Beyond the adiabatic regime, our understanding of quantum dynamics in coupled systems remains limited, and the choice of representation continues to obscure physical interpretation and simulation accuracy. Here we propose a natural and efficient basis for electron nuclear dynamics by drawing on the concepts of pointer and preferred states from decoherence theory, adapted to systems where electrons and nuclei interact strongly. Within this framework, we show that 1) the independent dynamics exploited by mixed quantum classical (MQC) methods is best understood as a manifestation of entanglement viewed in a preferred basis, rather than a consequence of decoherence, and 2) the adiabatic Born Oppenheimer states satisfy the conditions of an approximate preferred basis. This perspective reconciles widely used approximations with a more fundamental structure of the theory and provides a systematic route to more reliable MQC strategies. In effect, we revisit MQC methods through the lens of preferred states, clarifying when they succeed and how they can be improved.
Beyond the adiabatic regime, our understanding of quantum dynamics in coupled systems remains limited, and the choice of representation continues to obscure physical interpretation and simulation accuracy. Here we propose a natural and efficient basis for electron nuclear dynamics by drawing on the concepts of pointer and preferred states from decoherence theory, adapted to systems where electrons and nuclei interact strongly. Within this framework, we show that 1) the independent dynamics exploited by mixed quantum classical (MQC) methods is best understood as a manifestation of entanglement viewed in a preferred basis, rather than a consequence of decoherence, and 2) the adiabatic Born Oppenheimer states satisfy the conditions of an approximate preferred basis. This perspective reconciles widely used approximations with a more fundamental structure of the theory and provides a systematic route to more reliable MQC strategies. In effect, we revisit MQC methods through the lens of preferred states, clarifying when they succeed and how they can be improved.
Nuclear quantum phenomena beyond the Born-Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron-nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron-nuclear problem on near-term quantum computers, based upon the Nuclear-Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10-6 hartree and entanglement entropy to within 10-4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.
Quantum trajectory calculations for electrons are a useful tool in the field of molecular dynamics, e.g. to understand processes in ultrafast spectroscopy. They have, however, two limitation: On the one hand, such calculations are typically based on the Born-Oppenheimer approximation (BOA) and the electron dynamics for stationary nuclei is considered, thus neglecting quantum effects of the nuclei. On the other hand, even if the quantum nuclear motion would be taken into account, a BOA dynamics on a single potential energy surface would not provide any electron trajectories because the electronic part is treated as a stationary problem. By using the exact factorization method, we overcome these limitations and generalize the theory of electronic quantum trajectories to a fully quantum-mechanical treatment of the nuclei. After reviewing the time-dependent theory of quantum hydrodynamics and quantum trajectories, we show that the nuclei can be viewed as a quantum clock for the electronic motion and we develop a fully quantum-mechanical clock-dependent version of quantum hydrodynamics. This theory is used to obtain electronic trajectories for quantum nuclei, as is exemplified for a model system of a proton-coupled electron transfer dynamics. Our work generalizes the concept of quantum trajectories and lays the foundations for the development of trajectory-based simulation methods of electron dynamics beyond the BOA.
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.
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.
We present a novel quantum-classical approach to nonadiabatic dynamics, deduced from the coupled electronic and nuclear equations in the framework of the exact factorization of the electron-nuclear wave function. The method is based on the quasiclassical interpretation of the nuclear wave function, whose phase is related to the classical momentum and whose density is represented in terms of classical trajectories. In this approximation, electronic decoherence is naturally induced as an effect of the coupling to the nuclei and correctly reproduces the expected quantum behavior. Moreover, the splitting of the nuclear wave packet is captured as a consequence of the correct approximation of the time-dependent potential of the theory. This new approach offers a clear improvement over Ehrenfest-like dynamics. The theoretical derivation presented in this Letter is supported by numerical results that are compared to quantum mechanical calculations.
The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons and nuclei beyond the Born-Oppenheimer approximation. The case of crystalline solids is discussed in detail.