Abstract The present day quantum chemistry is mathematically based on von NEUMANN'S formulation of quantum mechanics and therefore not able to deal correctly with anomal quantum states. A general theory of large molecules and large molecular systems has to be founded on quantum field theory allowing for inequivalent irreducible representations. It is proposed to formulate quantum chemistry in the framework of the B*‐algebra of the observables. The paradox of molecular systems having an infinite number of degrees of freedom is discussed, and the possible significance of anomal molecular quantum states for molecular biology is pointed out.
Abstract The present day quantum chemistry is mathematically based on von NEUMANN'S formulation of quantum mechanics and therefore not able to deal correctly with anomal quantum states. A general theory of large molecules and large molecular systems has to be founded on quantum field theory allowing for inequivalent irreducible representations. It is proposed to formulate quantum chemistry in the framework of the B*‐algebra of the observables. The paradox of molecular systems having an infinite number of degrees of freedom is discussed, and the possible significance of anomal molecular quantum states for molecular biology is pointed out.
Abstract Quantum mechanical and classical aspects are of equal importance in chemistry, and theoretical chemistry must be able to deal with this situation. Classical quantities with which a chemist has to deal include, for example, the temperature and the chemical potential of a substance, and chirality, knot‐type and tertiary structures of molecules. Classical concepts of this type are no more consequences of traditional quantum chemistry than is the theory of complex chemical processes (for example adsorption, heterogeneous catalysis) and macroscopic systems. Algebraic quantum mechanics affords a general framework for the discussion of classical quantities and large systems. There are other independent approaches for the theoretical treatment of chemical problems; these can be summarized by using the keywords “fractals”, “chaotic motion”, “quasicrystals” and “knot theory”. The relationships between these four research topics and their connection with algebraic quantum mechanics, as well as their importance for theoretical chemistry, were discussed at a NATO Workshop in 1987 (see Ref. [1]). In an extended theoretical chemistry, the notion of “molecules” will not play a role as central as it does in quantum chemistry. In the theory of large systems in particular, the introduction of complementary approaches appears to be appropriate; the term “molecule” as used by the chemist does not appear either in the description of disordered systems by means of fractals or in the thermodynamic limit of algebraic statistical mechanics.
Abstract The molecular N‐electron problem is formulated in terms of the particle density 𝔔(r) = ∑σ(r − qn) and its canonically conjugated field. Weyl's representation of the canonical commutation relations is used and a characteristic functional of the molecular system is introduced as the expectation value of Weyl's functional exp {i(α, Q) + i(β, P)}. The Hamiltonian equation of motion for the characteristic functional is derived and its cumulant approximation is discussed. The first paper is restricted to a general discussion, independent of a special choice of the Hamiltonian. The density functional representation may be considered as a step towards a formulation of a renormalized quantum chemistry. The lowest order cumulants of the characteristic functional are accessible to actual precision measurements; on the other hand, most experiments are correctly represented by a characteristic functional of the Gaussian type. The significance of the higher order correlations is neither theoretically nor experimentally disclosed yet, their possible importance for biological systems is mentioned, and the question is raised of how far quantum mechanics is empirically confirmed for extremely complex systems.
Abstract A comprehensive review of the attempts to rephrase molecular quantum mechanics in terms of the particle density operator and the current density or phase density operator is given. All pertinent investigations which have come to our attention suffer from severe mathematical inconsistencies and are not adequate to the few‐body problem of quantum chemistry. The origin of the failure of these attempts is investigated and it is shown that a realization of a local quantum field theory of molecular matter in terms of observables would presuppose the solution of many highly nontrivial mathematical problems.
Abstract The classical concept of molecular structure (namely, a set of atoms with a well‐defined geometrical arrangement in space) originated well before the advent of quantum theory, and, contrary to widespread views, these two conceptual schemes (classical molecular structure and quantum theory) are not easy to reconcile. We begin with a preliminary discussion of molecular structure in terms of space and time correlations in the framework of quantum theory. Then we summarize the salient features of some critical views raised‐during recent years (essentially by Woolley) concerning the concept of molecular structure. The essential claim of this viewpoint is that “molecular structure” is not an intrinsic attribute of an isolated molecule, and appears rather as an effect of the environment (other molecules? vacuum electromagnetic field, according to quantum electrodynamics?). Afterwards, we discuss these issues in detail: we show that it is necessary to distinguish between a “quantum” or “potential” structure (which may be deduced in a straightforward way from the quantum treatment of the molecule), and the “classical” molecular structure (according to which the molecule is considered as a set of rotating and vibrating material points, corresponding to the atoms); we briefly discuss some aspects of the Born–Oppenheimer approximation. From our discussion, we deduce that the “classical structure” concept is indeed non‐trivial from a purely quantum‐mechanical point of view; in actual fact, it is related to the problem of the “classical limit” of quantum mechanics, and it appears that, at the present time, none of these classical concepts and behaviours is convincingly derivable from a strict quantum mechanical basis. Finally, we consider these same problems from the point of view of stochastic electrodynamics (SED), which is one of the stochastic models for microphysics proposed so far as a possible alternative or extension of usual quantum theory. In the framework of such a theory, the “classical limit” and, in particular, the “classical molecular structure” no longer poses a problem, but aside from these satisfactory qualitative aspects, several quantitative failures of SED prevent us from accepting it, at least in its present state, as a genuine physical alternative to quantum theory. We therefore conclude that “classical molecular structure”, and classical concepts in general, remain an open problem in the framework of present‐day quantum physics.
Bohm theory is a formulation of Quantum Mechanics that characterises the state of a quantum system according to both the wave function, as in the conventional formulation, and the coordinates (positions) of all the particles that evolve in time drawing quantum continuous trajectories. Furthermore, a statistical ensemble of all the possible trajectories, raising from the impossibility to know the initial position of all the particles, establishes the exact correspondence with the traditional Quantum Mechanics. From a computational point of view, Bohm theory has found many applications in Chemical Physics especially to develop new methodologies for solving the Schrödinger equation and to address semi-classical approximations of Quantum Mechanics. From a theoretical point of view, the most appealing feature of Bohm theory is its capability to supply a conceptual map between the quantum formalism and our representation of what a chemical system is. Chemical systems are composed of molecules, but the same idea of molecule requires a specific arrangement in the space of particles, i.e., the nuclei of the atoms. The statistical description of conventional Quantum Mechanics on the basis of wave function alone is insufficient to establish a clear correspondence with such a picture of molecules. Indeed, chemists employ usually Classical Mechanics in order to overcome this drawback of the standard quantum theory. On the other hand, if the particles position is included in the quantum formalism, as Bohm theory does, the map can be defined in a self-consistent way. In other words, Bohm theory appears to be the suitable quantum framework to represent molecules and their motion. The chemical representation of molecular systems finds a natural correspondence with a single Bohm trajectory, since it is always implicitly assumed that molecular components have specific spatial position independently of our knowledge about it. Consequently, we develop a quantum method whose fundamental assumption is that a single Bohm trajectory, i.e., a quantum molecular trajectory, describes the molecular systems and the molecular motion correctly. First of all, we examine the correspondence between a single Bohm trajectory and the conventional Quantum Mechanics, without using the ensemble of trajectories. We verify that such a correspondence exists through numerical simulations and we prove formally that the statistical properties of a single Bohm trajectory explain the probabilistic description of Quantum Mechanics. Once the consistency of this original approach has been established, we investigate the predicted properties. For instance, we take into account the constants of motion (such as the energy) corresponding to the time evolution of the coordinates and the behaviour of simple chemical systems, e.g., the vibrational motion of single molecules interacting with a resonant field. In this way, unexpected features of the molecular motion are found. Secondly, we tackle the challenge of describing many components systems (like the chemical systems in ordinary conditions). As a matter of fact, the computation of the Bohm trajectory and of the wave function is extremely demanding. However, the statistical properties of the Bohm trajectory allow the derivation of stochastic theories for examining the dynamics of open quantum systems, i.e., few molecules (or few degrees of freedom) interacting with their environment (the other molecules). One of the developed stochastic methods correlates the dynamics of the reduced density matrix, for the degrees of freedom of interest, to the evolution of the corresponding Bohm coordinates. In other words, the Bohm equation, determining the set of all the particles velocities according to the full wave function, is replaced with a stochastic one that approximates the velocity of a subset of coordinates according to the reduced density matrix. In such a way, the quantum fluctuations induced by the environment are taken into account. The advantage of this method concerns its capability of describing quantum systems, including open quantum systems, in terms of a quantum trajectory. This could allow the understanding of the molecular motion during a spectroscopical experiment. The possibility of investigating reactive systems, such as conformational changes, is particularly interesting. As a matter of fact, chemical reactions can be completely characterised only through the particles motion and we define the suit- able quantum methodology providing a self-consistent description of the molecular motion.
Abstract A mathematically rigorous reformulation of molecular quantum mechanics in terms of the particle density operator and a canonically conjugated phase field is given. Using a momentum cutoff, it is shown that the usual molecular Hamiltonian can be expressed in terms of the particle density operator and a rigorously defined phase operator. It is shown that this Hamiltonian converges strongly to the cutoff‐free Hamiltonian. In spite of the fact that this Hamiltonian is of second order in the phase operators, all hitherto published expressions are not correct. Unfortunately, the correct formulation destroys the intuitive appeal of using the particle density operator as a coordinate for the many‐body problems of quantum chemistry. Unless somebody provides an essential new and clever idea, we propose to resist the fascination of a local quantum field theory of molecular matter in terms of the particle density operator.