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Journal of Chemical Theory and Computation

Lossless and Lossy Characterization of the State of Perturbed Anharmonic Diatomics: An Information-Theoretic Compaction of Quantum Dynamics

James Russell Hamilton, Raphael David Levine

Published Feb 18, 2026• 0 citations
Publisher

Abstract

A lossless, exact compaction of the time-evolved state of the quantum dynamical system of a perturbed anharmonic molecule is demonstrated using dynamical symmetries. The density matrix of the anharmonic molecule is a linear combination of these symmetries, and it remains so as a time-dependent perturbation is applied. Accurate, unitary-but-approximate, and thereby irreversible compaction is further shown using fewer symmetries, and the fidelity of this lossy compaction is quantified. Perturbations are typically linear in the operators of a Lie algebra. For a Hamiltonian that is also linear, one knows well how to reversibly compact the state of a dynamical system. However, anharmonic vibrations have a finite number of unequally spaced energy levels, and a good description of their spectra typically requires an algebraic-type Hamiltonian that is bilinear in the operators of a Lie algebra. For a bilinear Hamiltonian we show how a matrix-based approach allows us to compact both the populations and the coherences, either exactly reversibly or inexactly irreversibly, with fewer symmetries. A forced Morse oscillator is used as an explicit analytical and numerical example covering the entire range of dynamics from the sudden to the adiabatic limits.

DOI
10.1021/acs.jctc.6c00025

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A lossless, exact compaction of the time-evolved state of the quantum dynamical system of a perturbed anharmonic molecule is demonstrated using dynamical symmetries. The density matrix of the anharmonic molecule is a linear combination of these symmetries, and it remains so as a time-dependent perturbation is applied. Accurate, unitary-but-approximate, and thereby irreversible compaction is further shown using fewer symmetries, and the fidelity of this lossy compaction is quantified. Perturbations are typically linear in the operators of a Lie algebra. For a Hamiltonian that is also linear, one knows well how to reversibly compact the state of a dynamical system. However, anharmonic vibrations have a finite number of unequally spaced energy levels, and a good description of their spectra typically requires an algebraic-type Hamiltonian that is bilinear in the operators of a Lie algebra. For a bilinear Hamiltonian we show how a matrix-based approach allows us to compact both the populations and the coherences, either exactly reversibly or inexactly irreversibly, with fewer symmetries. A forced Morse oscillator is used as an explicit analytical and numerical example covering the entire range of dynamics from the sudden to the adiabatic limits.
We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time point of interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using singular value decomposition, SVD, of this matrix we generate realistic approximations for the time-independent observables and their respective time dependent coefficients. This allows a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A nonstationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive analytical exact expression for the surprisal as a function of time and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, the H2 and I2 molecules. We further demonstrate the approach for a two coupled electronic states problem, the well studied non radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.
High Resolution Image Download MS PowerPoint Slide A practical approach is put forward for a compact representation of the time evolving density matrix of the forced Morse oscillator. This approach uses the factorized product form of the unitary time evolution operator, à la Wei-Norman. This product form casts the time evolution operator in the basis of operators that form a closed Lie algebra. The further requirement that the Hamiltonian of the system be closed within this Lie algebra is satisfied by restricting the dynamics to its sudden limit. One is thereby able to propagate in time both pure and mixed quantum states. As an example, for a thermal initial state, the time-evolved density matrix of maximum entropy is derived, and it is compacted to be described by only three explicit constraints: one time-dependent constraint, which is a dynamical symmetry, and two constants of the motion, with corresponding time-independent coefficients. This representation is a significant reduction from O ( j 2 ) constraints down to just three, where j is the number of bound states of the Morse oscillator.
We model an anharmonic diatomic molecule using deformed creation and annihilation operators such that the energy spectrum generated by a Hamiltonian of the harmonic oscillator's form written in terms of deformed operators is similar to that of a Morse potential. We construct an approximate time evolution operator and evaluate transition probabilities which are compared with those obtained by an expansion in a basis of Morse eigenfunctions. The algebraic results compare favorably with the numerical results.
Optical excitation in the dipole approximation and other probes couple two quantum states of an unaddressed finite quantum mechanical discrete system. Thereby the interaction of the system with the probe is bilinear in the coherence between the two states and in the time-dependent strength of the probe. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for a Lie algebra that can be represented as a direct product of individual two state systems each using the population and the coherence between two states. Dynamical symmetries can be used to advantage to describe the progress of such systems in time. They also offer a compact and efficient representation for a density matrix of maximal entropy that evolves in time. Using the factorization approach of Wei and Norman, (J. Wei and E. Norman, Lie Algebraic Solution of Linear Differential Equations, J. Math. Phys. 4, 575 (1963)), we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two state systems. Thereby one can propagate, to all orders in perturbation theory, both the wave function and the density matrix with special reference to dynamical symmetries.
We have studied quantum-mechanical behavior of a driven Morse oscillator coupled to a bath of harmonic oscillators. The purpose is to compare the quantum behavior of such a system with the classical solutions of a driven damped Morse oscillator. We start with the quantum Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an su(2) Lie algebra. This algebra and the Markovian approximation allow us to derive the generalized master equation (for the reduced density matrix of the Morse oscillator), which contains level-dependent energy and phase relaxation terms. We have numerically integrated the differential equations of the matrix elements to obtain the time evolution of the reduced density matrix and the energy of the Morse oscillator. We show that the energy of the Morse oscillator in general varies with time and eventually reaches an asymptotic oscillatory state. The mean energy value of the asymptotic oscillatory state is studied as a function of the relaxation rates and laser frequency and amplitude. Vibrational distributions have also been found as functions of laser frequency and amplitude. The shape of the distribution changes gradually as laser amplitude increases; it first peaks at the ground state at a small field amplitude, then it peaks at some excited state at a large field amplitude, and for an amplitude in between it can have two maxima. This bimodal vibrational distribution reflects the bistability observed in the classical and semiclassical models. Thus bistability exists in quantum results, not as a hysteresis loop, but as a bimodal distribution. Finally, we show that as laser intensity increases, the time series of the oscillator energy evolves from regular to quasiperiodic behavior, and eventually a chaotic-looking series appears.
In this paper, we consider the one-dimensional anharmonic oscillator, which represents well the anharmonic vibrations in diatomic molecules. For the description of the associate potential we use the Morse potential, which gives a good approximation of the experimentally observed vibrational modes of molecules and hence contributes to the realistic description of the spectrum of diatomic molecules. The generalized and Gaussian coherent states are thus constructed and compared in terms of the localization of the particle in those states. We apply these results to the example of the sodium chloride molecule, 1 H 35 Cl.