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A comparison of semiclassical and quantum versions of molecular light scattering theory at finite temperatures is presented. A general formulation of the semiclassical radiation model is developed to the point where its relationship to the corresponding QED formalism can be established: the classical scattered electric field is proportional to the same R-matrix element as that obtained from QED for the photon scattering amplitude. The result is valid for non-resonant scattering at T = 0. The semiclassical theory conventionally also inherits aspects of a classical molecular model, principally origin-dependent molecular multipole moments. Origin independent multipoles, and corresponding response functions can be defined if the theory is cast in terms of centre-of-mass and translation invariant internal coordinates. Such a choice of coordinates brings molecular light scattering theory into line with the theory of the molecular Schrodinger equation. This is illustrated for the case of a diatomic molecule. A s...

A novel Hamiltonian scheme for non–relativistic quantum electrodynamics in which the gauge arbitrariness of the field potential is kept explicit is used to study the gauge–dependence properties of various versions of perturbation theory when resonance and line–broadening effects are admitted. Time–dependent perturbation theory is shown to have severe gauge–dependence problems unless ad hoc modifications are made. A time–independent formulation of S –matrix theory is then studied. Far from resonance, the S –matrix is gauge invariant in all orders of perturbation theory due to a very precise cancellation of gauge–dependent terms which requires, among other things, complete sets of intermediate states; energy conservation also has a crucial role. However, an obvious separation of the S –matrix into a resonant (pole) and non–resonant background leads to incomplete cancellation of the gauge–dependent terms. The introduction of the Heitler damping matrix into an integral equation for the T –matrix leads to a gauge–invariant result. This provides the basis for a gauge invariant S –matrix theory of atoms and molecules interacting with electromagnetic radiation that encompasses resonance and damping effects.

A general account of the interaction of charges with electromagnetic radiation is presented; the charges are treated in the nonrelativistic approximation for simplicity (not an essential assumption). The two main approaches in the literature based on either the potentials of the electromagnetic field (minimal coupling) or the polarization fields of the charges (e.g., the multipolar Hamiltonian) are presented in a unified framework that characterizes them as simply different viewpoints of the same general structure. This structure is very suitable for a discussion of the gauge invariance of the theory. The final quantum mechanical Hamiltonian derived here has recently been shown to provide the basis for a proof of the gauge invariance of its associated S-matrix to all orders of perturbation theory, provided the S-matrix is taken on the energy shell. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 531–545, 1999

The quantum theory of the optical birefringence induced in molecular gases by external fields is discussed in a general setting. The scattering of light by molecules is expressed by the generalized Kramers-Heisenberg formula for the R-matrix element, and the Stokes parameter formalism is used to relate this scattering amplitude to the birefringence observables. Our approach is motivated by Van Vleck's quantum mechanical treatment of the Langevin-Debye formula; for a gas at finite temperature the relationship between the characteristics of the system perturbed by the external field and the unperturbed system can be obtained using thermodynamic perturbation theory. This is conveniently combined with the form of the scattering amplitude obtained in linear response theory, since the thermodynamic averaging and the effects of the external field can be treated in a unified way. We compare this Dyson series expansion method with the more traditional Rayleigh-Schrodinger perturbation theory approach and show how ...

The Hamiltonian for non-relativistic quantum electrodynamics can be expressed in terms of the usual dynamical variables for the field and charged particles and as a functional of a c-number vector field p(x;x') which is essentially arbitrary. This corresponds to the freedom to use the vector potential of he field in an arbitrary gauge since every p(x;x') determines a gauge condition for the potential. We study the S -matrix for two different choices of p(x;x') and calculate the difference. This is done directly in low-orders of perturbation theory and suggests a conjecture for the general formula for the difference, valid in any order of perturbation theory; we verify the validity of this conjecture by elementary means. On the energy shell the S- matrices are identical, i.e. gauge-invariant.

Physical observables associated with interactions between electromagnetic radiation and charged particles are required to be gauge invariant, i.e. independent of the gauge of the field potentials. Gauge invariance implies the involvement of the electric polarization field, P(x), in electrodynamics, but leaves undetermined the transverse component P(x)⊥ which occurs in the interaction terms coupling charges to radiation. We put the Hamiltonian for non-relativistic electrodynamics into a completely general form displaying the arbitrary polarization field. The well-known Coulomb gauge and ‘multipolar’ Hamiltonian formalisms then arise as special cases corresponding to particular choices of P(x)⊥. For practical calculations some choice of P(x)⊥ has to be made, and we present a particularly useful form for neutral collection of charges, not necessarily bound, but typically appropriate to atoms and molecules. Some results concerning the independence from the arbitrary quantities of physical observables are desc...

The cohesive energy per atom in a bulk transition metal is known to depend on the fractional d-orbital occupancy per atom and to be directly proportional to the square root of the dispersion of the d-electron density of states, and hence also to Z1/2, where Z is the average coordination number of an atom. By studying several model densities of states it is shown that the fractional orbital occupancy and Z1/2 should also be the relevant parameters to characterise the cohesive energies of cluster species. The model energy level schemes studied are: (i) finite numbers of degenerate and nondegenerate equally spaced levels, (ii) two subbands separated by an energy gap and (iii) energy levels derived from the tensor surface harmonic (TSH) model of cluster bonding.

A general account of the interaction of electric charges with electromagnetic radiation is presented, the charges being treated in the non-relativistic approximation. The two main approaches in the literature, based on either the potentials of the electromagnetic field (‘minimal coupling’) or the polarization fields of the charges (e.g., the ‘multipolar Hamiltonian’) are presented in a unified framework, that characterizes them as simply different viewpoints of the same general structure. This structure is very suitable for a discussion of the gauge invariance of the theory. The final quantum mechanical Hamiltonian presented here has recently been shown to provide the basis for a proof of the gauge invariance of its associated S-matrix to all orders of perturbation theory, provided the S-matrix is taken on the energy-shell. Some outstanding problems are described briefly.

Molecular structure is the cornerstone of chemistry; it is the remarkable synthesizing concept that makes chemistry intelligible. Why then should it appear as a topic in a volume concerned with ‘conceptual trends’ ? For most of us, molecular structure just is! We know enough about it to feel comfortable with it as the central idea in chemistry, and there is its extraordinary success in applications that is apt to shut off any motivation for further enquiry. All well and good, and yet from a modern theoretical perspective based on quantum theory, molecular structure is problematic. Quantum chemistry claims to account for the existence of chemical isomers (the key chemical question that molecular structure ‘solves’) in terms of states associated with the minima of potential energy surfaces. From the point of view of quantum theory this is at best an engineering solution, on two counts at least. Firstly, potential energy surfaces are classical constructions and beg the question as to what they can mean within quantum theory; secondly, granted the conventional quantum states of molecular species associated with individual minima of PE surfaces through the usual product forms of electronic, vibrational, rotational functions etc. why not form superpositions of states associated with different minima ? A pedestrian answer recognizes that the Rayleigh-Ritz variational principle guarantees a lowering in energy for the superposition, and that quest has motivated much of the history of quantum chemistry. More seriously the quantum formalism positively invites the construction of such superpositions which correspond to physically possible states. Why then are such superpositions not encountered in actual experiments ? The answer may reside in the part of modern quantum theory concerned with ‘decoherence’ and the emergence of classical properties.

The electronic structure of the transition-metal cluster moiety M/sub 6/(..mu../sub 3/-X)/sub 8/, with idealized O/sub h/ symmetry, is examined for X = chalcogen, halogen, and carbonyl ligands. The role of the metal d electrons in metal-metal bonding is emphasized, and the structure of the metal cluster d-band is described in detail. 54 references, 4 figures.

The aim of this review is to give an account of recent theoretical work (‘Ligand-Field Analysis’) that has both led to clarification of the basic structure of ligand-field theory and has facilitated the acquisition of information about chemical bonding in paramagnetic transition-metal complexes that form insulating crystals. The review begins with a justification in modern terms for the classical crystal-field approach based on a well defined dn-configuration for the metal ion; the central idea is that the ground and low-lying electronic states are based on localized electron wavefunctions because the interelectron repulsion energy is so important in these systems. The ligand-field theory is developed in two stages; first, the group product wavefunction method is used to construct a physically important subspace of many-electron wavefunctions for a transition-metal complex. The full n-electron hamiltonian is then studied in this basis using Lowdin partitioning and the chain formalism of Haydock; ...

It is shown that the on-energy-shell single-particle scattering amplitudes obtained from the minimal-coupling and the multipolar forms of the Hamiltonian in non-relativistic quantum electrodynamics are equal to order e2; both forms of the Hamiltonian thus lead to the same differential cross section for Kramers-Heisenberg scattering. The proof given is based directly on the relations which hold between the two transition matrices and is simpler than a previous proof which relied on sum rules derived from the canonical commutation relations. The proof is also more general in that the multipolar Hamiltonian is defined in terms of line integrals along paths which are not assumed to be straight lines but which can be chosen from a large class of possible curves.

The results of an investigation of austentic stainless steels fabricated in the United Kingdom are presented. All were welded with first choice filler materials. The weldment under test was representative of industrially fabricated assemblies. Even in the case of the 304L material, the filler was 308L. Results of testing show that welding degrades the corrosion resistance of all austentitic stainless steels investigated. In all materials, the weld bead microstructures were the most susceptible to corrosion. The HAZ microstructures were more corrosion susceptible than the wrought microstructure, but less so than the weld bead. Corrosion behaviour is dominated by the distribution of molybdenum across the y/8 interface. The authors feel that it is not possible to categorize corrosion behavior simply by the microstructural constituents because the formation mechanisms of the weld bead controls corrosion behavior.

The one-electron states in the polynuclear transition metal carbonyls of Ni and Pt with formula (M3(CO)6)n2- (n=1,2) are studied using the chemical pseudopotential method. The bonding in these systems is approached by first studying the molecular orbitals of the metal cluster (Mn) and carbonyl skeleton fragments (CO)k, before allowing them to interact: the orbitals of these 'prepared' fragments interact rather weakly and remain relatively independent. The main electronic interactions appear to involve metal s and d orbitals and the 'lone pair' (5 sigma ) molecular orbitals of the ligands. The molecular symmetry point group is used to classify the molecular orbitals and the symmetry-allowed (electric-dipole) electronic transitions are obtained and compared with the limited amount of experimental information available for solution absorption spectroscopy.

A new gauge condition which leads directly to the 'multipole' hamiltonian for the interaction of molecules with the electromagnetic field is studied within the context of quantum electrodynamics ie when the molecules and field are considered as a closed dynamical system. The new formalism which maintains the advantages of the usual Coulomb gauge theory but avoids the difficulties associated with the presence of unphysical terms in the propagation function of the Coulomb gauge vector potential, is thus obtained without recourse to the unitary transformation of the usual Coulomb gauge formalism which has previously been used to achieve this result.

Following the detailed discussion of the transition-metal cluster moiety M/sub 6/(..mu../sub 3/-X)/sub 8/ in the preceding paper, a more general account of the importance of the d electrons in transition-metal cluster chemistry is presented. The putative analogy with borane clusters (and their derivatives) is examined critically. Although an isolobal relationship exists between, e.g., BH and appropriate ML/sub n/ fragments (e.g. conical Fe(CO)/sub 3/), this does not imply that the BH and ML/sub n/ fragments behave in electronically similar ways when cluster formation occurs, even when structurally related clusters are formed. Nonidentical isolobal fragments have orbital differences that manifest themselves in interfragment resonance integrals and require a qualitative distinction to be drawn between the bonding modes and detailed electronic structures of (i) transition-metal cluster compounds and (ii) boranes, carboranes, and their metalla derivatives; an analysis developed in the electronic structure theory of transition-metal systems shows why this is the case. The isolobal principle and Wade's rule owe their generality and utility to being symmetry-based statements; the energetics and details of the electronic structure of cluster compounds however are a separate matter requiring appropriate methods of theoretical chemistry. 39 references, 3 figures.

The wave mechanical formulation of quantum electrodynamics is investigated in an explicitly gauge invariant form and this leads to the connection between manifest charge conservation and the Power-Zienau-Woolley transformation.

A discussion of the quantum mechanics of molecules with the emphasis on quantum mechanical principles is presented in the expectation that a better understanding of the assumptions tacitly embodied in conventional quantum chemistry will follow. The discussion centres mainly on the position of the chemical structure hypothesis in a quantum mechanical theory based on quantum states. Various different kinds of molecular states are identified according to characteristic properties of their wavefunctions, ranging from the completely delocalized (fully symmetrized) eigenstates to the “molecular structure states” of rigid molecules. I suggest that the adiabatic separation of electrons and nuclei can be avoided by working with the generator coordinate method (GCM) which promises to be a powerful non-adiabatic formalism for molecular quantum theory.

Muffin-tin orbital methods in electronic structure theory evaluate the Hamiltonian and overlap matrix elements by breaking up the integrals into contributions from non-overlapping cells about each atomic centre. The simplest choice of cells is to take maximally non-overlapping spheres about each atom enclosed in a large sphere, and one then has a single 'cell' for the interstitial region. The authors describe a fast computational procedure for calculating the cell projection operator theta (r) for this configuration.

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 comparison is made of the conventional quantum mechanical hamiltonian for the interaction of molecular systems with the electromagnetic field and the alternative multipole formulation given recently (Atkins & Woolley 1970). The conventional hamiltonian is first derived by using Dirac’s generalized hamiltonian theory in which the Coulomb gauge condition is introduced as a supplementary condition. We analyse further the interpretation of the unitary transformation that connects the two hamiltonians in terms of the arbitrariness of the phase of the wavefunctions of charged particles in the presence of the electromagnetic field, and finally examine the problem of exhibiting explicitly the binding energies of the molecular systems.

The Green function technique of Edwards (1962) is used to obtain the energy spectrum and density of states for an electron moving in a binary alloy with local order, that is, with a non random distribution of A and B atoms. The model consists of an electron with an effective mass, perturbed by a potential which fluctuates from site to site and has zero average value. The degree of local order in this fluctuating potential is expressed approximately in terms of a single parameter alpha . Calculations for a 50-50 alloy show that the reciprocal lifetime changes greatly as alpha moves from negative (anticlustering) through zero (random) to positive values (clustering). This change should be observable in resistance measurements.

The scattering matrix techniques developed in a previous series of papers (Atkins & Barron 1968 a , b ) are applied to the problem of formulating a theory of optical activity on the basis of Kirkwood’s independent group model. We restrict ourselves to the case of non-degenerate subsystems interacting in the dipole approximation, though those restrictions are convenient simplifications, not essential features. Retardation effects are explicitly evaluated and shown to be negligible over relevant intramolecular distances. The translation invariance of the theory is demonstrated. The unique feature of the theory is that unlike previous theories it is developed without reference to the Rosenfeld equation for the rotation angle. Furthermore, it provides a physically appealing interpretation of the ‘coupled oscillator’ terms in the expression.

A relationship between the work of Barron and Gray (see abstr. A10321 of 1973) on the multipole Hamiltonian and the usual theory based on a unitary transformation is demonstrated.

Over the last few years there has been a growing awareness that the traditional formulation of quantum chemistry does not exhaust the possibilities for the application of quantum theory to chemical problems. This awareness has come about through a re-examination of the foundations of theoretical chemistry to which many have contributed in the last decade. Ten years ago, I encountered considerable hostility to my suggestion that the programme of conventional quantum chemistry is not just a simple consequence of setting out a molecular quantum theory if one starts from the Schrodinger equation for a system of interacting electrons and nuclei; today that is a much less controversial statement, and it is now widely recognized that classical molecular structure is problematic for a quantum theory of molecules. It is of course true that most of us use the procedures of conventional quantum chemistry in most of our work; nevertheless under the impact of new experimental techniques, especially those based on the use of laser radiation sources and supersonic molecular beams, new theoretical methods have come into quantum chemistry. An overriding concern with the determination of molecular structures has receded somewhat, and some new theoretical techniques in chemical physics make little or even no reference at all to molecular structure. Perhaps this increase in abstraction is no more than an illustration of the process foreseen for the development of theoretical science by the late P. A. M. Dirac in the opening pages of his famous book, The Principles of Quantum Mechanics.

One of the new and increasingly important activities of chemists and solid state physicists is the search for novel materials with interesting physical properties. This has academic interest and, of course, potential technological value. For an effective collaboration between chemists and physicists it may be essential that both can refer to a common set of theoretical concepts. In this review I shall propose a general classification of the electronic structure found in chemical substances and materials, crystalline and amorphous, including liquids; it is linked to their ground-state properties — electrical conductivity, heat capacity, magnetism and so on. I do not claim that the scheme is rigorous for there are certainly detailed technical points that need to be clarified, but I do believe it has at least heuristic value. Certainly the usual material in chemistry teaching texts concerning bonding in condensed phases is seriously deficient, and there is a challenge for chemists and material scientists to update their treatment of chemical bonding and electronic structure in liquids and solids. The aim of the review is to bring the conventional account of chemical bonding, familiar from structural chemistry, into sharper focus. The discussion will be confined to bulk materials that are ‘3-dimensional’, considered at some given pressure, temperature and chemical composition. Low-dimensional systems have some special features because of their low dimensionality and are of considerable current interest, but perhaps we should get to know more conventional bulk materials first.

A model is described for the calculation of the principal magnetic susceptibilities of exchange-coupled binuclear cobaltn species. The two-centre exchange calculation is performed within a basis constructed from appropriately antisymmetrized products of the one-centre functions belonging to the 4 F + 4 P terms. The exchange parameters refer to interactions between separate one-centre orbitals, in the spirit of Anderson’s kinetic and potential exchange pathways, so bearing some resemblance to the angular overlap parameters used to describe the one-centre, ligand-field perturbation.

Recent discussions of the thermodynamics of the electromagnetic field interacting with two-level atoms demonstrate the existence of a phase transition. It is argued however that this only holds for field theories that do not satisfy the principle of gauge invariance.

The information about molecular structure that may be obtained from a study of the intensity-dependence of the optical rotation angle is discussed. It is shown that there are two contributions: one depends upon the relative disposition and anisotropies of the polarisability and gyration tensors, whilst the other reflects the contribution of excited states of the molecule. The non-linear contribution is compared with the effect that an externally applied electric field would have on the rotation angle, and the relative advantages and disadvantages are discussed.