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CHEMPHYSCHEM REVIEWS DOI: 10.1002/cphc.201200941 Trajectory-Based Nonadiabatic Dynamics with TimeDependent Density Functional Theory Basile F. E. Curchod, Ursula Rothlisberger, and Ivano Tavernelli*[a] 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1314 – 1340 1314 CHEMPHYSCHEM REVIEWS Understanding the fate of an electronically excited molecule constitutes an important task for theoretical chemistry, and practical implications range from the interpretation of attoand femtosecond spectroscopy to the development of lightdriven molecular machines, the control of photochemical reactions, and the possibility of capturing sunlight energy. However, many challenging conceptual and technical problems are involved in the description of these phenomena such as 1) the failure of the well-known Born–Oppenheimer approximation; 2) the need for accurate electronic properties such as potential energy surfaces, excited nuclear forces, or nonadiabatic cou- 1. A Short Historical Overview Ab initio molecular dynamics has led to a large number of theoretical predictions for molecules and solids.[1, 2] In the well-established mixed quantum/classical formulation,[3–5] and thanks to highly efficient electronic structure methods such as Kohn– Sham (KS) density functional theory (DFT),[6] simulations on molecular systems with up to thousands of atoms are nowadays feasible.[7–11] Initially restricted to a single adiabatic state, DFT-based ab initio molecular dynamics was recently extended to the nonadiabatic regime,[12–14] and this provided an important tool for the study of photophysical and photochemical processes.[15–24] Among the most commonly used nonadiabatic molecular dynamics schemes are Ehrenfest dynamics and Tully’s trajectory surface hopping[25, 26] (TSH, Table 1). In the first case, the nuclear dynamics is replaced by a single-point-like trajectory evolving in the mean-field potential derived from the time evolution of the electronic wavefunction. Differently, in TSH the nuclear wavepacket is represented by a swarm of independent classical trajectories, whereas the nonadiabatic couplings (NACs) induce hops between different electronic states that occur according to a stochastic algorithm. The classical approximation for the motion of the nuclei breaks down when interferences,[27] wavepacket bifurcation,[28] (de)coherence, or tunneling effects occur during the dynamics. A trajectorybased solution of the quantum dynamics able to describe theses phenomena was introduced by Wyatt.[29] The so-called quantum trajectory method (QTM) describes the time evolution of the nuclear wavefunction by means of the quantum hydrodynamics (Bohmian)[30–32] equations of motion.[33] In this approach, the nuclear wavepacket is split into fluid elements that represent volume elements of configuration space carrying information about the amplitude and the phase of the nuclear wavefunction. These fluids elements are then propagated [a] B. F. E. Curchod, Prof. Dr. U. Rothlisberger, Dr. I. Tavernelli Laboratory of Computational Chemistry and Biochemistry Ecole Polytechnique Fdrale de Lausanne 1015 Lausanne (Switzerland) E-mail: [email protected], corresponding author. Designed the research and wrote the bulk of this review article. Supporting information for this article is available on the WWW under http://dx.doi.org/10.1002/cphc.201200941. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.chemphyschem.org pling terms; and 3) the necessity of describing the dynamics of the photoexcited nuclear wavepacket. This review provides an overview of the current methods to address points 1) and 3) and shows how time-dependent density functional theory (TDDFT) and its linear-response extension can be used for point 2). First, the derivation of Ehrenfest dynamics and nonadiabatic Bohmian dynamics is discussed and linked to Tully’s trajectory surface hopping. Second, the coupling of these trajectory-based nonadiabatic schemes with TDDFT is described in detail with special emphasis on the derivation of the required electronic structure properties. Table 1. Selected cornerstones for trajectory-based nonadiabatic dynamics based on LR-TDDFT. Landmark Year Ref. Born–Oppenheimer approximation Ehrenfest theorem First nonadiabatic model (LZS)[a] Bohmian trajectories Born–Huang Ansatz DFT KS DFT TSH TDDFT Ab initio MD Fewest switches TSH LR-TDDFT Nuclear gradient within TDDFT NACs in TDDFT Nonadiabatic QTM DFT-based TSH Approximate “TDDFT”-based TSH LR-TDDFT-based TSH NACs between excited states 1927 1927 1932 1952 1954 1964 1965 1971 1984 1985 1990 1995 1999 2000 2001 2002 2005 2007 2010 [45] [46] [47–49] [30–32] [50, 51] [52] [6] [25] [53] [1] [26] [54] [55, 56] [57] [58] [12] [13] [14] [59] [a] Landau–Zener–Stueckelberg along Bohmian trajectories that differ from their classical counterpart by the presence of an additional potential called the quantum potential, which is responsible for all quantum effects. Other schemes have been proposed to approximate the nonadiabatic dynamics of the nuclear wavefunction, such as, for example, semiclassical approaches,[34, 35] extended surface hopping,[36, 37] quantum-classical Liouville approaches,[27, 38] linearized nonadiabatic dynamics (LAND-map),[39] and ab initio multiple spawning methods.[40] As an alternative to trajectory-based approaches, quantum dynamics methods use an exact treatment of both electronic and nuclear wavefunctions (e.g. see refs. [41–44] for a discussion of the multiconfiguration time-dependent Hartree method). However, the applicability of these methods is hampered by their high computational cost, which limits the number of accessible nuclear degrees of freedom. In addition, these methods usually require fitting of the relevant electronic potential energy surfaces (PESs) prior to propagation. This review is focused on the coupling of [linear response (LR-)]TDDFT to trajectory-based nonadiabatic schemes such as TSH and Ehrenfest dynamics. It is, however, important to note that also semiempirical and wavefunction-based methods have ChemPhysChem 2013, 14, 1314 – 1340 1315 CHEMPHYSCHEM REVIEWS www.chemphyschem.org been extensively used, especially in combination with TSH.[21, 23, 60–65] After a short description of those most relevant approaches able to describe the nonadiabatic dynamics of molecular systems in the unconstrained phase space, we will discuss their implementation within the framework of DFT/ (LR-)TDDFT, which allow for efficient “on-the-fly” calculation of the required electronic structure properties such as electronic energies (i.e. PESs), nuclear forces, and NACs. In an effort to extend the applicability of nonadiabatic molecular dynamics (MD) to larger systems of physical, chemical, and biological interest, we will also discuss the possibility of bridging different time and length scales, while keeping a valid description of the photoactive components. This is done through a hierarchical scheme, which combines the ultrafast dynamics of photo- excited electrons [purely quantum mechanical (QM) tier] with the nuclear dynamics of the excited molecule (mixed-quantum classical tier) and with the reorganization of the environment [fully classical, molecular mechanical (MM), level]. Basile F. E. Curchod was born in 1984 in Vevey, Switzerland. He earned his Bachelor (2007) and Master (2009) degrees in Chemistry at the cole Polytechnique Fdrale de Lausanne, Switzerland. Since May 2009, he has been working as a Ph.D. student in the Laboratory of Computational Chemistry and Biochemistry of Prof. Ursula Rçthlisberger under the supervision of Dr. MER Ivano Tavernelli. His research interests are nonadiabatic ab initio molecular dynamics, nonadiabatic quantum trajectories, and excitedstate properties of inorganic and organometallic molecules. Ivano Tavernelli was born in 1967 in Mendrisio, Switzerland. He studied Biochemistry (M.Sc., 1991) and then Theoretical Physics (M.Sc., 1996) at the Eidgençssische Technische Hochschule Zrich (ETHZ), where he also obtained his Ph.D. in 1999. In 2000, he received a Marie Curie Fellowship and went to work in the group of Prof. Michiel Sprik in the department of Chemistry at Cambridge University (UK), where he stayed until 2002. He then moved back to Switzerland to join the group of Prof. Ursula Rçthlisberger, first at ETHZ (2002–2003) and later at the cole Polytechnique Fdrale de Lausanne (EPFL, 2003 until now). In 2010, he earned the title of Matre d’Enseignement et de Recherche (MER) at EPFL. In the field of quantum-classical dynamics, Ivano Tavernelli has developed novel theoretical frameworks able to combine electronic structure techniques based on density [DFT and (LR-)TDDFT] with the calculation of nonadiabatic quantum and classical trajectories. His research interests comprise adiabatic and nonadiabatic molecular dynamics (Ehrenfest dynamics, trajectory surface hopping, and Bohmian dynamics) for the study of photochemical and photophysical processes in molecules, condensed phase, and biological systems. Ursula Rothlisberger was born in Switzerland and obtained her diploma in Physical Chemistry from the University of Bern. She earned her Ph.D. degree at the IBM Zurich Research Laboratory in Rschlikon. From 1992–1995, she worked as a postdoctoral fellow, first at the University of Pennsylvania in Philadelphia (USA) and then at the Max-Planck-Institute for Solid State Physics in Stuttgart, Germany. In 1996, she moved as a Profile 2 Fellow of the National Science Foundation to the ETH in Zurich. One year later, she became Assistant Professor of Computer-Aided Inorganic Chemistry at the ETH Zurich, and in 2002 she accepted a call for a position as Associate Professor at the cole Polytechnique Fdrale de Lausanne (EPFL). Since 2009, she has been working as a full Professor in Computational Chemistry and Biochemistry at the EPFL. In 2001, she received the Ruzicka Prize, and in 2005, the World Association of Theoretically Oriented Chemists (WATOC) awarded her the Dirac Medal for “the outstanding computational chemist in the world under the age of 40”. Ursula Rothlisberger is an expert in the field of density functional based mixed quantum mechanical/molecular mechanical molecular dynamics simulations in the ground and electronically excited states. She has published more than 200 original publications in peer-reviewed journals and various review articles in specialized journals and as book chapters. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 2. Nonadiabatic Dynamics The starting point of our derivation is the time-dependent Schrçdinger equation [Eq. (1)]: in which R = (R1,R2,…,RNn) is the collective vector of the Nn nuclear positions in R3Nn and r = (r1,r2,…,rNel) is the collective ChemPhysChem 2013, 14, 1314 – 1340 1316 CHEMPHYSCHEM REVIEWS www.chemphyschem.org vector for the Nel electrons. (A full list with the nomenclature used herein can be found in the Supporting Information.) In ^ mol is the molecular Hamiltonian [Eq. (2)]: Equation (1), H the nuclear positions. The propagation of this functional form for Y(r,R,t) implies a restriction of the Hilbert space accessible to the molecular wavefunction at later times. A more rigorous factorization of Y(r,R,t) was proposed by Hunter[67] and then further developed into an exact-factorization nonadiabatic MD scheme by Gross and co-workers.[68] The exponential part of Equation (4) is called the “phase term” and is defined by: and Y(r,R,t) is the total wavefunction of the nuclear (labeled with g and z) and electronic (labeled with i and j) degrees of freedom. In the following, atomic units will be used except for the reduced Planck constant h, which will be kept for clarity. In this first section, we will derive the equations of motion for the nuclear (slow) and electronic (fast) degrees of freedom by using what is known as a trajectory-based approach. In this framework, the electrons are described at a quantum mechanical level, whereas the nuclear wavepacket is discretized into an ensemble of points in phase space and then propagated along classical (or Bohmian quantum) trajectories that, as we will see, keep some essential quantum character including nonadiabatic effects. The first step in the derivation of the equations of motion for the combined electron-nuclear dynamics is the definition of a suitable representation of the total system wavefunction. Depending on the particular choice of this expansion, we can obtain different (approximated) solutions of the initial molecular Schrçdinger equation [Eq. (1)]. In the following, we will restrict ourselves to two main representations of the total molecular wavefunction that will give rise to two main trajectorybased nonadiabatic molecular dynamics solutions: mean-field Ehrenfest dynamics and surface hopping dynamics. Inserting the representation of the total wavefunction given in Equation (3) into the total Schrçdinger equation [Eq. (1)] and multiplying from the left-hand side by W*(R,t) and integrating over R we get [Eq. (5)]: P P Zg Z Z g;i R r þ g<z R gRz j j j g ij j g zj In a similar way, multiplying by F*(r,t) and integrating over r we obtain [Eq. (6)]: ^ RÞ ¼ in which Vðr; P 1 i<j r r i j Conservation of energy has also to be imposed through the ^ mol i d hH condition that dt 0. Note that both the electronic and nuclear parts evolve according to an average potential generated by the other component (in square brackets). These average potentials are timedependent and are responsible for the feedback interaction between the electronic and nuclear components. The set of Equations (5) and (6) forms the so-called mean-field time-dependent self-consistent field (TDSCF) method.[4] 2.1.1. The Time-Dependent Nuclear Equation In both approaches, the nuclei are described by classical trajectories (one in the mean-field case), and therefore, the methods belong to the class of mixed quantum/classical solutions. The following derivation is partially inspired by the articles of Tully (ref. [4]), the book of Marx and Hutter on ab initio molecular dynamics (ref. [2]), and Curchod et al. (ref. [66]). We start from the polar representation of the nuclear wavefunction [Eq. (7)]:[69, 70] in which the amplitude A(R,t) and the phase S(R,t)/h are real functions (see Figure 1 for a simple representation of these functions). Inserting Equation (7) into Equation (6) gives Equation (8): 2.1. Ehrenfest Dynamics In Ehrenfest dynamics, we make use of a single-configuration Ansatz for the total wavefunction [Eq. (3)]: in which F(r,t) describes the (time-dependent) electronic wavefunction and W(R,t) describes the (time-dependent) nuclear wavefunction. In the factorization in Equation (3), the electronic wavefunction does not depend (even parametrically) on 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1314 – 1340 1317 CHEMPHYSCHEM REVIEWS www.chemphyschem.org or, equivalently [Eq. (11)]: Figure 1. Complex nuclear wavefunction W(R) (top) and its representation in terms of amplitude and i phase A(R) and S(R) (bottom): W(R) = A(R)exp h SðRÞ . Each classical atom g moves according to the law of classical mechanics on a PES given by the expectation value of the electronic Hamiltonian, H^el , which depends on the position of all nuclei at time t. For practical reasons, the forces in Equation (11) are often computed by using the Hellmann–Feynman theorem as the expectation value of the gradient of the Hamil tonian,[72, 73] Fðr; tÞrg H^el ðr; RÞFðr; tÞ . Notably, multitrajectory extensions of Ehrenfest dynamics have also been proposed in the literature (e.g. see refs. [74, 75]). 2.1.2. The Time-Dependent Electronic Equation Collecting the terms that do not depend on h,[71] that is, taking the classical limit (h !0), we obtain [Eq. (9)]: Starting from Equation (5), we first take the classical limit of the nuclear degrees of freedom by replacing the nuclear density j W(R,t) j 2 with a product of delta-functions centered at the atomic positions R at time t [Eq. (12)]: This has the form of the “Hamilton–Jacobi” (HJ) equation of classical mechanics, which establishes a relation between the partial differential equation for S(R,t) in configuration space and the trajectories of the corresponding (quantum) mechanical systems.[70] The identification of S(R,t) with the “classical” action defines a point-particle dynamics with a Hamiltonian, Hcl, given by (minus) the right-hand side of Equation (9) and with momenta P = rRS(R). The solutions of this Hamiltonian system are represented by curves in the (R,t) space, which are extrema of the action S(R,t) for given initial conditions R(t0) and P(t0) = rRS(R) j R(t0) and are called the characteristics of the differential equation in Equation (9). In this new formulation and after some rearrangements,1 we obtain the Newton-like equation [Eq. (10)] for the nuclear trajectories corresponding to the HJ Equation (9): Note that R2R3Nn is a vector in the configuration space and Rg2R3 is the actual position of atom g. In this way, we recover the time-dependent Schrçdinger equation for the many-electron wavefunction [Eq. (13)]: 1 Starting from Equation (9) we first take the derivative with respect to Rz, rz Note that both the Hamiltonian H^el (r,R(t)) and the electronic wavefunction F(r;R(t),t) depend parametrically on the position of the classical nuclei. This ensures the coupling between the time-dependent propagation of the electronic and nuclear degrees of freedom. 2.1.3. The TDDFT-Based Ehrenfest MD Scheme The numerical implementation of the Ehrenfest MD scheme requires the simultaneous solution of the coupled differential equations for the electronic and nuclear dynamics [Eqs. (14) and (15)]: The first term becomes Rearranging these equations, one gets and, finally, using rzS/Mz = uz and the definition of the Lagrangian time derivative we obtain 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim in which “;R(t)” denotes the parametric dependence of the electronic Schrçdinger equation from the atomic positions. This is accomplished through the use of a Runge–Kutta algorithm as described in refs. [16, 76]. All quantities required in Equations (14) and (15) are, therefore, computed “on-the-fly” at the instantaneous potential generated by the nuclear positions (Rg(t)) and electronic wavefunctions F(r;R(t),t). ChemPhysChem 2013, 14, 1314 – 1340 1318 CHEMPHYSCHEM REVIEWS www.chemphyschem.org Until this point, the formulation was kept as general as possible and did not depend on a particular choice of the electronic structure method used to solve the time-dependent Schrçdinger equation for the electrons. In many cases, it is however convenient to map the set of Equations (14) and (15) into a DFT/TDDFT-based formulation, which allows efficient calculation of the molecular electronic structure and properties for systems composed of several hundreds of atoms at a moderate computational cost. By using the Runge–Gross theorem,[53] we can derive the equation of motion for the electron density from the variational principle applied to the action functional [Eq. (16)]2 : in which Exc is the DFT (ground-state) exchange and correlation energy density functional. This approximation is sufficient for most applications, but simulations beyond the AA are also present in the literature.[79–82] An alternative solution to the propagation of the electronic wavefunction within DFT/TDDFT is to introduce the representation of the TDKS orbitals in a linear combination of static KS orbitals, opt : ðr; RðtÞÞ , obtained from the diagonalization of the KS Hamiltonian at time t with effective potential us[1] j 1(r) 1(r,t) [Eq. (21)]: ! in which H[1(r,t)] is the Hartree energy functional. In the DFT and TDDFT context, r refers to a position in the Euclidean space (r2R3) and should not be confused with the collective electron coordinate used for the many-electron wavefunctions. Applying the variational principle to Equation (17) subject to P the constraint 1(r,t) = k jk ðr; tÞj2 results in the time-dependent KS (TDKS) equations [Eq. (18)]: Inserting this expansion into Equation (18) produces a set of differential equations for the coefficients ckp(t) that, when squared, can be interpreted as KS orbital occupations. Although this offers some additional information about the nature of the propagated wavefunction, the diagonalization of the KS Hamiltonian requires further computational costs that can be avoided by using the straightforward propagations in Equation (18) (see Appendix A in the Supporting Information). The mapping of the nuclear equation [Eq. (15)] into the DFT formalism is more straightforward and only requires the description of the forces rghH^el (r,R(t))i as a functional of the time-dependent density 1(r,t). Replacing the expectation value of the electronic Hamiltonian with the DFT energy evaluated with the exchange-correlation potential uxc[1] j 1(r) 1(r,t), the gradient with respect to the nuclear coordinates can be performed analytically as in the case of the adiabatic Born–Oppenheimer and Car–Parrinello[1] MD schemes (BO-MD and CP-MD, respectively).[2] ! in which Ĥee is the exact electron-electron interaction and F[1](t) is the time-dependent wavefunction associated to the time-dependent density 1(r,t).3 In the KS formulation of DFT, the action becomes a functional of the one-electron KS orbitals [Eq. (17)]: 2.1.4. Limitations and Use of Ehrenfest MD in which [Eq. (19)]: P Z in which uext ðr; tÞ ¼ g rRg þ duapp ðr; tÞ is the external poj gj tential, and uH(r,t) is the Hartree potential. The effective potential us also depends on the initial wavefunction at time t0 (F0(r,t0)) or, equivalently, the corresponding density 1(r,t0). To simplify the notation, in the following we remove the dependence from the initial value conditions. In Ehrenfest MD, the nuclei evolve in time according to forces computed as the gradient of the average energy, hH^el (r,R(t))i, by using the instantaneous electronic wavefunction F(r;R(t),t). Both quantum Hamiltonian and electronic wavefunction depend parametrically on R(t). The mean-field character of this dynamics is evident if we use the following expansion of the wavefunction F(r;R(t),t) in the adiabatic (static) base obtained from the solutions of the electronic time-independent Schrçdinger equation [Eq. (22); {F.(r;R(t))}, see Section 2.2 for more details]: The simplest approximation to the time-dependent exchange-correlation action functional Axc [1(r,t)] is the so-called adiabatic approximation (AA) [Eq. (20)]: 2 This form of the action functional violates causality because a potential of @v ðr;tÞ 3 xc the form has a functional derivative @1ðr 0 ;t 0 Þ, which issymmetric in time and, therefore, does not guarantee that information only travels forward in time. Different solutions to the causality problem exists,[77, 78] but they do not affect our derivation. In the remainder of this section we drop the explicit dependence of wavefunctions and operators on the nuclear position. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim The square of the time-dependent coefficients ci(t) describes the “occupation probability” of the different states that contribute to the nuclear forces. The validity of the “mean-field” approximation is restricted to the case in which the classical trajectories corresponding to the different states do not differ significantly.[83] In fact, after having left a region of strong nonadiabatic coupling (mixing), the system keeps evolving on the ChemPhysChem 2013, 14, 1314 – 1340 1319 CHEMPHYSCHEM REVIEWS average potential without collapsing to one of the adiabatic states, and for the case in which trajectories on different states would have different fates in the configuration space, the average Ehrenfest trajectory could lose its physical meaning (even though it could still provide acceptable expectation values for quantum observables). For this reason, the use of Ehrenfest dynamics in molecular computational physics and chemistry should be restricted to cases in which atomic rearrangements along the reaction channels associated to the different electronic states involved in the mean-field dynamics are not significantly different. Examples are the calculation of absorption spectra and dielectric functions of systems in the gas phase and in solution,[84, 85] for which the electron dynamics of the perturbed electronic structure relaxes on a timescale that is much shorter than the one of the nuclei (in many cases one can use frozen atomic positions[86]). Other applications include the investigation of ultrafast processes triggered by intense laser fields that produce ionized states followed by fast electronic rearrangement of the electronic structure such as photoionization and Coulomb explosion (when only dissociation channels are populated).[87–94] Another severe deficiency of Ehrenfest MD is the violation of detailed balance (microscopic reversibility), which is an essential property of the kinetics of physical and chemical processes. www.chemphyschem.org that is nonlocal in time and that depends on the path(s) of the classical nuclear system, R(t). This means that the forces on the nuclei at time t depend on the forward and backward propagation of the wavefunctions from time t’ and t’’, respectively. However, this can only be determined when the path is already known, which introduces a self-consistency issue in the calculation of the electronic and nuclear dynamics. Pechukas’ equations must, therefore, be solved iteratively, which limits the efficiency of all algorithms based on this approach. Concerning the observed nuclear trajectory, R(t), Ehrenfest dynamics generates, for any given initial state, a single classical mean-field path, whereas Pechukas’ mixed quantum–semiclassical approach gives no guarantees of convergence towards a global path connecting the initial and final states, but in many occasions it may produce multiple paths solutions. It could be, therefore, very hard to find a rigorous derivation of Ehrenfest dynamics from semiclassical arguments even though attempts in this direction were already proposed.[97] 2.1.6. Application: Coulomb Explosion in Condensed Matter The main approximations in Ehrenfest MD are the use of the single-configuration Ansatz for the electronic wavefunction in Equation (3) and the classical limit applied to the nuclear degrees of freedom. To judge the quality of these approximations, we can compare the results obtained for Ehrenfest MD with the ones derived from the stationary phase approximation of the path-integral formulation of quantum dynamics.[95, 96] This approach due to Pechukas is derived from the saddle-point approximation of the Feynman–Kac propagator and, therefore, does not rely on any a priori Ansatz for the nature of the many-electron wavefunction. As this semiclassical method produces a single (or few) nuclear trajectory(ies), it can be interesting to compare Pechukas’ equations of motion with the one derived from the Ehrenfest approximation. Following Pechukas’ derivation, the nuclear trajectories are the solution to Equation (23): “Hadron therapy” characterizes a type of radiotherapy in which highly charged ion beams are used to target deep-seated tumors.[98] This cancer therapy has shown promising results, despite the fact that its microscopic mechanisms are far from being understood and only few molecular theoretical studies are available.[99, 100] Studying the interaction of a highly charged and ultrafast cation (e.g. proton, C4 + or C6 + ) with biological media such as DNA base pairs or simply water molecules constitute an important step towards a better rationalization of the effects of hadron therapy. This interaction is expected to lead to Coulomb explosion of the targeted molecules due to extraction of one or more core electrons by the charged projectile. Ehrenfest dynamics constitutes, therefore, a method of choice to simulate the electronic reorganization after ionization and the subsequent nuclear dynamics. We highlight here recent work[92] in which the Coulomb explosion of uracil surrounded by QM water molecules is simulated. The nonadiabatic dynamics is initiated by removing two electrons from a molecular orbital localized on the C=O group of uracyl (Figure 2, left). Ehrenfest dynamics is then employed to observe the Coulomb explosion of solvated uracil2 + , which takes place after 30 fs with the ejection of the carbonyl oxygen atom (Figure 2, ^ in which U(t’’,t) is the time evolution operator. In this case, the initial and final electronic states Fj(r,R,t’) and Fi(r,R,t’’) as well as the initial and final time of the propagation, t’ and t’’, respectively, need to be known before starting the propagation. At difference to Ehrenfest MD, in Pechukas’ formulation the electronic quantum system exerts a potential on the nuclei Figure 2. TDDFT-based Ehrenfest dynamics of the Coulomb explosion of solvated uracil2 + . Left) Molecular orbital from which two electrons are removed. Right) Snapshot obtained 30 fs after the ionization. 2.1.5. Discussion of Ehrenfest MD 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1314 – 1340 1320 CHEMPHYSCHEM REVIEWS www.chemphyschem.org right). This atomic oxygen is then solvated by the water molecules surrounding the uracil. Interestingly, uracil rearrangement after the oxygen loss produces a H3O + molecule in the first solvation shell. Further results of the Coulomb explosion of uracil simulated with Ehrenfest dynamics have been successfully compared to experimental data.[93] 2.2.1. The Born–Oppenheimer Approximation: The Adiabatic Case 2.2. BO-MD and Its Nonadiabatic Extensions which only induce a shift of the electronic PESs Ejel ðRÞ felt by the nuclear wavefunctions [the second term of Eq. (30) is zero for j = i, when the F·(r;R) are real]. The BO-MD equations can be derived starting from the Born– Huang representation of the molecular wavefunction [Eq. (24)]:[50, 51] In this equation, {F.(r;R)} describes a complete set of orthonormal electronic wavefunctions, solutions of the time-independent Schrçdinger equation [Eq. (25)]: with hFj j Fii = dij and for which “;R” denotes the parametric dependence of the electronic Schrçdinger equation from the position of the atoms. Note that only the nuclear wavefunction depend explicitly on time, whereas H^el (r,R) and Fi(r;R) only depend on t through the implicit time-dependence of R(t). Inserting Equation (24) into the time-dependent Schrçdinger equation [Eq. (1)], we obtain (after multiplying by Fj* ðr; RÞ from the left-hand-side and integrating over dr) Equation (26): In the Born–Oppenheimer approximation, only the diagonal terms, F jj , are retained [Eq. (28)]: In this approximation, the nuclei move in the potential of a single electronic state, the PES Ejel ðRÞ, and the electronic [Eq. (25)] and nuclear [Eq. (26)] Schrçdinger equations become completely decoupled. The term F jj (R) is called Born–Oppenheimer diagonal correction[101, 102] and, depending on the nuclear mass, induces an isotope dependence[103–105] of the total energy, Ejel ðRÞ + F jj (R). However, this term is usually small and is neglected in the so-called Born–Oppenheimer adiabatic approximation.4 As in the derivation of the Ehrenfest equations of motion, we introduce the polar representation of the nuclear wavefunction Wj(R,t) [Eq. (29)]: Even though the same protocol as that in the Section 2.1.1 can be used to extract Born–Oppenheimer equations of motion, it is interesting here to follow another route. Inserting the polar representation into Equation (26) [with all F ji (R) = 0] and separating the real and the imaginary parts, we obtain Equations (30) and (31): The quantities F ji (R) [Eq. (27)]: are the NACs, with a contribution from the nuclear kinetic energy operator and a second from the momentum operator. In the most general case, the nondiagonal elements of F ji (R) are non-zero and induce a coupling between different electronic states due to the motion of the nuclei. In fact, the last term in Equation (26) brings (or gives) amplitude (F ji Wi(R,t)) from the “electronic state” with energy Eiel ðRÞ, which is the solution of the electronic time-independent Schrçdinger equation [Eq. (25)], to the actual state j, with energy Ejel ðRÞ. This interpretation of the nuclear wavefunction dynamics Equation (26) is at the basis of the surface hopping description of nonadiabatic dynamics. Taking the classical limit h !05 in Equation (30), we obtain a HJ equation for what can now be interpreted as the action function6 Sj(R,t) [Eq. (32)]: 4 5 6 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Note that Born–Oppenheimer approximation is often used for what is here called Born–Oppenheimer adiabatic approximation (see also refs. [44, 106, 107]). The classical limit proposed in this derivation is sometimes called the “canonical condition” for enforcing classical behavior. It is mainly a mathematical procedure with limited physical meaning. Alternative formulations with their physical implications can be found in different references.[70, 108, 109] In Bohmian for example, the classical limit is more properly defined by hdynamics, i rg2 Aj h2 P ! 0.[108] 2 g M1 g Aj As already stated in Section 2.1.1, in classical mechanics the action function S(R,t) also depends on the initial coordinates andR time and, therefore, should _ 0 ÞÞdt 0 , along be written as SR0,t0(R,t). It is defined as SR0,t0(R,t) = R~ LðRðt0 Þ; Rðt the extremal path R̃(t) connecting the points R0 at time t0 to R at time t, in which L(R(t’), Ṙ(t’)) is the Lagrangian. In this definition, SR0,t0(R,t) also corresponds to the Hamilton’s principal function.[110] ChemPhysChem 2013, 14, 1314 – 1340 1321 CHEMPHYSCHEM REVIEWS www.chemphyschem.org 2.2.2. Trajectory-Based Nonadiabatic Dynamics which correspond to a classical point-particle time evolution of the nuclei, and from Equation (31) a continuity equation7 for the propagation of the amplitude on the adiabatic state of interest, d/dt(sdRjWj(R,t) j 2) = 0. Comparing this result with the one obtained for the Ehrenfest dynamics [Eq. (9)], we observe that the potential acting on the nuclei is derived, in this case, from a static expectation value of the electronic Hamiltonian computed for the time-independent state Fj(r;R) solution of the electronic Schrçdinger equation Equation (25). Using the relation, rg Sj RðtÞ ¼ Pgj ðtÞ, we obtain a Newton-like equation of motion for the “classical” nuclei evolving in electronic state j (see footnote 1) [Eq. (33]: The Born–Oppenheimer approximation breaks down when electronic states approach in energy, which especially occurs when the dynamics is initiated in one of the electronically excited states of the system. This is the usual situation encountered in a pump-probe experiment, in which an initial pulse excites the system while a second one monitors its time-dependent relaxation towards the ground state (or a stable excited-state configuration). The starting point is the time-dependent Schrçdinger equation for the molecular system given by Equation (26) that we rewrite as Equation (36): In summary, the BO-MD equations can be described by the following system of coupled equations [Eqs. (34) and (35)]: in which [Eq. (37)]: are the first-order NACs (or NAC vectors), and [Eq. (38)]: in which only the second one describes an explicit time evolution. The electronic energies and the forces acting on the nuclei are computed statically by solving Equation (34) “onthe-fly” at each new position sampled along the trajectory R(t). Contrary to what is obtained for Ehrenfest dynamics, in BO-MD there is no explicit time dependence of the electronic degrees of freedom. It is important to further stress that, due to the assumption that F ji = 0, BO-MD always evolves on a single electronic PES, even in the case in which the system approaches regions of strong coupling between electronic and nuclear degrees of freedom. In practice, the state of interest is the ground state for which the adiabatic separation from all other states (excited states) holds in most (nonmetallic) cases. The combination of BO-MD with DFT for the “on-the-fly” calculation of the electronic structure properties (energies and forces) at each MD step is straightforward and can be found in many textbooks (e.g. ref. [2]). By using the Hohenberg–Kohn theorem, one first maps the electronic structure problem from the wavefunction space into the density space and then, within the KS formulation of DFT, the electronic ground-state energy functional, E0el ½1ðr; RÞ, and its gradients are computed. 7 are the second-order coupling elements. 2.2.3. The Nonadiabatic Bohmian Dynamics (NABDY) Equations Using the polar representation for the nuclear wavefunction Equation (29) in Equation (36) we obtain, after separating real and imaginary parts, Equation (39): and Equation (40): The continuity equation can be easily obtained by multiplying both sides of r S Equation (31) by 2 Aj and by using the relation Mgg j ¼ ugj . We therefore get the equation in which 1j ¼ A2j and jgj is the nuclear probability flux for g. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1314 – 1340 1322 CHEMPHYSCHEM REVIEWS in which both Sj(R,t) and Aj(R,t) are real fields and fij(R,t) = 1 h(Si(R,t)Sj(R,t)). Equations (39) and (40) correspond to the exact Schrçdinger equation for a nuclear wavefunction evolving in the potential of the different electronic surfaces determined by the time-independent Schrçdinger equation. In comparison with the single surface case or with the diabatic formulation, time evolution of phases and amplitudes involves firstand second-order coupling elements that mix contributions from other PESs. In particular, transfer of amplitude from one electronic state to another becomes possible thanks to the coupling terms, which make, however, the solution of the set of Equations (39) and (40) more involved. In the dynamics that emerges, the first equation is the equivalent of the classical HJ equation (first two terms) for the action S(R,t), augmented with two additional parts of quantum nature of order h and h2. The third term is the quantum potential Q(R,t), which describes all quantum effects in a single electronic state and introduces nonlocality.[29]8 Finally, the fourth to sixth terms constitute the nonadiabatic quantum potential iD(R,t), which describes interstate contributions. After applying the gradient with respect to the nucleus b on both sides of Equation (39) and moving to the Lagrangian frame,[33] we obtain a Newton-like equation of motion (using the HJ definition of the momenta rb Sj ðR; tÞRðtÞ ¼ Pbj ðtÞ [Eq. (41)]: which describes the time evolution (trajectory) of the Rb components of a fluid element with collective variable R(t) (d/dtj = @/@t + grgSj(R,t)/Mg·rg). Even though complicated, the NABDY equations [Eqs. (39), (40), and (41)] can be implemented in any DFT/LR-TDDFT code, which allows the calculations of energies (for ground and excited states), nuclear forces, and couplings as a functional of the electronic density for any given molecular geometry R(t)9. The most severe limitation in the implementation of NABDY is related to calculation of the derivatives in the configuration space such as the one of the amplitudes, rgAj(R(t),t), and of the phases rgSj(R(t),t). In fact, as the size of the system increases, it becomes very impractical to perform such derivatives on a real grid of dimensions 3Nn6. Possibilities to move “off grid” and compute the derivatives by using techniques such as Voronoi space partitioning,[111, 112] dimensional reduction,[113] or high-dimensional interpolations are under investigation. 8 9 The quantum potential is responsible for numerical instabilities associated with the formation of nodes (when the nuclear amplitude tends to zero).[29] Note that for any practical application of NABDY (or TSH, see below), the P summation over electronic states 1 i is truncated, and therefore, only a finite number of electronic states are included, that is, coupled. This constitutes an approximation, called the “group Born–Oppenheimer approximation”.[44, 107] 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.chemphyschem.org 2.2.4. The TSH Dynamics TSH can be seen as an approximation to the “in principle” exact NABDY formulation of the nuclear equations of motion. Starting from Equations (39) and (40) and keeping only firstorder terms in O(h) for the quantities Sj(R,t) and Aj(R,t) we get Equation (42): and Equation (43): At this point, we look for the trajectories in the (R,t) space, which are the extrema of the action function, dS(R,P,t) = 0, as we did in the NABDY case. However, we now consider a “classical” type of dynamics in which the different trajectories corresponding to different initial conditions are assumed to be independent from each other. In this independent trajectory approximation (ITA) the nuclear wavepacket density is given by the fraction of the total number of classical trajectories, Nt, that are found in the volume dV d3NnR [Eq. (44)]: in which N(R(t),dV,t) is the number of trajectories in the volume element dV. In TSH, the equation of motion for the nuclear trajectories is given by the terms in Equation (42), which are independent from the NAC vectors, dgji ðRðtÞÞ. This dynamics (formulated in the HJ framework) for what we call the classical acCL tions SCL j ðRðtÞ; tÞ and amplitudes Aj ðRðtÞ; tÞ is exactly equivalent to Newton’s equation (in the Lagrangian frame) [Eq. (45)]: in which Pj ðtÞ ¼ rSCL j ðR; tÞ RðtÞ . As far as the NACs are negligible, the trajectories evolve, therefore (independently), along the adiabatic PES, Ejel ðRðtÞÞ. However, when a region of strong coupling is encountered, the nuclear trajectories experience an additional force of quantum origin given by the gradient of the last term in Equation (42). At the same time, quantum amplitude is transferred from one electronic state to another according to the third term in Equation (43). The transfer of classical probability density (ACL(R(t),t))2 between electronic states is accomplished through a trajectory “hop” from one PES to the other. This occurs through the switching algorithm introduced by Tully[26] (see below). Both nonadiabatic forces and amplitude transfer have no classical counterpart and, therefore, ChemPhysChem 2013, 14, 1314 – 1340 1323 CHEMPHYSCHEM REVIEWS must be treated separately from the classical evolution of the trajectories. On the basis of Equations (42) and (43), we introduce a quanQM tum phase, SQM j ðRðtÞ; tÞand a quantum amplitude, Aj ðRðtÞ; tÞ governed by the following evolution equations [Eqs. (46) and (47)]10 : 1 QM QM with QM ij ðRðtÞ; tÞ ¼ h Si ðRðtÞ; tÞ Sj ðRðtÞ; tÞ . These two equations represent the polar form of Tully’s TSH equations for the state amplitudes [Eq. (48)]: h i i QM with Cj ðRðtÞ; tÞ ¼ AQM j ðRðtÞ; tÞ exp h Sj ðRðtÞ; tÞ . Note that in Equations (46), (47), and (48), the actions SQM j ðRðtÞ; tÞ, ampliðRðtÞ; tÞ, and coefficients C (R(t),t) are, strictly, not tudes AQM j j fields of the extended configuration-time space, but they are all restricted to the trajectory path, and therefore, they can be QM½a simply labeled with a trajectory index, ([a]), to give Sj ðtÞ, QM½a ½a Aj ðtÞ, and Cj ðtÞ, respectively. The dependence on R is recovered from the distribution of the trajectories in the configu 2 ration-time space as done for ACL ðRðtÞ; tÞ in Equation (44). j In TSH dynamics, Equations (45) and (48) are solved simultaneously with the assumption that after sampling a sufficient number of trajectories [Eq. (49)]: for each state j, which is the internal consistency criterion described in ref. [115]. This is achieved (approximately) by means of the fewest-switches TSH algorithm designed by Tully,[26]11 which gives the probability [Eq. (50)]: for a trajectory to “hop” from one reference electronic state, j, to another state, i, in the time interval Dt measured at time t. Transitions are decided by a Monte–Carlo-type procedure, in which a random number z2[0,1] is generated and a hop from state j to state i (j ¼ 6 i) is made if, and only if [Eq. (51)]: 10 11 A more detailed attempt to link the NABDY equations of motion with the TSH equations will be described in ref. [114]. In the following, we will mostly focus on the fewest-switches version of TSH. For a summary of other TSH methodologies, refer to refs. [116, 117]. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.chemphyschem.org In summary, within the quantum hydrodynamic formulation of the quantum nuclear dynamics we obtain an extended picture of the TSH dynamics. Starting from the exact NABDY equations, we reached TSH equations by using two main assumptions: 1) That in each electronic state j, the nuclear dynamics can be split into a classical and a quantum component governed, respectively, by the action SCL j ðRðtÞ; tÞ and the phase QM SQM j ðRðtÞ; tÞ with the amplitude Aj ðRðtÞ; tÞ. 2) That through ½a the hopping probability pi j [Eq. (50)], the square of the classiQM cal and quantum amplitudes, ACL j ðRðtÞ; tÞ and Aj ðRðtÞ; tÞ, coincide. 2.2.5. The TSH Algorithm: Advantages and Pitfalls In TSH dynamics, a series of independent trajectories are computed starting from a previously equilibrated population distribution of initial configurations sampled at a given temperature, T, in a chosen electronic state, j (better would be the sampling of the corresponding Wigner distribution, which, in general, is however more difficult to compute; see refs. [65, 118] for a discussion on initial conditions). All trajectories are classical in the sense that only classical forces need to be computed from the gradient of the selected PES. For a given independent trajectory [a] of the swarm, the quantum amplitudes are also evolved in time by using Equation (48) together with the nuclear coordinates propagated with Equation (45). In a region of coupling between two PESs j and i, this classical trajectory can eventually hop from one surface to another according to the ½a probability pi j. After a surface hop, the excess (deficient) energy due to the transition is redistributed into (extracted from) the motion along the direction of the NAC vectors, as suggested by the extra force obtained from the gradient of the third term on the left-hand side of Equation (42) and by Pechukas’ semiclassical interpretation of the nonadiabatic transitions.[36, 96] In this way, energy conservation is guaranteed along the entire trajectory. Frustrated hops occur when the quantum particles have insufficient kinetic energy to compensate the potential energy loss in an upward transition.[119, 120] This method has the advantage to be simple enough to be implemented into any existing wavefunction-based or DFT/LRTDDFT code that offers the possibility to compute efficiently the electronic energies (PESs), the classical forces, and the NACs. These quantities can be either precomputed or evaluated “on-the-fly” along the growing trajectory. However, the first approach is only suited in the case of relatively small systems made up of only a few atoms for which the multidimensional 3Nn6 potential energy hypersurfaces (or a restricted portion of them) can be computed for all relevant states. The “on-thefly” approach reduces dramatically the number of electronic structure calculations to the number of integration steps of the classical trajectory, and therefore, it can be used for the simulation of the nonadiabatic dynamics of large systems (made of up to hundreds of atoms[121]). ChemPhysChem 2013, 14, 1314 – 1340 1324 CHEMPHYSCHEM REVIEWS www.chemphyschem.org Due to the classical nature of the dynamics along the PESs, TSH cannot describe nuclear tunneling and nuclear dephasing processes.[117, 122] The only nuclear quantum effects reproducible with TSH dynamics are those related to wavepacket splittings induced by avoided crossings and conical intersections (regions of strong nonadiabatic couplings). Due to the ITA and the local nature of the transitions (in space and time), quantum decoherence effects between states are difficult to capture in standard TSH dynamics. However, variations of TSH algorithm to circumvent these limitations are available.[115, 120, 123–127] Figure 3. Nonadiabatic dynamics of NaI. Top) Electronic energy as a function of the interatomic distance. In the Compared to Ehrenfest MD, inset we report the full potential energy profiles (c) together with the nonadiabatic couplings (a). The nonthe trajectories in TSH evolve on adiabatic dynamics starts in the first excited state of NaI with a kinetic energy component pointing to the right. In adiabatic PESs (except for the in- TSH, all trajectories lie on an adiabatic PES, and therefore, they are only distinguishable at the instant in time at stantaneous transitions), which which a surface hop occurs (vertical lines). In contrast, Ehrenfest trajectory evolves on a mean-field PES that is a superposition of the two adiabatic states (b). Bottom) The two lower panels show the TSH trajectories distribusimplifies the physical interpreta- tion (histograms) and NABDY nuclear probability density (*) at four different times along the dynamics (State 1 is tion of the results and compari- the ground state and State 2 is the first excited state). son with experiments. In addition, TSH MD in its “fewestcontrast, in TSH the independent trajectories evolve on adiaswitches” implementation[26] obeys detailed balance approxibatic PESs (the two potential energy curves of Figure 3), and mately, with deviations that tend to vanish in the limits of population transfer among the states occurs through surface small adiabatic splitting and the limit of large NACs.[128] The hops described by Equation (50). By computing the population use of the ITA together with the representation of the nuclear densities at each instant of time according to Equation (44) we wavepacket amplitude by the density of trajectories offers can reconstruct approximate nuclear wavepackets on the two a simple (even though approximate) solution to the problem PESs (Figure 3, lower panels), which compare well with the of the high dimensional derivatives that we discussed in the “exact” quantum wavepackets obtained from the propagation case of the implementation of the NABDY equations. More of the Bohmian trajectories within the NABDY scheme. For specific information can be found about TSH in the following TSH, the time evolution of the energy and distance spreads of references.[5, 65, 97, 122, 125, 129–133] the wavepacket is shown in Figure 4, where the density of curves reproduces the squared amplitude of the wavepacket 2.3. Comparison between the Nonadiabatic Schemes (transitions between the two states are represented by vertical lines). In the lower panel, we can see how the single Ehrenfest As an illustrative example to show the main differences among path follows the evolution of the “center” of the TSH trajectory the nonadiabatic dynamics schemes proposed so far (Ehrenfest bundle. However, when the kinetic energy is small enough to dynamics, TSH, and NABDY), we discuss the case of photoexcitprevent the full depletion of the excited state, a wavepacket ed sodium iodide (NaI). The PESs of this diatomic molecule exsplitting is observed (inset), which can only be reproduced by hibit a characteristic avoided crossing between the ground using a multitrajectory-based approach such as TSH and state and the first excited state (Figure 3, inset) that, at the NABDY. equilibrium distance, have ionic and covalent character, respectively. When a nuclear wavepacket is prepared and launched in the first electronic excited state, part of its amplitude is transferred back to the ground state. The resulting dynamics de2.4. DFT/TDDFT-Based TSH Approaches and Their pends largely on the mixed quantum/classical method used Applications (see Figure 3). In Ehrenfest dynamics, the system evolves on Several implementations of “on-the-fly” DFT/TDDFT-based TSH a mean-field PES (gray dashed line in Figure 3) so that EquaMD are nowadays available in different software packages tion (22) is a weighted average of the ground-state and first such as CPMD,[134] Turbomole[135] (from version 6.4), and excited-state PESs. The degree of mixing depends on the size of the NACs in the region surrounding the avoided crossing. In NEWTON-X[136] (which proposes an interface with several elec 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1314 – 1340 1325 CHEMPHYSCHEM REVIEWS www.chemphyschem.org Figure 4. Nonadiabatic dynamics of NaI—Electronic energy and interatomic distances as a function of simulation time for the same dynamics reported in Figure 3. Top) Time evolution of the electronic energies for a swarm of TSH trajectories (black lines with different styles) and for the single Ehrenfest trajectory (b). Bottom) Time evolution of the intramolecular distances (same color code as in the top panel). Inset: Same plot but for a smaller initial momentum. In this case, splitting of the wavepacket between the two states is observed. Note that although TSH can describe the splitting of the wavepacket, the Ehrenfest dynamics proceeds along a single mean-field trajectory. transition probabilities according to fewest-switches criteria,[26] an additional orthonormalization of the electronic wavefunctions is required.[12] The ROKS surface hopping methodology[12, 137] can be use to study nonradiative decays between the first singlet excited state (S1) and the ground state of molecular systems (see refs. [138–146] for additional examples). In recent work,[147] the use of bridged azobenzene as a potential photoswitch was studied by ROKS nonadiabatic dynamics. Azobenzene is a wellknown molecular photoswitch that is used in many biological and material applications. Its photoexcitation leads to Z!E isomerization, which consequently stretches the molecule. It has been found that bridged azobenzene (Figure 5) exhibits tronic structure codes). They mainly differ in the way in which the electronic structure calculations are performed. 2.4.1. TSH/Restricted Open-Shell Kohn–Sham (ROKS) Among the DFT-based TSH MD implementations, the method by Doltsinis and Marx[12] is based on the restricted open-shell formulation of the first singlet DFT excited state (i.e. ROKS) and is available in the software package CPMD.[134] Briefly, in TSH/ROKS dynamics[12] the total electronic wavefunction of the system is represented by iR P Y ¼ j aj Fj exp h Ej dt in which aj are time-dependent expansion coefficients and Fj are KS-based many-electron wavefunctions with corresponding eigenvalues Ej. By using the closed-shell KS ground-state Slater determinant for F0 and the ROKS S1 excited-state wavefunction F1, the following set of coupled equations [Eqs. (52) and (53)] of motion for the expansion coefficients is obtained: R Ej in which pj ¼ iE h dt , Sij = hFi j Fji with S01 = S10 = S and D exp @ Sii = 1, Dij ¼ Fi @t Fj , Hii = Ei, and H01 = H10 = E0S. This set of equations can be easily integrated together with the equation of motion for the nuclei by using, for instance, a fourth-order Runge–Kutta scheme. For the calculation of the nonadiabatic 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Figure 5. Average value of the CNNC dihedral angle in the first excited state of azobenzene: bridged (E)-azobenzene (*), (E)-azobenzene (*), (Z)-azobenzene (~) for which the 1808 CNNC dihedral angle is plotted. Inset) Optimized structures of azobenzene (top) and bridged azobenzene (bottom) for the Z (left) and E (right) isomers. Figure reprinted with permission from M. Bçckmann et al., Angew. Chem. 2010, 122, 3454; Angew. Chem. Int. Ed. 2010, 49, 3382. Copyright 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. a much larger quantum yield for photoisomerization than the standard, unbridged (E)-azobenzene. This rather counterintuitive fact has been studied theoretically, and the enhanced isomerization quantum yield ( 50 %) of the bridged azobenzene has been precisely reproduced. In fact, this study revealed that the mechanism of photoisomerization of bridged (E)-azobenzene is similar to that of unconstrained (Z)-azobenzene, which involves specific prearrangement of the phenyl rings. ChemPhysChem 2013, 14, 1314 – 1340 1326 CHEMPHYSCHEM REVIEWS 2.4.2. TSH/TDKS More recently, Prezhdo et al. developed a TSH/MD scheme based on the time-dependent propagation of the KS orbitals, which are used to approximate the many-electron Slater-type wavefunctions for the calculation of the NACs.[13, 148] In short, the time-dependent KS orbitals fk(r,t) are expanded in the basis of the instantaneous adiabatic KS orbitals opt : ðr; RðtÞÞ [see Eq. (18)], which leads [when included into TDKS Equation (21)] to the following set of differential equations [Eq. (54)] for the expansion coefficients: D E @ opt _ KS in which dKS . Within this h opt pm ðRðtÞÞ R ðtÞ ¼ i p @t m framework, the many-electron adiabatic states are generated ad hoc by using excited Slater determinants of KS orbitals, whereas the NAC terms (and the corresponding surface-hopping transition probabilities) are computed from the corresponding matrix elements involving electron-hole (virtual/occupied) KS pairs. www.chemphyschem.org same trend as that of the population decay. Interestingly, holes with higher energy (closer to the Fermi level) show a slower relaxation decay (pluses in Figure 6 b). 2.4.3. TSH/LR-TDDFT In 2007, Tavernelli and co-workers[14] derived a LR-TDDFT-based TSH MD scheme in which all quantities required for the propagation of the trajectories [Eq. (45)] and amplitudes [Eq. (48)] are rigorously derived from TDDFT in the LR formulation (see Table 2) and expressed as a functional of the electronic density. The details of this derivation are given in Section 3 after a brief introduction to LR-TDDFT. TSH/LR-TDDFT has been applied to a multitude of different photophysical/photochemical processes of organic molecules (see refs. [14, 159–164] for a nonexhaustive list). The nonradiative relaxation of the photoexcited oxirane molecule is taken here as a prototypical example of TSH/LR-TDDFT dynamics.[20] Even though it is well known that photoexcitation of oxirane leads to its ring opening,[165] the mechanistic details of this photochemical reaction remained elusive for a long time. In ref. [14], we showed that the first electronic excited state S1, initially with the character 1(n,3s), is nonreactive, that is, it behaves as an energy reservoir during the first part of the dynamics ( 280 fs in a typical trajectory, Figure 7). Nevertheless, dramatic changes in the TSH dynamics are observed whenever This method is simple and efficient, but the description of excited states by means of excited KS Slater determinants is not rigorous, and therefore, the approximation done in the representation of the PESs and their couplings is not controllable and cannot be improved systematically. The adiabatic KS states are indeed seen as a zeroth-order approximation to the LR-TDDFT adiabatic states.[148] We highlight here an application of TSH/TDKS in the study of electron and hole relaxation dynamics in photoexcited (7,0) zigzag carbon nanotubes[149] (for further examples, see refs. [17, 150–152]). Interestingly, it was found that the electron relaxation follows a simple exponential, whereas the hole relaxes through more complicated pathways, which leads to multiple timescales (Figure 6). The nonadiabatic dynamics indicates, in addition, that the main relaxation pathways involve high-frequency longitudinal optical phonons. Holes seem to couple in addition to lower frequency radial breathing modes. FigFigure 6. Left) Electron and hole relaxation dynamics in (7,0) zigzag carbon nanotubes: Right) Population decay ure 6 b shows the electron and (a) and the average energies of the electrons and holes along the dynamics (b). Figure taken from ref. [149]. hole energy decay after photo- Figure reprinted with permission from B. F. Habenicht, C. F. Craig, O. V. Prezhdo, Phys. Rev. Lett. 2006, 96, 187401. excitation, and it exhibits the Copyright (2006) by the American Physical Society. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2013, 14, 1314 – 1340 1327 CHEMPHYSCHEM REVIEWS www.chemphyschem.org Table 2. Summary of the main electronic properties required in TSH/LRTDDFT[14] and references to their derivation and validation (the labels i and j refer to the electronic states). Electronic property Excited-state PESs Excited-state forces on the nuclei NAC vectors[a] NAC vectors[b] Transition dipole moments[c] LR-TDDFT response density[d] Ref. el j E Fj d0i dji mji d1j(r) [54, 153] [55, 154, 155] [14, 156, 157] [59] [158] [54, 155] carbonyl ligand upon photoexcitation.[170] The so-formed free coordination site can then be filled by solvent molecules in different ways.[171] Crespo-Otero et al. performed a theoretical study of the release of one carbonyl ligand from a chromium hexacarbonyl complex.[169] They used a swarm of 30 TSH trajectories starting from a MLCT (metal-to-ligand charge transfer) excited electronic state (either S1, S2, or S3) to investigate the possible reaction channels. Three main time windows dominate the relaxation process towards the ground state S0 (Figure 8). During the first 20 fs, the dynamics is confined in [a] Ground to excited state, based on the auxiliary many-electron wavefunction. [b] Excited to excited state, based on the auxiliary many-electron wavefunction. [c] The same for matrix elements of any one-body operator, based on the auxiliary many-electron wavefunction (see Section 3.3). [d] For QM/MM applications, for example. Figure 8. Potential energy lines as a function of time. Left) Typical TSH/LRTDDFT trajectory (top) and representative snapshots of the relevant structures (bottom). The dots indicate the state that drives the dynamics of the nuclei. Right) Schematic illustration of the energy profiles. Figure reprinted with permission from R. Crespo-Otero et al., J. Chem. Phys. 2011, 134, 164305. Copyright 2011, American Institute of Physics. Figure 7. 1D cut of the PESs along the reaction path computed with TSH/ LR-TDDFT: Energies computed along the most probable, reactive path (c) and low probability path that would continue on the nonreactive surface (a); S0 (c), S1 (c), S2 (c), S3 (c); running states are indicated by *). The geometries of the TSH trajectory are shown at time a) 243, b) 284, and c) 300 fs. Figure reprinted with permission from E. Tapavicza et al., J. Chem. Phys. 2008, 129, 124108. Copyright 2008, American Institute of Physics. a mixing between S1 [1(n,3s)] and S2 [1(n,3pz)] occurs (t > 280 fs in Figure 7; the probability to continue the dynamics on the nonreactive 1(n,3s) state through a surface hop is small). Population of the 1(n,3pz) state leads to the dissociation of a CO bond and, therefore, to ring opening. Interestingly, this constitutes the only photochemical rearrangement observed in the excited state, and any further dissociation into two radical moieties is expected to occur only once the system has relaxed into the ground state. Applications of TSH/LR-TDDFT to metal-based molecules are numerous (e.g. refs. [166–168]). In this paragraph, we present the nonadiabatic relaxation of chromium hexacarbonyl Cr(CO)6.[169] Metal hexacarbonyl (e.g. based on molybdenum, chromium, or tungsten) are known to efficiently release one 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim the manifold of MLCT states of the photoexcited Cr(CO)6 complex. Later, a splitting of the excited-state bands into two subbands is observed (region II in Figure 8), which is associated with the dissociation of a carbonyl ligand. This process leads to the formation of an excited Cr(CO)5 moiety and a free CO molecule within approximately 40 fs of dynamics. Finally, Cr(CO)5 reaches the ground state after an average time of about 100 fs. The time constants extracted for the photodissociation dynamics agree nicely with experimental data obtained by ultrafast optical spectroscopy. 3. TDDFT Quantities for Nonadiabatic Dynamics The nonadiabatic schemes presented in the previous section depend, in their implementation, on a number of electronic properties such as electronic energies (i.e. PESs), forces, NACs, and others (for a complete summary see Table 2). The efficient and reliable calculation of these terms is key to the development of “on-the-fly” nonadiabatic molecular dynamics schemes. Matrix elements between electronic states are straightforwardly obtained by means of wavefunction-based methods such as, for example, configuration interaction singles (CIS),[172] multireference configuration interaction (MR-CI),[173, 174] and equation-of-motion excitation energy coupled-cluster singles and doubles (EOMEE-CCSD).[175] However, the evaluation ChemPhysChem 2013, 14, 1314 – 1340 1328 CHEMPHYSCHEM REVIEWS of any matrix element within LR-TDDFT requires reformulation of these quantities as a functional of the electronic density or, equivalently, of the KS orbitals. This is necessary as DFT and TDDFT are wavefunction-free theories that are based exclusively on the electronic density and its representations. In this section, we will briefly present the main LR-TDDFT equations and introduce the concept of auxiliary many-electron wavefunctions, which is used to describe all matrix elements required in the nonadiabatic dynamics. For a short summary of the main LR-TDDFT equations in the so-called Casida formalism,[54, 176, 177] see Appendix B in the Supporting Information. www.chemphyschem.org one obtains, after matrix inversion, the LR-TDDFT equations in matrix form (see Appendix B in the Supporting Information) [Eq. (61)]: P in which d1ðr; wÞ ¼ ijs is ðrÞjs* ðrÞdPijs ðwÞ and e. are the KS orbital energies. Note that we use the same symbol for a function and its Fourier transform. To which one we are referring to is clear from the variable specified (t or w). 3.1.1. The LR Equation for TDDFT 3.1. The Casida Equations The excitation energies of a molecular system (in the gas phase as well as in the condensed phase) are given by the poles of the Fourier-transform in frequency space of the density–density response function defined as [Eq. (55)]:[178] in which d1(r,t) is the electron density variation, c(r,t,r’,t’) is the response function, and duapp(r’,t’) is the applied perturbation. Because c(r,t,r’,t’) depends only on the time difference tt’, Equation (55) is usually formulated in the Fourier (energy) space [Eq. (56)]: which is the starting point for the derivation of LR-TDDFT equations. Key to the development of Casida’s equations for LR-TDDFT is the formulation of the LR equation [Eq. (56)] in the basis of KS orbitals {fi(r)} with occupations fi and spin si [Eq. (57)]: Starting from the LR susceptibility for a system of noninteracting electrons (see Appendix C in the Supporting Information) [Eq. (58)]: As we just showed in TDDFT, the LR equation [Eq. (61)] is complicated by the fact that the self-consistent potential us[1](r,t) [see Eq. (19)] depends on the response of the density. Using Equation (A-15) from Appendix B in the Supporting Information we get [Eq. (62)]: in which, formally, c1 = ([cs]1K), and: By ordering the KS orbital basis according to the occupation, we can split Equation (62) into particle–hole (ph) and hole–particle (hp) sectors and rewrite Equation (62) in the following form (in the following i,j,k,… label occupied orbitals and a,b,c,… label virtual orbitals) [Eq. (63)]: in which [Eqs. (64) and (65)]: and using the matrix representation of the density–density response for the fully interacting system [Eq. (59)]: and for the corresponding (noninteracting) KS system (Appendix B in the Supporting Information) [Eq. (60)]: 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim The matrix on the left-hand side of Equation (63) is the inverse of the susceptibility, and therefore, the excitation energies wn can be obtained from the solution of the generalized eigenvalue equation [Eq. (66)]: ChemPhysChem 2013, 14, 1314 – 1340 1329 CHEMPHYSCHEM REVIEWS for which we now assume that the matrices A and B are frequency independent (AA for the TDDFT kernel, fxc). This equation has paired excitation (wn > 0) and de-excitation (wn < 0).[177] The usual normalization is [Eq. (67)]: 3.1.2. The Tamm–Dancoff Approximation (TDA) The TDA[179, 180] consists in setting B = 0 in Equation (66), which leads to the simplified LR-TDDFT equation [Eq. (68)]:[181, 182] Physically, this means that in the TDA all contributions to the excitation energies coming from the de-excitation of the correlated ground state are neglected. Even though it is an approximation, the TDA can improve the stability of the LRTDDFT calculations with most of the standard (approximated) functionals. In particular, by decoupling the DFT ground-state problem from the calculation of the LR-TDDFT excitation energies, TDA can provide better PESs, especially in the regions of strong coupling with the ground state.[20, 177, 183, 184] This is of crucial importance for all nonadiabatic MD schemes based on LRTDDFT PESs. 3.1.3. The Sternheimer Formulation of LR-TDDFT The Sternheimer time-dependent perturbative approach[185] offers an alternative way to calculate the LR-TDDFT energies and properties. Most importantly, this method does not require calculation of the unoccupied KS orbitals, as in the case of Casida’s equations, but it is based on the calculation of one response orbital for each occupied KS orbital. This may become computationally advantageous when looking for high-energy excited states. In the Sternheimer formulation of LR-TDDFT,[56, 155] the LR perturbation density is given by [Eq. (69)]: n o in which the LR orbitals fg satisfy Equation (70):[155] n;i ðrÞ P Q ¼ 1 Ni el ji ihi j, H0 , is the KS Hamiltonian, Eij is a Lagrangian multiplier ensuring the orthogonality of the ground-state orbitals, and dun,SCF(r,w) is the change of the self-consistent field (in the AA) upon perturbation [Eq. (71)]: www.chemphyschem.org 3.1.4. Current Approximations in LR-TDDFT: The Pitfalls Even though LR-TDDFT (in the AA, see Section 3.1.1) has been successfully applied to a multitude of molecular systems,[82, 177]12 its rise in the realm of quantum chemistry and quantum physics is still hampered by several failures rooted in the approximations of the xc functional and its functional derivatives. A complete account of those failures is far beyond the scope of this article and can be found in specialized books[82, 186] or reviews.[177, 187] Nevertheless, we find important to highlight here those deficiencies that are particularly deleterious in nonadiabatic dynamics and that add up to the ones already known for ground-state DFT. First, LR-TDDFT is not able to properly describe electronic states with strong (> 50 %, see ref. [82]) double excitation character.[188–192] This problem is related to the use of the AA for the TDDFT kernel, and unfortunately, at present only few potential solutions to this failure exist.[190, 193] Second, the inaccurate description of derivative discontinuities, the problem of self-interaction error, and the incorrect long-range properties of currently used xc potentials are at the heart of the most important problem faced in the application of LR-TDDFT in chemistry and physics, namely, the chargetransfer failure.[194–200] LR-TDDFT using standard local density approximation (LDA) or generalized gradient approximation (GGA) functionals is indeed inappropriate to describe excitations and charge-transfer excitations between spatially separated molecules. However, the use of long-range corrected functionals[201, 202] strongly improves the situation. Finally, it is important to stress that LR-TDDFT may have difficulties in describing conical intersections between the ground state and the first electronic state.[191] However, it has been shown that, at least in some cases, the use of the TDA improves the description of these critical points.[20, 186] 3.2. The Nuclear Forces from LR-TDDFT LR-TDDFT could become the method of choice for the calculation of excited-state PESs to be used in different nonadiabatic MD schemes. In this case, the calculation of nuclear forces within the LR-TDDFT formalism becomes an essential step. Among the different approaches developed for the calculation of analytic derivatives, the Lagrangian method[203] is of particular interest because of its numerical efficiency. However, the derivation of LR-TDDFT is technically involved, and as it does not bring any new physical insights, we simply refer the interested reader to the rich literature on this subject.[2, 155, 204–206] 12 Excited-states energies and densities are computed from the solutions of Equation (70). For more details see Appendix D in the Supporting Information and refs. [155, 156]. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim See also the following special issues: “Time-Dependent Density-Functional Theory”, Phys. Chem. Chem. Phys. 2009, 11, 4421–4688; “Time-Dependent Density-Functional Theory for Molecules and Molecular Solids”, J. Mol. Struct.: THEOCHEM 2009, 914, 1–138; “Open Problems and New Solutions in Time-Dependent Density Functional Theory”, Chem. Phys. 2011, 391, 1– 176. ChemPhysChem 2013, 14, 1314 – 1340 1330 CHEMPHYSCHEM REVIEWS www.chemphyschem.org 3.3. The Quest for Matrix Elements in TDDFT In Section 2, we showed that matrix elements such as the NAC vectors [Eq. (37)] or the second-order NACs [Eq. (38)] are key to the implementation of any nonadiabatic dynamics scheme [Section 4.2 will also highlight the importance of transition dipole moments, Eq. (114)]. However, such quantities are formally defined in terms of electronic wavefunctions, and the main challenge consists, therefore, in the reformulation of these quantities as a functional of the ground-state electronic density (or equivalently, of the KS orbitals). In the following, we will introduce the concept of auxiliary many-electron wavefunctions. Those are linear combinations of singly excited KS Slater determinants with the property that, when used in the calculation of matrix elements of one-body operators, they provide the formally exact results within LRTDDFT (formally exact in the sense that this approach would give the exact matrix elements if the exact xc functional and TDDFT kernel would be known). When used as approximations to the “real” electronic states, the auxiliary many-electron wavefunctions perform poorly [even though successfully applied in the context of the assignment of electronic transitions;[54] instructive in this context is the meaning of the ground-state KS Slater determinant (built from the occupied KS orbitals), which gives a poor representation of the groundstate wavefunction (less accurate than the corresponding Hartree–Fock Slater determinant)]. It is only when applied in the context of the calculation of matrix elements between ground state and excited states (as well as between pairs of excited states, see Section 3.3.4) that the auxiliary many-electron wavefunctions provide the correct answer within the linear response. The auxiliary many-electron wavefunctions have been used to calculate several electronic properties such as, for example, the NAC vectors (Section 3.5), the second-order NACs,[207] the adiabatic excited-state nuclear forces,[208] the transition dipole moments[158, 209] (Section 4.2), and the relativistic spin–orbit couplings,[210, 211] which, with different flavors, are already available within the LR-TDDFT formalism (e.g. refs. [212, 213]). In the following sections we outline the main steps in the derivation of the auxiliary many-electron wavefunctions. First, in Section 3.3.1 we drive the sum-over-states (SOS) representation of the density–density response function within LR-TDDFT. Then, by using the equivalent SOS formula derived in manybody perturbation theory (MBPT, Section 3.3.2) and by equating the LR-TDDFT and MBPT SOSs residue-by-residue, we finally obtain the desired auxiliary many-electron wavefunctions (Section 3.3.3) that are applied in the calculation of the matrix elements in LR-TDDFT. 3.3.1. The LR-TDDFT SOS Formula In this section we derive a SOS representation of the density– density response function within LR-TDDFT. The starting point is the matrix form of the TDDFT LR equations in frequency space [Eq. (63)]. For a real perturbation of the type du’(w) = 1/2(du(w) + du*(w)) Equation (63) leads to Equation (72):[54] 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim in which [Eq. (73)]: and [Eq (74)]: We can therefore define the susceptibility c(w) as [Eq. (75)]: Using the spectral representation of the (W(w)w2I)1, we can write [Eq. (76)]: in which Zn are the eigenvectors of W(w) [Eq. (77)]: and are related to the eigenvectors of Equation (66) according to [Eq. (78)]:[177] Note that in case of a frequency-dependent W(w), the eigenvectors Zn require the normalization [Eq. (79)]:[54] to enforce the residues in the spectral representation [Eq. (76)] to be equal to one. Finally, we obtain for the susceptibility c the following SOS representation [Eq. (80)]: Using Equation (80), we can now compute the response ^ dhOi(w) of any one-body operator O^ due to the action of a generic time-dependent perturbation û(r,t) = û’(r)E(t). Taking the usual representation of one-body operators in terms of the KS creation and annihilation operators [Eqs. (81) and (82)]: R R ^ *is ðrÞjs ðrÞ and u0 ijs ¼ dr^ with oijs ¼ drOðrÞ u0 ðrÞ*is ðrÞjs ðrÞ, ^ ^ we first get for dO(w) = dhOi(w) = sdrO(r)d1(r,w) [Eq. (83)]: ChemPhysChem 2013, 14, 1314 – 1340 1331 CHEMPHYSCHEM REVIEWS www.chemphyschem.org in which, according to d1(r,w) = sdru’(r’)E(w)c(r,r’,w) [Eq. (84)]: Therefore, from Equation (80), the final LR-TDDFT SOS equation reads [Eq. (85)]: dm As an example, the polarizability defined as aE1,E2(w) = E2 2ðtÞ in which E1 and E2 are two Cartesian directions in space measures the change in dipole dmE1 = E1·r induced by a perturbation of the form u’(r,w) = E2·rE(w). From Equation (84) we get (for E1 = x̂ and E2 = ẑ) [Eq. (86)]: 1 For finite-sized systems in which we can construct a discrete set of “quasiparticle” one-electron orbitals to be used in Slatetype many-electron wavefunctions (Hartree–Fock or KS orbitals in most cases), we can introduce the matrix representation [Eq. (91)]: in which [Eq. (92)]14 : 2 in which the factor 2 comes from the fact that (for symmetry reasons) the summation over ij can be restricted to terms with (fisfjs) > 0. P P ^yis a ^js , u ^yis a ^js . Using the symmetry ^0 ¼ ijs u0 ijs a and O^ ¼ ijs oijs a property [Eq. (93)]: we finally obtain the SOS formula [Eq. (94)]: 3.3.2. The Matrix Elements in LR-TDDFT The derivation of general formulae for the evaluation of matrix elements of one-particle operators in LR theory [Eq. (87)]: is based on a direct comparison with the same quantity derived by using MBPT. In Equation (87), the states j F0i and j Fni describe the ground-state and nth electronic excited state wavefunctions, respectively. Comparing the residues of LR-TDDFT response function Equation (85) with the residues of the MBPT response function Equation (94) at equal energy wn, we obtain the following identity [Eqs. (95) and (96)]15 : We start, therefore, with a short outline of the main LR equations in MBPT. From the definition of the retarded density–density response function [Eq. (88)]13 : the change of an observable O, under the influence of a perturbation ûext(r’,t’) in the LR regime is given by [Eq. (89)]: This equation was derived by Casida[54] and then applied by Tavernelli et al.[14, 156, 157] and Hu et al.[214, 215] for the calculation of the NAC vectors between the ground state and an excited state. A similar equation was also given in ref. [216]. 3.3.3. The Auxiliary Many-Electron Wavefunction (here we consider an interaction of the form ûext(r’,t’) = û’(r’)E(t’)). If c depends only on the difference (tt’), the Fourier transform in time gives [Eq. (90)]: 13 It may be useful at this point to investigate the possibility to further simplify the definition and the calculation of matrix elements within LR-TDDFT by means of the definition of a set of “auxiliary” multideterminantal many-electron wavefunctions 14 For the special case of o(r) = 1, we get the LR density: 15 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim With no Hartree–Fock exchange contribution in the functional, (AB) is diagonal and becomes:[177] ChemPhysChem 2013, 14, 1314 – 1340 1332 CHEMPHYSCHEM REVIEWS based on KS orbitals. This route was first explored by Casida[54] to solve the assignment problem of the LR-TDDFT excitedstate transitions and then further developed by Tavernelli et al.[14] in relation to the calculation of matrix elements in the linear and second-order response regimes.[59, 156, 157] In ref. [157], we showed that defining the ground-state many-electron wavefunction as a Slater determinant of all ocel cupied KS orbitals fi gNi¼1 [Eq. (97)]: and the excited-state wavefunction corresponding to the excitation energy wn as [Eq. (98)]: we obtain for any one-body operator of the form O^ = P ^yps a ^qs (in which p and q are general indices) the corpqs opqs a rect LR expression for the matrix element hF0 j O^ j Fni. Equation (98) is derived from Equation (96) for which now the index i runs over all occupied KS orbitals and the Eindex a runs over ~ as denotes a singly the unoccupied (virtual) KS orbitals and F is excited Slater determinant defined by the transition is!as. This theory was then successfully extended to the case of the calculation of matrix elements between two excited-state wavefunctions, hFn j O^ j Fmi, as shown in the section on the calculation of NAC vectors (Section 3.5). It is important to further stress the fact that both auxiliary functions introduced in Equations (97) and (98) have only a physical meaning when used withinDLR-TDDFT Efor the calcu~ 0 O^F ~ n and eventulation Eelements of the type F D of matrix ^ ~ ~ ally Fn OFm (see Section 3.3.4). The use of these representations of the ground-state and excited-state KS many-electron wavefunctions in other contexts is not justified. www.chemphyschem.org ðiÞ Fn and mðiÞ ¼ Fn mðiÞ Fm . with mðiÞ n ¼ F0 m nm The TDDFT second-order density–density response function can also be obtained (see refs. [217, 220]) and then converted into a SOS representation by setting up a classical system of coupled harmonic oscillators (bosons) that share the same linear and second-order response properties as those of TDDFT.[217, 219, 220] It is beyond the scope of this review to describe the derivation of the second-order matrix elements following this approach in detail. It is, however, important to mention that this solution is an approximation, and in particular, it does not account for contributions to the matrix elements produced by the de-excitation of the correlated ground state j F0i.[59] In ref. [59], Dit was shown E that second-order matrix elements ~ n O^F ~ m can be evaluated. This solution is of the type F more symmetric with respect to the ph and hp transitions than the one derived in refs. [217, 219, 220], but only considers terms first-order in Z in the expansion of the KS energy functional around the equilibrium (unperturbed) value. The TDDFT second-order matrix elements are, therefore, evaluated according to [Eq. (100)]: and match exactly the leading LR contributions obtained by using the second-order coupled electronic oscillator approach of Mukamel and Tretiak and co-workers.[217, 219, 220] In the TDA[59, 179, 180] the two approaches coincide. In terms of Casida’s eigenvectors (Xn,Yn), Equation (100) becomes [Eq. (101)]: 3.3.4. The Second-Order Response Matrix Elements The same procedure used in the derivation of the matrix elements between the ground and excited states can be followed for theDcalculation E of the second-order response matrix ele ments Fn O^Fm , in which both states n and m are LR excited states. We will, therefore, evaluate the SOS second-order density–density response function in both wavefunction-based MBPT and TDDFT, and then equate the residues corresponding to equal poles term-by-term. Within the many-body formulation of quantum mechanics in second quantization, the SOS second-order density–density response function is obtained by using a perturbative approach applied to the molecular Hamiltonian and reads [Eq. (99)]:[59, 217–219] 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim This equation will be used to evaluate NAC vectors along nonadiabatic trajectories produced with the TDDFT-based TSH approach.[14] To this end, the general operator O^ is replaced by the nuclear gradient of the electronic Hamiltonian (see Section 3.5). 3.4. The LR-TDDFT Response Density As it will be discussed in Section 4.1, QM/MM nonadiabatic dynamics requires the evaluation of the electrostatic interaction between the classical environment and the quantum system, ChemPhysChem 2013, 14, 1314 – 1340 1333 CHEMPHYSCHEM REVIEWS which comprises the excited density 1n(r,t) = 10(r) + d1n(r,t). Within Casida’s formulation, the density response d1n(r,t) can be expanded in the auxiliary many-electron wavefunctions of Equation (98) [Eq. (102)]: www.chemphyschem.org 3.5.1. The Couplings with the Ground State We start from an alternative definition of the NAC vectors[222] (see also Chapter 5 of ref. [223] for a complete discussion) between the ground (0) state and the nth excited state for a molecular system characterized by nuclear coordinates R in the configuration space (R3Nn) [Eq. (107)]:16 which in first-order becomes [Eq. (103)]: where g is an atomic label, H^el is the (many-body) electronic Hamiltonian, and @gH^el = @H^el /@Rg. Applying the results of Sections 3.3.2 3.3.3 Eon the evaluD and ation of matrix elements of the form F0 O^Fn in LR-TDDFT to the NAC vector gives directly the desired expression [Eq. (108)]: in which f̃p = (1fp). Using the LR-TDDFT equation for the transition density 1’(r) = pqfpf̃q(Xn+Yn)pqfp(r)yq(r) and the definition of the coeffin [14, 156, 177] cients Cpq we finally get, in agreement with refs. [177, 221] [Eq. (104)]: in which [Eq. (105)]: and x1 ðrÞ; . . . ; xNel ðrÞ; xNel þ1 ðrÞ; xNel þ2 ðrÞ; . . . ¼ 1 ðrÞ; . . . ; Nel ðrÞ; y1 ðrÞ; y2 ðrÞ; . . . . Within the Sternheimer framework (see Section 3.1.3), the response density matrix elements in TDA are given by ref. [155] [Eq. (106)]: R in which hgijs ¼ dr@g H^el is* ðrÞjs ðrÞ. This formula for the NAC vectors within LR-TDDFT has been derived several times in the literature by using slightly different formalisms. The first derivation was by Chernyak and Mukamel[57] who used classical Liouville dynamics for the single-electron density matrix. Later, Tavernelli et al.[14, 156, 157] and Hu et al.[214, 215] arrived at the same result [Eq. (108)] by using the most widely used formulation based on Casida’s LR-TDDFT equations.[54] Concerning the numerical implementation of Equation (107), several approaches have also been proposed that differ mainly in the choice of the basis set and in the way in which the implicit dependence of the pseudopotentials on the nuclear positions is treated. Due to the technical nature of this subject, we will not go through the numerical details but instead refer to the literature, which is very rich on this subject.[156, 214, 224–227] 3.5.2. The Finite Difference Formulation of the NACs For a detailed account of the implementation see the Appendix D in the Supporting Information and refs. [155, 156]. 3.5. The NAC Vectors in LR-TDDFT Traditionally, the computation of the nonadiabatic coupling vectors is carried out by using wavefunction-based ab initio quantum chemistry approaches, which, however, are not well suited for condensed-phase applications and become computationally unaffordable when the dynamics of large molecular systems is considered. Using the auxiliary many-electron wavefunction approach outlined in Section 3.3, we present here an approach for the calculation of nonadiabatic vectors within LR-TDDFT. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim In TSH dynamics, the NAC terms appear as a scalar product of the NAC vectors with the particle velocities, s0n = d0n·Ṙ, in the time evolution of the amplitudes associated to the different states. Starting from an equivalent definition of the NAC vectors [Eq. (109)]: we, therefore, obtain [Eq. (110)]:[228] 16 Note that with explicit dependencies, Equation (107) would read: ChemPhysChem 2013, 14, 1314 – 1340 1334 CHEMPHYSCHEM REVIEWS in which j F0i and j Fni are evaluated at subsequent time t the auxiliary many-electron wavefunctions, E andEt + dt by using F ~ 0 and F ~ n [Eqs. (97) and (98)], inqthe ffiffiffiffiffiffiffiffiTDA approximation ea ei [14, 225] obtained through the replacement Because wn ! 1. s0n j t + dt/2 is the only quantity needed in the evaluation of the surface hopping probability in the [t,t + dt] beE time E interval ~ 0 and F ~ n , it is numerically tween the electronic states F more efficient to use Equation (110) instead of the cumbersome evaluation of d0n followed by the product d0n·Ṙ. However, at the position of the “surface hops”, the calculation of the full NAC vectors is required for the redistribution of the potential energy difference to guarantee energy conservation. 3.5.3. NACs between Excited States In the excited-state nonadiabatic dynamics of molecular systems we need to compute NAC vectors between pairs of excited states. These are beyond the reach of LR theory (see Section 3.3.4) and, therefore, cannot be evaluated by using Eq. (96). A second-order response Efor the evaluation of D theory matrix elements of the form Fn O^Fm within TDDFT was first derived by Mukamel and co-workers[217, 219] by using an approximate mapping of the original electronic problem into a boson system sharing the same response properties. The use of the “auxiliary” electronic wavefunctions introduced in Section 3.3.4 offers a valid alternative to this approach and produces second-order matrix elements that include contributions from the de-excitation of the correlated ground state,[59] which are neglected in the derivation given in ref.[219] In fact, the two approaches coincide in the TDA up to terms second-order in Zm.[59] For the calculation of the ENAC vectors between a pair of ex E ~ m , in the LR-TDDFT-based TSH dy~ n and F cited states, F namics of Section 2.4.3 we, therefore, use the expression given in Equation (101) with O^ replaced by @gH^el . The quality of these matrix elements was assessed in ref. [59] 4. Coupling with the Environment In this section we will outline the implementation of a coupling scheme that allows the simulation of nonadiabatic processes coupled to external static and time-dependent fields. The first case is suited for the investigation of all solvent effects that do not involve charge transfer from the molecule of interest to the solvent (or vice versa) or (photo)chemical reactions between solute and solvent molecules. The coupling with timedependent fields offers instead the possibility to investigate the interaction of matter with any form of electromagnetic radiation from solar light to laser fields. www.chemphyschem.org molecule surrounded by solvent molecules or embedded in a matrix could be strongly altered by electrostatic effects, spatial confinement, or photochemical reactions with the solvent (e.g. see refs. [15, 229–237]). The most straightforward solution for describing explicit condensed-phase effects in simulations would be to include directly solvent molecules in the ab initio simulation (ab initio means, in this context, any theory that describes, even approximately, the quantum nature of the electrons). Even though this would have the benefit of fully describing the possible electronic coupling effects between solute and solvent, the computational cost of full (solute and solvent) TSH/LR-TDDFT calculations is nowadays still prohibitive and only few solvent molecules can be added (to the QM part) to get a first indication of the role of solvent-induced nonradiative pathways.[238] Instead of treating solvent molecules as part of the quantum setup, classical force fields can be used for their description within a MM approach. This is at the hearth of the QM/MM method.[239–242] The molecule of interest is still modeled at a QM level (QM part), whereas the solvent (MM part) is described at a classical level and no charge transfer between the two subsets is possible. In this way, complete solvation of the QM part can be achieved at very low computational cost. In the so-called additive scheme,[242] the point charges associated to each atom of the solvent molecules polarize the electronic density of the QM part, and if a polarizable force field is used, the QM part can eventually influence the MM region in return. When the system of interest is coupled to its environment in a QM/MM setup, the electrostatic interaction between the QM subsystem (treated at TDDFT or LR-TDDFT level) and the MM subsystem (classically described through an empirical Hamiltonian) is described by [Eq. (111)]:[168] in which d1i(r,t) is the electron density perturbation induced by the transition into the ith excited state and is evaluated according to Equation (104) together with Equation (105) or, equivalently, Equation (106). The potential usC(jRgrj) is a screened Coulomb potential generated by the atom at Rg and modified at short range to avoid spurious overpolarization effects.[241] In a QM/MM setup we, therefore, need, in addition to the calculation of the (LR-)TDDFT energies, forces, and couplings, an explicit evaluation of the (LR-)TDDFT perturbed densities. When coupled to QM/MM, the TSH/LR-TDDFT/MM dynamics allow the photophysics and photochemistry of molecules in their natural environment to be described (e.g. refs. [168, 238, 243, 244]). 4.1. Classical Environment and TDDFT/MM Coupling The nonadiabatic schemes presented so far can be applied to the study of the photochemistry and photophysics of molecules in the gas phase, a condition that is, for example, encountered in experiments performed under vacuum or in molecular jets. However, the nonradiative decay of a photoexcited 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 4.2. External TD Fields The coupling of nonadiabatic MD with an external TD electric field is given by the interaction Hamiltonian (with no spin– magnetic field contributions) [Eq. (112)]: ChemPhysChem 2013, 14, 1314 – 1340 1335 CHEMPHYSCHEM REVIEWS www.chemphyschem.org second-order quantities that transcend LR-TDDFT as we saw in Section 3.3.4. in which the classical vector potential A(ri,t) is related to the 1 @A actual electric field by E = c @t . The summation in Equation (112) is over all electrons, whereas the interaction with the nuclei is treated at the fully classical level and is, therefore, not included in the following derivation. Ĥint can be directly added to the Hamiltonian that governs the Ehrenfest dynamics [Eqs. (14) and (15)].[84, 245] In TDDFT-based Ehrenfest dynamics, the coupling is explicit and can be included by replacing the square of the momentum operator p̂2 = h2r2 with the term (p̂e/cA(r,t))2 in Equation (18).[71] In TSH nonadiabatic MD cases,[158, 209] one needs to evaluate the radiation field coupling matrix element [Eq. (113)]:17 in which A(t) = A(t)El and [Eq. (114)]: is the transition dipole vector, and wji = (EjEi)/h. Once more, the use of auxiliary many-electron wavefunctions within LRTDDFT simplifies the calculation of the transition dipoles, which can be computed by using Equation (114) after replacing j Fii by j F̃ii and hFj j by hF̃j j . As discussed in Section 3.3.4, this equation can also be used to evaluate transition dipole vectors between pairs of excited states, which is an approximate solution that becomes exact when the TDA is invoked. Finally, the differential equation for the TSH coefficients Equation (50) in the presence of an external radiation field interacting with the system[24, 158, 167, 243, 247–253] becomes [Eq. (115)]: in which the last term is responsible for the amplitude transfer among the different electronic states. The coupling of TSH dynamics with explicit time-dependent external fields within LR-TDDFT was the subject of a series of recent publications.[158, 167, 209, 243] Even though everybody agrees on the form of the coupling, in some cases no mention is made on the meaning and the derivation of the LR-TDDFT many-electron wavefunctions used in the calculation of the matrix elements; this is particularly deceptive in the case of the coupling between pairs of excited states, which are, strictly, 17 The derivation of these coupling matrix elements implies the Coulomb gauge ri·A(ri,t) = 0 and several approximations such as the “weak-field limit” [ne1 glect of the term 2c2 A(ri,t)·A(ri,t) in Eq. (112)] and the dipole approximation. See, for example, refs. [158, 246] for more details on those approximations. 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 4.2.1. Local Control and Pulse Design Of particular interest is the possibility to design laser pulses able to induce specific physical or chemical modifications to a molecular system of interest. To this end, several optimal control schemes have been developed in both the theoretical and experimental communities with the aim to control chemical reactions. In this section, we will briefly describe a pulseshaping algorithm, which is intimately coupled to the nonadiabatic dynamics schemes discussed in previous sections. In particular, TSH dynamics offers an interesting opportunity for the implementation of what is known as local control theory.[254–262] The aim of local control theory is to compute the electric field that maximizes (or minimizes) the TSH population associated to a selected electronic state “on-the-fly” by using Equation (115). Starting from the time evolution of an arbitrary operator B^ [Eq. (116)]: ^ which is only true in and assuming that Ĥmol commutes with B, the absence of NACs, we obtain [Eq. (117)]: (note that in the presence of NACs, Equation (117) will contain an additional time-dependent term). To control the population of a particular electronic state j Fii, we introduce the projector operator P^i = j FiihFi j . The time evolution of the state population is simply given by [Eq. (118)]: Using the TSH Ansatz for a classical trajectory a [Eq. (119)]: Equation (118) becomes [Eq. (120)]: It is now evident that choosing the electric field as [Eq. (121)]: ChemPhysChem 2013, 14, 1314 – 1340 1336 CHEMPHYSCHEM REVIEWS D E will ensure that P^i½a ðtÞ increases (or decreases) at all times. In conclusion, when a targeted chemical process can be associated to a well-defined excited-state transition, local control theory coupled to TSH dynamics offers a valuable tool for the optimization of the external TD electric field able to efficiently trigger the desired reaction.[209] 5. Conclusions In this review, we presented a thorough description of several trajectory-based nonadiabatic molecular dynamics schemes based on TDDFT for “on-the-fly” calculation of all required electronic structure quantities (e.g. energies, nuclear forces, NACs, and coupling with external fields). In particular, mean-field Ehrenfest MD and TSH MD emerge as best candidates for the description of nonadiabatic effects in DFT/TDDFT-based ab initio MD. Alternative solutions based on a more rigorous description of the nuclear quantum dynamics have also been explored, and the NABDY approach represents, in our opinion, a very promising alternative with the potential to include quantum effects (such as tunneling, entanglement, and decoherence) in an efficient and accurate way.[40, 58, 66, 263, 264] What makes these trajectory-based nonadiabatic schemes particularly efficient is the use of TDDFT for “on-the-fly” calculation of electronic properties. Being a density-based theory, TDDFT avoids dealing with the complex realm of wavefunction-based electronic structure methods. However, despite its apparent formal simplicity, the physical interpretation of TDDFT remains intrinsically difficult to grasp. Although the many-body solution of the many-electron problem could appear cumbersome at first glance, its physical content is transparent and does not require further investigation. In contrast, TDDFT is a density-based theory that relies on an “existence theorem” and that describes response quantities as contracted two-point functions [Eq. (56)] instead of the four-point functions obtained from the solution of the Bethe–Salpeter equation[265] when “physical” nonlocal potentials are present. Therefore, rigorous derivation of most electronic properties within LR-TDDFT can become more involved than when standard perturbation theory is used, as is traditionally done in quantum chemistry and MBPT (based on Green’s functions techniques. This is the reason why, in this review, we have devoted a large section to the derivation of all TDDFT-based electronic structure properties essential to nonadiabatic dynamics. In particular, we introduced the concept of “auxiliary manyelectron wavefunction” in LR-TDDFT to emphasize their physical origin and to avoid the misleading parallel with real (Hartree–Fock and post-Hartree–Fock) many-electron wavefunctions. Despite the success of LR-TDDFT in physics and chemistry, the quality of the results still strongly depends on the quality of the exchange and correlation functionals and of the LR kernel used in the calculations. Fortunately, recent developments in the design of better-performing functionals have im 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.chemphyschem.org proved the description of key physical and chemical processes such as charge transfer, many-electron excitations, and Rydberg-type excitations and of the topography of the PESs that are, in particular, close to conical intersections. In conclusion, we believe that, thanks to the most recent theoretical advances, TDDFT-based nonadiabatic dynamics MD will become the method of choice for the investigation of photochemical and photophysical processes of complex molecular systems in the gas and condensed phases. By combining the efficiency of “on-the-fly” trajectory-based techniques with the formal simplicity of TDDFT, these nonadiabatic MD approaches have an enormous potential that is far from being fully exploited. Acknowledgements The authors thank Giovanni Ciccotti, Sara Bonella, Thomas J. Penfold, Jrme F. Gonthier, and Felipe Franco de Carvalho for useful discussions. Pascal Miville is acknowledged for graphical representations. This work was supported by the COST Action CM0702, the NCCR-MUST interdisciplinary research program, and the Swiss National Science Foundation Grants Nos. 200020130082 and 200021-146396. Financial support for CADMOS and the Blue Gene/P system is provided by the Canton of Geneva, Canton of Vaud, Hans Wilsdorf Foundation, Louis-Jeantet Foundation, University of Geneva, University of Lausanne and cole Polytechnique Fdrale de Lausanne. 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