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Born-Oppenheimer Reduction of Quantum Evolution of Molecules

André Martínez, Vania Sordoni

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case where the electronic Hamiltonian admits a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order.

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We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case where the electronic Hamiltonian admits a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order.
We generalize the standard Born-Oppenheimer approximation to the case of open quantum systems. We define the zeroth-order Born-Oppenheimer approximation for an open quantum system as the regime in which its effective Hamiltonian can be diagonalized with fixed slowly changing variables. We then establish validity and invalidity conditions for this approximation for two types of dissipations---the spin relaxation and the dissipation of center-of-mass motion. As an example, the Born-Oppenheimer approximation for a two-level open system is analyzed.
A semiclassical method is discussed for propagating wave packets that incorporates nonlinear classical dynamical information through the use of multiple reference trajectories. We apply the theory to obtain an analytic approximation for the autocorrelation function in the Coulomb potential which is extremely accurate in many circumstances in spite of the Coulomb singularity. All the detailed quantum behavior is reproduced including the long time revivals and the spectrum. The evolving phase space structure of the classical analog reveals how well a particular case will be approximated.
A new approach to the quantum mechanical fundamentals of molecular physics is presented. We start by introducing the interaction representation for molecular systems in which the interaction between molecular fermions and bosons leads to the time evolution of molecular harmonic oscillators. Such a linear harmonic oscillator determines the metric tensor in the two-dimensional Riemann space and its time evolution occurs on a self-time curve which is a geodetic in the Riemann space. The molecular bosons in this self-time representation have the energy dispersion and the superfluid properties. The molecular spectra in the self-time representation acquire the line shape which results from nonlinearity of the geodetic and are related to dephasing between the nuclear and electronic motions. Dephasing between the nuclear and electronic motions leads to the retarded electrodynamic interactions in the form of the Liénard–Wiechert potentials. In the limit of the Born–Oppenheimer approximation, the electrodynamic interactions reduce to the purely electrostatic ones and the molecular line shape function to the Dirac δ function.
We analyze the entanglement between electronic and nuclear motions in molecular wave functions widely used by theoretical chemists, namely, (i) Born-Oppenheimer factorization in the adiabatic picture, (ii) the transformation into a diabatic picture, (iii) the use of a Born-Huang expansion, and (iv) the eigenfunction of the full molecular Hamiltonian. Our showcase is based on two one-electron one-dimensional molecular Hamiltonians (H2+ and the Shin-Metiu model). We find that within the Born-Oppenheimer approximation, any molecular state (although factorizable) is always entangled, and its entanglement content may be assessed by the variation of the electronic wave function along the different nuclear geometries, with the nuclear wave function indeed playing the role of a tester. The presence of avoided crossings among the adiabatic potential energy curves brings about dramatic changes in the entanglement content of the wave function: sharp avoided crossings favor a diabatic picture (real crossings between potential energy curves), while in broad avoided crossings, the adiabatic picture prevails. The total eigenfunction of the molecular Shin-Metiu Hamiltonian indicates that nuclear densities accommodate well within the diabatic curves for strong adiabatic couplings but within adiabatic curves for weak ones. Consequently, we find that the electron-nuclei entanglement content is a valid witness to unveil strong or weak nonadiabatic couplings in molecules. In terms of entanglement, we also find that the Born-Huang expansion, based on Born-Oppenheimer adiabatic electronic states, does not provide a correct trend of entanglement compared with that of the total molecular eigenfunction, thus indicating a very slow convergence of this expansion.
An expression is obtained for the electronic decoherence time of the reduced density electronic matrix in mixed quantum-classical molecular-dynamics simulations. The result is obtained by assuming that decoherence is dominated by the time dependence of the overlap of minimum-uncertainty packets and then maximizing the rate with respect to the parameters of the wave packets. The expression for the decay time involves quantities readily available in non-Born-Oppenheimer molecular-dynamics simulations, and it is shown to have a reasonable form when compared with two other formulas for the decay time that have been previously proposed.