Landau-Zener Type Surface Hopping Algorithms

Landau–Zener type surface hopping algorithms

Andrey K. Belyaev∗ Department of Theoretical Physics, Herzen University, St. Petersburg 191186, Russia

Caroline Lasser† and Giulio Trigila‡ Zentrum Mathematik, Technische UniversitätMünchen,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 Schrödingerequation 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 Schrödingerequation 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 nonadiabatic 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 probability (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- Schrödingeroperator 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 are fined, and diabatic representations obtained by the same ~2 ~2 procedure in two-state and in multiple-state cases may T = − ∆nuc,Tel = − ∆el, deviate substantially45. Lastly, a determination of LZ 2m 2mel parameters is often troublesome in practical applications where ∆nuc and ∆el denote Laplacians with respect to of the conventional LZ formula. the nuclear and electronic coordinates. Moving to atomic Two novel formulas have recently been proposed units (~ = mel = e = 1) and defining for nonadiabatic transition probabilities within the √ ε = 1/ m, LZ model: the diabatic multi-dimensional formula26 and the adiabatic-potential-based transition probabil- the molecular Hamiltonian rewrites as ity formula44 adapted for classical trajectory surface- ε2 1 hopping studies. The former was derived when math- Hmol = − ∆nuc − ∆el + Vel + Vnuc + Vattr ematically analysing effective dynamics through conical 2 2 intersections38 and tested on the two-state three-mode 1 Let Hel(q) = − ∆el + Vel + Vnuc(q) + Vattr(·, q) be the model of pyrazine27 by means of the single switch clas- 2 electronic Hamiltonian for a given position q ∈ Rd of the sical trajectory surface-hopping algorithm, while the lat- nuclei. Let ter was applied to inelastic multi-channel atomic colli- 44 ± sions by means of the branching classical trajectory U (q) ∈ σ(Hel(q)) 3

+ − be two adiabatic potential energy surfaces (PES), that scalar-valued functions ψt and ψt . We use the Wigner is, two eigenvalues of the electronic Hamiltonian. We as- functions of the scalar components sume that each eigenvalue is of multiplicity one and that ± both are well separated from the rest of the electronic W (ψt )(q, p) = spectrum. Then, by time-dependent Born–Oppenheimer Z (2πε)−d eiy·p/εψ±(q − 1 y)ψ±(q + 1 y)∗y, theory48,49, the effective nuclear quantum motion is gov- t 2 t 2 . erned by the time-dependent nuclear Schrödinger equation which map phase space points (q, p) ∈ R2d to the real numbers. The ε-scaling of the Wigner function allows the iε∂tψt = (T + V )ψt, (1) direct relation to the position, momentum, and kinetic energy operators, where the potential V takes values in the real symmetric d 2 × 2 matrices and is given in a global diabatic represen- ε2 1 X qˆ = q, pˆ = −iε∇,T = − ∆ = pˆ2 tation as 2 2 j j=1 v (q) v (q) V (q) = 11 12 . v12(q) v22(q) for the nuclear degrees of freedom. Indeed, up to nor- malizing factors, we obtain the corresponding expecta- We note that by the definition of a diabatic matrix, tion values of the upper and the lower adiabatic surface the potential energy surfaces are its eigenvalues. In the as present work, we assume the existence of a conical inter- Z ± ± ± section, that is, hψt | qˆ | ψt i = q W (ψt )(q, p)(.q, p), + − Z {q | U (q) = U (q)} ± ± ± hψt | pˆ | ψt i = p W (ψt )(q, p)(.q, p), is a manifold of codimension two of the nuclear configu- Z hψ± | T | ψ±i = 1 |p|2 W (ψ±)(q, p)(q, p). ration space. Then, one has to account for nonadiabatic t t 2 t . transitions between the eigenspaces associated with the potential energy surfaces. More general expectation values for the Weyl quantiza- For notational convenience, we write the diabatic ma- tion Aˆ of a phase space function A are accordingly writ- trix as the sum of a centric dilation and its trace-free ten as part, Z hψ± | Aˆ | ψ±i = A(q, p)W (ψ±)(q, p)(q, p). t t t . v1(q) v2(q) V (q) = v0(q) + , v2(q) −v1(q) These are the observables whose dynamics can be approximated by surface hopping algorithms. and express the two potential energy surfaces as ± ˆ ∓ Cross-term quantities like hψt | A | ψt i, which require relative phase information of the upper and the lower ± p 2 2 U (q) = v0(q) ± v1(q) + v2(q) . surface components cannot be obtained, since the Wigner + − functions W (ψt ) and W (ψt ) determine the functions Their gap is denoted by + − ψt and ψt only up to a global phase factor. Z(q) = U +(q) − U −(q). B. The general algorithmic scheme The corresponding eigenvectors of V (q) satisfy V (q)χ±(q) = U ±(q)χ±(q). They are uniquely determined up to a phase and are singular at conical The class of surface hopping algorithms to be investi- intersection points. Our choice for the phase is gated is determined by the following steps: (i) Sampling of the initial condition: We choose phase ± ± ± ± cos(α(q)) − sin(α(q)) space points (q1 , p1 ),..., (qN , pN ) so that χ+(q) = , χ−(q) = , 0 0 sin(α(q)) cos(α(q)) N0 ± ˆ ± 1 X ± ± 1 hψ0 | A | ψ0 i ≈ A(qj , pj ) with mixing angle α(q) = arctan(v2(q)/v1(q)). N 2 0 j=1

for the observables A of interest. This is achieved by A. The observables Monte Carlo or Quasi-Monte Carlo sampling of the initial ± Wigner functions W (ψ0 ). We note that an unreﬁned We write the wave function at time t as a linear com- sampling of the initial Husimi functions deteriorates the + + − − 50,51 bination of the eigenvectors ψt = ψt χ + ψt χ with approximation . 4

(ii) Classical trajectory calculations: The chosen phase Contrary to the diabatic formula, the building blocks of space points are evolved along the trajectories of the cor- the adiabatic one are accessible also in cases when a dia- responding classical Hamiltonian system batic potential matrix is missing, which is often the case for the simulation of polyatomic systems. A simple cal- q˙ = p, p˙ = −∇U ±(q). culation reveals the connection between the two LZ formulas: Depending on the potential energy surface U ± Since the observables of interest are computed by phase guiding the classical motion, we have space averaging, these classical equations of motion s should be discretized symplectically as e.g. by the 1 Z(q )3 Z(q )2 c = c Störmer–Verlet method or by higher order symplectic 2 d2 p 2 ± 2 Z(q(t)) |t=tc 4 |v.(qc)pc| + v(qc) · w (qc, pc) Runge–Kutta schemes52. dt (iii) Surface hopping: Whenever the eigenvalue gap be- with comes minimal along an individual classical trajectory a 2 ± D v1(q)p · p ± nonadiabatic transition occurs. Let t 7→ (q(t), p(t)) be a w (q, p) = 2 − v.(q)∇U (q) classical trajectory associated with the upper or the lower D v2(q)p · p surface. Whenever the function t 7→ Z(q(t)) attains a where D2v(q) denotes the Hessian matrix of v(q). Since local minimum, a transition to the other eigenspace is ± 1 ± |v(q)·w (q, p)| ≤ 2 |w (q, p)| Z(q), the difference between performed according to a Landau–Zener transition prob- the two formulas is dominated by the gap size and negligi- ability. In the following sections §III and §IV, different ble for trajectories with small minimal gap. Nevertheless, Landau-Zener formulas and transition schemes will be we observe a notable difference. The diabatic formula has discussed in more detail. the same functional form for transitions originating from (iv) Evaluation of the observables: At some time t, the the upper or the lower surface, while the adiabatic for- surface hopping algorithm has resulted in phase space mula implicitly depends on the surface with which the points (q±(t), p±(t)),..., (q± (t), p± (t)). Then, the ex- 1 1 Nt Nt hopping trajectory is associated. pectation values of interest are approximated according to IV. TRANSITION SCHEMES Nt ± ˆ ± X ± ± ± hψt | A | ψt i ≈ A(qj (t), pj (t))wj (t), (2) j=1 One can algorithmically interprete nonadiabatic transitions with LZ probabilities either in a deterministic way ± with branching trajectories or probabilistically with sur- where the individual weight wj (t) depends on the initial sampling and the employed transition scheme, see §IV. face hopping trajectories. The probabilistic method is appealing, since it is less costly from the computational point of view. The deterministic branching process has 38 III. TWO LANDAU–ZENER PROBABILITIES been mathematically analysed and has been proven to be asymptotically correct in the semiclassical limit ε → 0, provided that at the same time t and in the same phase We compare two recent formulas for nonadiabatic tran- c point (q , p ) only upper or lower surface trajectories ini- sition probabilities. Both of them are applied whenever c c tiate nonadiabatic transitions. This restriction is due to the eigenvalue gap becomes minimal along an individ- the neglect of relative phase information for the upper ual classical trajectory t 7→ (q(t), p(t)), that is, when the and the lower wavefunctions ψ+ and ψ−. The numeri- function t 7→ Z(q(t)) attains a local minimum. We de- t t cal experiments presented later confirm the mathematical note corresponding critical times and phase space points assessment of the algorithm’s properties. by tc and (qc, pc) respectively. The first formula is a multi-dimensional Landau–Zener formula derived from a global diabatic representation of A. Deterministic transitions the potential matrix26,38 26,44 2 For the deterministic branching process , at a crit- LZ π Z(qc) Pd = exp − , (3) ical point (qc, pc) of minimal gap a trajectory splits into ε 4|v.(qc)pc| two, and a new branch is created on the other surface. The weight of the new trajectory is equal to the old where v(q) denotes the 2×d gradient matrix of the vector . weight multiplied by the Landau–Zener probability P LZ, v(q) = (v (q), v (q)) defining the trace-free part of the 1 2 while the weight of the trajectory remaining on the same diabatic potential matrix V (q). The second formula is surface is multiplied by 1 − P LZ. At time t, we are the purely gap and trajectory based, adiabatic formula44 left with a certain number of classical trajectories dis- tributed along the upper and the lower surface. Let us s 3 ! π Z(qc) P LZ = exp − (4) indicate with Nt the number of trajectories on the up- a d2 + + 2ε 2 Z(q(t)) |t=t per surface and with w (t), . . . , w (t) the corresponding dt c 1 Nt 5

+ weights. Each weight wj (t) is the product of the initial V. NUMERICS sampling weight 1/N0 and a certain number of Landau– Zener probabilities. Corresponding expectation values We now compare the two Landau–Zener transition are computed according to (2). We note that the num- probabilities for the Schr¨odingerequation (1) associated ber of trajectories may rapidly increase as time evolves, with a linear Jahn–Teller matrix demanding more memory storage and increasing compu- q q tational costs. V (q) = γ|q|2 + 1 2 , q2 −q1

where the quadratic term confines the motion around the B. Probabilistic transitions conical intersection located at the origin. The strength of the confinement is chosen to be γ = 3, while for the In constrast to the deterministic method, the proba- semiclassical parameter we set ε = 0.01. These values are bilistic version of the surface hopping algorithm keeps the comparable with those obtained by ab initio electronic number of trajectories constant during all the simulation structure calculations for the triangular silver molecule53. time. In this case, once a classical trajectory attains a The initial wave packet is localized entirely on the up- local gap minimum, we compute the LZ probability P LZ + + per surface and is given by ψ0 = ψ0 χ where and compare it with a pseudo random number ξ generated from a uniform distribution on [0, 1]. If ξ ≤ P LZ + −1/2 1 2 ψ0 (q) = (πε) exp − 2ε |q − q0| . the trajectory hops on the other surface, otherwise it √ √ continues along the same surface. The weights in the The initial position center is q0 = (5 ε, 0.5 ε), so that expectation summation (2) are all equal to 1/N0. the wave packet is localized close to the conical intersection. The final simulation time tf = 5.34 roughly corre- sponds to 129.3 fs and allows the wavefunction to pass C. Momentum adjustment the conical intersection four times. As previously developed for the computation of expectation values via the Wigner function52, the initial In both methods described above, a choice has to be Wigner function made regarding the point in phase space at which a trajectory appears on the other energy surface at the mo- W (ψ+)(q, p) = (πε)−2 exp− 1 (|q − q |2 + |p|2) (5) ment of a nonadiabatic transition. For clarity, let us con- 0 ε 0 sider a transition from the upper to the lower surface. We is sampled by a quasi-Monte Carlo technique so that denote with (q+, p+) the point on the upper surface, in Z which the trajectory attains a local gap minimum, and + ˆ + + with (q−, p−) the point on the lower surface, in which a hψ0 | A | ψ0 i = A(q, p)W (ψ0 )(q, p)d(q, p) − + new trajectory is initiated. Our choice is q = q = qc N 1 X0 for the position, and we rescale the momentum according ≈ A(q+, p+) − + N j j to p = kp with k > 0 to ensure conservation of energy. 0 j=1 The value of k is computed by simply imposing

for the observables Aˆ of interest. We have used N0 = 1 + 2 + 1 + 2 − + + + + 2 |p | + U (qc) = 2 |kp | + U (qc), 1296 Halton points (q , p ),..., (q , p ), which de- 1 1 N0 N0 terministically approximate the uniform distribution on p + 2 2d leading to k = 1 + 2Z(qc)/|p | . Analogously for tran- the unit cube [0, 1) , and have mapped them by the sitions from the lower to the upper surface, we have inverse of the cumulative distribution function of 2d p − 2 k = 1 − 2Z(qc)/|p | , where we neglect the transition one-dimensional Gaussian distributions to approximate if the trajectory does not have enough kinetic energy to the 2d-dimensional Gaussian distribution given in equa- compensate the difference in the potential energy. tion (5). The corresponding convergence rate for the ap- 2d This particularly simple momentum adjustment seems proximation of expectation√ values scales as (log N0) /N0 natural from the classical trajectory point of view, since compared to the 1/ N0 scaling characterizing plain it treats each newly generated trajectory as a contin- Monte Carlo. uation of its generator, while ensuring that both tra- The numerical integration of the classical trajectories is jectories have the same classical energy. Moreover, it implemented using a 4th order symplectic Runge-Kutta only depends on the adiabatic surfaces and their gap. time-stepping method, while, in order to estimate the With respect to the direction of the momentum adjust- second derivative d2Z(q(t))/dt2 for the evaluation of the ment, the literature contains various other, more com- adiabatic LZ probability, a 4th order accurate central plicated choices, such as the normal direction to a pre- finite difference scheme is used. defined surface of (avoided) intersection19,42 or the di- The simulations presented below are performed both rection of the nonadiabatic coupling vector22 defined as in the deterministic and the probabilistic setting. For + − + − (χ (q) · ∂1χ (q), . . . , χ (q) · ∂dχ (q)). each numerical experiment, we compute the values of the 6 surface populations and the expected values of the mo- the initial momentum expectation equals zero, and the ± ˆ ± mentum and position, that is, hψt | A | ψt i for A = 1, potential energy surfaces are radially symmetric. How- A = p and A = q, respectively. These expected val- ever, this expectation is not met, which can be explained ues are compared with reference solutions computed by by the surface hopping approximation: during the time solving the Schrödingerequation (1) via a numerically interval [17fs, 51fs], the wave packet is almost entirely converged Strang splitting scheme using the fast Fourier located on the lower surface. In this case, the few tra- transform for the computation of the Laplacian. This jectories on the upper surface, initially sampled from the ref 50 grid-based reference solution ψt approximates the so- tail of the Wigner distribution, gain relative weight with lution ψt of the Schrödingerequation (1) with an accu- respect to the trajectories that have initiated nonadia- ref ref −12 racy of hψt − ψt | ψt − ψt i ≈ 10 . batic transitions. Because points sampled from the tail of the distribution are more likely to be arranged in a non symmetric way with respect to the origin in momen- A. Time evolution tum space, the average momentum does not point into the direction of the conical intersection. We compare the time evolution of the above mentioned In Fig. 3 we show the absolute deviation with respect observables when using the two different LZ formulas for to the reference solution for the expectation values of nonadiabatic transition probabilities. In Fig. 1 we show position, momentum and population referal to the upper the population of the upper and lower surfaces calculated surface. The differences are larger when the wave func- in the deterministic setting. As suggested by our pre- tion is mostly located on the lower surface namely, for vious analysis, the curves associated with probabilities the time intervals [17fs, 51fs] and [77fs, 108fs]. In partic- computed by the two different LZ formulas are almost ular, during the second time interval, the deviation on indistinguishable and in good agreement with the refer- the three observables is amplified due to interference ef- ence. The slight deterioration of both surface hopping fects. The wave packet relative to the upper and lower approximations after the third and fourth nonadiabatic surface arrive simultaneously at the conical intersection passage (around time = 80fs and = 100fs, respectively) (see Fig. 2 at time = 75 fs). It is also important to no- are due to unresolved interference effects between the up- tice that the curves describing the difference of the two per and the lower wave packet components. This effect Landau-Zener transition probabilities overlap quite well is also visible in Fig. 3 later on. on the scale of the deviation with respect to the reference solution; hence, our experiments confirm the closeness of 0 1 2 3 4 5 6 the two LZ probabilites for small values of the gap. 1 Upper level 0.8

0.6 B. LZ probabilities

0.4

Population 0.2 In Fig. 4, we analyze the diﬀerence between the LZ

0 probabilities (3) and (4) computed simultaneously for 0 20 40 60 80 100 120 140 Time(fs) each trajectory at each local minimum of the gap. In 0 1 2 3 4 5 6 particular, we notice that the magnitude of the diﬀer- 1 ence between the two transition probabilities is mostly −3 −3 0.8 10 or smaller and increases up to 5 × 10 as the value

0.6 of the gap function increases. The two branches shown in the lower panel of Fig. 4 are explained by the speciﬁc 0.4 form of LZ probabilities for the linear Jahn–Teller case: Population 0.2 Lower level We have 0 0 20 40 60 80 100 120 140 2 Time(fs) LZ π |qc| Pd = exp − ± , ε |pc | FIG. 1: Population of the upper and lower surface, respectively. The blue markers refer to the simulation obtained us- and obtain by the calculation of section §III LZ LZ ing Pd while the red markers refer to Pa ; the black curve   represents the reference solution. The label on the x axis at 2 LZ π |qc| the bottom of each panel indicates the time in femtoseconds, Pa = exp − q  , (6) ε ± 2 2 while the axis located on top of the panel indicates the time |pc | − 2γ|qc| ∓ |qc| in units that are consistent with the equation (1). where the plus and minus sign refer respectively to tran- In Fig. 2 we show the expected position as a func- sitions from the upper level to the lower and vice versa. LZ tion of time. Ignoring nonadiabatic eﬀects, one might In the linear Jahn–Teller situation, we have Pd < LZ expect that the average position of the wave packet fol- Pa for transitions from the upper to the lower surface LZ LZ lows a straight line through the conical intersection, since and Pd > Pa for transitions from the lower to the 7

time=12.9 fs time=28.4 fs

0.05 0.05

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−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 time=43.9 fs time=59.5 fs

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−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 time=75 fs time=90.5 fs

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−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 time=106 fs time=129.3 fs

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FIG. 2: Average position for the upper and the lower surfaces at diﬀerent times. The coordinates axis of each panel are q1 and q2. The red marker represents the average position, while the coloured curve shows positions visited at earlier times. The colour code is such, that positions visited more recently are in red compared to those visited at earlier times appearing in blue. 8

Absolute Errors the diabatic and the adiabatic formulas (3) and (4) re- 0.2 spectively, the Jahn–Teller speciﬁc analytic version of the adiabatic formula (6) and the intermediate probability 0.1   Position π |q |2 0 LZ c 0 20 40 60 80 100 120 140 P0 = exp − q  , Time(fs) ε ± 2 2 0.4 |pc | − 2γ|qc|

0.2 which lacks the surface dependent term ∓|qc| of (6). In Fig. 6, we show the corresponding information for a typ-

Momentum 0 ical lower level trajectory. 0 20 40 60 80 100 120 140 Time(fs) 0.2 2

1.5 0.1 1

Population 0 0.5 0 20 40 60 80 100 120 140

Time(fs) Eigenvalues 0

−0.5 0 20 40 60 80 100 120 140 FIG. 3: Absolute deviation of the ﬁrst component of the posi- Time(fs) tion expectation (upper panel), momentum expectation (mid- 0.67 dle panel) and population (lower panel) of the upper surface with respect to the reference solution. The blue markers are 0.669 relative to the simulation obtained using the diabatic LZ for- 0.668 mula, those in red to the adiabatic one. 0.667

0.666

Transition Probability 0.665 0.4 0 20 40 60 80 100 120 140 Time(fs) 0.3

0.2 FIG. 5: Eigenvalues and transition probabilities of a trajectory located on the upper surface when using the determin- 0.1 istic method. Upper panel: eigenvalues relative to the upper Distribution level (red) and lower level (black), respectively. Lower panel: 0 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 transition probabilities in correspondence to the local min- + x 10−3 ima of t 7→ Z(q (t)); the red circles refer to the adiabatic LZ x 10−3 5 probability, while the blue crosses refer to the diabatic one. LZ Black and green markers represent formula (6) and P0 re-

erence spectively. The slight diﬀerence between the red and green

ﬀ 0 markers is due to the numerical error when computing the second derivative of t 7→ Z(q+(t)). −5

Probability di −10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Figs. 5 and 6 depict that the differences between tran- Gap sition probabilities calculated by means of the different LZ formulas are small. It is worth emphasizing that FIG. 4: Difference between the transition probabilities. Up- these deviations are within the accuracy of the LZ ap- LZ LZ per panel: distribution of Pa − Pd , the average value and proximation, since the conventional LZ formula is ob- standard deviation are µ = 3.82 × 10−4 and σ = 1.2 × 10−3, tained assuming a constant and high momentum, that LZ LZ 2 respectively. Lower panel: Pa − Pd vs the gap value Z(qc). is, |pc| Z(qc) = 2|qc|. The formulas derived in the present section for the linear Jahn-Teller case clearly show that the diabatic and the adiabatic formulas co- incide in the high-energy regime, while the corrections upper surface, which explains the two branches in the are mainly due to an acceleration and are of the order of lower panel. the energy gap. Next we monitor individual trajectories located on the Moreover, Figs. 5 and 6 also show that local nonadi- lower and the upper surface respectively. The upper abatic regions along classical trajectories have the form panel of Fig. 5 represents the values of the two eigenval- of an avoided crossing, although the global nonadiabatic ues U + and U − along a typical upper surface trajectory. region is formed by conically intersecting adiabatic en- For each local minimum of t 7→ Z(q+(t)) we compute ergy surfaces. Since classical trajectories passing ex- the transition probability in four different ways: We use actly through a conical intersection point are very rare, 9

2 Upper level 1.5 0.2

1 0.15

0.5 0.1 Error Eigenvalues 0 0.05 −0.5 0 20 40 60 80 100 120 140 Time(fs) 0 0 20 40 60 80 100 120 140 0.812 Time(fs) Deterministic vs Probabilistic 0.05 0.8115 0.04

0.03 0.811 0.02 Deviation Transition Probability 0.8105 0.01 0 20 40 60 80 100 120 140 Time(fs) 0 0 20 40 60 80 100 120 140 Time(fs) FIG. 6: Eigenvalues and transition probabilities along a trajectory located on the lower surface when using the determin- FIG. 7: Upper panel: absolute error of the upper surface istic method. The colors and markers correspond to the ones population with respect to the reference solution. The blue in Fig. 5. LZ markers refer to the simulation obtained using Pd , those in LZ red to Pa . Lower panel: difference, in the absolute value, of the level population between the deterministic and the probabilistic approach. The dotted line indicates the confidence almost all local nonadiabatic regions along trajectories interval. have the form of one-dimensional avoided crossings. The adiabatic LZ formula (4) has been applied to the one- dimensional case of several avoided crossings in atomic 44 Na+H collisions , and a good approximation of the nonadiabatic regions, see e.g. Fig. 5 of Ref.55. The de- quantum results has been found. The same is true for terministic approach with its branching classical trajec- the diabatic formula (3), which has also been success- 54 tories simulates this situation. Moreover, the determin- fully applied to one-dimensional avoided crossings . istc method has been mathematically analysed26. On The above findings are also valid for different values the other hand, in some cases the deterministic approach of the semiclassical parameter ε. In particular analogous may produce too many trajectories, so that memory re- distributions for the difference of P LZ −P LZ are obtained a d quirements and computing times become unfeasible and when using ε = 0.05 ( mean and standard deviation being −4 −3 the probabilistic method with its constant number of tra- µ = 9.78 × 10 and σ = 1.3 × 10 respectively) and jectories is preferable. ε = 0.001 (µ = 4.57 × 10−4 and σ = 1.3 × 10−3).

C. Probabilistic VS Deterministic VI. CONCLUSION In this section we compare the previous results with those obtained by the probabilistic approach. The simu- We have investigated a class of surface hopping algo- lations presented in this section are obtained taking the rithms, which perform nonadiabatic transitions for each average over 10 runs with the same initial trajectories classical trajectory individually. Nonadiabatic transi- used for the deterministic approach. tions are allowed, when the surface gap attains a local The results obtained are analogous to those displayed minimum along an individual trajectory. We have com- in the deterministic case. In particular, in the lower panel pared two recent Landau–Zener formulas for the prob- of Fig. 7 we compare the population for the upper sur- ability of nonadiabatic transitions, one of them requir- face obtained by the deterministic and the probabilistic ing a diabatic representation of the potential matrix, the approach. other one only depending on the adiabatic potential en- The final Fig. 8 shows the time evolution of a sample of ergy surfaces. Our numerical experiments confirm the typical surface hopping trajectories. As expected, their expected affinity of both LZ probabilities as well as the positions fit with the position density of the reference good approximation of reference values, that have been solution. obtained by a grid based quantum solver. We have vi- Comparing the deterministic and probabilistic ap- sualized position expectations and superimposed surface proaches, we point out that probability currents com- hopping trajectories with reference position densities for puted by a quantum method split when passing through an enhanced understanding of the effective dynamics. 10

Acknowledgments 00163-a). All three authors have been supported by the German Research Foundation (DFG), Collaborative Re- AKB gratefully acknowledges supports from the Rus- search Center SFB-TRR 109. We thank Diane Clayton- sian Foundation for Basic Research (Grant No. 13-03- Winter for her reading of the manuscript.

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time=12.9 fs time=28.4 fs

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−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 time=43.9 fs time=59.5 fs

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−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 time=75 fs time=90.5 fs

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−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 time=106 fs time=129.3 fs

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FIG. 8: Sample of trajectories in the probabilistic setting together with the modulus of the projected wave function obtained by solving the Schr¨odingerequation. Each panel of the ﬁgure refers to the upper and lower surface at a given time. The trajectories are represented with a colored curve ending with a red marker representing the position at the time indicated on each panel. The colour code is so that positions visited more recently are in red compared to those appearing in blue referring to positions visited earlier in the past.