Landau{Zener type surface hopping algorithms Andrey K. Belyaev∗ Department of Theoretical Physics, Herzen University, St. Petersburg 191186, Russia Caroline Lassery and Giulio Trigilaz Zentrum Mathematik, Technische Universit¨atM¨unchen,Germany (Dated: September 5, 2018) A class of surface hopping algorithms is studied comparing two recent Landau-Zener (LZ) formulas for the probability of nonadiabatic transitions. One of the formulas requires a diabatic representation of the potential matrix while the other one depends only on the adiabatic potential energy surfaces. For each classical trajectory, the nonadiabatic transitions take place only when the surface gap attains a local minimum. Numerical experiments are performed with deterministically branching trajectories and with probabilistic surface hopping. The deterministic and the probabilistic approach confirm the affinity of both the LZ probabilities, as well as the good approximation of the reference solution computed by solving the Schr¨odingerequation via a grid based pseudo-spectral method. Visualizations of position expectations and superimposed surface hopping trajectories with reference position densities illustrate the effective dynamics of the investigated algorithms. I. INTRODUCTION first introduced by Bjerre and Nikitin 17 , though with reduced dimensionality. They proposed to branch a clas- A great variety of physical processes and chemical re- sical trajectory into two trajectories after traversing a actions occurs due to nonadiabatic transitions between nonadiabatic region which had to be specified before- adiabatic electronic states, often mediated by conical hand. The more systematic classical trajectory surface- 18 intersections1{5. Nonadiabatic transitions are of quan- hopping approach was proposed by Tully and Preston tum nature and, in general, should be described with based on the deterministic (\ants") procedure or/and on quantum mechanical theory. the probabilistic (\anteater") method. In the latter, clas- While for small molecular systems the nonadiabatic sical trajectories remain unbranched and a random de- effects can indeed be investigated in detail with quan- cision is made whether to hop or not, depending on a tum mechanical methods, for larger systems these meth- hopping probability. In both papers, nonadiabatic tran- ods are computationally too expensive. Example of such sition probabilities were estimated in an approximate way 39{41 systems are biomolecules, atomic and molecular clusters, within the Landau-Zener (LZ) model , with param- 18 molecular complexes and condensed matter. eters calculated beforehand. Tully and Preston have For this reason, more approximate classical or semi- also used semiclassical methods to investigate the hop- classical computational methods are becoming an impor- ping probability and to prove LZ model usage. Later, 20 tant alternative because their lower computational cost Kuntz et al. proposed a probabilistic approach in which scales more favorably with the system size. Moreover, hopping points were not specified beforehand but de- these methods can provide intuitive insight into the dy- termined during the trajectory propagation, based on a namics of a chemical reaction6. Particularly interest- maximum of the nonadiabatic time-derivative coupling ing for practical purposes are mixed quantum{classical matrix element. Approaches to localize nonadiabatic re- approaches which treat the electronic motion quantum- gions by a local minimum for an adiabatic splitting have 42 mechanically and the nuclear motion classically. been proposed by Miller and George as well as Stine and Muckerman 19 . In contrast to these approaches, Several quasi-classical methods exist for the treat- 22 ment of the nonadiabatic nuclear dynamics. Well-known where nonadiabatic transitions are localized, Tully examples are the semiclassical initial-value representa- proposed the fewest-switches approach, which extended tion (IVR)7{10, the Ehrenfest dynamics method11{15, the the classical trajectory surface-hopping method to an ar- frozen Gaussian wave-packet method16, the propagation bitrary number of states and to situations in which tran- arXiv:1403.4859v2 [physics.chem-ph] 19 Jun 2014 of classical trajectories with surface hopping17{28, as well sitions can occur anywhere, not just at localized regions. as the multiple-spawning wave-packet method29{32; see This is achieved by a solution of the time-dependent the leading Perspective33 of the special issue dedicated to Schr¨odingerequation along classical trajectories in com- nonadiabatic nuclear dynamics. Also the mathematical bination with the probabilistic fewest-switches algorithm literature provides rigorous analytical results on nonadi- that decides at each integration time step whether to abatic nuclear dynamics34{38. switch the electronic state. Since then, many variants of One of the most widely used mixed quantum-classical the classical trajectory surface-hopping approaches have approaches for simulating nonadiabatic dynamics is the been proposed and applied to different physical phenom- classical trajectory surface-hopping method with its ena and processes. The main differences between differ- many variants. To the best of our knowledge, the com- ent classical trajectory surface-hopping versions are in bination of classical trajectories and surface hopping was two features: (i) how a nonadiabatic region (a seam) is 2 defined, and (ii) when and how a hopping probability and the branching probability current algorithms46. The is determined. The present paper is addressed to these formula derived in Ref.44 is easy implemented in prac- questions in connection with a conical intersection case. tice as it only requires the information about adiabatic The simplicity of the classical trajectory surface- potentials (see below). It should be mentioned that Zhu hopping technique renders it attractive for the study of and Nakamura 47 have derived the formula for the LZ high-dimensional quantum systems which are difficult or transition probability written in terms of several param- unreachable for quantum treatments. Today, trajectory eters that are expressed via adiabatic potentials, but the surface-hopping calculations are widely employed in the Zhu-Nakamura formula is different from the adiabatic- context of so-called ab initio molecular dynamics, that is, potential-based formula44. Tully and Preston 18 , Stine the forces for the trajectory calculation and the nonadia- and Muckerman 19 , Voronin et al. 24 have calculated tran- batic couplings are computed ”on-the-fly” with ab initio sition probabilities by means of the conventional LZ for- or semiempirical electronic-structure methods, see, e.g., mula with diabatic LZ parameters determined from adi- Ref.43. Nevertheless, many surface-hopping methods abatic potentials along a trajectory. Their approaches have been derived and tested for one- or two-dimensional are also different from the one of Ref.44. The adiabatic- cases. In the present paper, we treat a two-dimensional potential-based formula has been applied so far to nona- two-state model for studying nonadiabatic transitions in diabatic transitions in atomic collisions44,46. the vicinity of a conical intersection by different classical Thus, the main goal of the present work is to study trajectory approaches. different versions of classical trajectory surface-hopping A classical trajectory surface-hopping simulation of algorithms based on the novel formulas for nonadiabatic nonadiabatic dynamics involves the following steps: (i) transition probabilities within the Landau-Zener model sampling of the initial condition, (ii) performing clas- in their applications to a two-state two-dimensional sical trajectory calculations on multi-dimensional adia- model for a conical intersection. In addition, we test batic potential energy surfaces (PES), (iii) accounting probabilistic versus deterministic versions of the algo- for nonadiabatic effects through surface hopping accord- rithm and study simulation performance for several con- ing to specified criteria, and (iv) evaluation of the ob- secutive nonadiabatic transition phases. We also explore servables of interest from the ensemble of trajectories. the possibilities of visualizing nonadiabatic dynamics by The important feature distinguishing different surface- surface-hopping simulations. hopping approaches is the way of calculating nonadia- batic transition probabilities. There are several solutions II. SURFACE HOPPING WITH WIGNER to this problem, many of them based on the LZ model, FUNCTIONS see, e.g.,17,18,20,24{26,44. Although the LZ model provides the simple formula for a nonadiabatic transition proba- bility (see below), it is formulated as a one-dimensional Molecular quantum motion is governed by the problem in a two-state diabatic representation. In prac- Schr¨odingeroperator tical applications to polyatomic systems however, nona- Hmol = T + Tel + Vel + Vnuc + Vattr; diabatic transitions occur in a multi-dimensional space and quantum-chemical data are usually provided in an where Vel and Vnuc denote electronic and nuclear re- adiabatic representation, for example, for an on-the-fly pulsion, respectively, and Vattr the attraction between study. Moreover, often only adiabatic PESs are available, electrons and nuclei. By a rescaling of the nuclear co- not nonadiabatic couplings. As is well known, in contrast ordinates, we can assume that all nuclei have identical to adiabatic states, diabatic states are not uniquely de- mass m. Then, the kinetic energy operators