linear birefringence phenomenon in terms of R matrix elements between plane-polarized base states. Expressions in terms of molecular quantities are listed for the R matrix elements involved in some typical natural and induced birefringence effects. This description of birefringence is related to the infinitestimal unitary transformations of the density matrix and rotations in Poincare space.

The numerical integration of the time-dependent Schrodinger equation for a 2-level system is described with particular reference to spin-polarisation processes. Non-linear trajectories, re-encounters, and perturbations time-dependent in the strong mixing region can be investigated. Some typical situations are illustrated and the ‘anomalous’ effect of re-encounters described.

The procedure devised by Dirac for the canonical quantization of systems described by degenerate lagrangians is used to construct the hamiltonian for molecules interacting with the electromagnetic field. The hamiltonian obtained is expressed in terms of the gauge invariant field strengths and the electric and magnetic multipole moments of the molecules. The Coulomb gauge is introduced but other gauge conditions could be used. Finally, a physical interpretation of the unitary transformation that may be used to generate the multipole hamiltonian is given.

A simple extension of the model of a radical pair is described in which the relative translational motion of the radicals is described by an exponential translational correlation function for departure and re-encounter. A variety of polarisations can be accounted for without invoking re-encounters, although re-encounters can be important in particular situations.

The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.

The extension of the polarization density matrix description of optical birefringence phenomena involving multiple interactions between molecules and the electromagnetic field is described. This extension of the previous theory [1, 2] enables non-linear phenomena to be incorporated consistently into the description of birefringence.

The magnetic hyperfine interaction in Hund’s case ( c ) diatomic molecules is investigated by the method of spherical tensors and time-reversed angular momenta. Explicit expressions for the matrix elements are given, so too are expressions for the first and second order energies for an |Ω| = 1/2 state. The equations are applied to the electronic spectrum of BiO (Barrow, Gissane & Richards 1967) and the observed line widths are accounted for in terms of a magnetic hyperfine interaction with the 209 Bi nucleus. The electric quadrupole hyperfine interaction in BiO is also investigated and shown to be incapable of accounting for the observed effects.

The effect of a level crossing on the population may be positive or negative, and re-encounters may augment or diminish the population transferred at an earlier crossing. Multiple random re-encounters may have a net null effect on populations. The direction of population transfer is related to the gradient of the population at the start of the brief crossing region.

The biradical S-T −1 CIDNP magnetic field dependence can be treated as a spectral line shape problem in which the S-T −1 levels are tuned by the magnetic field; the shape is related to fluctuations in the through-space electron exchange interaction, J , modulated by conformational changes in the biradical. A density matrix treatment is applied to a two-conformer model of a polymethylene biradical.

Rules for constructing operator equivalents off-diagonal in the angular momentum quantum number are described. Hyperbolic angular momentum operators are used in conjunction with the usual (spherical) angular momentum operators to obtain equivalents for all values of Δj and Δm. Tables of operator equivalents for harmonics of rank 0–6 are given.

The scattering matrix technique in conjunction with the Stokes’s operator formalism for characterizing the polarization of a light beam which was developed in a previous paper (Atkins & Barron 1968) is applied to the description of the optical birefringence induced in fluid materials by electrical fields. The fields considered are both static (the Kerr effect), arise from a second, intense light beam (the optical Kerr effect) or from the measuring beam itself. The last phenomenon causes the polarization ellipse to rotate and affects the angle of rotation induced by optically active molecules.

The theoretical dependence of optical birefringence phenomena on the statistical nature of the light is investigated. Photon S-matrix calculations lead to expressions relating the change induced in the statistical nature of the beam to the molecular properties of the target sample. A hypothetical generalized optical birefringence experiment is analysed: in the experiment the coincidences of photons arriving at spatially separated detectors sensitive to different polarizations are observed with and without the sample in the beam, and the difference in the photon correlations depend on the initial statistical composition and polarization of the beam and the molecular properties of the sample. When the beam is generated by a thermal source the difference of the signals (with and without the sample) depends on the product of the polarizability and optical rotation tensors. When the light is generated by a single-mode coherent source the statistical properties of the light are such as to eliminate the second-o...

The scattering of polarized light by molecules is discussed in terms of a polarization density matrix. Particular attention is paid to scattering out of the direction of the incident beam, and the formalism enables the polarization characteristics of the non-forward scattered beam to be predicted for any polarization condition of the incident beam. Both Rayleigh and Raman scattering processes are contained in the formalism, although most attention is given to the former. Explicit expressions for the polarization of the scattered beams are obtained in terms of the polarizability and gyration tensors of the molecules by the application of diagrammatic perturbation theory, and it is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information which is additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not...

The Dirac equation for the electron is solved to terms that are second order in the applied field by the application of the Foldy-Wouthuysen transformation. In the process this generates all the terms that are necessary to ensure that the g tensor is gauge-invariant to the third order. An explicit expression is given for the third-order g tensor and the result is compared with the formula given by Tippins [1]. A term that reflects the velocity dependence by the Bohr magneton gives a significant contribution especially for atoms of high atomic number. This same term is additional to the extra term obtained by Stone [2] and is of the same magnitude: it must be invoked to account for the g factor of atomic hydrogen.

The inverse Faraday effect, in which a magnetization is induced in a solution through which is passed a polarized light beam of arbitrary ellipticity, is discussed on the basis of the S-matrix formulation of optical birefringence. It is shown that the Faraday effect and the inverse Faraday effect are topologically identical problems of diagrammatic perturbation theory and so it follows automatically that the magnetization should be proportional to the Verdet constant. The optical Faraday effect is the circular birefringence induced by an intense circularly polarized beam of light propagated colinearly with the weak measuring beam: the electric vector of the circularly polarized beam interacts with the molecule in a way that resembles the interaction of a static magnetic field. The interrelations of these two effects and the normal Faraday effect the self-rotation of the polarization ellipse of an intense beam are discussed.