Non-Born-Oppenheimer molecular dynamics of Na⋯FH photodissociation

Frustrated charge transfer in vibrationally inelastic Ar++N2 collisions via hard collision glory scattering

Vibrational energy transfer in collisions between ions and neutrals is a fundamental process in interstellar media, planetary atmospheres, and plasmas. The conventional wisdom is that glancing collisions with large impact parameters are forward-scattered with low vibrational excitation, while hard collisions with small impact parameters are sideway- or backward-scattered with relatively high vibrational excitation. Here, we report experimental observations with a three-dimensional velocity-map imaging crossed-beam apparatus in the inelastic scattering process Ar++N2(v'' = 0, J'')→Ar++N2(v', J'), where all the vibrationally excited N2 products are dominated by forward scattering, contradicting the textbook model. Trajectory surface hopping calculations not only reproduced the experimental observation qualitatively, but also revealed that the vibrational excitation mainly occurs through a transient charge-transfer process. The hard collision glory mechanism, which has so far only been observed in inelastic rotational energy transfer between neutrals, is shown to play a major role for vibrational excitation in the inelastic Ar++N2 collision, via the frustrated charge transfer process.

Quantum state-to-state nonadiabatic dynamics of the charge transfer reaction H+ + NO(X2Π) → H + NO+(X1Σ+): Influence of ro-vibrational excitation of NO

Quantum state-to-state nonadiabatic dynamics of the charge transfer reaction H+ + NO(X2Π, vi = 1, 3, ji = 0, 1) → H + NO+(X1Σ+) has been studied based on the recently constructed diabatic potential energy matrix. It was found that the vibrational excitation of reactant NO inhibits the reactivity, while the rotational excitation of reactant NO has little effect on the reaction probability. These attributes were also observed in the semi-classical trajectory calculations employed in the adiabatic representation. Such an inhibitory effect of the vibrational excitation of reactant NO was owing to lower accessibility of the conical intersection and avoided crossing regions, which are located in the wells with respect to the Π diabat, as evidenced by the analysis of the population of the time-independent wave functions. Calculated vibrational state distributions of the product show that the decrease of the reaction mainly leads to the less formation of low vibrational states (vf < 6), and the product vibrational state distributions are more evenly populated for vi = 1 and 3, suggesting a non-statistical behavior. However, the overall shapes of the product rotational distributions remain unchanged, indicating that the redistribution of energy into the rotation of product NO is sufficient in the charge transfer process between H+ and NO. While the reaction is dominated by the forward and backward scattering in differential cross sections (DCSs), consistent with the complex-forming mechanism, a clear forward bias in the DCSs appears, indicating that the occurrence of the reaction is not sufficiently long to undergo the whole phase space of the interaction configurations.

Imaging the state-to-state charge-transfer dynamics between the spin-orbit excited Ar+(2P1/2) ion and N2

Ar++N2 → Ar+N2+ has served as a paradigm for charge-transfer dynamics studies during the last several decades. Despite significant experimental and theoretical efforts on this model system, state-resolved experimental investigations on the microscopic charge-transfer mechanism between the spin-orbit excited Ar+(2P1/2) ion and N2 have been rare. Here, we measure the first quantum state-to-state differential cross sections for Ar++N2 → Ar+N2+ with the Ar+ ion prepared exclusively in the spin-orbit excited state 2P1/2 on a crossed-beam setup with three-dimensional velocity-map imaging. Trajectory surface-hopping calculations qualitatively reproduce the vibrationally dependent rotational and angular distributions of the N2+ product. Both the scattering images and theoretical calculations show that the charge-transfer dynamics of the spin-orbit excited Ar+(2P1/2) ion differs significantly from that of the spin-orbit ground Ar+(2P3/2) when colliding with N2. Such state-to-state information makes quantitative understanding of this benchmark charge-transfer reaction within reach.

Investigations of the reaction mechanism of sodium with hydrogen fluoride to form sodium fluoride and the adsorption of hydrogen fluoride on sodium fluoride monomer and tetramer

CONTEXT

The reaction between Na and HF is a typical harpooning reaction which is of great interest due to its significance in understanding the elementary chemical reaction kinetics. This work aims to investigate the detailed reaction mechanisms of sodium with hydrogen fluoride and the adsorption of HF on the resultant NaF as well as the (NaF)4 tetramer. The results suggest that the reaction between Na and HF leads to the formation of sodium fluoride salt NaF and hydrogen gas. Na interacts with HF to form a complex HF···Na, and then the approaching of F atom of HF to Na results in a transition state H···F···Na. Accompanied by the broken of H-F bond, the bond forms between F and Na atoms as NaF, then the product NaF is yielded due to the removal of H atom. The resultant NaF can further form (NaF)4 tetramer. The interaction of NaF with HF leads to the complex NaF···HF; the form I as well as II of (NaF)4 can interact with HF to produce two complexes (i.e., (NaF)4(I-1)···HF, (NaF)4(I-2)···HF, (NaF)4(II-1)···HF and (NaF)4(II-2)···HF), but the form III of (NaF)4 can interact with HF to produce only one complex (NaF)4(III)···HF. These complexes were explored in terms of noncovalent interaction (NCI) and quantum theory of atoms in molecules (QTAIM) analyses. NCI analyses confirm the existences of attractive interactions in the complexes HF···Na, NaF···HF, (NaF)4(I-1)···HF, (NaF)4(I-2)···HF, (NaF)4(II-1)···HF and (NaF)4(II-2)···HF, and (NaF)4(III)···HF. QTAIM analyses suggest that the F···Na interaction forms in the HF···Na complex while the F···H hydrogen bonds form in NaF···HF, (NaF)4(I-1)···HF, (NaF)4(I-2)···HF, (NaF)4(II-1)···HF and (NaF)4(II-2)···HF, and (NaF)4(III)···HF complexes. Natural bond orbital (NBO) analyses were also applied to analyze the intermolecular donor-acceptor orbital interactions in these complexes. These results would provide valuable insight into the chemical reaction of Na and HF and the adsorption interaction between sodium fluoride salt and HF.

METHODS

The calculations were carried out at the M06-L/6-311++G(2d,2p) level of theory which were performed using the Gaussian16 program. Intrinsic reaction coordinate (IRC) calculations were carried out at the same level of theory to confirm that the obtained transition state was true. The molecular surface electrostatic potential (MSEP) was employed to understand how the complex forms. Quantum theory of atoms in molecules (QTAIM) and noncovalent interaction (NCI) analysis was used to know the topology parameters at bond critical points (BCPs) and intermolecular interactions in the complex and intermediate. The topology parameters and the BCP plots were obtained by the Multiwfn software.

Nonadiabatic Dynamics Simulations for Photoinduced Processes in Molecules and Semiconductors: Methodologies and Applications

Nonadiabatic dynamics (NAMD) simulations have become powerful tools for elucidating complicated photoinduced processes in various systems from molecules to semiconductor materials. In this review, we present an overview of our recent research on photophysics of molecular systems and periodic semiconductor materials with the aid of ab initio NAMD simulation methods implemented in the generalized trajectory surface-hopping (GTSH) package. Both theoretical backgrounds and applications of the developed NAMD methods are presented in detail. For molecular systems, the linear-response time-dependent density functional theory (LR-TDDFT) method is primarily used to model electronic structures in NAMD simulations owing to its balanced efficiency and accuracy. Moreover, the efficient algorithms for calculating nonadiabatic coupling terms (NACTs) and spin-orbit couplings (SOCs) have been coded into the package to increase the simulation efficiency. In combination with various analysis techniques, we can explore the mechanistic details of the photoinduced dynamics of a range of molecular systems, including charge separation and energy transfer processes in organic donor-acceptor structures, ultrafast intersystem crossing (ISC) processes in transition metal complexes (TMCs), and exciton dynamics in molecular aggregates. For semiconductor materials, we developed the NAMD methods for simulating the photoinduced carrier dynamics within the framework of the Kohn-Sham density functional theory (KS-DFT), in which SOC effects are explicitly accounted for using the two-component, noncollinear DFT method. Using this method, we have investigated the photoinduced carrier dynamics at the interface of a variety of van der Waals (vdW) heterojunctions, such as two-dimensional transition metal dichalcogenides (TMDs), carbon nanotubes (CNTs), and perovskites-related systems. Recently, we extended the LR-TDDFT-based NAMD method for semiconductor materials, allowing us to study the excitonic effects in the photoinduced energy transfer process. These results demonstrate that the NAMD simulations are powerful tools for exploring the photodynamics of molecular systems and semiconductor materials. In future studies, the NAMD simulation methods can be employed to elucidate experimental phenomena and reveal microscopic details as well as rationally design novel photofunctional materials with desired properties.

Use of QM/MM Surface Hopping Simulations to Understand Thermally Activated Rare-Event Nonadiabatic Transitions in the Condensed Phase

We implement a rare-event sampling scheme for quantifying the rate of thermally activated nonadiabatic transitions in the condensed phase. Our Quantum mechanics/molecular mechanics (QM/MM) methodology uses the recently developed Interface for NonAdiabatic QM/MM in Solvent (INAQS) package to interface an elementary electronic structure package and a popular open-source molecular dynamics software (GROMACS) to simulate an electron transfer event between two stationary ions in a solution of acetonitrile solvent molecules. Nonadiabatic effects are implemented through a surface hopping scheme, and our simulations allow further quantitative insight into the participation ratio of a solvent and the effect of ion separation distance as far as facilitating electron transfer. We also demonstrate that the standard gas-phase approaches for treating frustrated hops and velocity reversal must be refined when working in the condensed phase with many degrees of freedom. The code and methodology developed here can be easily expanded upon and modified to incorporate other systems and should provide a great deal of new insight into a wide variety of condensed phase nonadiabatic phenomena.

Imaging of the charge-transfer reaction of spin-orbit state-selected Ar+(2P3/2) with N2 reveals vibrational-state-specific mechanisms

Charge-transfer reactions are ubiquitous and play important roles in various gaseous environments, but, despite a long history of extensive research, our understanding of their dynamics at the quantum state-to-state level is still lacking. Here we report quantum-state-resolved experiments for the paradigmatic charge-transfer reaction Ar+ + N2 → Ar + N2+ using a three-dimensional velocity-map imaging crossed-beam apparatus with the Ar+ beam prepared exclusively in the spin-orbit state 2P3/2. High-resolution scattering images show strong dependence of rotational and angular distributions on the vibrational quantum number of the N2+ product. Trajectory surface-hopping calculations, which semi-quantitatively reproduce the experimental observations, support the existence of two distinct charge-transfer mechanisms. One of these, in the dominant N2+(v' = 1) channel, is the well-known long-distance harpooning mechanism. However, the highly rotationally excited products in the forward direction are attributed to a hard-collision glory scattering mechanism, which occurs on account of the strong attraction between the collisional partners counterbalanced by the short-range repulsive interaction.

New ab initio potential energy surface of NaFH (1A') system and quantum dynamics studies for the Na + HF (v, j) → NaF + H reaction

A global potential energy surface (PES) for the electronic ground state of the Na + HF reactive system is constructed by three-dimensional cubic spline interpolation of 37 000 ab initio points obtained using the multireference configuration interaction method including the Davidson's correction (MRCI + Q) with auc-cc-pV5Z basis set. The endoergicity, well depth and properties of the separated diatomic molecules are in good agreement with experimental estimations. Quantum dynamics calculations have been performed and compared with those of the previous MRCI PES as well as experimental values. The better agreement between theory and experiment indicates the accuracy of the new PES.

New Gradient Correction Scheme for Electronically Nonadiabatic Dynamics Involving Multiple Spin States

It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.

Direct Nonadiabatic Dynamics of Ammonia with Curvature-Driven Coherent Switching with Decay of Mixing and with Fewest Switches with Time Uncertainty: An Illustration of Population Leaking in Trajectory Surface Hopping Due to Frustrated Hops

Mixed quantum-classical nonadiabatic dynamics is a widely used approach to simulate molecular dynamics involving multiple electronic states. There are two main categories of mixed quantum-classical nonadiabatic dynamics algorithms, namely, trajectory surface hopping (TSH) in which the trajectory propagates on a single potential energy surface, interrupted by hops, and self-consistent-potential (SCP) methods, such as semiclassical Ehrenfest, in which propagation occurs on a mean-field surface without hops. In this work, we will illustrate an example of severe population leaking in TSH. We emphasize that such leaking is a combined effect of frustrated hops and long-time simulations that drive the final excited-state population toward zero as a function of time. We further show that such leaking can be alleviated-but not eliminated-by the fewest switches with time uncertainty TSH algorithm (here implemented in the SHARC program); the time uncertainty algorithm slows down the leaking process by a factor of 4.1. The population leaking is not present in coherent switching with decay of mixing (CSDM), which is an SCP method with non-Markovian decoherence included. Another result in this paper is that we find very similar results with the original CSDM algorithm, with time-derivative CSDM (tCSDM), and with curvature-driven CSDM (κCSDM). Not only do we find good agreement for electronically nonadiabatic transition probabilities but also we find good agreement of the norms of the effective nonadiabatic couplings (NACs) that are derived from the curvature-driven time-derivative couplings as implemented in κCSDM with the time-dependent norms of the nonadiabatic coupling vectors computed by state-averaged complete-active-space self-consistent field theory.

Decoherence and Its Role in Electronically Nonadiabatic Dynamics

Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.

Nonadiabatic Reactive Quenching of OH(A2Σ+) by H2: Origin of High Vibrational Excitation in the H2O Product

The nonadiabatic dynamics of the reactive quenching channel of the OH(A²Σ⁺) + H₂/D₂ collisions is investigated with a semiclassical surface hopping method, using a recently developed four-state diabatic potential energy matrix (DPEM). In agreement with experimental observations, the H₂O/HOD products are found to have significant vibrational excitation. Using a Gaussian binning method, the H₂O vibrational state distribution is determined. The preferential energy disposal into the product vibrational modes is rationalized by an extended Sudden Vector Projection model, in which the h and g vectors associated with the conical intersection are found to have large projections with the product normal modes. However, our calculations did not find significant insertion trajectories, suggesting the need for further improvement of the DPEM.

Semiclassical Trajectory Studies of Reactive and Nonreactive Scattering of OH(A 2S+) by H2 Based on an Improved Full-Dimensional Ab Initio Diabatic Potential Energy Matrix

We present a new full-dimensional diabatic potential energy matrix (DPEM) for electronically nonadiabatic collisions of OH( A 2 Σ + ) with H 2 , and we calculate the probabilities of electronically adiabatic inelastic collisions, nonreactive quenching, and reactive quenching to form H 2 O + H. The DPEM was fitted using a many-body expansion with permutationally invariant polynomials in bond-order functions to represent the many-body part. The dynamics calculations were carried out with the fewest-switches with time uncertainty and stochastic decoherence (FSTU/SD) semiclassical trajectory method. We present results both for head-on collisions (impact parameter b equal to zero) and for a full range of impact parameters. The results are compared to experiment and to earlier FSTU/SD and quantum dynamics calculations with a previously published DPEM. The various theoretical results all agree that nonreactive quenching dominates reactive quenching, but there are quantitative differences between the two DPEMs and between the b = 0 results and the all- b results, especially for the probability of reactive quenching.

Coupled- and Independent-Trajectory Approaches Based on the Exact Factorization Using the PyUNIxMD Package

We present mixed quantum-classical approaches based on the exact factorization framework. The electron-nuclear correlation term in the exact factorization enables us to deal with quantum coherences by accounting for electronic and nuclear nonadiabatic couplings effectively within classical nuclei approximation. We compare coupled- and independent-trajectory approximations with each other to understand algorithms in description of the bifurcation of nuclear wave packets and the correct spatial distribution of electronic wave functions along with nuclear trajectories. Finally, we show numerical results for comparisons of coupled- and independent-trajectory approaches for the photoisomerization of a protonated Schiff base from excited state molecular dynamics (ESMD) simulations with the recently developed Python-based ESMD code, namely, the PyUNIxMD program.

Full-dimensional quantum stereodynamics of the non-adiabatic quenching of OH(A2Σ+) by H2

The Born–Oppenheimer approximation, assuming separable nuclear and electronic motion, is widely adopted for characterizing chemical reactions in a single electronic state. However, the breakdown of the Born–Oppenheimer approximation is omnipresent in chemistry, and a detailed understanding of the non-adiabatic dynamics is still incomplete. Here we investigate the non-adiabatic quenching of electronically excited OH(A²Σ⁺) molecules by H₂ molecules using full-dimensional quantum dynamics calculations for zero total nuclear angular momentum using a high-quality diabatic-potential-energy matrix. Good agreement with experimental observations is found for the OH(X²Π) ro-vibrational distribution, and the non-adiabatic dynamics are shown to be controlled by stereodynamics, namely the relative orientation of the two reactants. The uncovering of a major (in)elastic channel, neglected in a previous analysis but confirmed by a recent experiment, resolves a long-standing experiment–theory disagreement concerning the branching ratio of the two electronic quenching channels.

The breakdown of the Born–Oppenheimer approximation is omnipresent in chemistry and detailed understanding of non-adiabatic dynamics is still incomplete. Now, the non-adiabatic quenching of electronically excited OH(A²Σ⁺) molecules by H₂ has been investigated using full-dimensional quantum dynamics calculations and a high-quality diabatic-potential-energy matrix, providing insight into the branching ratio of the two electronic quenching channels.

Direct coherent switching with decay of mixing for intersystem crossing dynamics of thioformaldehyde: The effect of decoherence

We evaluate the effect of electronic decoherence on intersystem crossing in the photodynamics of thioformaldehyde. First, we show that the state-averaged complete-active-space self-consistent field electronic structure calculations with a properly chosen active space of 12 active electrons in 10 active orbitals can predict the potential energy surfaces and the singlet-triplet spin-orbit couplings quite well for CH₂S, and we use this method for direct dynamics by coherent switching with decay of mixing (CSDM). We obtain similar dynamical results with CSDM or by adding energy-based decoherence to trajectory surface hopping, with the population of triplet states tending to a small steady-state value over 500 fs. Without decoherence, the state populations calculated by the conventional trajectory surface hopping method or the semiclassical Ehrenfest method gradually increase. This difference shows that decoherence changes the nature of the results not just quantitatively but qualitatively.

Quantum Dynamics Calculations of Na (32S, 32P) + HF → NaF + H Reactions

The dynamics of the reactive scatterings of the ground (Na(3S)) and first excited (Na(3P)) state sodium atoms from hydrogen fluoride (HF) molecules is investigated by performing comprehensive Coriolis-coupled quantum wave packet calculations on the recent ab initio potential energy surface (PES). In the Na(3S) + HF reaction, the nonadiabatic effect is found to be negligibly small, and the reactivity is significantly promoted by the initial vibrational excitations being in line with the available experimental result. Excellent quantitative agreement between theory and experiment is also achieved for the initial state specified integral cross sections and rate constants for v ≥ 3 vibrational states. However, the calculated rate constant for v = 2 significantly underestimates the experimental result. The possible cause for the disagreement is discussed in detail. For the Na(3P) + HF scattering, which can lead to the formation of either the ground state NaF + H product or Na(3S) + HF reactant via the harpooning process, the calculated result for the integral cross section shows excellent agreement with the available experimental result indicating the reasonable accuracy of the interstate coupling term of the employed PES.

Recent Progress in Surface Hopping: 2011-2015

Developed 25 years ago, Tully's fewest switches surface hopping (FSSH) has proven to be the most popular approach for simulating quantum-classical dynamics in a broad variety of systems, ranging from the gas phase, to the liquid and solid phases, to biological and nanoscale materials. FSSH is widely adopted as the fundamental platform to introduce modifications as needed. Significant progress has been made recently to enhance the accuracy and efficiency of the surface hopping technique. Various limitations of the standard FSSH-associated with quantum nuclear effects, interference and decoherence, trivial or "unavoided" crossings, superexchange, and representation dependence-have been lifted. These advances are needed to allow one to treat many important phenomena in chemistry, physics, materials, and related disciplines. Examples include charge transport in extended systems such as organic solids, singlet fission in molecular aggregates, Auger-type exciton multiplication, recombination and relaxation in quantum dots and other nanoscale materials, Auger-assisted charge transfer, nonradiative luminescence quenching, and electron-hole recombination. This Perspective summarizes recent advances in the surface hopping formulation of nonadiabatic dynamics and provides an outlook on the future of surface hopping.

Scaling relationships for nonadiabatic energy relaxation times in warm dense matter: toward understanding the equation of state

The dependence of nonadiabatic ion-electron energy transfer rates in warm dense aluminum on the mass density and temperature with decoherence changing this relationship qualitatively.

Does Nonadiabatic Transition State Theory Make Sense Without Decoherence?

We analyze thermal rate constants as computed with surface hopping dynamics and resolve certain inconsistencies that have permeated the literature. On one hand, according to Landry and Subotnik (J. Chem. Phys. 2012, 137, 22A513), without decoherence, direct dynamics with surface hopping overestimates the rate of relaxation for the spin-boson Hamiltonian. On the other hand, according to Jain and Subotnik (J. Chem. Phys. 2015, 143, 134107), without decoherence, a transition state theory with surface hopping underestimates spin-boson rate constants. In this Letter, we resolve this apparent contradiction. We show that, without decoherence, direct dynamics and transition state theory should not agree; agreement is guaranteed only with decoherence. We also show that, even though the effects of decoherence may be hidden for isoenergetic reactions, these decoherence failures are exposed for exothermic reactions. We believe these lessons are essential when interpreting surface hopping papers published in the literature without any decoherence corrections.

Fewest Switches Surface Hopping in Liouville Space

The novel approach to nonadiabatic quantum dynamics greatly increases the accuracy of the most popular semiclassical technique while maintaining its simplicity and efficiency. Unlike the standard Tully surface hopping in Hilbert space, which deals with population flow, the new strategy in Liouville space puts population and coherence on equal footing. Dual avoided crossing and energy transfer models show that the accuracy is improved in both diabatic and adiabatic representations and that Liouville space simulation converges faster with the number of trajectories than Hilbert space simulation. The constructed master equation accurately captures superexchange, tunneling, and quantum interference. These effects are essential for charge, phonon and energy transport and scattering, exciton fission and fusion, quantum optics and computing, and many other areas of physics and chemistry.

Improving the efficiency of Monte Carlo surface hopping calculations

A surface hopping method with a Monte Carlo procedure for deciding whether to hop at each step along the classical trajectories used in the semiclassical calculation is discussed. It is shown for a simple one-dimensional model problem that the numerical efficiency of the method can be improved by averaging over several copies of the sections of each trajectory that span the interaction regions. The use of Sobol sequences in the selection of the initial momentum for the trajectories is also explored. It is found that accurate results can be obtained with relatively small trajectory samples.

Investigation of the molecular mechanisms of electronic decoherence within a quinone cofactor

The notion of decoherence is particularly adapted to discuss the quantum-to-classical transition in the context of chemical reactions. Decoherence can be modeled by computing the time evolution of nuclear wave packets evolving on distinct potential energy surfaces, here using density functional theory (DFT) and Born–Oppenheimer molecular dynamics simulations. We investigate a redox cofactor of biological interest (tryptophan tryptophylquinone, TTQ) found in the enzyme methylamine dehydrogenase. We also report the first systematic comparison of semi-empirical DFT (tight-binding DFT) and classical force field approaches for estimating decoherence in molecular systems. In the TTQ cofactor, we find that decoherence combines structural and dynamical aspects: it is initiated by the divergent motions of few atoms and then propagates dynamically to the remaining atoms. It is the mass effect of all the atoms that leads to decoherence within a few femtosecond.

Adiabatic and nonadiabatic contributions to the energy of a system subject to a time-dependent perturbation: complete separation and physical interpretation

When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.