Strong Influence of Decoherence Corrections and Momentum Rescaling in Surface Hopping Dynamics of Transition Metal Complexes

Strong Influence of Decoherence Corrections and Momentum Rescaling in Surface Hopping Dynamics of Transition Metal Complexes Felix Plasser, Sebastian Mai, Maria Fumanal, Etienne Gindensperger, Chantal Daniel, Leticia González

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Felix Plasser, Sebastian Mai, Maria Fumanal, Etienne Gindensperger, Chantal Daniel, et al.. Strong Influence of Decoherence Corrections and Momentum Rescaling in Surface Hopping Dynamics of Transition Metal Complexes. Journal of Chemical Theory and Computation, American Chemical Society, 2019, 15 (9), pp.5031-5045. 10.1021/acs.jctc.9b00525. hal-03037918

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, , Felix Plasser,∗ † ¶ Sebastian Mai,† Maria Fumanal,‡ Etienne Gindensperger,‡ Chantal Daniel,‡ and Leticia González†

Institute for Theoretical Chemistry, Faculty of Chemistry, University of Vienna, † Währingerstr. 17, 1090 Vienna, Austria Laboratoire de Chimie Quantique, Institut de Chimie Strasbourg, UMR7177 CNRS/Université‡ de Strasbourg 4 Rue Blaise Pascal BP296/R8, F-67008 Strasbourg, France Department of Chemistry, Loughborough University, Loughborough, LE11 3TU, U.K. ¶ E-mail: [email protected]

Abstract portance of the often neglected parameters in surface hopping and shows that there is still The reliability of different parameters in the need for simple, robust, and generally applica- surface hopping method is assessed for a vi- ble correction schemes. bronic coupling model of a challenging transition metal complex, where a large number of electronic states of different multiplicities are 1 Introduction met within a small energy range. In particular, the effect of two decoherence correction schemes Many important processes in photochemistry and of various strategies for momentum rescal- and electrochemistry are governed by nonadi- 1–6 ing and treating frustrating hops during the dy- abatic transitions between electronic states, namics is investigated and compared against an at which the Born-Oppenheimer approximation accurate quantum dynamics simulation. The breaks down, meaning that electronic and nu- results show that small differences in the sur- clear degrees of freedom can no longer be sepa- 7 face hopping protocol can strongly affect the rated. The surface hopping dynamics method results. We find a clear preference for momen- has become a popular approach to describe tum rescaling along the nonadiabatic coupling nonadiabatic processes due to its conceptual vector and trace this effect back to an enhanced simplicity and the intuitive interpretation of the number of frustrated hops. Furthermore, reflec- results in a quasi-classical picture. As a result, tion of the momentum after frustrated hops is surface hopping is widely applied in many dif- shown to work better than to ignore the com- ferent application areas and and a large body 3,8–15 pletely. The study also highlights the impor- of recent work exists. The simplicity of tance of the decoherence correction but neither the classical picture is deceptive as it never- of the two methods employed, energy based de- theless needs to mimic non-trivial underlying coherence and augmented fewest switches sur- quantum processes, such as (i) the branching of face hopping, performs completely satisfactory. the wavepacket onto different electronic states, More generally, the study emphasises the im- (ii) the loss of electronic coherence due to interactions with the nuclei or the environment, (iii)

1 the exchange of energy and momentum between of surface-hoping could be only assessed on sim- electronic and nuclear degrees of freedom, and ple systems where accurate reference values are (iv) classically forbidden transitions. available. In contrast, a reference for realistic Point (i) is treated by the surface hopping large systems is much more difficult to obtain. algorithm itself meaning that rather than de- In this paper, we introduce a new and gen- scribing the whole wavepacket branching onto erally applicable strategy to assess the qual- different potential energy surfaces (PES), one ity of surface hopping on complex large sys- of the surfaces is selected by using an stochas- tems using high-dimensional, many-state vi- tic algorithm and only this branch is further bronic coupling models. Since their introduc- propagated; an ensemble of trajectories follow- tion in the 80s,38 vibronic coupling models have ing the different branches is then needed to re- been very successful39–44 in reproducing ex- semble a bifurcating quantum wavepacket. The perimental work, particularly in combination fact that only one branch is propagated, auto- with the multiconfigurational time-dependent matically means that it is not possible to model Hartree (MCTDH) method.45–47 Recently, we the interactions between different branches and implemented an algorithm to perform surface their eventual loss of coherence (ii), and this hopping based on vibronic coupling models,48 has led to the introduction of decoherence cor- and showed that it can be extremely cheap com- rections on top of the surface hopping algo- putationally while still capturing the main fea- rithm.16–20 An exchange of energy and momen- tures of a variety of photophysical processes. tum (iii) should occur during surface hops and Here, we shall use a linear vibronic coupling different schemes of redistributing energy and (LVC) model to compare the results of sur- momentum have been developed. Here, a new face hopping against an accurate MCTDH ref- + complication (iv) comes into play if the quan- erence for [Re(im)(CO)3(phen)] (im = imida- tum and classical descriptions lead to incom- zole, phen = phenanthroline, see Fig. 1). patible results and the quantum propagation The choice of a transition metal complex as requires a classically forbidden hop, also called a test bed is purposely, as such systems fea- a "frustrated hop".18,21 In order to deal with ture a high number of excited electronic states the above-mentioned formal problems as well of different multiplicities in a limited domain as additional numerical problems22 a number of of energy and the description of its dynam- different flavours of the surface hopping method ics represent a particularly challenging case have been developed,9,12,14,23 able to work un- for spin-vibronic models.49 Moreover, the dy- + der different circumstances. namics of [Re(im)(CO)3(phen)] is particularly While surface hopping simulations can de- rich, as due to the presence of an intermedi- 3 pend strongly on the electronic structure ate intra-ligand triplet state IL (T3) that cou- method employed for the underlying on-the- ples strongly with the initially populated second fly calculations,24,25 it is often forgotten that singlet metal-to-ligand charge transfer 1MLCT changes in the surface hopping algorithm can (S2) state, spin-orbit coupling (SOC) effects are also have its consequences. The reliability of dominant at the early time of the dynamics (< surface hopping algorithms has been tested 50 fs) while vibronic effects lead to populate 3 particularly on idealized model systems, such the lowest MLCT (T1) state by exchange with 26–28 ? as spin-boson models, a quantum os- T3. Depending on the character and relative cillator,29 a two-level system in a classical positions of the low-lying states the early time bath,30 or on low-dimensional scattering prob- spin-vibronic mechanism will be driven essen- lems31–33 and the one-dimensional LiH sys- tially either by vibronic effects43,50 or by SOC tem.34 There also exist a few studies using effects.51 realistic high-dimensional PES via on-the-fly Studies based on the LVC model performed dynamics,18,35–37 but in this case it is more on a series of rhenium (I) carbonyl α-diimine challenging to find an accurate reference to complexes revealed the dominant normal modes compare with. Generally speaking, the validity and associated (spin) vibronic couplings that

2 drive the ultra-fast decay (< 200 fs) within 2.1 Wavefunction representations the low-lying singlet and triplet states.43,50–53 An important ingredient of SHARC is the op- Previous simulations on [Re(im)(CO) (phen)]+ 3 timal use of the possible representations of the showed that the key normal modes are the phen electronic wavefunctions.9 To ease the discus- and carbonyl vibrations whereas the imidazole sion, we establish here the name conventions ligand is a spectator.54? Here we employ a employed,9,55,56 see Figure 2. Most quantum model of 2 singlet and 4 triplet states, vibron- chemistry codes work with an electronic Hamil- ically coupled via 15 normal modes, as well tonian that includes molecular Coulomb inter- as three additional models with a reduced set actions but neither external fields nor SOC. of electronic states. In each case, 13 different We label this operator the molecular Coulomb surface hopping protocols are compared to the Hamiltonian (MCH) and its eigenfunctions MCTDH reference. The 200 trajectories prop- form the MCH basis (Figure 2 (b)). In this rep- agated over 500 fs are equivalent to a total of resentation, states possessing distinct multiplic- more than ten million formal electronic struc- ity are labelled as S ,S ,...,T ,T ,.... States ture computations. Doing this would be hardly 1 2 1 2 of the same spin-multiplicity do not cross in a feasible with on-the-fly dynamics but it requires one-dimensional picture whereas states of dif- negligible computational effort with our new ferent multiplicities do. The MCH states can implementation of LVC48 in the SHARC (sur- be transformed as to minimise nonadiabatic face hopping including arbitrary couplings)55,56 interactions, leading to states of almost con- dynamics package.57 stant character, known as diabatic38,58,59 (Fig- ure 2 (a)). These diabatic states are labelled according to their state character, e.g. 11MLCT and 21MLCT. The Hamiltonian including SOC is termed the "total Hamiltonian" and its eigenfunctions, generally possessing mixed spin, are the basis of what we call the "diagonal" repre- sentation55 (Figure 2 (c)). These states do not cross in a one-dimensional picture. It is important to realise that while the diabatic and MCH pictures feature a single PES for every triplet state, the diagonal representation considers explicitly the three individual surfaces composing the triplet state. The LVC model (see Section 2.2) is con- Figure 1: Chemical structure of the structed in a diabatic basis so that it can be di- + 47 [Re(im)(CO)3(phen)] complex studied within rectly used in MCTDH. In contrast, SHARC this work. expects input in the MCH representation and propagates the wavefunction in the diagonal picture. It is, thus, necessary to transform the LVC states into the MCH representation before 2 Methods feeding this data into SHARC, as described in Ref. 48. The output from SHARC can be trans- Here we review essential aspects of surface hop- formed back into any of the three pictures. In ping, such as the representations for the elec- this way, it is possible to perform a one-to-one tronic wavefunctions, the LVC approximation, comparison between SHARC and MCTDH de- and the methodological details of the surface spite the fact that different representations are hopping algorithm investigated in this work, i.e. used for the wavefunction propagation. decoherence corrections, momentum rescaling and frustrated hops.

3 Diabatic MCH Diagonal matrix.

(a) (b) S2 (c) E3 E5 X (n) 3 E2 E4 IL T1 Wnn = n + κi Qi (4) i 1 Energy 1 2 MLCT X (m,n) 1 MLCT ,Wmn = λ Qi. (5) S1 E1 i i Coordinate MCTDH SHARC input SHARC propagation The n are the vertical excitation energies. The κ(n) and λ(m,n) are termed intrastate and Figure 2: Wavefunction representations used i i interstate vibronic coupling constants.38 Here in this work: (a) the diabatic representation, these parameters were constructed from gradi- which is the basis for the LVC model and used ents and Hessian matrices, as described else- for MCTDH dynamics, (b) the MCH represen- where,51,52 while we have also shown that wave- tation, which is used by typical quantum chem- function overlaps can be used effectively for this istry codes and is the input for SHARC, and purpose.53 In addition, diabatic SOC constants (c) the diagonal representation, which is used were included as off-diagonal coupling terms, for SHARC propagations. as outlined in Ref. 51. All quantities required by the SHARC dynamics program can be con- 2.2 The linear vibronic coupling structed on-the-fly by means of straightforward model matrix operations, as detailed in Ref. 48. Within a vibronic coupling model, the PES are constructed in the diabatic representation, cf. 2.3 Decoherence corrections Fig. 2, as Decoherence is a fundamental concept in our V = V01 + W, (1) understanding of how a system governed by the laws of quantum mechanics can effectively be- where V0 is the ground state potential and the 60,61 W matrix collects the state-specific vibronic have classically. In the context of surface coupling terms. hopping dynamics decoherence comes into play The ground state potential is harmonic and whenever the electronic wavepacket splits into given as two different PES. For illustration, let us con- X hω¯ i sider an electronic wavepacket propagating on V = Q2. (2) 0 2 i two coupled PES with different nuclear gradi- i ents, see Fig. 3. Initially, the components on Here, Qi is a dimensionless mass-frequency the upper and lower surfaces start in the same scaled normal coordinate (cf. Ref. 58) defined region in space. However, while the component as r of the wavepacket on the upper surface moves ωi X p Q = K M r , (3) at constant speed, the part on the lower sur- i h¯ αi α α α face accelerates. As a consequence, the two parts of the wavepacket no longer occupy the where ωi is the frequency of normal mode i, Mα same region of space, leading to loss of coher- is an atomic mass, and Kαi denotes the orthog- onal conversion matrix between mass-weighted ence. If the system is simulated through sur- Cartesian and normal coordinates. face hopping dynamics, only one branch of the Within the current work, a linear vibronic wavepacket, for example the one on the upper coupling model (LVC) is considered, which con- surface, is explicitly propagated for each indi- tains the following state-specific terms in the W vidual trajectory. The nuclear coordinates on the second surface, indicated by empty squares in Fig. 3, are artificially fixed to match those of the first branch. As a consequence, in standard surface hopping decoherence is not treated cor-

4 rectly and a decoherence correction is usually ously damp the coefficients cα of all non-active included. In this work, we examine the effect states in each time step. To this aim, cα is re- −∆t/ταλ of two types of decoherence corrections. One placed by cαe and the coefficient of the is the energy based decoherence (EDC) scheme active state is then rescaled such that its phase of Grannucci et al.18 –based on earlier work is kept and the total population of all states is from Truhlar and co-workers16,17 – which only 1. requires information about energies at the cur- In the simplified version20 of the AFSSH for- rent time step. The other is a somewhat more malism the decoherence rate is computed as involved formalism, denoted augmented fewest switches surface hopping (AFSSH) and intro- 1 (Fαα Fλλ) (δRα δRλ) 20,62 = − · − duced by Subotnik and co-workers. The ταλ 2¯h − essence of the AFSSH method is that it ex- 2 Fαλ (δRα δRλ) | · − | (7) plicitly propagates auxiliary trajectories on the h¯ potential surfaces that are not active in the dy- ˆ namics. where Fαλ is defined as Ψα H Ψλ and − h | ∇ | i δRα is the position of the auxiliary trajectory belonging to state α. These auxiliary trajectories are propagated in a diabatic picture using a force that is proportional to the state popu- 2 lation cα as described in Ref. 20. The second term in| Eq.| (7) requires the evaluation of nonadiabatic coupling terms at every time step. In Energy order to lift this requirement, we project the term onto the nuclear velocity v and discretize the derivative to obtain Nuclear Coordinates 2 Hαλ v (δRα δRλ) | × · − | (8) Figure 3: Depiction of an electronic wavepacket ∆t h¯ v 2 propagating on two coupled potential energy × | | surfaces. The solid circles represent the true Here, Hαλ is an element of the locally diabatic behaviour of the system: the part of the Hamiltonian23 that is already used for wave- wavepacket on the lower surface moves faster function propagation in SHARC.9 Note that than the part on the upper surface and there- Eq. (8) is slightly modified with respect to Ref. fore the two branches of the wavepacket lose 20. coherence. The empty squares on S1 represent AFSSH proceeds by computing the decoher- the artificially overcoherent state present in sur- ence times for every inactive state and collaps- face hopping dynamics and illustrate the need ing its amplitude to zero according to a stochas- for applying a decoherence correction. tic algorithm.20 In addition, two ad hoc criteria are introduced to cause a reset of the auxiliary The EDC method18 proceeds by defining a trajectories without decoherence: (i) after every decoherence time surface hop and (ii) according to a reset-time derived from the first term of Eq. (8).20 h¯ C As final point, it is important to realise that ταλ = 1 + (6) Eα Eλ Ekin these decoherence corrections only help to de- | − | scribe how the wavepacket divides into inde- where Eλ and Eα are the potential energies of pendent branches. Should these branches meet the active surface λ and any other state α, Ekin again later, there is no way to describe their is the kinetic energy, and C is an adjustable pa- interference correctly with independent trajec- ? rameter usually set as C = 0.1 H.Sebastian: tories. The decoherence time ταλ is used to continu-

5 2.4 Momentum rescaling and vectors, which are not as readily available as frustrated hops energies and gradients do. Whereas momentum conservation (p) during A surface hopping algorithm ultimately has a hop is always possible and energy conserva- to describe the post Born-Oppenheimer ex- tion (E) is almost always possible for a large change of energy between nuclear and electronic system as the one considered here, it is often not degrees of freedom. Practically, this occurs possible to conserve both at the same time ful- through the momentum rescaling process that filling either the Eph or Epg conditions. Such is associated with surface hops. A number of cases, where a surface hop should occur accord- ? different strategies have been devised to this ing to the electronic Schrödinger equation but purpose. Here, we will consider four of these momentum rescaling is not possible, are termed possibilities, depending which quantity is con- frustrated hops. It has been argued that frus- 33 served in the hop. It is possible to impose trated hops are required to maintain quantum conservation of the total energy E, the nuclear detailed balance, i.e., the statistical ratio of momentum p, or both. Further, if both quanti- up and down transitions between different en- ties are conserved, then one has to allow at least ergy surfaces.29,30,65 Only by rejecting upward one degree of freedom where the momentum can hops is it possible to assure that the lower en- change; this can be done along the nonadiabatic ergy state has increased population in agree- coupling (NAC) vector hαλ = Fαλ/(Eα Eλ) or 29,30 − ment with the Boltzmann distribution. the gradient difference vector gαλ = Fαα Fλλ. − In the case of a frustrated hop the active state The four different momentum rescaling schemes does not change. There are two options about are summarised in Table 1. In the E scheme the what to do with the momentum and these op- energy is conserved and the the full momentum tions will be denoted + and –. In the + method vector is rescaled along the momentum. In the the momentum is left unaltered, i.e. nothing p scheme, the momentum is conserved, which at all happens after a frustrated hop. In the – means that no rescaling is performed at all. In method a portion of the momentum is reflected, the Eph and Epg schemes, the momentum is which effectively means that the trajectory ob- rescaled along the hαλ and gαλ directions, re- tains a second chance to pass through the cou- spectively. pling region and perform a hop to the upper surface. Table 1: Methods for momentum rescaling in- Specifically, we reflect the momentum paral- vestigated in this work. lel to the h or g vectors for the Eph and Epg Conserved quantity Rescaling methods, respectively. Following Refs 20,21 along this reversion is only done if the following two E Energy Momentum conditions are fulfilled: p Momentum None (Fλ fλα)(Fα fλα) < 0 (9) Eph Energy and momentum NAC · · Ep Energy and momentum Grad. diff. (Fα fλα)(p fλα) < 0 (10) g · · Based on formal arguments7,63,64 and numer- Here λ is the current state of the dynamics and ical results on model systems,29,65 a number α is the state the trajectory would have reached through the frustrated hop; fλα refers to hλα or of authors have concluded that Eph is the most rigorous option. Here We shall investi- gλα for the Eph and Epg methods, respectively. gate whether this conclusion holds for a larger The reversal proceeds by reversing the velocity for the atoms individually, thus conserving p system containing many degrees of freedom, as | | + and E in the frustrated hop. [Re(im)(CO)3(phen)] , and how strong the ef- + fect is. From a practical viewpoint it is worth new: We denote the resulting protocols Eph , Ep−, Ep+, and Ep−. In the case of E, it would noting that Eph is the only protocol that re- h g g quires the availability of nonadiabatic coupling in principle also be possible to apply the – pro-

6 tocol but we only evaluate + here (i.e. E implic- MCTDH method.45–47 The multiconfiguration itly means E+) for two reasons. First, the num- nuclear wave-function is expressed as a lin- ber of frustrated hops is small anyway. Second, ear combination of the Hartree products of an application of the E− method would imply the time-dependent basis functions, known to revert the full momentum, which means that as single-particle functions. The wavepacket the whole trajectory simply proceeds in reverse ansatz adapted to the present non-adiabatic after the frustrated hop. problem corresponds to the multiset formula- tion. The mode combination, number of prim- 2.5 Computational Details itive basis and single particle functions used in the simulations is the same as in Ref. 52. Vibronic coupling parameters are obtained Harmonic-oscillator basis sets were employed. from electronic structure calculations per- The initial wavepacket corresponds to the har- formed using the B3LYP functional (as defined monic vibrational ground state of the electronic by Frisch and coworkers66) and all electron ground state, promoted at time zero to the triple-ζ basis set.67 The scalar relativistic ef- 21MLCT absorbing state. The calculations fects were taken into account within the zeroth- are done with the Heidelberg MCTDH Package order regular approximation (ZORA).68 The (version 8.4.10).76 vertical transition energies for 2 singlets and 4 All surface hopping dynamics simulations triplet states were computed with TDDFT69,70 were performed in the diagonal representation at the same level described above under the [Fig. 1 (c)] meaning that SOC is included in Tamm-Damcoff approximation.71 The non- the PES. For the analysis, the results were equilibrium solvation within the linear-response transformed back into the diabatic represen- TD-DFT with a high-frequency dielectric con- tation as explained in Ref. 77, summing over stant of 1.77 for water was used. The SOC the three triplet components. The initial con- effects were introduced according to a simpli- ditions for the dynamics were created according fied relativistic perturbative TD-DFT formal- to a Wigner distribution of the zero-point vibra- ism.72,73 These electronic-structure calculations tional wavefunction within a harmonic approxi- were done with the ADF2013 code.74 mation.78 The electronic wavefunction was pre- The model multi-dimensional PES are built pared in the diabatic 21MLCT state and the ini- from the vibrational normal modes of the sin- tial active state of the trajectory was the diag- glet electronic ground state. From the 108 nor- onal surface most closely resembling this state. + mal modes of [Re(im)(CO)3(phen)] , 15 were Note that this is a non-standard option within selected as the most important ones involved SHARC requiring manual adjustment of the ini- in the excited state decay starting at 21MLCT, tial wavefunction coefficients. A nuclear time as described in Ref. 51. The resulting 15-mode step of 0.5 fs was chosen and the trajectories model Hamiltonian accounts for 12 a’ modes at were propagated during 500 fs. A locally di- 93, 235, 439, 498, 552, 637, 1174, 1336, 1444, abatic propagation for the wavefunctions was 1554, 1623, 1660 cm−1, and for 3 a" modes at chosen23 using 25 substeps per time step. For 90, 475 and 626 cm−1, which correspond mainly the decoherence correction we used the EDC to metal-carbonyl modes and vibrations local- method [Eq. (6), C=0.1 H], the simplified AF- ized on the phenanthroline ligand.51 The model SSH method,20 or no correction. Momentum parameters corresponding to the excited state rescaling was performed according to the four energies, SOC values, and intrastate and in- options presented in Tab. 1 and the + and – terstate coupling constants associated to those versions for treating frustrated hops as defined modes are reported in Ref. 52. All parameters in Section 2.4 were used. used are supplied along with the output data The quality of the dynamics was gauged by via an external repository.75 computing a time-averaged absolute error of The time-dependent Schrödinger equation the diabatic populations computed with surface for the nuclei was solved by employing the hopping against the MCTDH reference, defined

7 as Table 2: Singlet and triplet vertical excitations + of the [Re(im)(CO)3(phen)] complex: energies tmax Nst ∆t X X ref and state labels in the MCH and diabatic rep- tmax = pα(t) pα (t) . (11) tmax | − | resentations. t=∆t α=1 Here, p t is the population of state α at time MCH Diabatic ∆E (eV) α( ) 1 t and pref t corresponds to the reference value S1 1 MLCT 3.12 α ( ) 1 computed with MCTDH. In the case of triplet S2 2 MLCT 3.40 3 states, the p t value corresponds to the sum T1 1 MLCT 2.98 α( ) 3 over all three components of the state. Two T2 2 MLCT 3.07 3 values of t are considered: 5 fs to evaluate T3 1 IL 3.24 max 3 the accuracy during the verly early dynamics, T4 3 MLCT 3.42 and 500 fs. Accordingly, the maximum possible value for the error is 2, which would be ob- vertical excitation, the system was prepared in tained if there is no coincidence at all between the diabatic 21MLCT state and propagated for the states populated in the two runs. 500 fs. In Fig. 4 (a) dynamics computed at the As an alternative error measure, time con- MCTDH level of theory is presented. The first stants were obtained by fitting first-order ki- process observed is an ultrafast transfer of pop- netics models to the population data. Errors of ulation from 21MLCT to the almost quaside- the time constants were obtained with the boot- generate 33MLCT state mediated via SOC and 79 strapping method, using 1000 bootstrapping after only 12 fs the population of 21MLCT has samples for each ensemble. already decayed to 0.5. The 33MLCT state reaches a maximum at only 24 fs and sub- sequently decays while populating the other 3 Results triplet states and also transferring back some of the population to 21MLCT. After 500 fs about We start by discussing the states involved in the 3 dynamics and their main dynamical features, 60% of the population is in 1 MLCT while the remaining part is about evenly split between as determined by the MCTDH method. Sub- 3 3 sequently, we shall evaluate the performance of 2 MLCT and 1 IL. different surface hopping methods on the full Figure 4 (b) shows the evolution of the same model describing [Re(im)(CO) (phen)]+ and dynamics obtained with surface hopping and 3 selecting the EDC/E − option. At this level various simplified models. ph In line with previous work on this com- the agreement between surface hopping and plex,51,52 we will consider two singlet states MCTDH is very good and all the features men- and four triplet states (i.e. a total of 14 in- tioned above are well reproduced. The use of dividual spin-orbit coupled states). All but one SHARC allows us to easily connvert these re- are of predominant MLCT character with the sults into the MCH representation, which is remaining one of IL character. In Table 2, shown in Fig. 4 (c). The MCH populations the vertical excitation energies of these states closely resemble the diabatic ones with the ex- are presented. Two types of labels are given ception that the lower energy states have larger to account for the MCH (labelled by energy, populations than their corresponding diabatic states; for example, the rise of T4 in the early i.e. S1,S2,T1 T4) and diabatic (labelled by − 1 dynamics is somewhat lower than the rise of character, e.g. 2 MLCT for the second singlet 3 MLCT state) representations (cf. Fig. 2). Ta- 3 MLCT and the population of T1 at the end of the simulated period is somewhat larger than ble 2 shows that the six states occupy a narrow 3 energy range of only 0.5 eV, suggesting rapid the population of 1 MLCT. This is neverthe- nonadiabatic transitions. less expected as high-energy diabatic states mix To simulate the dynamics of the complex after with lower-lying energy states. moved up: We also want to briefly address the question of how

8 1 much the truncation of the model to 15 modes 11MLCT (a) MCTDH (15 modes) 1 affects the overall dynamics within the LVC 0.8 2 MLCT 13MLCT approximation. For this purpose, we recom- 0.6 23MLCT 3 puted the surface hopping dynamics including 0.4 1 IL + 33MLCT all 108 normal modes of [Re(im)(CO)3(phen)] 0.2 and the corresponding linear vibronic coupling Diabatic pop. 0 1 constants. The results, determined at the 1 − 1 MLCT (b) SHARC (15 modes) 1 EDC/Eph level, are presented in Fig. 4 (d). 0.8 2 MLCT This figure closely represents the correspond- 13MLCT 0.6 23MLCT ing results for the 15-mode model (Fig. 5 (b)) 3 0.4 1 IL with only a few exceptions, e.g. the rise of 33MLCT 0.2 33MLCT in the early dynamics is somewhat Diabatic pop. 0 less pronounced and there is no second rise of 1 21MLCT. We therefore conclude that the 15- S1 0.8 (c) SHARC (15 modes) S2 mode model is a reasonable approximation of T1 0.6 T2 the overall complex, at least within a harmonic T3 approximation. 0.4 T4 We are now in the position to address the MCH pop. 0.2 main concern of this work: How well are these 0 1 processes reproduced by using different approx- 11MLCT (d) SHARC (all modes) 1 imations within the surface hopping method? 0.8 2 MLCT 13MLCT For this purpose, we have evaluated the sur- 0.6 23MLCT 3 face hopping dynamics using 13 different lev- 0.4 1 IL 33MLCT els of theory, where the decoherence correction, 0.2 the mode of momentum rescaling and the treat- Diabatic pop. 0 ment of frustrated hops are varied. The results 0 100 200 300 400 500 are summarised in Fig. 5, see also the individ- Time (fs) ual results in Figures S1-S13 of the supporting Figure 4: Time-evolution of the state popu- information. The bars of Fig. 5 are computed lations of the full model considering (a) di- as floating averages on a logarithmic time scale abatic populations at the MCTDH level (15 considering the following intervals 0-1 fs, 1-2 modes), (b) diabatic populations (15 modes), fs, 2-4 fs, 4-8 fs until 256-500 fs, and coloured (c) MCH populations (15 modes), (d) diabatic according to their electronic and spin charac- populations (all modes) at the surface hopping ter. The upper left panel displays compactly (EDC/Ep−) level. the MCTDH results from Fig. 4 (a). Here, one h can see the initial population of 21MLCT, which + − is the dominant state until the fifth bar (the in- the EDC/Eph and EDC/Epg methods (49% terval 8-16 fs). Then, the 33MLCT state domi- each). nates (16-32 fs) while in the final interval more To provide a more quantitative discussion of than 50% of the population is in the 13MLCT the deviations we compute the mean-absolute state. error tmax [Eq. (11)] for the times tmax=5 Gratefully, all the methods considered show and 500 fs. When considering the whole time an appropriate time scale for the 21MLCT de- span of 500 fs, the lowest error (marked by − cay and correctly predict the intermediate rise in Fig. 5) is obtained for EDC/Eph with 3 ∗ + of 3 MLCT. However, most of them fall short 500 = 0.178 followed, again, by EDC/Eph and − when describing the outcome at the end of EDC/Epg . Our implementation of the AF- − SSH method performs surprisingly bad for this the dynamics and only the EDC/Eph correctly places more than 50% of the popula- system and none of the 500 values are below tion in the 13MLCT state, closely followed by 0.500. Not applying any decoherence correc-

9 + + EDC/E EDC/p EDC/Eph EDC/Eph− EDC/Epg EDC/Epg− MCTDH (0.079, 0.479) (0.080, 0.438) (0.076, 0.246) (0.077, 0.178) (0.144, 0.361) (0.147, 0.297) 1. 11MLCT 21MLCT 0.5 13MLCT ∗ 23MLCT 13IL 0. 33MLCT

0.5 Diabatic population

1. + + none/E AFSSH/E AFSSH/p AFSSH/Eph AFSSH/Eph− AFSSH/Epg AFSSH/Epg− (0.030, 0.660) (0.029, 0.617) (0.030, 0.631) (0.028, 0.564) (0.029, 0.541) (0.119, 0.581) (0.119, 0.534) 1. 11MLCT 21MLCT 0.5 13MLCT • 23MLCT 13IL 0. 33MLCT

0.5 Diabatic population

1. Figure 5: Diabatic electronic populations for the full model as a function of time considering different surface hopping algorithms plotted on a logarithmic time scale for 0-500 fs after photoexcitation. Mean-absolute errors computed for the first 5 fs and for the whole dynamics are given in parentheses (5, 500) and the best performers over these two timescales are marked by and , respectively. • ∗ tion “none/E” performs worst over the full time this purpose, we use the same set of parameter- scale with an error of 0.660. The picture is re- ized 15 normal modes but a smaller number of versed when just the first 5 fs are considered. In states. First, we consider only the two singlet this case all the EDC methods have errors over states 11MCLT and 21MCLT. As seen in the 0.075 while the AFSSH methods, with the ex- upper left panel of Fig. 6, the interconversion ception of Epg, have errors below 0.030 and the between these two states happens on the time + best result ( ) is obtained for Eph . Interest- scale of a few hundred femtoseconds and for the ingly, not applying• any decoherence correction last two bars, i.e. the time after 128 fs, more performs well for the first 5 fs indicating that than 50% of the population is in 11MCLT. It at this short time scale a coherent propagation can be readily seen that the interconversion is of the electron wavefunction is adequate. Al- too slow with all surface hopping protocols as though a detailed discussion will be done in the none of them has two bars below the line in- next section, we emphasise now that the errors dicating a population of 0.5. Interestingly, the introduced by the different surface hopping al- EDC protocols have significantly enhanced er- gorithms are non-negligible and that apparent ror bars when compared to their AFSSH coun- − unimportant algorithmic details do affect the terparts. For this model, the AFSSH/Eph results strongly. method gives the best performance considering Noting the challenges involved in the full both 5 and 500. + model of [Re(im)(CO)3(phen)] , where the in- We systematically increase the complexity of teractions of 14 spin-orbit coupled states have the model, considering the interaction between to be modelled correctly, we want now to get a the 21MCLT and 13IL states that gives rise to deeper insight into the effect of the different im- four spin-orbit coupled states. The upper left plementations by studying simpler models. For panel of Fig. 7 shows that intersystem crossing

10 + + EDC/E EDC/p EDC/Eph EDC/Eph− EDC/Epg EDC/Epg− MCTDH (0.028, 0.533) (0.029, 0.499) (0.035, 0.482) (0.028, 0.427) (0.054, 0.515) (0.054, 0.515) 1. 21MLCT 11MLCT 0.5

0.5 Diabatic population

1. + + none/E AFSSH/E AFSSH/p AFSSH/Eph AFSSH/Eph− AFSSH/Epg AFSSH/Epg− (0.018, 0.704) (0.018, 0.311) (0.018, 0.365) (0.018, 0.196) (0.018, 0.164) (0.050, 0.248) (0.050, 0.252) 1. 21MLCT 11MLCT 0.5

0.5 Diabatic population

1. • • • • •∗ Figure 6: Diabatic electronic populations considering the singlet MCLT states as a function of time considering different surface hopping algorithms plotted on a logarithmic time scale for 0-500 fs after photoexcitation. Mean-absolute errors computed for the first 5 fs and for the whole dynamics are given in parentheses (5, 500) and the best performers over these two timescales are marked by and , respectively. • ∗ should occur on a similar time scale to the inter- sult over the full time scale ( ) is obtained for ∗ − nal conversion of the previous case. However, EDC/p, closely followed by EDC/Eph while − the underlying physics is clearly different as for the first 5 fs, AFSSH/p and AFSSH/Eph 21MCLT and 13IL are coupled via SOC whereas are in the lead. Interestingly, the p method, the singlet states in Fig. 6 are connected by meaning that no momentum rescaling is per- non-relativistic vibronic coupling. In addition, formed at all after a hop, performs very well the position of the minimum of the 13IL poten- here. tial surface is significantly altered when com- Finally, we consider the 33MLCT and 13IL pared to the MLCT states.51 As a consequence states, which gives rise to 9 states interacting of these changes in parameters, strongly varying via SOC and vibronic coupling, see Fig. 8. In outcomes are observed for the different surface this case the interconversion occurs somewhat hopping methods. In general, improved results faster than in the previous cases and after 42 fs are obtained with respect to the previous case half the population is already in the 13IL state and most methods show acceptable error bars. according to the MCTDH reference. Most of Surprisingly, the only severe outliers are the the surface hopping protocols perform reason- + + Eph and Epg protocols, which exhibit strongly able well in this case with 5 values below 0.01 enhanced errors through overestimating the in- and 500 below 0.20. However, in this case the tersystem crossing rate when compared to all Epg methods are significantly away from the other methods. Again, this underlines how others with long-time errors above 0.40. The a seemingly innocuous methodological detail, best performing methods over 5 fs ( ) and 500 fs + • + such as the +/– treatment of frustrated hops ( ) are AFSSH/Eph and EDC/Eph , respec- can have severe consequences. The best re- tively.∗

11 + + EDC/E EDC/p EDC/Eph EDC/Eph− EDC/Epg EDC/Epg− MCTDH (0.011, 0.071) (0.010, 0.050) (0.030, 0.250) (0.024, 0.053) (0.035, 0.237) (0.015, 0.086) 1. 21MLCT 13IL 0.5

0.5 Diabatic population

1. ∗ + + none/E AFSSH/E AFSSH/p AFSSH/Eph AFSSH/Eph− AFSSH/Epg AFSSH/Epg− (0.003, 0.452) (0.005, 0.175) (0.002, 0.182) (0.007, 0.367) (0.002, 0.194) (0.008, 0.320) (0.007, 0.177) 1. 21MLCT 13IL 0.5

0.5 Diabatic population

1. • • Figure 7: Diabatic electronic populations considering the 21MLCT and 13IL states as a function of time considering diﬀerent surface hopping algorithms plotted on a logarithmic time scale for 0-500 fs after photoexcitation. Mean-absolute errors computed for the ﬁrst 5 fs and for the whole dynamics are given in parentheses (5, 500) and the best performers over these two timescales are marked by and , respectively. • ∗

A more compact form of the results is pre- were considered sented in Table 3, where the 5 and 500 values of the four types of dynamics are averaged. Ac- S2 + S1 )* T4 + T3 + T2 + T1 (12) − cordingly, one finds that the EDC/Eph method S2 S1 (13) clearly outperforms all other protocols, closely → S2 T3 (14) followed by AFSSH/E − and EDC/E +. If → ph ph T )* T (15) high precision is required in the early part of 4 3 the dynamics, then EDC somewhat overcor- and the data fitted appropriately. In all these rects and one of the AFSSH protocols is prefer- cases, the decay times for the forward reac- able where the lowest errors are obtained for tion were compared although for Eq. (12) and AFSSH/Eph in its + and – versions. In gen- Eq. (15) we also fitted the backwards reac- eral, we observe that the most rigorous, yet the tion rate. The obtained times are plotted in computationally most involved, Eph method Fig. 9 on a logarithmic scale. The dotted lines clearly outperforms the others. We also observe indicate the reference values computed at the a clear trend in the way that frustrated hops are MCTDH level and the shaded areas indicate treated where – outperforms +. The compar- 25% error windows. corresponding to val-± ison between EDC and AFSSH is not so clear ues between 3/4 and 4/3 times the ref- but there is a slight preference for EDC. erence time. In the case of the full model, A complementary perspective of the dynam- we fitted the overall intersystem crossing as a ics can be achieved by fitting time constants. reversible interconversion between singlets and For this purpose, the following kinetic models triplets [cf. Eq. (12)]. The reference time

12 + + EDC/E EDC/p EDC/Eph EDC/Eph− EDC/Epg EDC/Epg− MCTDH (0.003, 0.141) (0.005, 0.204) (0.033, 0.044) (0.003, 0.049) (0.024, 0.395) (0.022, 0.433) 1. 33MLCT 13IL 0.5

0.5 Diabatic population

1. ∗ + + none/E AFSSH/E AFSSH/p AFSSH/Eph AFSSH/Eph− AFSSH/Epg AFSSH/Epg− (0.012, 0.644) (0.009, 0.147) (0.010, 0.191) (0.002, 0.080) (0.006, 0.068) (0.046, 0.405) (0.047, 0.412) 1. 33MLCT 13IL 0.5

0.5 Diabatic population

1. • Figure 8: Diabatic electronic populations considering the 33MLCT and 13IL states as a function of time considering different surface hopping algorithms plotted on a logarithmic time scale for 0-500 fs after photoexcitation. Mean-absolute errors computed for the first 5 fs and for the whole dynamics are given in parentheses (5, 500) and the best performers over these two timescales are marked by and , respectively. • ∗ for the forward intersystem crossing was deter- tion. mined as 16.0 fs. All the surface hopping simu- Figure 9 evidences that none of the surface lations stay well within the indicated error win- hopping protocols places all four time constants dow and deviate less than 2 fs from this result. within the given error window; the only pro- The 21MLCT/11MLCT system decays with a tocol that keeps at least three out of the four time constant of 235 fs for MCTDH. Interest- time constants within the specified error win- − ingly, all the surface hopping protocols over- dow is AFFSH/Eph . Moreover, one sees that − shoot this value and AFFSH/Eph is the only the errors are generally not uniform as the time method within the specified error window. The scales are sometimes overestimated and some- 21MLCT/13IL interconversion occurs on a very times underestimated, showing that there is similar time scale to the previous case at the probably no simple solution. An interesting MCTDH level (239 fs). As opposed to the pre- trend is the fact that all the + methods de- vious case, most of the employed surface hop- cay on shorter time scales than the – methods. ping methods underestimate this time and four This may be understood in the following simple methods are within the error window: EDC/E, picture: The – protocol redirects the trajectory − − EDC/p, EDC/Eph , and Epg . The intercon- into the crossing region after a frustrated hop. version between 33MLCT and 13IL occurs with Thus, the trajectory obtains a second chance to a time constant of 57 fs at the MCTDH level. hop up into the higher state, which leads to the This time constant is generally overestimated fact that the overall decay to the lower state is by surface hopping, with the best time con- slowed down. + stants provided by the Eph protocol using ei- Finally, it is of interest to discuss whether the ther the EDC or AFSSH decoherence correc- results are statistically significant, i.e. whether

13 Table 3: Time-weighted mean absolute errors averaged over the four types of trajectories computed for the initial 5 fs and the full 500 fs of the dynamics.

+ − + − tmax E p Eph Eph Epg Epg 5 fs EDC 0.030 0.031 0.043 0.033 0.064 0.059 AFSSH 0.016 0.016 0.015 0.015 0.057 0.057 none 0.017 500 fs EDC 0.306 0.298 0.255 0.177 0.377 0.333 AFSSH 0.313 0.343 0.302 0.242 0.389 0.344 none 0.616

200 trajectories are enough to provide almost of a smaller number of well-separated states as converged decay times. This is assessed using usually found in small organic molecules. The ? a bootstrap algorithm that estimates the challenge of applying the Eph protocol is that error related to the finite number of trajectories. it requires the availability of nonadiabatic cou- The results are presented as error bars in the plings, which are not straightforward to calcu- EDC/E panel shown on the left of Fig. 9. As late with most of the electronic structure codes these errors are significantly smaller than the available. With this hurdle in mind, we have fluctuations between the different methods, it attempted the same protocol only replacing the is fair to assume that none of the conclusions nonadiabatic coupling with the gradient differ- would change if a significantly larger number of ence (Epg). However, we found that this intro- trajectories were run. duces significant errors and in many respects even performs worse than a simple rescaling along the momentum. A more pragmatic ap- 4 Discussion proach to solve the problem may be to introduce a new ad hoc criterion to reduce the num- The disparity of the data presented above illus- ber of upward hops, e.g. limiting the maximal trates the importance of the algorithmic details, hopping energy. But this option is out of the and while it makes it difficult to draw a final scope of this paper. conclusion, a number of clear trends emerge. Considering that the number of frustrated For all the models considered and ways to eval- hops is the main feature that sets the Ep and uate the results, it is found that momentum h Ep methods apart from the others, we have rescaling along the nonadiabatic coupling vec- g evaluated different options of what to do after a tor (E ) is superior to the three other eval- ph frustrated hop in these two cases and found that uated methods. It has been argued previously reflection (–) generally outperforms completely that this is the most rigorous way of momen- ignoring the hops (+). The – protocol reflects tum rescaling.7,63,64 The main effect seen by the trajectory back into the crossing region and the application of E is an enhanced num- ph provides a second chance of hopping into the ber of frustrated hops, which lead to the fact higher state. As a consequence, we find that the that the trajectory correctly remains trapped net transfer to the lower state is slowed down in the lower state. This is similar to the re- whenever the – protocol is used. As discussed sults of Refs 29,30,65 which discuss this effect above, the – protocol does not make much sense in terms of detailed balance. This effect is par- for simple rescaling along the momentum (E−) ticularly drastic whenever a large number of and was not investigated here. states are present in a narrow energy window Unfortunately, the question of decoherence as is the case for transition metal complexes correction does not have a straightforward an- but will probably play a smaller role in the case swer. While we generally find that not applying

14 Full model 21MLCT/11MLCT 21MLCT/13IL 33MLCT/13IL

640

320

160

40 Decay time (fs)

+ + + + E p − g − E p − g − E p h p h p p g p h p h p p g E E E E E E E E EDC/ EDC/ none/ EDC/ EDC/ EDC/ EDC/ AFSSH/ AFSSH/ AFSSH/ AFSSH/ AFSSH/ AFSSH/

Figure 9: Fitted decay times for the diﬀerent models and computational methods plotted on a logarithmic scale. The dashed lines and highlighted areas indicate MCTDH reference values (note that the reference values for 21MLCT/11MLCT and 21MLCT/13IL almost coincide and are, therefore, not distinguishable in this plot).

a decoherence correction leads to incorrect re- consistent with respect to the density of states sults, none of the EDC and AFSSH protocols in the system as the addition of non-interacting provided satisfactory solutions. EDC overcor- electronic states to any model system will lower rected in the short time scale while the AFSSH the decoherence rate through an enhanced num- method generally underestimated decoherence. ber of surface hops. This problem might be The problem of these methods may be under- solved by using a more sophisticated implemen- stood in the sense that neither of them is size- tation of the AFSSH algorithm that treats triv- consistent in the present implementation. The ial crossings differently from true surface hops. EDC method [Eq. (6)] depends on the overall [Sebastian: One could suggest that 1 mi- kinetic energy Ekin of the system. Adding non- nus the hop-inducing probability could be interacting atoms to the system will increase used as a reset probability, but if you do the overall kinetic energy and thus reduce the not want to add such speculations, it’s decoherence time. In the case of AFSSH the ar- also fine to me.] However, for the present gument is more subtle. Following Ref. 20, the study we are left to conclude that AFSSH does positions and momenta of the auxiliary trajec- not produce enhanced results compared to EDC tories are reset after every surface hop. In the despite its more involved formalism. full model of the complex considered there are On the optimistic side, we find that for any 14 states present within as little as 0.5 eV. As a of the models considered, there is at least one consequence, an exorbitant number of hops oc- surface hopping protocol that produces satis- cur, mostly related to trivial crossings, meaning factory results, revealing that errors are not due that the auxiliary trajectories never have the to non-local quantum effects –which would in- time to properly build up to induce decoher- validate the trajectory approximation as such– ence and as a consequence the decoherence rate but rather to small methodological implementa- is strongly underestimated. More generally, we tions of the surface hopping method. This work can say that the AFSSH algorithm is not size- then illustrates that it is important to obtain

15 a deeper understanding of the effect of these Time-dependent populations of the dynam- methodological details. Conversely, it would ics simulations for all methods and models be interesting to see if improved results could (Figure S1-S52). The research data support- be obtained by other on-the-fly dynamics ap- ing this publication can be accessed via DOI: proaches such as ab initio multiple spawning,1 10.17028/rd.lboro.c.4493135.v1.75 This mate- the coupled-trajectory mixed quantum-classical rial is available free of charge via the Internet scheme,37 and variational multiconfigurational at http://pubs.acs.org/. Gaussians.80 References 5 Conclusion (1) Martínez, T. J. Acc. Chem. Res. 2006, 39, The purpose of this study was to evaluate the 119–126. reliability of surface hopping dynamics in the challenging case of transition metal complexes, (2) Plasser, F.; Barbatti, M.; Aquino, A. J. A.; which are typically characterised by the pres- Lischka, H. Theor. Chem. Acc. 2012, 131, ence of a high density of electronic states and a 1073. large number of crossings among them. For this (3) Nelson, T.; Fernandez-Alberti, S.; Roit- purpose, we constructed a LVC model of the berg, A. E.; Tretiak, S. Acc. Chem. Res. + [Re(im)(CO)3(phen)] complex including 15 vi- 2014, 47, 1155–1164. brational normal modes. Simulations were run on the full set of relevant states (2 singlets and (4) Long, R.; Prezhdo, O. V.; Fang, W. Wiley 4 triplets) as well as on smaller subsets. Surface Interdiscip. Rev. Comput. Mol. Sci. 2017, hopping simulations were run by varying three 7, e1305. parameters in the algorithm –the mode of mo- (5) Oberhofer, H.; Reuter, K.; Blumberger, J. mentum rescaling, the treatment of frustrated Chem. Rev. 2017, 117, 10319–10357. hops, and the decoherence correction– and then compared against an MCTDH reference. It was (6) Curchod, B. F. E.; Martínez, T. J. Chem. clearly found that momentum rescaling along Rev. 2018, 118, 3305–3336. the coupling vector outperforms all other methods. Also a preference for reflecting the momen- (7) Tully, J. C. J. Chem. Phys. 1990, 93, tum after a frustrated hop was found. Neither 1061–1071. of the two decoherence corrections applied were (8) Kubar, T.; Elstner, M. Phys. Chem. completely satisfactory but we found better re- Chem. Phys. 2013, 15, 5794. sults for the simple and robust EDC method. Acknowledgement FP, SM and LG grate- (9) Mai, S.; Marquetand, P.; González, L. Int. fully acknowledge funding from the Aus- J. Quantum Chem. 2015, 115, 1215–1231. trian Science Fund – Austria (FWF) within (10) Tavernelli, I. Acc. Chem. Res. 2015, 48, project I2883. MF, EG and CD thank the 792–800. Labex CSC (ANR-10-LABX- 0026_CSC) and the French/Austrian ANR-15-CE29-0027-01 (11) Subotnik, J. E.; Jain, A.; Landry, B.; Pe- DeNeTheor. The electronic structure calcu- tit, A.; Ouyang, W.; Bellonzi, N. Annu. lations were performed at the Vienna Scientific Rev. Phys. Chem. 2016, 67, 387–417. Cluster, the High Performance Computer Cen- (12) Wang, L.; Akimov, A.; Prezhdo, O. V. J. tre (HPC) of the University of Strasbourg and Phys. Chem. Lett. 2016, 7, 2100–2112. nodes cluster of the Laboratoire de Chimie Quantique (CNRS/University of Strasbourg). (13) Spencer, J.; Scalfi, L.; Carof, A.; Blum- Supporting Information Available: berger, J. Faraday Discuss. 2016, 195, 215–236.

16 (14) Crespo-Otero, R.; Barbatti, M. Chem. (30) Parandekar, P. V.; Tully, J. C. J. Chem. Rev. 2018, 118, 7026–7068. Phys. 2005, 122 .

(15) Filatov, M.; Min, S. K.; Kim, K. S. J. (31) Topaler, M. S.; Hack, M. D.; Alli- Chem. Theory Comput. 2018, 14, 4499– son, T. C.; Liu, Y.-P.; Mielke, S. L.; 4512. Schwenke, D. W.; Truhlar, D. G. J. Chem. Phys. 1997, 106, 8699–8709. (16) Zhu, C.; Nangia, S.; Jasper, A. W.; Truh- lar, D. G. J. Chem. Phys. 2004, 121, (32) Topaler, M. S.; Allison, T. C.; 7658–7670. Schwenke, D. W.; Truhlar, D. G. J. Chem. Phys. 1998, 109, 3321–3345. (17) Zhu, C.; Jasper, A. W.; Truhlar, D. G. J. Chem. Theory Comput. 2005, 1, 527–540. (33) Jasper, A. W.; Hack, M. D.; Truh- lar, D. G. J. Chem. Phys. 2001, 115, (18) Granucci, G.; Persico, M. J. Chem. Phys. 1804–1816. 2007, 126, 1–11. (34) Mignolet, B.; Curchod, B. F. E. J. Phys. (19) Jaeger, H. M.; Fischer, S.; Prezhdo, O. V. Chem. A 2019, 123, acs.jpca.9b00940. J. Chem. Phys. 2012, 137, 22A545. (35) Plasser, F.; Lischka, H. J. Chem. Phys. (20) Jain, A.; Alguire, E.; Subotnik, J. E. J. 2011, 134, 34309. Chem. Theory Comput. 2016, 12, 5256– 5268. (36) Worth, G. A.; Hunt, P.; Robb, M. A. J. Phys. Chem. A 2003, 107, 621–631. (21) Jasper, A. W.; Truhlar, D. G. Chem. Phys. Lett. 2003, 369, 60–67. (37) Min, S. K.; Agostini, F.; Tavernelli, I.; Gross, E. K. U. The J. Phys. Chem. Let- (22) Plasser, F.; Granucci, G.; Pittner, J.; Bar- ters 2017, 8, 3048–3055. batti, M.; Persico, M.; Lischka, H. J. Chem. Phys. 2012, 137, 22A514. (38) Koppel, H.; Domcke, W.; Ceder- baum, L. S. Adv. Chem. Phys. 1984, 57, (23) Granucci, G.; Persico, M.; Toniolo, A. J. 59–246. Chem. Phys. 2001, 114, 10608–10615. (39) Raab, A.; Worth, G. A.; Meyer, H.-D.; (24) Plasser, F.; Crespo-Otero, R.; Peder- Cederbaum, L. S. J. Chem. Phys. 1999, zoli, M.; Pittner, J.; Lischka, H.; Bar- 110, 936–946. batti, M. J. Chem. Theory Comput. 2014, 10, 1395–1405. (40) Krawczyk, R. P.; Malsch, K.; Hohlne- icher, G.; Gillen, R. C.; Domcke, W. (25) Mai, S.; Atkins, A. J.; Plasser, F.; Chem. Phys. Lett. 2000, 320, 535–541. González, L. submitted for publication 2019, (41) Faraji, S.; Meyer, H.-D.; Köppel, H. J. Chem. Phys. 2008, 129, 074311. (26) Müller, U.; Stock, G. J. Chem. Phys. 1997, 107, 6230–6245. (42) Tamura, H.; Burghardt, I. J. Am. Chem. Soc. 2013, 135, 16364–16367. (27) Landry, B. R.; Subotnik, J. E. J. Chem. Phys. 2012, 137, 22A513. (43) Eng, J.; Gourlaouen, C.; Gin- densperger, E.; Daniel, C. Acc. Chem. (28) Chen, H. T.; Reichman, D. R. J. Chem. Res. 2015, 48, 809–817. Phys. 2016, 144, 094104. (44) Fumanal, M.; Daniel, C. J. Comput. (29) Käb, G. J. Phys. Chem. A 2006, 110, Chem. 2016, 3, 2454–2466. 3197–3215.

17 (45) Meyer, H.-D.; Manthe, U.; Cederbaum, L. (57) Mai, S.; Richter, M.; Ruckenbauer, M.; Chem. Phys. Lett. 1990, 165, 73–78. Plasser, F.; Oppel, M.; Marquetand, P.; González, L. Sharc: Surface Hopping (46) Beck, M. H.; Jäckle, A.; Worth, G. A.; Including Arbitrary Couplings – Pro- Meyer, H.-D. Phys. Rep. 2000, 324, 1– gram Package for Non-Adiabatic Dynam- 105. ics; available from http://sharc-md.org. (47) Meyer, H.-D., Gatti, F., Worth, G. A., 2014. Eds. Multidimensional Quantum Dynam- (58) Worth, G. A.; Cederbaum, L. S. Annu. ics: MCTDH Theory and Applications; Rev. Phys. Chem. 2004, 55, 127–158. Wiley-VCH, 2009. (59) Schuurman, M. S.; Yarkony, D. R. J. (48) Plasser, F.; Gómez, S.; Menger, M. F. Chem. Phys. 2007, 127, 094104. S. J.; Mai, S.; González, L. PCCP 2019, 21, 57–59. (60) Tegmark, M. Phys. Rev. E - Stat. Physics, Plasmas, Fluids, Relat. Interdiscip. Top. (49) Penfold, T. J.; Gindensperger, E.; 2000, 61, 4194–4206. Daniel, C.; Marian, C. M. Chem. Rev. 2018, 118, 6975–7025. (61) Zurek, W. H. Rev. Mod. Phys. 2003, 75, 715–775. (50) Harabuchi, Y.; Eng, J.; Gindensperger, E.; Taketsugu, T.; Maeda, S.; Daniel, C. J. (62) Subotnik, J. E.; Shenvi, N. J. Chem. Phys. Chem. Theory Comput. 2016, 12, 2335– 2011, 134, 30–45. 2345. (63) Herman, M. F. J. Chem. Phys. 1984, 81, (51) Fumanal, M.; Gindensperger, E.; 754–763. Daniel, C. J. Chem. Theory Comput. 2017, 13, 1293–1306. (64) Coker, D. F.; Xiao, L. J. Chem. Phys. 1995, 102, 496–510. (52) Fumanal, M.; Gindensperger, E.; Daniel, C. Phys. Chem. Chem. Phys. (65) Carof, A.; Giannini, S.; Blumberger, J. J. 2018, 20, 1134–1141. Chem. Phys. 2017, 147 .

(53) Fumanal, M.; Plasser, F.; Mai, S.; (66) Stephens, P. J.; Devlin, F. J.; Cha- Daniel, C.; Gindensperger, E. J. Chem. balowski, C. F.; Frisch, M. J. J. Phys. Phys. 2018, 124119. Chem. 1994, 98, 11623–11627.

(54) Fumanal, M.; Gindensperger, E.; (67) Van Lenthe, E.; Baerends, E. J. Jour- Daniel, C. The Journal of Physical Chem- nal of Computational Chemistry 2003, 24, istry Letters 2018, 9, acs.jpclett.8b02319. 1142–1156.

(55) Mai, S.; Marquetand, P.; González, L. Wi- (68) van Lenthe, E.; van Leeuwen, R.; ley Interdiscip. Rev. Comput. Mol. Sci. Baerends, E. J.; Snijders, J. G. Inter- 2018, 8, e1370. national Journal of Quantum Chemistry 1996, 57, 281–293. (56) Mai, S.; Plasser, F.; Marquetand, P.; González, L. In Attosecond Molecular (69) Runge, E.; Gross, E. K. Physical Review Dynamics; Vrakking, M. J. J., Lep- Letters 1984, 52, 997–1000. ine, F., Eds.; Theoretical and Computa- (70) Petersilka, M.; Gossmann, U. J.; Gross, E. tional Chemistry Series; The Royal Soci- K. U. Physical Review Letters 1996, 76, ety of Chemistry, 2019. 1212–1215.

18 (71) Peach, M. J.; Tozer, D. J. Journal of Phys- ical Chemistry A 2012, 116, 9783–9789.

(72) Wang, F.; Ziegler, T.; van Lenthe, E.; van Gisbergen, S.; Baerends, E. J. The Journal of Chemical Physics 2005, 122, 204103.

(73) Wang, F.; Ziegler, T. The Journal of Chemical Physics 2005, 123, 154102.

(74) ADF, SCM, Theoretical Chem- istry, Vrije Universiteit,Amsterdam, The Netherlands, 2013, online at https://www.scm.com/Downloads/.

(75) Supporting research data available: Pa- rameters (SHARC input format) and time-dependent populations from the dynamics simulations for all the models evaluated. https://doi.org/10.17028/ rd.lboro.c.4493135.v1.

(76) Meyer, H.-D. The MCTDH Package, version 8.4, University of Heidelberg, Heidel- berg, Germany, 2007.

(77) Mai, S.; Marquetand, P.; González, L. J. Chem. Phys. 2014, 140, 204302.

(78) Dahl, J. P.; Springborg, M. J. Chem. Phys. 1988, 88, 4535–4547.

(79) Nangia, S.; Jasper, A. W.; Miller, T. F.; Truhlar, D. G. J. Chem. Phys. 2004, 120, 3586–3597.

(80) Richings, G. W.; Polyak, I.; Spinlove, K. E.; Worth, G. A.; Burghardt, I.; Lasorne, B. Int. Rev. Phys. Chem. 2015, 34, 269–308.

19 Graphical TOC Entry

Decoherence Momentum rescaling Frustrated hops

Quantum dynamics Surface hopping