^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Performing molecular dynamics in electronically excited states requires the inclusion of nonadiabatic effects to properly describe phenomena beyond the Born-Oppenheimer approximation. This article provides a survey of selected nonadiabatic methods based on quantum or classical trajectories. Among these techniques, trajectory surface hopping constitutes an interesting compromise between accuracy and efficiency for the simulation of medium- to large-scale molecular systems. This approach is, however, based on non-rigorous approximations that could compromise, in some cases, the correct description of the nonadiabatic effects under consideration and hamper a systematic improvement of the theory. With the help of an

Traditionally,

Among all nonadiabatic AIMD schemes, Tully’s fewest switches trajectory surface hopping [

Alternative schemes have been proposed for the description of the nonadiabatic dynamics of the nuclear degrees of freedom, among which we quote semiclassical approaches [

Despite the success of the nonadiabatic trajectory-based approaches, there are many quantum mechanical phenomena that cannot be entirely captured within this framework, namely nuclear quantum effects, like wave packets interference [

One possible way to account for quantum nuclear effects within a trajectory-based framework consists in the use of quantum (or Bohmian) trajectories [

In this article, we review a number of trajectory-based nonadiabatic molecular dynamics schemes together with our recent work on nonadiabatic Bohmian dynamics. Our aim is to provide a unified picture of the field by trying to “derive” the different approaches starting from a common framework, namely the quantum hydrodynamics reformulation of the molecular time-dependent Schrödinger equation. In particular, we propose a classification of the different trajectory-based approaches based on the choice of the initial expansion of the molecular wavefunction (that depends on both the nuclear and the electron degrees of freedom) into a sum or a single product of electronic and nuclear wavefunctions. Finally, we propose a rationalization of the TSH equation of motion based on our exact nonadiabatic Bohmian dynamics scheme, showing by means of tests on two simple model systems the origin of some typical failures of TSH.

In this Section, we briefly review the theoretical background of the different nonadiabatic molecular dynamics schemes that we have selected for this study. The selection is based on the fact that all these trajectory-based approaches can be classified according to the way the molecular wavefunction is represented in terms of the electronic and nuclear components.

Starting from the Born-Huang representation of the total molecular wavefunction, we first introduce the Born-Oppenheimer molecular dynamics (BO-MD), which is based on the adiabatic separation of the electronic and nuclear dynamics, the latter being described by a single classical trajectory. Nonadiabaticity is then reintroduced following different strategies. In trajectory surface hopping (TSH), when the classical trajectories enter a region of strong coupling between different PESs, they are allowed to

In the second part of this review, we discuss nonadiabatic AIMD approaches that can be derived from a single product ansatz for the total molecular wavefunction. Two of these methods will be investigated, namely the approximated Ehrenfest dynamics and the exact solution, named “Exact Factorization”, which has recently been proposed by Gross and coworkers.

We begin by introducing the time-dependent Schrödinger equation (TDSE) for a molecular system, which, neglecting the nuclear and electronic spins, is given by:
_{1}_{k}, . . . ,_{Nel}_{el}_{1}_{γ}. . . ,_{Nn}_{n}_{γ}_{el}

The Born-Huang expansion gives an exact expression for the total wavefunction [

The total wavefunction, Ψ(_{el}_{i}_{j}_{ji}_{j}_{j}

The BO approximation consists in neglecting all off-diagonal terms, _{ji}_{j}_{j}_{jj}_{j}_{j}_{j}_{j}_{γ}S_{j}_{j}

One of the most successful methods for nonadiabatic dynamics is Tully’s trajectory surface hopping [

In this section, we only give a brief introduction to TSH, while a more detailed description of the method is given in Section 3, where we attempt a “derivation” of TSH, starting from the nonadiabatic Bohmian dynamics equations of motion.

The main ansatz in TSH is given by the following description of the molecular wavefunction [

A surface hop between two PESs,

Full Multiple Spawning (FMS) [

In the FMS method, the nuclear wavefunction in electronic state _{i}_{i}

The time-evolution of the expansion coefficients, ^{i}_{ii}_{ii}_{ij}_{el}_{R}_{R}_{r}

The spawning procedure takes place when a region of nonadiabaticity is detected along a trajectory (by monitoring the strength of nonadiabatic couplings in the adiabatic representation) and allows for the generation of new Gaussian basis functions (children), placed in the newly populated electronic state, according to physical rules (like position or momentum conservation [

The spawning procedure, therefore, limits the number of Gaussian basis function used in the calculation by defining precisely where and when they are needed. Moreover, the FMS method offers a numerically exact [

While keeping a trajectory-based formalism, FMS fully incorporates nuclear quantum effects that are missing in methods like TSH. Furthermore, the nuclear propagation can be performed on-the-fly, by computing any electronic structure property needed, like electronic energies (
_{i}_{R}_{j}_{r}_{iKjK′}

Just as for the previous three methods, nonadiabatic Bohmian dynamics (NABDY) is also based on the propagation of trajectories. However, this time, the trajectories evolve under the action of additional quantum potentials (adiabatic and nonadiabatic), which make the dynamics exact in principle. In other words, this approach is able to capture all adiabatic and nonadiabatic nuclear quantum effects through the propagation of a sufficiently large (

The derivation of the NABDY equations of motion starts from the insertion of the polar representation of the nuclear wavefunction in ^{2}). The third term,

_{j}^{2}, with corresponding probability density flux

The two equations for the phases and the amplitudes are coupled, and they therefore need to be solved simultaneously. Instead of solving complex differential equations for the two fields, (_{j}_{j}

The equation of motion that drives Ehrenfest dynamics (EHD) is derived from a simpler ansatz for the total wavefunction than the Born-Huang expansion (

In EHD, the molecular wavefunction is described by the simple product:

The exponential in

Following the derivation proposed by Tully [^{*}(

Applying an analogous procedure, we can also derive the equation of motion for Ω(

Using the relations [_{el}_{R}_{R}

Once again, we obtain a classical Hamilton-Jacobi equation, which can be transformed into a Newton equation of motion given by:
_{el}

The equation of motion for the electronic amplitudes, _{γ}

Recently, Gross

The factorization of Ψ(^{2} as the conditional probability of finding an electron in volume element, ^{2} is an electronic probability density function. According to the standard interpretation of quantum mechanics, Φ(

The time evolution of the wavefunctions, Φ(

When it comes to nonadiabatic molecular dynamics, TSH is probably the most popular simulation scheme. As stated in Section 2, it relies on the description of nuclear wave packets by means of a swarm of classical trajectories. A complex coefficient,

In this Section, starting from

The following steps were reported in [

The nuclear wave packet dynamics is discretized into a swarm of classical trajectories. Within the _{j}_{ji}_{ji}

For the description of the nonadiabatic components of the dynamics (the three last terms in

Neglecting the second-order nonadiabatic couplings, _{ji}^{γ} = −_{γ}

In the derivation of the equation of motion for the nuclear amplitude coefficients, we start by assigning delta-like wave packets (denoted as the TSH wave packet in the following) to each trajectory, ^{[}^{α}^{]}(^{λ}^{[α]}(_{λ→∞}^{λ}^{[}^{α}^{]}(^{[}^{α}^{]}(_{traj}^{λ}^{[}^{α}^{]}(^{λ}

Since we are working in the Lagrangian frame, we need only consider the explicit time-dependence of the amplitudes and phases. As a consequence, the TSH nuclear wave packet evolving in electronic state ^{λ}^{[}^{α}^{]}) (where ^{[}^{α}^{]} is the position vector in the Lagrangian frame), and is described by

If we substitute Ω_{j}^{[}^{α}^{]} are the nuclear velocities at time

Notice that

We have described until now the dynamics of TSH nuclear wave packets following a single classical trajectory, _{traj}

In TSH, the balance described in

In the

The switching probability is obtained from quantum mechanical arguments [_{j}_{av}

In summary, starting from the

While TSH is an elegant compromise between accuracy and efficiency for the simulation of nonadiabatic phenomena, its accuracy (either in its fewest-switches version or with additional corrections) has been challenged several times in the literature (see [

In this model, a Gaussian wave packet launched from _{1}). Right after this nonadiabatic event occurs, the two potential energy curves will diverge, one exhibiting a strong positive slope (S_{1} state), the other a negative one (GS). The wave packet contribution in each electronic state will therefore be spatially split and eventually recombined in a second nonadiabatic region at _{1} after the second nonadiabatic region strongly depends on the spatial decoherence between the nuclear wave packets. However, such peculiar decoherence is hardly captured by TSH, due to the _{1}, but does not improve it substantially [

We further investigate the effects of overcoherence on the TSH dynamics by means of a second model system consisting of two coupled harmonic potentials, as depicted in the upper inset of _{1}) at _{0} = 40 a.u.. In this model system, a single nonadiabatic region is located at

The S_{1} wave packet enters the strong coupling region at _{1} inverts the direction of its propagation and rapidly returns towards the nonadiabatic region at _{1}) are spatially separated. As for the first transition through the nonadiabatic region at _{1} wave packet is transferred almost entirely to the other electronic state (now, the GS, _{1}, couple coherently, because they share the same support (same position in space for any time _{1} increases back to ∼78% of the

The description of the nonadiabatic dynamics of molecular systems is a challenging task for theory, due to the difficulty of providing both the electronic structure of a system

We are grateful to Jiří Vaníček and Tomàš Zimmermann for providing us a version of their exact propagation code. COSTactions CM0702 and CM1204 and Swiss National Science Foundation grants 200021-137717 and 200021-146396 are acknowledged for funding and support.

The authors declare no conflicts of interest.

^{+}with D

_{2}

The double arch model in the adiabatic representation. The ground state (GS) (S_{1}) potential energy curve is represented with a red (dashed) line and nonadiabatic coupling with a blue dotted line. The initial nuclear wave packet is displayed in grey.

Nonadiabatic dynamics for the double arch system. (_{j}^{2}, and trajectory surface hopping (TSH) histograms for _{0} = 45 a.u. (lower panel = GS; upper panel = S_{1}). The adiabatic potential energy curves are given in red, while the nonadiabatic coupling vectors are shown in blue. (_{1} from an exact nuclear wave packet propagation obtained with TSH and nonadiabatic Bohmian dynamics (NABDY), for different initial momenta (“TSH”: initial conditions sampled from a Gaussian distribution for positions and momenta, 1,500 trajectories; “TSH”: same initial conditions, momentum and position, for all 1,500 trajectories; “NABDY” is based on a maximum total number of 162 trajectories). The maximum total number of quantum trajectories used in NABDY is 162.

Nonadiabatic dynamics on two coupled harmonic potential energy curves. Population in the first excited state (S_{1}) along the dynamics for 3,444 TSH trajectories (green) and an exact propagation (red). (_{1}) potential energy curve is represented with a continuous (dashed) black line and the nonadiabatic coupling with a blue dotted line. The initial nuclear wave packet is displayed in grey. (_{1}, then jumps to the GS after the first coupling and, finally, hops back to S_{1} after it reaches back to the coupling region. This representation highlights that the model describes two nonadiabatic events with a single nonadiabatic region.