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Molecular Physics

The Schrödinger group in molecular quantum mechanics: beyond the Born–Oppenheimer approximation

Andrzej Witkowski

Published Oct 17, 2011• 3 citations
Publisher

Abstract

The Galillei transformation invariant form of molecular quantum mechanics is obtained, which is not restricted by the Born–Oppenheimer approximation. The molecular Hamiltonian takes then the form of a linear combination of operators of the Schrödinger group, which define the new space–time molecular symmetry properties (e.g. helicity). The puzzling homochirality of the hydrogen bonded biomolecular systems appears then as a simple result of the molecular space-time symmetry.

DOI
10.1080/00268976.2011.617319

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The Galillei transformation invariant form of molecular quantum mechanics is obtained, which is not restricted by the Born–Oppenheimer approximation. The molecular Hamiltonian takes then the form of a linear combination of operators of the Schrödinger group, which define the new space–time molecular symmetry properties (e.g. helicity). The puzzling homochirality of the hydrogen bonded biomolecular systems appears then as a simple result of the molecular space-time symmetry.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations keep the Heisenberg inequalities invariant and form a group. They are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schrödinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction.
Abstract Philosophers have claimed that: (a) Born–Oppenheimer approximation methods for solving molecular Schrödinger equations violate the Heisenberg uncertainty relations; therefore, (b) ‘quantum chemistry’ is not fully quantum; and (c) therefore chemistry does not reduce to physics. This paper analyses the reasoning behind Born–Oppenheimer methods and shows that they are internally consistent and fully quantum mechanical, contrary to (a)–(c). Our analysis addresses important issues of mathematical rigour, physical idealization, reduction, and classicality in the quantum theory of molecules, and we propose an agenda for the philosophy of quantum chemistry more grounded in scientific practice.
The Schrödinger equation iℏ ∂ₜψ = Ĥψ, with non‑relativistic Hamiltonian Ĥ = ℏ²|p̂|²/(2m) + V(x̂), has been quantum mechanics’s central dynamical statement for nearly a century. It governs atomic spectra, molecular bonding, semiconductors, neutron interferometry, and countless other empirical settings. The question is why it has the form it does. In standard QM, the Schrödinger equation is a postulate. The Hilbert space is given. The momentum operator p̂ = −iℏ∇ is given. The kinetic form |p̂|²/(2m) is inherited directly from classical mechanics — quantum theory simply adopts the classical T = p²/(2m) structure without deeper justification. The first‑order time derivative is chosen because empirically it works. Each piece is accepted because the resulting equation matches experiment. This walkthrough shows how the Event Density framework derives the Schrödinger equation, in its full standard form, from a smaller set of structural commitments about what reality is made of at the substrate level. Every element — the Hilbert space, the momentum operator, linearity, first‑order time structure, the kinetic‑plus‑potential form, the factor of 1/(2m) — emerges as a forced consequence rather than a separate postulate. The derivation runs through Stone’s theorem applied twice — once on spatial translation (forcing p̂ = −iℏ∇) and once on time translation (forcing existence and uniqueness of a self‑adjoint generator with automatic linearity and first‑order structure) — closed by Galilean Lie algebra integration that identifies the time‑translation generator with the kinetic‑plus‑potential operator. The factor of 1/(2m) emerges as the chain‑rule Jacobian of the Galilean commutator [Ĥ, K̂ᵢ] = −iℏ p̂ᵢ, not as a borrowing from classical mechanics. The first‑order time derivative is a consequence of operator exponentials, not a chosen feature.
Galilei transformations are formulated in a 5-dimensional form and on such basis the non-relativistic quantum mechanics is reconstructed. Our approach has the following advantages over others'. (1) The Schrödinger wavefunctions are taken to be vector (not projective) representations of the Galilei group, so that (2) the problem here is considerably simplified. (3) The formulation proceeds in a way quite parallel to that of relativistic quantum mechanics, and what is most important, (4) Galilei covariance is manifest throughout.
Quantum chemistry aims to understand and predict molecular structure and properties by solving the time-independent Schr\"odinger equation within fixed nuclei (Born-Oppenheimer) approximation. The Hamiltonian may have continuous symmetries (e.g., spin) or discrete symmetries (for example, point group). Mean-field wave functions may break some or all of these symmetries. By taking the symmetry-adapted pieces of these broken symmetry states, one can restore the physical character of the wave function while also treating relevant strong correlations. On the other hand, mean-field methods lack dynamic correlation arising from short-range interactions. Still, single-reference methods such as configuration interaction or coupled cluster theory can handle these correlations efficiently. Ideally, we would like to combine these approaches: performing configuration interaction or coupled cluster theory on a broken symmetry reference while restoring the desired symmetries. This work introduces a new approach to integrating symmetry projection and coupled cluster theory. By borrowing techniques from previous research, the new method can restore continuous symmetries like UUU(1) and SUSUSU(2) and discrete symmetries like point group and time reversal. This method achieves accurate results in challenging systems and has the same computational scaling as traditional coupled cluster theory. Jordan-Wigner transformation is a potent tool in spin models, but its application is challenging due to the inherent strings. The same techniques we use to combine symmetry projection and coupled cluster theory can be used to combine Jordan-Wigner transformation and coupled cluster. The strings are the ' symmetry projection ' operator when constructing the Hartree-Fock and coupled cluster wave functions for a Jordan-Wigner transformed Hamiltonian. Leveraging the method of symmetry projection, we obtain excellent results for both 1-D and 2-D spin models.