Letter pubs.acs.org/JPCL
Mixed Quantum-Classical Dynamics Using Collective Electronic Variables: A Better Alternative to Electronic Friction Theories Ilya G. Ryabinkin*,† and Artur F. Izmaylov*,‡ †
Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
‡
S Supporting Information *
ABSTRACT: An accurate description of nonadiabatic dynamics of molecular species on metallic surfaces poses a serious computational challenge associated with a multitude of closely spaced electronic states. We propose a mixed quantum-classical scheme that addresses this challenge by introducing collective electronic variables. These variables are defined through analytic block-diagonalization applied to the time-dependent Hamiltonian matrix governing the electronic dynamics. We compare our scheme with a simplified Ehrenfest approach and with a full-memory electronic friction model on a 1D “adatom + atomic chain” model. Our simulations demonstrate that collective-mode dynamics with only a few (two to three) electronic variables is robust and can describe a variety of situations: from a chemisorbed atom on an insulator to an atom on a metallic surface. Our molecular model also reveals that the friction approach is prone to unpredictable and catastrophic failures.
R
dozens of states were sufficient for converged dynamics on a subpicosecond time scale.15 Despite all of the merits, this approach requires prior knowledge of the hybridization function, which has to be deduced from the Newns−Anderson model Hamiltonian,17 making this approach unsuitable for onthe-fly simulations. A more traditional approach to systems with a dense electronic spectrum is to eliminate the explicit electronic dynamics entirely. Typically, the electronic subsystem is described as a bath of harmonic oscillators.18,19 However, realistic electron−nuclear couplings do not allow for analytic integration of the electronic equations of motion (EOM), and further simplifications become inevitable. These usually involve the constant-coupling approximation, which leads to the Markovian limit and an assumption that the density of states around the Fermi level is smooth.19 Several such electronic friction theories have been presented over the years.19−27 All of them were capable of capturing experimental trends when applied to selected realistic systems.15,25,28−39 Surprisingly, none of these frictional theories have been scrutinized on models where the Ehrenfest method can be used to gauge the adequacy of introduced dynamical approximations. To the best of our knowledge, only ref 15 made a direct comparison of the Head-Gordon and Tully19 friction model with the FSSH-style dynamics. Authors point out that the friction model predicts somewhat faster relaxation at short times (t < 150 fs) as compared with its surface-hopping counterpart. There are other indications that friction models
ecent progress in theory and implementation has made fully atomistic first-principles nonadiabatic simulations of large molecular systems increasingly more accessible. Many of these simulations are performed with the aid of mixed quantum-classical (MQC) methods,1−8 which treat nuclei classically, whereas electrons are treated quantum-mechanically. Computational efficiency of MQC methods hinges upon a large spacing between electronic states. Only a small subset of strongly coupled states needs to be considered, and their explicit propagation poses no computational difficulty. The situation becomes more complicated for nonadiabatic dynamics on metallic and semiconductor surfaces. Tracing a plethora of electronic states is computationally intensive.9 Even in the simplest MQC Ehrenfest method,1,2 a representation chosen for the electronic wave function cannot contain fewer variables than the number of accessible states, and hence hundreds or even thousands of electronic degrees of freedom (DOF) must be considered for molecular dynamics on a metallic surface. Computing this number of excited states and nonadiabatic couplings (NACs) may not be a daunting task per se. For example, within the Kohn−Sham density functional theory and the independent-particle approximation, the excitation frequencies are only orbital energy differences,10 and the NACs can be computed from derivatives of the corresponding orbitals.11−14 A challenge, therefore, is how to avoid solving the timedependent Schrödinger equation for all electronic variables. This problem has already received some attention. An effective electronic picture has been proposed by Shenvi et al.15 for the fewest-switches surface hopping (FSSH) method. A metallic continuum was discretized through a Gaussian quadrature, whose nodes and weights were derived from a molecule−metal coupling (hybridization) function.16 Several © 2016 American Chemical Society
Received: November 19, 2016 Accepted: December 30, 2016 Published: December 30, 2016 440
DOI: 10.1021/acs.jpclett.6b02712 J. Phys. Chem. Lett. 2017, 8, 440−444
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The Journal of Physical Chemistry Letters
adopted: (1) the adiabatic excitation frequencies are positionindependent, ℏωk0(R) = Ek(R) − E0(R) = constk, and (2) the ground electronic state is weakly coupled to the excited ones, which translates into the condition |c0(t)|2 ≈ 1. Under these assumptions, the electronic dynamics can be integrated out, and eqs 5 and 6 are converted into
may exaggerate nuclear energy dissipation: for example, refs 28 and 29 report the divergence of the friction coefficient (“infinite stopping power”) for the H on Cu(111) system. However, it is not clear to what level of approximation these issues must be attributed. We carefully assess the friction approximation and propose a computationally superior scheme that employs a superadiabatic-like transformation.40,41 This transformation introduces approximate dynamical decoupling of electronic DOF and allows us to reduce the number of retained electronic variables to only few. With the new formalism we aim at the description of systems with a dense but potentially highly nonuniform electronic spectrum, such as chemi- and physisorbed atoms, radicals, or molecules on metallic surfaces.42 Ehrenfest and Electronic Friction. The Ehrenfest method can be derived19,43 from a semiclassical energy functional W=
∑ α
MαṘ α2 + ⟨Ψ|Ĥe|Ψ⟩ 2
Mα R ̈ α +
∑ cj(t )e−iϕ (t)|Φj[R(t )]⟩ j
(2)
where ϕj(t) = ℏ−1∫ t0 Ej[R(t′)]dt′ are dynamical phases, {Ej[R(t)]} and {|Φj[R(t)]⟩} are adiabatic potential energy surfaces (PESs) and wave functions, respectively, Ĥ e |Φj[R(t)]⟩ = Ej[R(t)]|Φj[R(t)]⟩, and {cj(t)} are complex time-dependent coefficients. Using the electronic time-dependent Schrödinger equation projected onto {|Φj[R(t)]⟩}, one can derive the electronic EOM ck̇ = −∑ cjeiϕkjτkj (3)
j
where τkj(t) = ⟨Φk[R(t)]|Φ̇j[R(t)]⟩ are the time-derivative NACs. Conservation of W, Ẇ = 0, gives the nuclear EOM2 Mα R ̈ α +
∑ |ck(t )|2 k
∂Ek = ∂R α
∑ ck*cjeiϕ f kjα kj
k≠j
(4)
where f αkj(t) = −⟨Φk|∂RαĤ e|Φj⟩ are nonadiabatic forces along the trajectory and ϕkj(t) = ϕk(t) − ϕj(t). Two approximations are used for deriving friction theories: (1) the average force −∑k |ck(t)|2∂Ek/∂Rα is replaced with a derivative of the ground-state PES, −∂E0/∂Rα, in eq 4, and (2) all excited-excited couplings τ kj are considered to be insignificant. These approximations are justified by accounting a predominant role of the ground state and low population of the excited states. They result in the simplified Ehrenfest (SE) equations c ̇ = −Tc, Tkj = (δk 0 + δj0)τkjeiϕkj Mα R ̈ α +
∂E0 = ∂R α
∑ ck*cjeiϕ f kjα
+ αβ(t , t ′)Ṙβ(t ′) dt ′ + 9 α(t )
⎛ d ̇ ⎞ ⎛ 0 − z ⎞⎛ d ⎞ 0 ⎜ 0⎟ = ⎜ ⎟⎜⎜ ⎟⎟ ⎜ ̇ ⎟ ⎝ z iw11 ̃ ⎠⎝ d1 ⎠ ⎝ d1 ⎠
(8)
where w̃ 11 = z−2∑k=1 ωk0τ2k0 is the collective frequency, which originated from N. The nonadiabatic force matrix {fαkj(t)}k,j≥0, projected onto the interaction space, {U†fαU}2×2, has only one ̃ = z−1∑k=1 f αk0τk0. Using this projection, we nonzero element, fα10 rewrite the nuclear Ehrenfest EOM (eq 6) as
(5)
kj
k≠j
β
0
t
where + αβ(t , t ′) = −2ℏ ∑k > 0 f kα0 (t )fkβ0 (t ′) cos[ωk 0(t ′ − t )]/ωk 0 is the friction kernel, and 9 α(t ) represents a random force associated with generally unknown initial conditions for the electronic subsystem. In what follows we omit this term as for a pure initial electronic state it is zero. We integrate the Langevin eq 7 exactly; details of the integration scheme can be found in the Supporting Information (SI). It is the most accurate frictional approach possible because of the absolute minimum number of introduced approximations. Any other models with additional approximations can only be more accurate due to fortuitous cancellation of errors related to eliminating the electronic DOF, approximating the memory kernel and electronic structure. Two Collective Mode (2CM) Model. As an alternative to the frictional approach, we introduce a procedure that instead of eliminating electronic DOF in the SE eqs 5 and 6 reduces their number to not more than three. We start with an approximate decoupling of elementary electronic variables {cj} via a collective transformation similar in nature to the superadiabatic transformation.40,41 The T matrix in eq 5 can be blockdiagonalized analytically so that the only nonzero block of the resulting matrix Z = U−1TU is its upper-left corner, Z2×2= −iσy z, where σy is the Pauli matrix and z = (∑k=1 τ2k0)1/2. Z is rankdeficient and acts nontrivially only within a 2D subspace (d0,d1) of transformed electronic variables d = U−1c. We will refer to the (d0,d1)-subspace as the interaction space. d0 coincides with the probability amplitude of the ground adiabatic state c0 and will be used to track the ground-state population evolution. d1 is the only collective mode coupled to ground state via z, and the remaining variables {di}i=2,... define the spectator space because they are not coupled to the interacting space via regular NACs. The transformation matrix U is time-dependent; therefore, the electronic EOM in new variables, ḋ = −(U−1U̇ + Z)d, acquires a new nonadiabatic term N = U−1U̇ .40,41 N has a quite involved explicit form, but its 2 × 2 upper block is quite simple. Therefore, we introduce our first approximate 2CM method by restricting the electronic dynamics to the interaction space only
(1)
j
∑∫
(7)
where R ≡ {Rα} is a collection of nuclear classical DOF, a dot stands for the full time derivative, |Ψ⟩ is a time-dependent electronic wave function, and Ĥ e is the electronic Hamiltonian that operates on electronic variables and is a parametric function of R. In the interaction picture |Ψ⟩ is2 |Ψ⟩ =
∂E0 = ∂R α
(6)
To obtain a microscopic frictional theory, we follow HeadGordon and Tully.19 Note, however, that, in accordance with the original work, two additional approximations have been
Mα R ̈ α + 441
∂E0 α = (d0*d1 + d1*d0)f10̃ ∂R α
(9) DOI: 10.1021/acs.jpclett.6b02712 J. Phys. Chem. Lett. 2017, 8, 440−444
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The Journal of Physical Chemistry Letters Three Collective Mode (3CM) Model. The 2CM model treats all of the original electronic DOF in a democratic fashion in the collective frequency ω̃ 11 and coupling z. For a chemisorbed species on a surface, however, one coupling, which corresponds to molecular binding, can be appreciably larger than others. Interaction-space variables in this case may be biased toward a description of that particular electronic DOF, and the 2CM dynamics will miss contributions from weakly coupled states. Therefore, one needs a procedure to select the most important DOF from the spectator space. The spectator space interacts with (d0, d1)-space only through the matrix N. In the presence of one dominant coupling, which we can take without restricting generality as τ10, the ratios pk = τk0/τ10, k > 1, and their time derivatives ṗk are small. Therefore, the matrix N can be simplified by neglecting ṗk and off-diagonal quadratic terms pkpj N≈N
(a)
⎛ 0 0 0⎞ ⎜ ⎟ = idiag{0, w11 ̃ , w22 ̃ , ...} − i⎜ 0 0 π ⎟ ⎜ † † ⎟ ⎝0 π 0 ⎠
Figure 1. Potential energy surfaces for an adatom-metallic chain model, 10 states per layer; other parameters are given in the SI. The lowest energy adiabatic PESs for each layer (including the ground-state one) are solid blue thick lines. Adiabatic PESs within a layer are solid thin gray lines. Diabatic PESs are dashed red lines. E(0) is the initial energy; a black cross marks the initial position for the trajectory.
detailed in the SI.) All electronic states are organized in nine layers, so that the off-diagonal couplings are large for states in different layers and small for states within the same layer. Diagonalization of Ĥ e determines the adiabatic potentials {Ek(R)}, NACs {τkj(R)}, and forces {f kj(R)}. The resulting adiabatic PESs in a multiple-state-per-layer case are sketched in Figure 1. τkj and f kj are set to zero once both k,j ≥ 1 simultaneously. The nuclear kinetic energy is TN = MṘ 2/2 with M = 2000me, multiplied by the identity matrix of the appropriate dimension. We consider five dynamical approaches: SE (eqs 5 and 6), the full-memory friction (eq 7), 2CM (eqs 8 and 9), 3CM (eqs 11 and 12) models, and the Born-Oppenheimer (BO) approximation (eq 6 with the zero right-hand side). We start with a set of molecular parameters that correspond to weak NAC between layers (see the SI). Nonadiabatic effects are quantified by the deviation R(t) − RBO(t) of a given trajectory from the BO counterpart. The initial energy is taken to be low so that the nonadiabatic dynamics takes place within a groundstate layer (Figure 1). Nonadiabatic effects are small overall: The deviations are fractions of bohr (a0) as compared with the total trajectory length of 50a0 (Figure 2). With one state per layer, an insulator-like surface, both collective mode models work very well, almost exactly reproducing the SE dynamics. Despite the fact that this setup favors the friction approach as the electronic wave function remains close to the ground state, the friction model still deviates because it accounts for different excitations mostly by the strength of the corresponding couplings but not according to their actual importance determined by available energy. With more states per layer, the ground-state depletion becomes more pronounced (Figure 2d,f). For 10 states per layer, the ground-state weight drops to ca. 50%, and the 2CM model breaks down due to the large number of energetically accessible (albeit weakly coupled) electronic states. On the contrary, the 3CM model closely follows the SE nuclear dynamics (Figure 2e) and reproduces on average the electronic evolution (Figure 2f). The nuclear dynamics is good because all PESs within a layer have almost identical profiles, and hence the nucleus feels the same force, so that the electronic population within a layer needs to be accurate only on average. The friction model shows moderate deviations from the SE dynamics (Figure 2e,f) and may still be perceived as a fairly good approximation.
(10)
where w̃ kk = wk0 − τ2k0ωk1/z2, k > 1, and π = (π2, ...) with πk = pk 2 τ10 ωk1/z2. In the interaction representation for N(a) the diagonal matrix of collective frequencies (the first term in eq 10) is transferred to the corresponding dynamical phases. This reveals the structure of N(a), which is similar to that of the matrix T (cf. eq 5) with only zero first row and column. Thus we repeat the block-diagonalizing transformation and separate one additional mode from the spectator space to obtain a new 3CM model with the following EOM ⎛ 0 f ̃α f ̃α ⎞ 10 20 ⎟ ⎜ ∂E0 ⎜ ⎟ α † Mα R ̈ α + = d ⎜ f10̃ 0 0 ⎟d ∂R α ⎜⎜ α ⎟⎟ ̃ ⎝ f20 0 0 ⎠
(11)
⎛ 0 −z 0 ⎞ ⎜ ⎟ ̃ −iz ̃ ⎟d ḋ = ⎜ z iw11 ⎜ ⎟ ̂ ⎠ ⎝ 0 −iz ̃ iw22
(12)
where d = (d0, d1, d2) are new collective electronic variables, z̃ = ̃ = z̃−1z−1 ∑k=2 (fα10τk0 (∑k=2π2k )1/2, ŵ 22 = z̃−2 ∑k=2 w̃ kkπ2k , and fα20 α − f k0 τ10)πk. The recursive nature of the procedure should be apparent now. One can proceed by separating more modes from the spectator space to obtain higher level collective-mode models. The appearance of combination frequencies (e.g., ωk1 = (Ek − E1)/ℏ) warrants, however, that neglecting the excited−excited couplings, which are dynamical partners of these frequencies, is no longer justified. Moreover, the transformed forces fαk0̃ = 0, k ≥ 3, and hence, higher collective modes do not contribute to the nuclear dynamics directly, although they are still coupled in the electronic space. That is, the 3CM model constitutes a natural stopping point for the collective-mode transformation. Molecular Model. We assess different nonadiabatic approaches on a model with several electronic and a single nuclear DOF. The model is intended to represent an atom chemisorbed on a 1D chain of atoms modeling the surface. We parametrize the electronic Hamiltonian Ĥ e in the diabatic representation as Ĥ e = ∑ij|φi⟩Vij(R)⟨φj|, where Vij(R) are either constants for i ≠ j or harmonic potentials. (All definitions are 442
DOI: 10.1021/acs.jpclett.6b02712 J. Phys. Chem. Lett. 2017, 8, 440−444
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excited, this leads to overestimated energy transfer and eventual trapping. This failure emphasizes the fact that the friction formalism even with the full memory can give qualitatively incorrect results due to intrinsically flawed representation of electronic DOF as harmonic oscillators. On the contrary, dynamics in both collective-mode models match very well that of the Ehrenfest method for both setups. In conclusion, we have derived a new MQC dynamical model that is fully ab initio: The collective variables are constructed on-the-fly from the nonadiabatic data available from electronic structure calculations. The electronic dynamics in our approach is unitary and properly accounts for electronic back-reaction and memory. Our numerical simulations show that the collective variable approach provides good and reliable description of nonadiabatic dynamics in a variety of situations. The 2CM variant is accurate for “an atom on an insulator-like surface” cases, whereas the 3CM model is applicable to metallic surfaces as well. One of the collective variables coincides with the ground-state amplitude and can be used to draw a conclusion of whether the nonadiabatic effects are important. Application of the introduced collective modes can be extended to the surface-hopping framework, which has certain advantages compared with the Ehrenfest method.1,15 Comparison of the collective-mode models with the fullmemory ab initio electronic friction model shows the superiority of the former. The main drawback of the friction theory is insufficient energy back-flow from the electronic subsystem. That can cause nuclear reflection or trapping if certain and largely unpredictable resonance conditions are met. Our observation supports an idea that the similar failure reported in refs 28 and 29 is caused by the friction approach itself and is unrelated to the level of electronic structure theory.
Figure 2. Trajectory dynamics in low-energy, weak-coupling regime. Top row (a,b): one state per layer. Middle row (c,d): five states per layer. Bottom row (e,f): 10 states per layer (see Figure 1). The left column shows deviations of trajectories in different methods from that of the Born−Oppenheimer (BO) approximation; the right column displays ground-state weight |c0|2(t) dynamics.
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Breakdown of the Friction Model. Considering the molecular model in Figure 3a with two possible initial energies, we found qualitative breakdown of the friction model. In a low-energy regime (Figure 3b), the friction trajectory experiences a reflection at around R = 2a0. This reflection occurs because the nuclear kinetic energy barely exceeds the ground-state potential barrier, and larger energy dissipation in the friction model leads to the reflection. However, even with a large excess of energy, a particle can still be trapped, as follows from Figure 3c. In this case, we run into a resonance, when the nuclear energy is quickly pumped into one of the oscillators representing an electronic DOF. Since the harmonic oscillator in contrast with the true electronic DOF can be multiply
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b02712. Explicit form of all matrices, parameters for the molecular models, and the details of numerical integration scheme. (PDF)
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AUTHOR INFORMATION
Corresponding Authors
*I.G.R.: E-mail:
[email protected]; *A.F.I.: E-mail:
[email protected].
Figure 3. Molecular model has five levels per layer; all other parameters are detailed in the SI. (a) Adiabatic and diabatic potentials (same as in Figure 1). Horizontal lines labeled as ① and ② show two different initial energies. Trajectories for the setups ① and ② are in panels b and c, respectively. 443
DOI: 10.1021/acs.jpclett.6b02712 J. Phys. Chem. Lett. 2017, 8, 440−444
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(22) Echenique, P. M.; Nieminen, R. M.; Ritchie, R. H. Density functional calculation of stopping power of an electron gas for slow ions. Solid State Commun. 1981, 37, 779−781. (23) Puska, M. J.; Nieminen, R. M. Atoms Embedded in an Electron Gas: Phase Shifts and Cross Sections. Phys. Rev. B: Condens. Matter Mater. Phys. 1983, 27, 6121−6128. (24) Juaristi, J. I.; Arnau, A.; Echenique, P. M.; Auth, C.; Winter, H. Charge State Dependence of the Energy Loss of Slow Ions in Metals. Phys. Rev. Lett. 1999, 82, 1048−1051. (25) Askerka, M.; Maurer, R. J.; Batista, V. S.; Tully, J. C. Role of Tensorial Electronic Friction in Energy Transfer at Metal Surfaces. Phys. Rev. Lett. 2016, 116, 217601. (26) Dou, W.; Nitzan, A.; Subotnik, J. E. Frictional Effects Near a Metal Surface. J. Chem. Phys. 2015, 143, 054103. (27) Dou, W.; Subotnik, J. E. Electronic Friction Near Metal Surfaces: A Case Where Molecule-Metal Couplings Depend on Nuclear Coordinates. J. Chem. Phys. 2017, 146, 092304. (28) Trail, J. R.; Graham, M. C.; Bird, D. M.; Persson, M.; Holloway, S. Energy Loss of Atoms at Metal Surfaces Due to Electron-Hole Pair Excitations: First-Principles Theory of Chemicurrents. Phys. Rev. Lett. 2002, 88, 166802. (29) Trail, J. R.; Bird, D. M.; Persson, M.; Holloway, S. ElectronHole Pair Creation by Atoms Incident on a Metal Surface. J. Chem. Phys. 2003, 119, 4539−4549. (30) Head-Gordon, M.; Tully, J. C. Vibrational Relaxation on Metal Surfaces: Molecular-Orbital Theory and Application to CO/Cu(100). J. Chem. Phys. 1992, 96, 3939−3949. (31) Head-Gordon, M.; Tully, J. C. Molecular-Orbital Calculations of the Lifetimes of the Vibrational Modes of CO on Cu(100). Phys. Rev. B: Condens. Matter Mater. Phys. 1992, 46, 1853−1856. (32) Tully, J. C.; Gomez, M.; Head-Gordon, M. Electronic and Phonon Mechanisms of Vibrational Relaxation: CO on Cu(100). J. Vac. Sci. Technol., A 1993, 11, 1914−1920. (33) Kindt, J. T.; Tully, J. C.; Head-Gordon, M.; Gomez, M. A. Electron-Hole Pair Contributions to Scattering, Sticking, and Surface Diffusion: CO on Cu(100). J. Chem. Phys. 1998, 109, 3629−3636. (34) Juaristi, J. I.; Alducin, M.; Muiño, R. D.; Busnengo, H. F.; Salin, A. Role of Electron-Hole Pair Excitations in the Dissociative Adsorption of Diatomic Molecules on Metal Surfaces. Phys. Rev. Lett. 2008, 100, 116102. (35) Monturet, S.; Saalfrank, P. Role of Electronic Friction During the Scattering of Vibrationally Excited Nitric Oxide Molecules from Au(111). Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 075404. (36) Füchsel, G.; Klamroth, T.; Monturet, S.; Saalfrank, P. Dissipative Dynamics Within the Electronic Friction Approach: the Femtosecond Laser Desorption of H2/D2 from Ru(0001). Phys. Chem. Chem. Phys. 2011, 13, 8659−8670. (37) Blanco-Rey, M.; Juaristi, J. I.; Díez Muiño, R.; Busnengo, H. F.; Kroes, G. J.; Alducin, M. Electronic Friction Dominates Hydrogen Hot-Atom Relaxation on Pd(100). Phys. Rev. Lett. 2014, 112, 103203. (38) Rittmeyer, S. P.; Meyer, J.; Juaristi, J. I.; Reuter, K. Electronic Friction-Based Vibrational Lifetimes of Molecular Adsorbates: Beyond the Independent-Atom Approximation. Phys. Rev. Lett. 2015, 115, 046102. (39) Novko, D.; Blanco-Rey, M.; Juaristi, J. I.; Alducin, M. Ab Initio Molecular Dynamics with Simultaneous Electron and Phonon Excitations: Application to the Relaxation of Hot Atoms and Molecules on Metal Surfaces. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 201411. (40) Deschamps, M.; Kervern, G.; Massiot, D.; Pintacuda, G.; Emsley, L.; Grandinetti, P. J. Superadiabaticity in Magnetic Resonance. J. Chem. Phys. 2008, 129, 204110. (41) Lim, R.; Berry, M. V. Superadiabatic Tracking of Quantum Evolution. J. Phys. A: Math. Gen. 1991, 24, 3255−3264. (42) van Santen, R. A.; Tranca, I. How Molecular Is the Chemisorptive Bond? Phys. Chem. Chem. Phys. 2016, 18, 20868− 20894. (43) Minnhagen, P. Excitation Probability in a Time-Dependent External Potential. J. Phys. C: Solid State Phys. 1982, 15, 2293.
Artur F. Izmaylov: 0000-0001-8035-6020 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS A.F.I. acknowledges funding from a Sloan Research Fellowship and the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants Program.
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REFERENCES
(1) Tully, J. C. Mixed Quantum-Classical Dynamics. Faraday Discuss. 1998, 110, 407−419. (2) Doltsinis, N. L.; Marx, D. First Principles Molecular Dynamics Involving Excited States and Nonadiabatic Transitions. J. Theor. Comput. Chem. 2002, 1, 319−349. (3) Tully, J. C. Perspective: Nonadiabatic Dynamics Theory. J. Chem. Phys. 2012, 137, 22A301. (4) Kapral, R. Quantum Dynamics in Open Quantum-Classical Systems. J. Phys.: Condens. Matter 2015, 27, 073201. (5) Wang, L.; Akimov, A.; Prezhdo, O. V. Recent Progress in Surface Hopping: 2011−2015. J. Phys. Chem. Lett. 2016, 7, 2100−2112. (6) Pittner, J.; Lischka, H.; Barbatti, M. Optimization of Mixed Quantum-Classical Dynamics: Time-Derivative Coupling Terms and Selected Couplings. Chem. Phys. 2009, 356, 147−152. (7) Barbatti, M.; Ruckenbauer, M.; Plasser, F.; Pittner, J.; Granucci, G.; Persico, M.; Lischka, H. Newton-X: A Surface-Hopping Program for Nonadiabatic Molecular Dynamics. WIREs Comput. Mol. Sci. 2014, 4, 26−33. (8) Fischer, S. A.; Chapman, C. T.; Li, X. Surface Hopping with Ehrenfest Excited Potential. J. Chem. Phys. 2011, 135, 144102. (9) Saalfrank, P. Theory of Photon- and STM-induced Bond Cleavage at Surfaces. Curr. Opin. Solid State Mater. Sci. 2004, 8, 334− 342. (10) Savin, A.; Umrigar, C. J.; Gonze, X. Relationship of Kohn-Sham Eigenvalues to Excitation Energies. Chem. Phys. Lett. 1998, 288, 391− 395. (11) Hammes-Schiffer, S.; Tully, J. C. Proton Transfer in Solution: Molecular Dynamics with Quantum Transitions. J. Chem. Phys. 1994, 101, 4657−4667. (12) Craig, C.; Duncan, W.; Prezhdo, O. Trajectory Surface Hopping in the Time-Dependent Kohn-Sham Approach for Electron-Nuclear Dynamics. Phys. Rev. Lett. 2005, 95, 163001. (13) Shenvi, N.; Tully, J. C. Nonadiabatic Dynamics at Metal Surfaces: Independent Electron Surface Hopping with Phonon and Electron Thermostats. Faraday Discuss. 2012, 157, 325−335. (14) Ryabinkin, I. G.; Nagesh, J.; Izmaylov, A. F. Fast Numerical Evaluation of Time-Derivative Non-Adiabatic Couplings for Mixed Quantum-Classical Methods. J. Phys. Chem. Lett. 2015, 6, 4200−4203. (15) Shenvi, N.; Roy, S.; Tully, J. C. Nonadiabatic Dynamics at Metal Surfaces: Independent-Electron Surface Hopping. J. Chem. Phys. 2009, 130, 174107. (16) Shenvi, N.; Schmidt, J.; Edwards, S.; Tully, J. Efficient Discretization of the Continuum Through Complex Contour Deformation. Phys. Rev. A: At., Mol., Opt. Phys. 2008, 78, 022502. (17) Newns, D. M. Self-Consistent Model of Hydrogen Chemisorption. Phys. Rev. 1969, 178, 1123−1135. (18) Meyer, H.-D.; Miller, W. H. A Classical Analog for Electronic Degrees of Freedom in Nonadiabatic Collision Processes. J. Chem. Phys. 1979, 70, 3214−3223. (19) Head-Gordon, M.; Tully, J. C. Molecular Dynamics with Electronic Frictions. J. Chem. Phys. 1995, 103, 10137−10145. (20) Persson, M.; Hellsing, B. Electronic Damping of Adsorbate Vibrations on Metal Surfaces. Phys. Rev. Lett. 1982, 49, 662−665. (21) Hellsing, B.; Persson, M. Electronic Damping of Atomic and Molecular Vibrations at Metal Surfaces. Phys. Scr. 1984, 29, 360−371. 444
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