Semiclassical Bohmian Dynamics

Home , Crossing (physics), Quantum hydrodynamics, Quantum nonlocality, Semiclassical physics, Wave function

CHAPTER 6

Sophya Garashchuk,a Vitaly Rassolov,a and Oleg Prezhdob a Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina bDepartment of Chemistry, University of Rochester, Rochester, New York

INTRODUCTION

Quantum mechanics lies at the heart of chemistry. It is impossible to under- stand the structure of the Periodic Table, chemical bonding patterns, free energies of chemical reactions, reaction rates and branching ratios, and other chemical phenomena without a quantum-mechanical (QM) description. In particular, the dynamics of molecular systems often involve QM effects such as zero-point energy, tunneling, and nonadiabatic transitions. QM effects are essential for accurate description and understanding of reactions in complex chemical envi- ronments. For example, the zero-point energy stored in the vibrational modes of chemical reactants, products, and transition state species modiﬁes reaction energy barriers. Reaction rates and branching ratios can be affected greatly by such changes. QM tunneling can be critical in proton transfer reactions. However, the conventional methods of solving the time-dependent Schrodinger¨ equation1 scale exponentially with the system size. In addition, the relevant dynamics occur on a long time-scale.2 Therefore, it is extremely difﬁcult to model QM tunneling in condensed phase chemical systems.3–6 Nonadiabatic dynamics involving transitions between different electronic or vibrational energy levels

287 288 Semiclassical Bohmian Dynamics is ubiquitous in photochemistry.7–11 In this process, a chemical system is excited by light, undergoes a nonequilibrium evolution, and ultimately relaxes to the ground state. QM tunneling can be viewed as a particular kind of nonadiabatic transition.12 Several multidimensional quantum approaches have been proposed including those using basis set contractions,13–15 coherent state representations, 16,17 and mixed quantum-classical strategies.18–24 A trajectory representation of large molecular systems carries a special appeal because of several favor- able factors: (1) the initial conditions of a trajectory simulation can be sampled with Monte Carlo techniques. This allows the exponential scaling of the exact wave function or density matrix with system size to be circumvented. (2) Var- ious degrees of freedom involving light and heavy particles can be treated on equal footing, and quantum-classical separation issues can be avoided.25 (3) Wave functions are highly oscillatory close to the classical → 0 limit. As a result, a trajectory description of heavy particles, such as nuclei, is often more appropriate than a grid or a basis set representation. (4) Classical equations of motion are simple to solve. Numerous molecular dynamics techniques26 for propagating classical trajectories are applied routinely to chemical systems composing hundreds of thousands of atoms. Incorporating the dominant QM effects caused by wave function localization27,28 constitutes a challenge for the trajectory methods. Such effects can be considered naturally by representing wave functions in terms of ensembles of trajectories. In comparison, semiclassical methods commonly use independent trajectories.29–31 A great interest in the Bohmian interpretation of quantum dynamics has been witnessed during the last decade. In particular, its potential to generate computational tools for solving the time-dependent Schrodinger¨ equation has attracted considerable attention. The Bohmian formulation of quantum dynamics promises a better than exponential scaling of the computational effort with system dimensionality. It also offers a convenient approach for mixed quantum- classical descriptions of large chemical systems. The Madelung-de Broglie– Bohm formulation of the time-dependent Schrodinger¨ equation has a long his- tory dating back to the birth of quantum mechanics.32,33 It gained a wider recognition after David Bohm34 used it to develop an alternative interpretation of quantum mechanics. This review describes the semiclassical methodologies inspired by the Bohmian formulation of quantum mechanics. These methods are designed to represent the complex dynamics of chemical systems. The review is constructed as follows: the next section introduces the Madelung-de Broglie–Bohm formalism. This is done by drawing an analogy with classical mechanics and explicitly highlighting the non-classical features of the Bohmian mechanics. The nonclassical contributions to the momentum, energy, and force are introduced. The fundamental properties of Bohmian quantum mechanics—the conservation and normalization of the QM probability, the computation of the QM expectation values, properties of stationary states, and behavior at nodes—are discussed. Several ways to obtain the classical limit The Formalism and Its Features 289 within the Bohmian formalism are considered. Mixed quantum-classical dynamics based on the Bohmian formalism is derived and illustrated with an example involving a light and a heavy particle. At this point, the Bohmian representation is used as a tool to couple the quantum and classical subsystems. The quantum subsystem can be evolved by either Bohmian or traditional techniques. The Ehrenfest approach is the most straightforward and common quantum- classical approach, and it is the starting point for other quantum-classical for- mulations. The Bohmian formulation of the Ehrenfest approach is used to derive an alternative quantum-classical coupling scheme. This resolves the so-called quantum backreaction problem, also known as the trajectory branching problem. Next, the partial hydrodynamic moment approach to coupling classical and quantum systems is outlined. The hydrodynamic moments provide a connection between the Bohmian and phase-space descriptions of quantum mechanics. The “Independent Trajectory Methods” Section describes approaches based on independent Bohmian trajectories. It includes a discussion of the derivative propagation method, the Bohmian trajectory stability approach, and Bohmian trajectories with complex action. Truncation of these hierarchies at the second order reveals a connection to other semiclassical methods. The focus then shifts toward Bohmian dynamics with globally approximated quantum potentials. Separate subsections are devoted to the global energy-conserving approximation for the nonclassical momentum, approximations on subspaces and nonadiabatic dynamics. Each approach is introduced at the formal theoretical level and then is illustrated by an example. The Section “Towards Reac- tive Dynamics in Condensed Phase” deals with computational issues including numerical stability, error cancellation, dynamics linearization, and long-time behavior. The numerical problems are motivated and illustrated by consid- ering speciﬁc quantum phenomena such as zero-point energy and tunneling. The review concludes with a summary of the semiclassical and quantum- classical approaches inspired by the Bohmian formulation of quantum mechanics. The three appendices prove the quantum density conservation, introduce quantum trajectories in arbitrary coordinates, and explain optimization of simulation parameters in many dimensions.

THE FORMALISM AND ITS FEATURES

The Trajectory Formulation For simplicity, let us start with a derivation of the Bohmian equations in one spatial dimension x for a particle of mass m. For notation clarity, let us use ∇ to denote differentiation with respect to x. Arguments of functions will be omitted where unambiguous. Differentiation (or Derivatives) with respect to a variable other than x will be indicated as a subscript; for example, ∇c denotes differentiation with respect to c. The multidimensional generalization to an 290 Semiclassical Bohmian Dynamics arbitrary coordinate system is given in Appendix B. The conventional form of the time-dependent Schrodinger¨ equation is 2 ∂ (x, t) − ∇2 + V (x, t) = ı [1] 2m ∂t

After Madelung, the complex time-dependent wave function is represented in polar form as = ı (x, t) A(x, t) exp S(x, t) [2] where A(x, t) and S(x, t) are real functions. Substitution of Eq. [2] into Eq. [1], division by (x, t), separation into real and imaginary parts, and a few simple manipulations results in a system of two equations

∂S(x, t) 1 =− (∇S(x, t))2 − V − Q [3] ∂t 2m ∂(x, t) 1 =−∇ ∇S(x, t)(x, t) [4] ∂t m

In Eq. [3], the term Q denotes what Bohm called the “quantum mechanical potential,”

2 ∇2A(x, t) Q =− [5] 2m A(x, t)

The quantum potential Q enters the equation on par with the external “classical” potential V = V(x, t), which is generally a function of x and t as well as Q = Q(x, t). In Eq. [4] (x, t) is the wave function density

(x, t) = A2(x, t) [6]

With identiﬁcation of the probability ﬂux as 1 ∗ j(x, t) = (x, t) ∇S(x, t) = (x, t)∇ (x, t) [7] m m

Equation [4] becomes the usual continuity equation. Analogy with fluid mechanics suggests the name “hydrodynamic” formulation of the Schrodinger¨ equation. Note that Eqs. [3] and [4] are formally equivalent to the original Schrodinger¨ Eq. [1] except that the polar form Eq. [2] is problematic at the nodes of the wave function. At the nodes, the phase S is undefined, A(x, t) = 0, The Formalism and Its Features 291 and Q is generally singular. The singularity in Q cancels for excited eigenstates as will be explained. Equations [3] and [4] describe the flow of the probability fluid through the stationary points x. Transition to the trajectory framework is made by identifying

p(x, t) =∇S(x, t) [8] and switching to the Lagrangian frame of reference

d ∂ p = + ∇ [9] dt ∂t m

Subsequently, the subscript t will be used to deﬁne trajectory-dependent quantities, such as xt and pt, for the trajectory position and momentum at time t. The subscript 0 will denote the initial values of these quantities at time t = 0. The action function and density along the trajectory (xt,pt) will be denoted as S(xt) and (xt). Variables without subscripts will refer to functions of coordinate x and time t. For example, S(x, t) is the phase of a wave function at time t, whereas S(xt) is the action function computed along the trajectory described by the position xt and momentum pt. Differentiation of Eq. [3] with respect to x gives Newton’s equations of motion for a trajectory characterized by momentum pt and position xt

dx p t = t [10] dt m dpt =−∇(V + Q) = [11] dt x xt

In the Lagrangian frame of reference, Eq. [3] becomes the quantum Hamilton- Jacobi equation

2 dS(xt) pt = − (V + Q) x=x [12] dt 2m t

As easily seen, Eqs. [10]–[12] are the standard equations of classical mechanics. These equations fully define the evolution of the wave function once the initial momenta of the quantum trajectories are defined according to Eq. [8]. The quantities xt and pt fully define the quantum trajectory. Quantum effects are incorporated into its behavior through the nonlocal quantum force. The force depends on the wave function amplitude and its derivatives up to the 292 Semiclassical Bohmian Dynamics third order. Often it is useful to consider an additional function attributable to the quantum trajectory, namely, the nonclassical momentum component r(x, t),

∇A(x, t) r(x, t) = [13] A(x, t)

Formally, it is complementary to the classical component p(x, t) because both result from the action of the QM momentum operator on the wave function given in the polar form [2] ∇A(x, t) pˆ = −ı +∇S(x, t) = (−ır(x, t) + p(x, t)) [14] A(x, t)

The quantum potential expressed in terms of r is

2 Q =− r2(x, t) +∇r(x, t) [15] 2m

The average value of Q can be termed the “quantum energy,” which using differentiation by parts in Eq. [15], is equal to

2r2 Q= [16] 2m

In particular, Q is one half of the zero-point energy for the ground state of the harmonic oscillator. The time-dependence of r can be derived from Eq. [4]. Combined with Eqs. [11] and [5], it gives the following evolution equations, which emphasize the common structure of the differential operator on the right-hand-side (RHS): ∇ dpt 2 m∇V x=x + m = rt + ∇rt [17] t dt 2 drt ∇ −m = rt + ∇pt [18] dt 2

Features of the Bohmian Formulation Conservation of Probability and Normalization Transformation of Eq. [4] into the Lagrangian frame of reference gives the evolution of the wave function density d ∂ p (x ) = + t ∇ (x ) =−∇p × (x ) [19] dt t ∂t m t t t The Formalism and Its Features 293

It follows from Eq. [19] that, in closed systems, the probability of ﬁnding a particle in the volume element dxt associated with each quantum trajectory, the trajectory “weight,” remains constant in time,35

dw(x ) w(x ) = (x )dx , t = 0 [20] t t t dt

This is consistent with the standard “classical” continuity equation. The corresponding multidimensional derivation is given in Appendix A. Therefore, all QM effects in the evolution of quantum trajectories result from the quantum force, Fq =−∇Q, acting on the trajectories in addition to the classical force, Fcl =−∇V. The quantum force is responsible for wave-packet delocalization, tunneling, over-the-barrier reflection, resonances, interference, and zero-point energy in bound systems. Conservation of the trajectory “weight” implies that the Bohmian trajectories define the most efficient grid representation for the wave function with time-dependent grid points. The wave function density will remain negligible at these time-dependent grid points provided that it was negligible at time t = 0. Equation [20] also helps to interpret the wave function using the time-dependence of the trajectory positions; because (x, t)is single-valued, the quantum trajectories cannot cross. Wide separation of initially equidistant trajectories indicates regions of low wave function density. Conversely, closely spaced trajectories correspond to high wave function density. A Gaussian wave packet evolving in a harmonic potential provides a simple illustration of quantum trajectory dynamics. The time-dependence of (x, t) is analytic,36 and the trajectories can be constructed easily. The center of the wave packet moves purely classically, whereas the time dependence of the overall wave packet is influenced by the quantum force, as illustrated in Figure 1. A comparison of the trajectories propagated with and without the quantum potential Q demonstrates the noncrossing rule—classical trajectories cross at focal points, whereas the quantum trajectories do not cross. The noncrossing rule is a manifestation of the single-valued wave function and of the Heisenberg uncertainty principle. An initially wide Gaussian wave packet with the initial width equal to 25% of the coherent value, Figure 1b, produces a small quantum potential Q. In this case, the dynamics of the quantum and classical trajectories are very similar except at the focal points. In contrast, an initially narrow wave packet with the initial width equal to 200% of the coherent value, Figure 1a, produces a large quantum potential Q. As a result, the quantum and classical dynamics differ at all times. The quantum force acts to make the wave packet “flat.” This is the infinite time limit for a Gaussian wave packet evolving in a constant classical potential. For more complicated classical potentials, with ∇V =/ 0, there will be an 294 Semiclassical Bohmian Dynamics

10 (a)

0 Position

-10

0246

10 (b)

0 Position

-10

0246 Time Figure 1 Quantum (dash) and classical (solid line) trajectories in the harmonic potential describing initially (a) narrow and (b) wide Gaussian wave-packets.

intricate interplay between the classical and quantum forces. The interplay gives rise to all QM effects.

The Quantum Trajectory Ensemble, Expectation Values, and Energy To solve the time-dependent Schrodinger¨ equation using trajectories, or sim- ply to visualize the wave function dynamics, an ensemble of trajectories is initialized at t = 0. For each initial position x = x0, the trajectory weight, 2 w = A (x, 0)dx0, and the classical momentum, p0 =∇S(x, 0), are determined from the initial wave function. The trajectories are propagated in time according to Eq. [10] in combination with either Eq. [11] or Eq. [8]. Equation [11] requires evaluation of A(x, t) and its derivatives through the third order, whereas Eq. [8] needs ∇S(x, t). Once the wave function is represented in terms of trajectories, the expectation values of x-dependent operators can be computed readily. This is done The Formalism and Its Features 295 by integrating over the time-dependent trajectory positions. In a discretized (i) trajectory representation, xt , the integration is replaced with the summation = = (i) oˆ (x) t o(x)(x, t)dx o(xt )wi [21] i where the index i enumerates the trajectories. It can be seen easily that the normalization of the wave function is conserved; in the Eq. [21], this corresponds to the unit operator oˆ = 1. Operators dependent on p can be evaluated in the same fashion. For example, the kinetic energy is 1 (i) 2 Tˆ t = (p ) w [22] 2m t i i

Note that the total energy of the wave function, or equivalently of the quantum trajectory ensemble, is conserved. However, the energies of individual quantum trajectories generally do not remain constant. The quantum potential is responsible for the energy redistribution within the quantum trajectory ensemble.

Stationary States and Behavior at the Nodes A stationary state is a special solution of the time-dependent Schrodinger¨ equation. It can be written as a product of the spatial and temporal factors. Substi- tuting the polar form of the wave function, = − ı (x, t) exp S(t) A(x) [23] into the time-dependent Schrodinger¨ equation, the division of Eq. [23] by , and the separation of variables gives the usual time-independent Schrodinger¨ equation

dS(t) = E [24] dt

2 ∇2A(x) − + V = Q + V = E [25] 2m A(x) where E is the energy eigenvalue. The solution for the phase is S(t) = S(0) + Et. In the quantum trajectory language, an eigenfunction is a wave function with a particular condition on its amplitude. The amplitude of an eigenfunction gives the quantum potential Q which differs from the negative of classical potential −V only by a constant. Therefore, the quantum force corresponding to an eigenfunction exactly cancels the classical force. Given zero initial momenta, (i.e., ∇S = 0) the trajectory positions do not change with time. 296 Semiclassical Bohmian Dynamics

This picture of stationary trajectories also applies to excited eigenstates implying that at the nodes where A(x) = 0 the singularity in the quantum potential, given by Eq. [5], always cancels. For example, for the eigenstates of the harmonic oscillator, V = mω2x2/2, the ground-state wave function, mω (x) = exp − x2 [26] 0 2 substituted into Eq. [25] gives E0 = ω/2. The ﬁrst excited state wave function 1(x) = x0(x) gives E1 = 3ω/2 and so on. Note that the wave function normalization factors have been omitted for clarity. In contrast, for nonstationary wave functions with nodes, the singularities in Q do not cancel. In general, this makes the direct numerical solution of Eqs. [4] and [12] impractical. Nevertheless, the behavior of the quantum trajectories is very intuitive; the trajectories “ﬂow” coherently, avoid the nodes, and never cross. The trajectories corresponding to the time-evolution of a linear combination of 0 and 1 are shown on Figure 2 around the density node at t = 2.5. Avoiding the nodal region, Figure 2a, is accomplished by rapid changes in the trajectory momenta, Figure 2b. As a consequence, numerical implementation of such unstable dynamics is very expensive. Many examples of the quantum trajectory dynamics can be found in the book by Holland.37

(a) (b)

5 1

0 0 Position Momentum

-1

-5

12341234 Time Time Figure 2 Bohmian dynamics in the presence of the density nodes: (a) position of the trajectories as a function of time; (b) the corresponding momenta for selected trajectories. Position and momentum of a trajectory are shown with the same line styles on both panels. The Formalism and Its Features 297

The Classical Limit of the Schrodinger¨ Equation and the Semiclassical Regime of Bohmian Trajectories The Bohmian form of the time-dependent Schrodinger¨ equation gives a straightforward route to classical mechanics. In the heavy particle limit, m → ∞, or equivalently, when a typical action becomes large compared with Planck’s constant, → 0, the quantum potential Q vanishes. This is consistent with the representation of a particle in terms of a localized wave function (i.e., a wave- packet). In the classical limit, the center of the wave packet moves along a classical trajectory, and the changes in the wave-packet width can be neglected. An alternative connection between Bohmian, classical, and semiclassical dynamics is through the Wentzel–Kramers–Brillouin (WKB)38 treatments that are based on the expansions of the exponentiated wave function, as in Eq. [60], which is described later. The traditional semiclassical condition is based on the WKB approximation to solutions of the time-independent Schrodinger¨ equation. This condition states that the action function must be much larger than Plank’s constant. The classical momentum p entering the action function is deﬁned as pWKB = 2m(E − V), pdx [27]

Identiﬁcation of the classical and nonclassical momenta, Eqs. [8] and [13] respectively, suggests a similar criterion that is applicable to time-dependent wave functions: |r| |p| [28] or in terms of the energy given in Eq. [15],

p2 |Q| [29] 2m

The momentum condition of Eq. [28] is more convenient than the energy condition Eq. [29] the former is expressed in terms of simple quantities, which are linear in the semiclassical picture of a moving particle (i.e., for Gaussian wave-packets). In the context of trajectory dynamics, the momentum semiclassical condition Eq. [28] is more general than the WKB expression Eq. [27]. This is because it is not based on a particular approximate solution to the Schrodinger¨ equation. Moreover, the condition of Eq. [28] is more convenient because it is expressed in terms of r and p, which are natural attributes of quantum trajectories. For semiclassical systems with small Q, the Bohmian momentum and the WKB momentum are close to each other. Therefore, the momentum semiclassical condition and the WKB expression are related closely. 298 Semiclassical Bohmian Dynamics

According to Eq. [28], the semiclassical approximation breaks down near wave function nodes where A(x, t) = 0. This so-called “node problem” leads to singular forces acting on quantum trajectories and causing numerical insta- bilities. A similar breakdown of the WKB approximation occurs near classical turning points. In the context of purely classical trajectories, this problem was dealt with by developing uniform semiclassical methods.30,39 A general-purpose semiclassical method based on quantum trajectories also must satisfy Eq. [28] in a uniform sense (i.e., for all points in the coordinate space). This requirement motivates the development of approximate quantum potentials outlined in the section, “Global Energy Conserving Approximation of the Nonclassical Momentom”. Approximate quantum potentials are deﬁned through the linearization of the nonclassical momentum via averaging over the wave function density. The relevant semiclassical condition becomes

|r| |p| [30]

The singularities in r have negligible contributions to the dynamics because of vanishing wave function density. A general semiclassical method must satisfy the semiclassical condition at all times during time evolution; therefore, it cannot be defined for any specific form of the initial wave function. In practice, this means that approximations must be made only for quantities that are negligible in the m →∞or → 0 limits. In particular, removing singularities in the quantum potential by con- straining density, phase, or momentum generally would violate the semiclassical condition. The approximate quantum potential method of “Global Energy- Conserving Approximation of the Nonclassical Momentum” constrains the functional form of the nonclassical momentum r(x, t), which enters Eq. [28] with the prefactor. At the same time, the density itself remains unconstrained. The traditional semiclassical methods, such as WKB and the Van Vleck– Gutzwiller propagator,38,40,41 as well as the independent Bohmian trajectory methods, are defined through the expansion of the solution to the Schrodinger¨ equation. Independent Bohmian trajectory methods, such as the derivative propagation method, the Bohmian trajectory stability method, and Bohmian mechanics with complex action, are discussed in the section on “The Inde- pendent Trajectory Methods.” The -expansion converges to the exact result. Methods that are not based on analytic solutions, such as the approximate quantum potential method, can be considered semiclassical if they can be improved systematically in the limit of large mass for an arbitrary physically reasonable initial wave function and kinetic energy density. The kinetic energy density is defined as −2∇2 /(2m ). “Improved systematically” implies that there is a general, unambiguous numerical prescription for convergence toward the exact solution. The approximate quantum potential approach can be improved systematically if the linearization of the nonclassical momentum is accomplished over subspaces,42 or if r(x, t) is represented in terms of a complete basis. The Formalism and Its Features 299

Using Quantum Trajectories in Dynamics of Chemical Systems Conceptually, the quantum trajectory formalism has been extended to nonadiabatic dynamics,43–45 the phase-space representation, and the density matrix approaches.46–53 There exist developments on imaginary time propagation,54 complex space Bohmian trajectories,55 and dynamics that are based on the bipolar rather than polar decomposition of wave functions.56,57 On the practical side, quantum trajectories have been used in theoretical and computational chemistry for three distinct purposes: (1) to interpret the wave function computed by the conventional wave function propagation techniques; (2) to monitor the wave function density flow for dynamical grid adjustments; and (3) to solve the time-dependent Schrodinger¨ equation, and to obtain (x, t) or quantities of interest, directly from the trajectories. Quantum trajectories have been used to interpret and to draw quantum- classical analogies in the area of surface scattering. Phenomena including Fresnel and Fraunhofer regimes, rainbow scattering, the quantum Talbot effect, and others have been explored.58–61 Some ideas from the Bohmian dynamics lead to the moving grid techniques62 in which positions of grid points are time-dependent but are different from the Bohmian trajectories. The numerical goals of moving grid techniques are to gain stability of the dynamics of grid points and to improve accuracy in derivative evaluations. Instead of moving the grid points, grids can be optimized by adding or eliminating the grid points by reconstructing Bohmian trajectories at grid edges.63 A similar criterion is used in the ab initio wave-packet dynamics of Iyengar to optimize the wave-packet representation and to minimize ab initio evaluations of classical forces.64–66 Using quantum trajectories as a practical way of solving the multidimensional time-dependent Schrodinger¨ equation is an exciting prospect. Several high-dimensional applications of the exact quantum trajectory method, including up to 200 degrees of freedom, have been reported.67,68 There, the quantum force is evaluated on the fly with the moving least-squares fitting of the wave function amplitude. However, a general implementation of the exact numerical Bohmian trajectory technique is difficult, even for low-dimensional systems. Complications develop as a result of the singularities in the quantum potential. 62,69–71 These problems motivated the “independent trajectory” implementations based on the Taylor expansion of the equations of motion truncated at a low order. Both real-valued54,72 and complex-valued trajectories55,73 have been used. The approximate quantum potential (AQP) approach35,74,75 involves propagation of the trajectory ensemble together with a global evaluation of the quantum force from the moments of the trajectory distribution. This “mean-field” type of approximation gives quantum force for all trajectories simultaneously. 300 Semiclassical Bohmian Dynamics

In the remainder of this review, the focus will be on the semiclassical and approximate implementations of Bohmian mechanics that have the greatest potential for high-dimensional chemical applications.

BOHMIAN QUANTUM-CLASSICAL DYNAMICS

Dynamics of most chemical reactions are typically very complex for a fully quantum-mechanical analysis. Fortunately often, it is possible to distin- guish between particles, such as electrons and protons, that require a quantum description and particles, such as heavy nuclei, which can be described accurately using classical mechanics. When quantum particles remain in the same quantum state throughout the reaction, the Born–Oppenheimer (adiabatic) approximation is invoked. The quantum state merely provides an external potential for the classical dynamics leading to adiabatic molecular dynamics.8,76–79 The Born-Oppenheimer approximation is valid, for example, for thermally acti- vated nuclear rearrangements proceeding in the ground electronic state. Many other types of chemical reactions involve several quantum states. Examples include photochemical reactions,80,81 transfers of electrons,11,82 protons,83,84 spins,85 energy86–88 and quantum phase,89 electron-vibrational relaxation processes,9,10,90,91 and solvation dynamics.92,93 These phenomena extend beyond the Born–Oppenheimer approximation and are modeled by the nonadiabatic generalizations of molecular dynamics.94–104 Coupling between quantum and classical degrees of freedom constitutes the key question in mixed quantum-classical approaches. It raises central issues that do not admit unique solutions. Numerous coupling schemes have been proposed ranging from formal mathematical solutions20,105–110 to specific al- gorithms that have been applied to many problems in chemistry, physics, and biology.9–11,80–84,87–92 Historically, the first and the most straightforward of the quantum-classical approaches is based on the Ehrenfest theorem. The theorem states that the equations of motions for the average values of the quantum position and momentum operators coincide with the classical equations of motion.99,111 This leads to the mean-field approximation in which the classical variables are coupled to the expectation values of the quantum observables. If the quantum system remains in a single state, then the Ehrenfest approach reduces to adiabatic molecular dynamics. In general, the quantum system forms a superposition of several states, and the classical dynamics evolve in the mean- field quantum potential. The average Ehrenfest trajectory is inappropriate when several reaction channels exist and involve substantially different potential energy surfaces.94,96 In a corresponding quantum description, the wave packets split and follow different reaction channels. The branching difficulty in the coupling of quantum and classical mechanics is known as the quantum backreaction problem. Most often, it is resolved by surface hopping94–97,100,101 in Bohmian Quantum-Classical Dynamics 301 which classical trajectories are designed to branch according to a specific algorithm. Other, more computationally demanding, quantum-classical approximations dealing with the trajectory branching include the multiconfiguration mean-field theory,20,98,103 partial Wigner transform dynamics,20,108 and semiclassical treatments.104 The Bohmian interpretation of quantum mechanics34 provides an alternative means of achieving the branching of the classical trajectories.24,25 By correlating each classical trajectory with an individual particle, an ensemble of trajectories can be generated. Trajectories associated with different quantum states are represented by different Bohmian particles. Tra- jectories evolve independently and branch as in the fully quantum-mechanical description. Subsequently, we describe the Ehrenfest and Bohmian quantum- classical approaches. Their properties are illustrated by a model representing scattering of a light particle off a surface containing slow phonon modes. The photoinduced electron transfer from a molecule to a semiconductor surface in dye-sensitized semiconductor solar cells11,82 is an example of such a process.

Mean-Field Ehrenfest Quantum-Classical Dynamics Consider a mixed quantum (x) classical (X) system. The quantum Hamil- tonian H(x; X) depends parametrically on the positions of classical particles

2 H(x; X) =− ∇2 + V(x; X) [31] 2m x

The classical subsystem generates an external ﬁeld contributing to the potential V(x; X) that governs the motion of the quantum subsystem. The total quantum- classical energy is the sum of the quantum-mechanical expectation value of the Hamiltonian [31] with the purely classical kinetic and potential W(X) energies 2 − ∗ MX˙ Eq cl = Eq + Ecl = d3x (x)H(x; X)(x) + + W(X) [32] 2

The wave function (x) evolves according to the time-dependent Schrodinger¨ equation ∂(x) 2 i = − ∇ 2 + V(x; X(t)) (x) [33] ∂t 2m x in which the potential V depends on time through the dynamics of classical variables X(t). The evolution of the classical coordinates obeys the Newton equation

q MX¨ =−∇XW(X) + F [34] 302 Semiclassical Bohmian Dynamics which contains the quantum force Fq in addition to the ordinary classical force −∇XW(X). The deﬁnition of the quantum force constitutes the quantum backreaction problem.20,105–110 The quantum force of the Ehrenfest approach is given by the quantum mechanical expectation value of the gradient of the quantum Hamiltonian q 3 ∗ F =− d x (x) [∇XH(x; X)] (x) [35]

The Ehrenfest force conserves the total quantum-classical energy in Eq. [32], as established by the time-dependent Hellmann–Feynman theorem.112 The qualitative features of the quantum-classical Ehrenfest approximation are illustrated in Figure 3. For a given wave function of the quantum subsystem, the Ehrenfest force deﬁnes a unique classical trajectory, Figure 3b. This feature of the Ehrenfest method is both its major advantage and its dis- advantage. Consider the case in which the quantum-mechanical wave packet, corresponding to the classical subsystem in the Ehrenfest approach, remains localized throughout the time of an experiment. Here, the Ehrenfest force of Eq. [35] generates an optimal, classical description of the wave packet. On the other hand, if the wave packet branches, as illustrated in Figure 3a, then the Ehrenfest approach fails to capture the branching. It asymptotically cannot describe the distinct reaction channels of the classical subsystem associated with different quantum states.

Quantum-Classical Coupling via Bohmian Particles The branching of the classical subsystem is reproduced by the Bohmian quantum-classical approach. This is done by generating an ensemble of classical trajectories correlated with different members of the Bohmian ensemble of quantum particles. Consider the Bohmian formulation of the Ehrenfest force of Eq. [35]. The polar form of the wave function (x) = R(x) exp(ıS(x)/) leads to the following expression for the quantum mechanical expectation value of the Hamiltonian Eq. [31]: 2 ∗ d3x (x) − ∇2 + V(x; X) (x) = d3xR2(x) 2m x (∇ S(x))2 × x + Q(x) + V(x; X) [36] 2m where Q(x) is the quantum potential

2 ∇ 2R(x) Q(x) =− x [37] 2m R(x) Bohmian Quantum-Classical Dynamics 303

Figure 3 Schematic representation of the evolution of a heavy particle moving in the potential created by light particles. For example, the X coordinate can represent the bond length of a diatomic molecule. The black solid lines depict the potential energy surfaces for the molecular ground and excited electronic states. (a) A quantum-mechanical wave-packet describing the heavy particle and shown by the grey dashed line is pro- moted from the ground electronic state to the excited state, as indicated by the grey solid line directly above the dashed line. While in the excited state, the wave-packet moves to infinite X, and the diatomic dissociates. The nonadiabatic coupling between the electronic states causes transfer of a fraction of the wave-packet back to the ground state potential energy surface. This part of the wave-packet returns to the initial state, representing a bound diatomic. Thus, a quantum-mechanical particle branches into several components, corresponding to different outcomes of the excitation dynamics. (b) In the Ehrenfest approximation, the classical particle cannot split and evolves on a single trajectory. The mean-field potential energy surface (grey line) is an average of the ground and excited state potentials. The particle evolution corresponds to neither of the quantum-mechanical outcomes; the diatomic can be artificially trapped between the bound and dissociated states. (c) In the Bohmian version of quantum-classical dynamics, the quantum subsystem, e.g. the electrons in the diatomic, is represented by an ensemble of classical-like particles. Each Bohmian particle is coupled independently to the heavy particle. This coupling generates an ensemble of classical trajectories evolving on different potential energy surfaces. This treatment mimics the quantum-mechanical branching of the wave-packet describing the heavy particle.

The quantum probability distribution is R2(x) = ∗(x)(x). As a result, in Bohmian mechanics, the quantum energy Eq. [36] is interpreted as the energy of an ensemble of particles with the probability distribution R2(x). The energy of 2 each particle is equal to [(∇xS(x)) /2m + Q(x) + V(x; X)]. The Ehrenfest force, 304 Semiclassical Bohmian Dynamics generated by the quantum subsystem and acting on the classical subsystem, takes the following form in the Bohmian representation: q 3 2 F =− d xR (x)∇XV(x; X) [38]

It can be viewed as the average of the forces −∇XV(x; X) resulting from the ensemble of Bohmian particles with the probability distribution R2(x). The ensemble averaged Bohmian force Eq. [38] is identical to the Ehrenfest force of Eq. [35], as indicated by the time-dependent Hellmann-Feynman theorem.112 The theorem states that the time derivative of the expectation value of the quantum energy involves only the derivative of the quantum Hamiltonian, provided that the wave function evolves according to the time-dependent Schrodinger¨ Eq. [33]. The Bohmian quantum-classical approach solves the branching problem by moving the d3xR2(x) ensemble averaging outside the quantum-classical dynamics. The initial conditions for the Bohmian ensemble of quantum particles are sampled from R2(x). The averaging is performed only at the ﬁnal time. In particular, the quantum force acting on the classical subsystem at any given instance does not involve the integration over R2(x) as in Eq. [38]. The quantum force is calculated for a single representative of the Bohmian ensemble. This treatment of the quantum-classical coupling generates a distribution of classical trajectories correlated with different Bohmian particles, Figure 3c. As a result, the classical trajectories evolve differently depending on whether the correlated Bohmian particles correspond to the excited state wave function, Figure 3c, or to the ground state. The Bohmian quantum-classical simulation runs according to the following algorithm. First, initial conditions for the wave function and classical trajectories are chosen in the usual manner. Positions x of Bohmian particles are sampled from the initial distribution R2(x). For each initial coordinate X of the classical particle, an ensemble of initial coordinates x for the Bohmian particles is sampled from R2(x). Each Bohmian particle is correlated with a separate copy of the classical subsystem. Second, for each member of the quantum-classical ensemble, the wave function is propagated by the time-dependent Schrodinger¨ Eq. [33]. Simultaneously, the classical trajectory is evolved by the Newton Eq. [34] with the quantum force

q F =−∇XV(x; X) [39]

Note that the quantum force depends on the position of the Bohmian particle x. The trajectory of the Bohmian particle is propagated either using the Newton equation, including the quantum potential Eq. [37]

mx¨ =−∇x[Q(x) + V(x; X)] [40] Bohmian Quantum-Classical Dynamics 305 or, equivalently, directly by

x˙ =∇xS/m [41]

The ﬁrst option can be used with the semiclassical schemes for propagating the Bohmian ensemble described in the sections “The Independent Trajectory Methods” and “Dynamics with the Globally Approximated Quantom Poten- tial” of this review. The second option is advantageous if the time-dependent wave function is available by a direct quantum-mechanical propagation. Third, the results are averaged over the ensemble of the Bohmian quantum-classical trajectories. The Bohmian quantum-classical method is deﬁned by Eqs. [33], [34], [39], and [40]. The fully classical limit for both subsystems is achieved easily by setting → 0 in Eq. [37] and eliminating the quantum potential from Eq. [40]. An alternative derivation of this approach is given in Refs. 24 and 113. The derivation starts with a fully quantum description of both subsystems. The quantum potential then is dropped from the equation of motion for the classical subsystem.

Numerical Illustration of the Bohmian Quantum-Classical Dynamics The Bohmian quantum-classical approach is illustrated here with a model intended originally as a simpliﬁed representation of gaseous oxygen interacting with a platinum surface.114,115 Alternatively, it can be viewed as a model for the photoinduced electron transfer in a molecular chromophore adsorbed on a surface of a solid-state bulk material. Systems of this type form the basis for dye-sensitized semiconductor solar cells, also known as Gratzel¨ cells,11,82 and various molecular electronics devices.116 The model consists of a light particle x with mass m colliding with a heavier particle X with mass M. The heavy particle is bound to an immobile surface, Figure 4. In the molecule-bulk electron transfer process, the light particle can be viewed as the electron coming from the molecule and scattering off the bulk surface containing a phonon mode. The total Hamiltonian for the system is given by

H(x; X) = T1(x) + V1(x) + T2(X) + V2(X) + V(x, X), with [42a] −2b(x−c) −b(x−c) V1(x) = a[e − 2e ] [42b] 1 V (X) = M2X2 [42c] 2 2 − − V(x, X) = Ae B(x X) [42d] where T1 and T2 are the kinetic energy operators. The harmonic potential V2 describes the interaction of the heavy particle with the surface. The Morse 306 Semiclassical Bohmian Dynamics

Figure 4 A model illustrating the advantages of the Bohmian quantum-classical dynamics over the Ehrenfest approach. The top panel depicts the system. The light particle approaches and scatters off a surface which contains a phonon mode involving the heavy particle. The Hamiltonian and its parameters are given in Eq. [42a] and Table I, respectively. The bottom panels show the time-dependent probability for the light particle to move a certain distance away from the surface following the scattering event. The data shown in the two panels differ in the initial kinetic energy of the light particle E. Quan- tum mechanically, the light particle has a 100% probability to leave the surface after a sufficiently long time. The light particle wave-packet splits, and a part of it remains temporarily trapped with the heavy particle. In the exact solution, the trapped part of the wave-packet follows the scattered part and eventually decays. The Ehrenfest approach errs both in the scattering onset and the asymptotic scattering probability, as highlighted by the boxes in the middle panel. The classical treatment of the phonon mistreats zero- point energy and allows transfer of the phonon energy to the light particle, accelerating the scattering. At longer times, the lack of branching creates an artificial constraint on the energy exchange between the light and heavy particles. The excess energy transferred from the heavy to the light particle during the early evolution leaves the heavy particle with insufficient energy to continue promoting the scattering of the light particle. Bohmian Quantum-Classical Dynamics 307

Table 1 Parameters Used in Simulation of the Scattering Problem, Eq. [42a] and Figure 4 m 1 amu a 700 kJ/mol M 10 amu b 5.0 A˚ −1 4 × 1014 s−1 c 0.7 4 A10kJ/mol x0 6.0 A˚ B 4.25 A˚ −1 0.5 A˚

potential V1 describes the interaction of the light particle with the surface. The two particles interact by the exponentially repulsive potential V. Parameters particular to the simulation are provided in Table 1 and are the same as in Refs. 25, 115 and 117. Initially, the light particle is moving toward the heavy particle. The light particle is described by a Gaussian wave-packet

ik x (x − x )2 (x, t = 0) = exp 0 exp − 0 [43] 2 located√ 6 A˚ away from the surface. The initial momentum of the light particle k0 =− 2mE0 corresponds to an incident energy E0 shown in Figure 4. The initial conditions of the heavy particle represent the ground state of the harmonic oscillator. Quantum mechanically, the heavy particle is described by the ground- state wave function of the harmonic potential V2, Eq. [42a]. In the Ehrenfest and Bohmian quantum-classical approaches, the ground state of the harmonic oscillator is represented by an ensemble of classical particles. The particle ensemble is generated microcanonically at the energy of the quantum-mechanical ground state. The Bohmian ensemble representing the light particle is sampled from the density R2(x) = ∗(x)(x) generated for the Gaussian Eq. [43]. The coordinate of the Bohmian particle is evolved in time according to Eq. [41] and is used to compute the force on the heavy particle using Eq. [39].

←

(Continued) Similar to the Ehrenfest scheme, the quantum-classical Bohmian approach errs at the early times by mistreating zero-point energy. However, it correctly reproduces the asymptotic value of the scattering probability, since the interactions of the immediately scattered and transiently trapped parts of the light particle with the heavy phonon are not averaged as in the Ehrenfest approach but are treated independently. 308 Semiclassical Bohmian Dynamics

The simulation is characterized by the time-dependent scattering probability for the light particle to leave the region within x0 = 6 A˚ of the surface ∞ 2 Ps(t) = |(x, t)| dx [44] x0

The results of the Ehrenfest and Bohmian quantum-classical approaches are similar to the fully quantum-mechanical simulation data at high-incident energies (Figure 4). The disagreement increases at lower energies. The Ehrenfest method deviates from exact quantum mechanics both at short and long times. Asymptotically, the failure of the Ehrenfest method to reproduce the complete scattering of the wave packet stems from its mean-field nature. In the fully quantum description, the wave packet describing the light particle splits. Part of the wave packet remains temporarily trapped with the heavy particle near the surface. In the exact solution, the trapped part of the wave packet follows the scattered part and eventually decays. The trapped and scattered portions of the wave packet become correlated with different parts of the wave-packet describing the heavy particle. The heavy particle wave packet also branches into several components. A single mean-field trajectory of the heavy particle produced by the Ehrenfest approach cannot describe this correlation. The energy exchange between the light and the heavy particles is treated incorrectly; the trapped part of the light particle never decays.115 At short times, the scattering probability in the Ehrenfest method increases too rapidly, relative to the fully quantum scattering probability. This occurs because the energy of the heavy particle, including zero-point energy, is fully available for exchange with the light particle. Quantum treatment of the phonon results in preservation of zero-point energy. However, when the phonon is described classicly, zero-point energy is transferred to the light particle. The excess energy transfer from the heavy to the light particle during the early evolution causes the light particle to accelerate too fast. In addition, the scattering probability in Figure 4 increases too rapidly. Later, the heavy particle is left with insufficient energy to continue promoting the scattering of the light particle at longer times. This leads to the erroneous scattering asymptotes in Figure 4. The quantum-classical Bohmian approach uses an altered treatment of the interactions of the scattered and transiently trapped parts of the light particle with the heavy particle. The asymptotic value of the scattering probability is reproduced correctly by this approach. Interactions are treated separately in the quantum-classical Bohmian approach, whereas the interactions are averaged in the Ehrenfest approach. The interaction explicitly depends on the position of a randomly selected Bohmian particle representing the light particle subsystem. An ensemble of heavy particle trajectories is generated for a single evolution of the light particle wave function. The energy associated with a single quantum- classical Bohmian trajectory is not required to be conserved. This feature of the Bohmian approach is analogous to the quantum-mechanical property that the Bohmian Quantum-Classical Dynamics 309 energy associated with different branches of a wave-packet is not conserved. As a result, the energy can continue to flow from the heavy to the light particle in the trajectories describing the transiently trapped part of the light particle, and scattering proceeds to full completion. Because the heavy particle is described classicly in the Bohmian approach, it does not preserve its zero-point energy. As in the case of the Ehrenfest method, (Figure 4) the initial scattering proceeds too fast. Thus, in the example considered here, the Bohmian quantum-classical approximation improves over the Ehrenfest approach in the properties associated with the branching of the trajectories of the heavy particle. The conservation of zero-point energy cannot be resolved properly with classical dynamics and requires semiclassical corrections, such as in the quantized Ehrenfest approach.89,102,117

Properties of the Bohmian Quantum-Classical Dynamics The Bohmian interpretation of quantum mechanics provides a classical- like view on the evolution of quantum particles and creates an alternative route for generating quantum-classical approximations to the dynamics of complex systems. In particular, Bohmian mechanics offers a solution to the trajectory branching problem in the quantum-classical simulation. The problem is resolved by creating a new type of the quantum back-reaction acting on the classical subsystem. The Bohmian back-reaction is defined uniquely and is computationally simple. It provides a straightforward connection to the full classical limit. The branching of quantum-classical trajectories is achieved in the Bohmian approach by coupling of the classical subsystem to a single quantum particle in the Bohmian ensemble. An ensemble of quantum-classical trajectories is generated for a single, initial quantum-mechanical wave function. This is in contrast to the quantum-classical Ehrenfest approximation in which a single average classical trajectory is generated. The Bohmian quantum-classical approach succeeds where the Ehrenfest method fails. In particular, it succeeds for cases in which distinct classical trajectories are required for different states of the quantum subsystem. Distinct Bohmian quantum-classical trajectories emerge for different quantum states. Traditionally, the branching problem is solved by modification of the Ehrenfest approach with a surface-hopping procedure.94–97,100,101 Although the Bohmian quantum-classical method produces results that are similar to surface hopping, the latter procedure is ad hoc and varies between implementations. In contrast, the Bohmian approach is uniquely defined. Bohmian quantum-classical dynamics can be implemented easily by a mi- nor modification of a standard Ehrenfest code. In the Ehrenfest method, the quantum force operator is averaged across the whole quantum-mechanical density distribution. In contrast, the Bohmian force is computed for a single point in the quantum-mechanical density function, avoiding the averaging. The 310 Semiclassical Bohmian Dynamics quantum subsystem can be evolved using the Bohmian equations of motion for the ensemble of Bohmian particles (Eqs. [37] and [40]). Alternatively, the evolution of the Bohmian particles can be obtained directly from the time-dependent wave function (Eq. [33]). The latter route eliminates calculation of the quantum potential [37] and its derivative, which may be ill-behaved in the regions of low quantum density. The total quantum-classical energy is not conserved along a single quantum-classical Bohmian trajectory. This is expected because the energy of a branch of a quantum-mechanical wave packet also has no conservation requirement. Potentially, the energy nonconservation can present a problem in a quantum-classical simulation, although applications considered thus far find no such problem.24,25,113,118–121 In addition to resolving the branching problem in a unique way, the Bohmian approach to the coupling of quantum and classical mechanics provides a new opportunity that is not attainable by the traditional approaches such as Ehrenfest and surface hopping. Consider a situation, such as in Refs. 122 and 12, in which a molecular complex AH-B is coupled to a solvent. The proton H has to be treated quantum mechanically, whereas both A and B atoms and the solvent can be treated classicly. In the traditional approaches, the quantum- classical coupling of the proton to the solvent is formulated in exactly the same way as the proton coupling to the atoms A and B. If the proton tunnels between A and B but does not tunnel into the solvent, then the quantum behavior of the proton depends only on the interaction with A and B. In the Bohmian quantum- classical method, it is conceivable to couple the proton to A and B quantum mechanically and to the solvent purely classically. The evolution of the proton will be determined by both classical V and quantum Q potentials, (Eq. [40]), such that the classical potential will contain the proton-A,B and proton-solvent coupling terms. The time-dependent Schrodinger¨ equation, which determines the quantum potential, will be decoupled from the solvent and will depend only on A and B. To couple the quantum-mechanical proton to the classical solvent, the traditional methods must include solvent-dependent terms in the Schrodinger¨ equation for the proton. The Bohmian quantum-classical approach can provide significant savings by ignoring the multiple solvent terms in the Schrodinger¨ equation. At the same time, solvent terms must be included in the Newton Eq. [40] for the proton. In addition to the approach described here,24,25,113 several related schemes for quantum-classical dynamics based on the Bohmian interpretation of quantum mechanics have been proposed as well.123–125 In particular, Ref. 123 uses the Bohmian formulation to achieve a classical limit for the wave- packet motion on coupled potential energy surfaces. Refs. 124 and 125 combine the Bohmian description for the quantum subsystem with a Liouville-space description for the classical subsystem. The Bohmian description is recast in the language of hydrodynamics, partial hydrodynamic moments are introduced, and a hierarchy of equations is derived using closures as in quantized Hamilton dynamics in Refs. 102, 117 and 89. Hybrid Bohmian Quantum-Classical Phase–Space Dynamics 311

To recapitulate, the Bohmian quantum-classical approach described is capable of reproducing quantum effects that are crucial in the simulation of chemical processes. The approach is computationally simple and is particularly suitable for studies of large and complex systems.

HYBRID BOHMIAN QUANTUM-CLASSICAL PHASE–SPACE DYNAMICS

The previous section presented a mixed quantum-classical scheme that collapses to a single trajectory in the classical limit for the heavy particle. This is the simplest and often adequate quantum-classical description. A more accurate representation can be obtained in a phase–space picture. The doubling of the number of independent variables improves the accuracy of the classical picture. For example, a purely classical propagation of a distribution function that corresponds to a Gaussian wave function gives an exact description of quantum harmonic motion.37,126 Therefore, the mixed quantum-classical description in the phase–space representation is particularly relevant for semiclassical systems. The coupling of approximate quantum and phase–space classical descriptions is explored in many publications.51,127–129 One challenge of this approach is the phase–space description of the quantum part. The expansion of quantum equations in powers of Plank’s constant suffers from a convergence problem. 130 The coupling of the Bohmian and phase–space descriptions is a promising alternative. Particularly relevant to this review is the work of Burghardt et al. 124,125,131 summarized subsequently. Quantum mechanics in phase space132,133 is governed by the quantum Liouville equation

∂ i W =−(H, ) = (H exp /(2i) − exp /(2i)H) ∂t W qu qp W W qp [45]

126 In Eq. [45] W = W (q, p, Q, P) is the Wigner distribution corresponding to the wave function (x, Q, P). The Wigner transform of the wave function is taken with respect to the quantum coordinate x. The dependence on the classical variables (Q, P) is parametric. The Hamiltonian Hˆ = Hˆ q(q, p) + Vint(q, Q) + Hˆ Q(Q, P) involves the Hamiltonians of the quantum and classical subsystems, Hˆ and Hˆ , respectively, together with the interaction potential V (q, Q). q Q ←− −→ ←− −→ int qp is the Poisson bracket operator qp = ∇ p ∇ q − ∇ q ∇ p. In the lowest order, the Poisson bracket operator reduces to the classical Poisson bracket {H, W }=1/2(H QPW − W QPH). The imbalance between the classical Poisson bracket and the quantum Poisson bracket operator in Eq. [45] leads to problems with operator ordering. These problems result in a violation of the Jacobi identity. Recent work by one of us demonstrates that it is possible to 312 Semiclassical Bohmian Dynamics restore proper behavior by performing all calculations quantum mechanically and taking the → 0 limit at the very end of the calculations.110 In contrast to the full phase space distribution, the Bohmian picture provides a unique momentum p = pq associated with every position q. The connection between the Bohmian and phase space descriptions is established via partial hydrodynamic moments n n n p =p W (q, Q, P) = dp p W (q, p, Q, P) [46]

By associating the ﬁrst partial hydrodynamic moment with the Bohmian momentum p =p and transforming into the Lagrangian frame of reference, see “Introduction”, the following set of equations is obtained:

p q˙ = [47] m ∂ p˙ =− (V (q) + V (q, Q)) + F (q, Q, P) [48] ∂q q int hyd P Q˙ = [49] M ∂ P˙ =− (V (Q) + V (q, Q)) [50] ∂q Q int

Here, the “quantum force” is deﬁned by the higher partial hydrodynamic moments through the hydrodynamic variance

−1 ∂ (q, Q, P) F (q, Q, P) = [51a] hyd m1 ∂q p2 (q, Q, P) =p2− [51b] 1 with 1=p0. The variance reﬂects the width of the phase-space distribution W in the p dimension for given values of (q, Q, P). The corresponding spatial variance with respect to the quantum position q gives rise to the quantum force. Unlike traditional Bohmian mechanics formulated in the coordinate space, the set of Eqs. [47–48] is not closed. The quantum force depends on p2, and its evolution depends on the higher order partial hydrodynamic moments. Closures are possible in two special cases. For pure quantum states that are not interacting with the classical subsystem, the hydrodynamic force reduces46 to the negative gradient of the quantum potential (Eq. [3]). For harmonic systems, the higher order partial hydrodynamic moments can be expressed through the lower ones. The latter case leads to a straightforward assessment of the more The Independent Trajectory Methods 313 general expression for the hydrodynamic force (Eq. [51a]) compared with the classical hydrodynamics force given in the coordinate space representation. An oscillator system that consists of light and heavy particles with bilinear cou- pling134 can be described exactly within the presented phase space formalism. Semiclassical closures lead to other related approximations, notably quantized hamilton dynamics.102,135,136

THE INDEPENDENT TRAJECTORY METHODS

This section describes approaches based on independent Bohmian trajectories and derived from the hierarchy of time-evolution equations for the wave function phase and amplitude as well as their spatial derivatives. For practical reasons, the hierarchy is truncated by setting higher order derivatives to zero. Truncation at the second order reveals a connection to other semiclassical methods.

The Derivative Propagation Method The derivative propagation method is based on the hierarchy of equations for the evolution of the wave function phase, amplitude, and their derivatives. The wave function is represented as an exponent with the real and imaginary functions as its argument = + ı (x, t) exp C(x, t) S(x, t) [52]

This form of the wave function is substituted into the time-dependent Schrodinger¨ equation as before. Separation into the real and imaginary parts,

∂C 1 =− [∇2S + 2∇C ×∇S] [53] ∂t 2m

∂S 1 2 =− ∇S ×∇S + [∇2C +∇C ×∇C] [54] ∂t 2m 2m and subsequent differentiation with respect to x gives an infinite hierarchy of equations for the derivatives. The equations are coupled because the equations for the lower order derivatives contain higher order derivatives. The terms with C in the RHS of Eq. [54] constitute the quantum potential. By truncating the Taylor expansion of S and C around position x at a certain order, one obtains a nonlinear system of evolution equations for the trajectory-specific expansion coefficients. For example, truncation at the second order results in a system of six equations involving the coefficients, classical potential V, and its 314 Semiclassical Bohmian Dynamics

ﬁrst and second derivatives. Truncation through the second order is exact for Gaussian wave packets evolving in a quadratic potential. The equations can be transformed into the Lagrangian frame of reference in which the trajectories and their associated systems of equations are propagated independently. The advantage of the derivative propagation method is that it is insensitive to the trajectory crossings, unlike the exact implementations. In addition, a few or even one trajectory may be sufﬁcient for some applications. However, the independence of the trajectories comes at the cost of requiring high derivatives of V. Moreover, the scaling on the number of equations with dimensionality makes implementation beyond the quadratic order prohibitively expensive for high-dimensional systems. We refer you to the book of Wyatt137 for more details and applications of the derivative propagation method. Of conceptual interest is the calculation of the energy-resolved transmission probability over the Eckart barrier. The calculation is carried out with the second and third order of the derivative propagation method. The second-order method shows a “typical semiclassical” accuracy that is surprisingly similar to the results of Grossmann and Heller138 obtained with the Van Vleck–Gutzwiller propagator. This result suggests a formal connection to the semiclassical theories. The third-order method improves agreement with the exact QM result.

The Bohmian Trajectory Stability Approach. Calculation of Energy Eigenvalues by Imaginary Time Propagation A different independent quantum trajectory scheme has been developed by Liu and Makri.139 Similar to the derivative propagation method, it forms an inﬁnite hierarchy of equations. Unlike the derivative propagation method, it explicitly uses conservation of the wave function probability given by Eq. [20]. According to wave function probability conservation, the wave function density (x, t) is related to the initial density by the stability matrix element of a trajectory, namely the Jacobian computed along the trajectory

∂x0 ∂xt (xt) = (x0) ,J(xt,x0) = [55] ∂xt ∂x0

The elements of the trajectory stability or monodromy matrices

⎛ ⎞ ∂pt ∂pt ⎜ ⎟ ⎜ ∂p0 ∂x0 ⎟ M(t) = ⎝ ⎠ [56] ∂xt ∂xt ∂p0 ∂x0 The Independent Trajectory Methods 315 appear in the semiclassical propagators, such as the WKB, Van Vleck– Gutzwiller, and initial value representations based on classical trajectories. 29,30,140,141 The difference is that the Bohmian trajectories are inﬂuenced by the quantum potential in addition to the classical potential. Evolution of the stability matrix ⎛ ⎞ ∂2H ⎜ 0 − ⎟ dM(t) ⎜ ∂x2 ⎟ = ⎝ ⎠ M(t) [57] dt ∂2H 0 ∂p2 gives J; however, it is coupled to the higher derivatives of J. Differentiation of Eq. [57] with respect to x results in an inﬁnite hierarchy of equations which in practice, are truncated at a low order. In general, the features of the Bohmian trajectory stability approach are similar to those of the derivative propagation method. An important difference is that, formally the stability method does not employ exponentiation of the amplitude, which is problematic at the nodes. Nor does the stability method rely on the Taylor expansion around trajectories, although smoothness is probably necessary for convergence. Estimation of the energy eigenvalues coupled with extension of the Bohmian formulation into imaginary139 time constitutes a very promising application of the stability matrix approach.54 It is known that transformation from real to imaginary time, t →−ı , converts the time-dependent Schrodinger¨ equation into the diffusion equation

∂ − (x, ) = Hˆ (x, ) [58] ∂

This equation offers a convenient way to obtain low-energy eigenvalues. Any wave function propagated under Eq. [58] converges to the eigenstate of the same symmetry, as in the diffusion Monte Carlo method.142 The same substitution can be made in the quantum trajectory Eqs. [12] and [19] implying

(x, ) = A(x, ) exp(−S(x, )/) [59]

The result is the quantum trajectory equations with the inverted, V + Q → −V − Q, potentials. This formulation generally results in singular trajectory dynamics. Nonetheless, Liu and Makri show that one can ensure smooth behavior of the trajectories by repartitioning A and S at each time step, because (x, ) is real and division into A and S is arbitrary. The authors proceed to compute the ground-state energies for HCN and H2O in two and three dimensions respectively. Accurate ground-state energies were obtained with the fourth- and sixth-order of the Bohmian trajectory stability propagation method. 316 Semiclassical Bohmian Dynamics

Bohmian Mechanics with Complex Action Bohmian mechanics with complex action55,73,143 is an extension of the time-dependent Schrodinger¨ equation into the complex plane. This constitutes a drastic departure from the methods discussed, yet the implementation of the complex action technique is related directly to the derivative propagation and the Bohmian trajectory stability methods discussed. The starting point is the exponentiation of the wave function = ı (x, t) exp S(x, t) [60] substituted into the time-dependent Schrodinger¨ equation. The result is an equation of the Hamilton-Jacobi type, now containing a complex action function S(x, t),

∂S mv2 ı =− − V + ∇v [61] dt 2 2 where mv(x, t) =∇S(x, t). Transformation into the Lagrangian frame of reference is accomplished formally as before

dx ∇S(x, t) = v(x, t) = [62] dt m

The solutions of Eq. [62] are complex trajectories characterized by complex action functions, velocities, and positions because of the last term in Eq. [61]. Differentiation of Eq. [62] gives a Newtonian equation

dv ı m =−∇V + ∇2v [63] dt 2

Similar to the derivative propagation and Bohmian trajectory stability methods, Eq. [63] involves an unknown spatial derivative of the phase. Thus, Eq. [63] is differentiated with respect to x and so forth to arrive at an infinite hierarchy of equations. For practical reasons, the hierarchy is truncated at a low order. The first order is equivalent to the complex Gaussian wave-packet method of Huber and Heller.144 The second order is equivalent to the complex WKB method.73 The complex quantum force acting on a Gaussian wave packet is zero, which is simpler than the linear force of the real-space formulation. The simplification comes at a price. First, analytic continuation of all quantities, including V and its gradients, Hessians and so on, into the complex plane is needed. Second, a trajectory root search is required for reconstruction of (x, t) on the real axes. For each t, one has to find the complex initial conditions of all trajectories whose xt will be real. In general, there is more than one root Dynamics with the Globally Approximated Quantum Potential (AQP) 317 trajectory, which is a challenge in itself. Remarkably, the multiple roots allow a “semiclassical” description of interference through multiple paths.143 The bipolar decomposition of the wave function, presented in a series of papers by Poirier,56,145–149 also describes interference but within the real Bohmian trajectory framework. Bohmian mechanics with complex action provides a new angle on the semiclassical description of quantum effects and representation of nonlocality. It creates insightful connections between several semiclassical methods. The need for analytic continuations and the rapid scaling of the efforts with system dimensionality constitute major challenges to numerical applications for chemical systems.

DYNAMICS WITH THE GLOBALLY APPROXIMATED QUANTUM POTENTIAL (AQP)

The independent trajectory methods described in the previous section provide a semiclassical description of quantum effects already at the second order of the hierarchy truncation. This feature can be interpreted as a “local” description of quantum nonlocality based on the first and second derivatives of the wave function at point x, through the phase, amplitude, classical, and quantum potentials. After all, if the wave function is smooth, one can define it accurately in a large region of space by knowing its value and high-order derivatives at a single point. In this section, we will describe a different approach for estimating nonlocal quantum effects in semiclassical dynamics in an “average” sense from the trajectory ensemble. We will use atomic units with = 1 below and will drop unless it is needed for a classical or semiclassical analysis. The ultimate goal of dynamics with the AQP is to have an inexpensive, robust, semiclassical method that can be applied to high-dimensional systems. “Inexpensive” means that the scaling with the system size is polynomial, ideally linear, such that the computation of the quantum force is a small addition to the classical trajectory propagation. The polynomial scaling allows the method to be applied to hundreds of degrees of freedom. “Robust” implies numerical stability. In particular, the numerical propagation should be insensitive to trajectory crossings and should switch to classical propagation if the quantum force becomes inaccurate. “Semiclassical” means that the method should describe the leading QM effects. However, it should not aim for exact QM dynamics. The latter requirement is a necessary trade-off for a better than exponential scaling with system dimensionality. In addition, it is desirable for a method to have an accuracy criterion and a systematic procedure for achieving the exact QM description. The first development of this kind of method was based on expansion of the wave function density (x, t) in terms of Gaussian functions with optimized parameters.35,150 The number of Gaussian functions 318 Semiclassical Bohmian Dynamics can vary but must remain small. The procedure was expensive as a result of nonlinear parameter optimization and was unstable toward the addition of new Gaussians in the course of dynamics. This was the proof of concept for the AQP idea and the first demonstration of quantum trajectory dynamics in the semiclassical implementation. A more numerically efficient and conceptually intuitive approximation based on the nonclassical momentum is presented subsequently.

Global Energy-Conserving Approximation of the Nonclassical Momentum Theory An analytical representation of the AQP, Q˜ is important for efficiency and accuracy of the quantum force computation. One physically appealing way of constructing Q˜ is to consider the nonclassical component, r(x, t), of the QM momentum operator defined by Eq. [13]. The representation of this single object within a basis set, r(x, t) ≈ r˜(x, t) = c(t) × f (x) [64] then can be used to obtain both Q˜ , 1 Q˜ =− (r˜2 +∇r˜) [65] 2m and the analytical quantum force Fq =−∇Q˜ . A detailed derivation for multidimensional systems is given in Ref. 74. The vector c contains the time-dependent basis expansion coefficients. The coefficients c are found from the minimization of the average deviation between r(x, t) and r˜(x, t) 2 2 I =(r − r˜) t = I0 + ∇r˜(x, t) + r˜ (x, t) (x, t)dx [66]

Differentiation by parts was used to obtain Eq. [66]. I0 denotes the term that does not depend on c. By changing the integration variable to xt, using Eq. [20], and replacing the integration with summation over discrete trajectories, one obtains = + ∇ + 2 = + ∇ (i) + 2 (i) I I0 r˜(xt) r˜ (xt) (xt)dxt I0 r˜(xt ) r˜ (xt ) wi i [67]

The fact that neither (x, t) nor its derivatives appear in Eq. [67] is of central importance for efﬁcient implementation. The algorithm scales linearly with Dynamics with the Globally Approximated Quantum Potential (AQP) 319 respect to the number of trajectories, as evidenced by the single summation over the trajectories. Minimization of I with respect to the expansion coefﬁcients that are, the elements of c, gives a system of linear equations

∇cI = 2Sc + b = 0 [68]

Here, S is the time-dependent basis function overlap matrix with elements sjk, and b is the vector of averages of the spatial derivatives of the basis functions df df = | = = j = j sjk fj fk wifjfk = (i) ,bj wi = (i) [69] x xt dx dx x xt i i

The solution in vector form is

1 − c =− S 1b [70] 2

One also can obtain the evolution equations for the expansion c in terms of various moments of the trajectory distribution (an interested reader is referred to Ref. 151). As shown in Ref. 75, Q˜ evaluated at the optimal values of c, ∇cI = 0, conserves energy for time-independent classical potentials,

dE p dp dx ∂Q˜ dc = +∇(V + Q˜ ) × + =∇Q˜ = 0 [71] dt m dt dt ∂t c dt

˜ The energy is conserved because Q is proportional to I − I0. Because of the latter property, the optimization procedure becomes a variational determination of Q˜ . The smallest physically meaningful basis f consists of just two functions f = (1,x). For this case, the approximation can be written in a particularly transparent form

x −x r˜ =− [72] 2(x2−x2)

The wave function is normalized to 1, and the nonclassical momentum is approximated with a linear function centered at the average position of the wave packet and inversely proportional to its variance, =x2−x2. This functional form of r˜ results in a quadratic Q˜ and, consequently, a linear quantum force. Hence, the approximation is termed the linearized quantum force (LQF). Note that Fq is inversely proportional to 2; therefore, Fq quickly vanishes for delocalized wave packets. 320 Semiclassical Bohmian Dynamics

From the physical point of view, the linear nonclassical momentum exactly corresponds to a Gaussian wave packet evolving in time in a locally quadratic potential. For general potentials, it can describe the dominant quantum effects in semiclassical systems, such as wave-packet bifurcation, wave-packet spread in energy, and moderate tunneling. Zero-point energy effects are reproduced for short times, including a few vibrational periods and depending on the anharmonicity of the system. This is adequate for direct gas phase reactions as demonstrated in the next section. A linear scaling of the variational procedure, defined by Eqs. [67] and [68], with the number of trajectories has been achieved. This was done by explicitly using the trajectory weights Eq. [20] instead of propagating the wave function density (xt) along the trajectories according to Eq. [19]. Therefore, to extract useful information, one has to reconstruct (x, t) through interpolation or fitting. In this way, information other than expectation values, for instance, correlation functions involving time-dependent overlaps of wave functions or internal eigenstate projections given by integrals of the type (0)| (t), can be obtained. Note that the reconstruction procedure is not part of the propagation process; therefore, it does not affect the accuracy of the dynamics. If (x, 0) is localized, then one can use the cheapest strategy and approximate (x, t)asa complex Gaussian. Parameters for the Gaussian are found from the moments of the trajectory distribution using |(x, 0)| as a weighting function.150 If more accurate information about (x, t) is required, then one can find (x, t) using a convolution with a narrow Gaussian with the width parameter, ˇ. For each trajectory position xt, the wave function phase S(xt) and its gradient pt are known. The wave function density can be determined at any position x as

ˇ (x) = exp −ˇ(x(i) − x)2 w [73] t i i

Another alternative is to ﬁnd the projections approximately using the Wigner transformation approach described in Ref. 152.

Application: Photodissociation Cross Section of ICN Let us consider photodissociation of ICN treated in the Beswick–Jortner model 153 following Refs. 154 and 155. A wave packet representing an ICN molecule is excited by a laser from the ground to the excited electronic state. In the excited state, the ICN molecule dissociates into I and CN. The Hamiltonian and the Jacobi coordinates (x, y) are described in Ref. 155. An initial wave function

2 (x, y, 0) = (˛ ˛ − ˛2 )1/4 exp −˛ (y − y¯)2 11 22 12 11 2 − ˛22(x − x¯ ) + 2˛12(y − y¯)(x − x¯ ) [74] Dynamics with the Globally Approximated Quantum Potential (AQP) 321 is deﬁned as the lowest eigenstate of the ground electronic surface with zero momentum. The ground-state potential consists of two harmonic potentials in CN and CI stretches. Thus, (x, y, 0) is a correlated Gaussian wave packet located on the repulsive wall of the excited surface. The photodissociation cross section is computed from the Fourier transform of the wave packet autocorrelation function C(t),

∞ (ω) = ω C(t) exp(ıωt)dt ,C(t) = (0)| (t) [75] −∞

The physical value of the repulsion parameter of the excited potential surface yields a rather simple dissociation dynamics; C(t) decays on the time scale of about one and a half oscillations of the CN stretch. The LQF spectrum is in excellent agreement with the quantum result.74 For a more challenging test, the value of the repulsion parameter three times larger than its physical value, yielding a predissociation process (system II in Ref. 155), also has been considered. Propagation of an initially real wave function makes calculation of the spectrum especially simple, = − | = ∗ | = (i) C(2t) ( t) (t) (t) (t) exp 2ıS(xt ) wi [76] i

The need to reconstruct (t) has been eliminated, and the propagation time has been reduced by a factor of two, which improves the propagation accuracy. For a two-dimensional system, the nonclassical momentum is a vector, r = (r(x),r(y)). In the LQF treatment, both components of r are approximated in a linear basis f = (1,x,y) with the component-speciﬁc expansion coefﬁcient vectors r˜(j) = c(j) × f for j = x, y. The minimization procedure is directly analogous to the one-dimensional case,

(x) (x) 2 (y) (y) 2 ∇xI =∇yI = 0,I=(r − r˜ ) +(r − r˜ ) [77]

The only difference is that in Eq. [70], the vectors c and b are replaced with (x) (y) matrices C = (c , c ), and B = (∇xf,∇yf ). The full derivation can be found in Ref. 75. The LQF parameters are shown in Figure 5. Panel (a) shows the average position of the wave packet and the initial positions of some of the trajectories sampling | (0)|2. An ensemble of 167 trajectories has been propagated for three vibrational periods of the CN stretch. The average position, plotted as y versus x, illustrates the distance between I and the center of mass of CN. The distance increases in the course of dissociation as the CN stretch undergoes three (x) vibrations. Panel (b) shows the optimal expansion parameters: c2 represents (y) the changing width of the wave packet in the CN mode; c3 describes the 322 Semiclassical Bohmian Dynamics

(a)

I-CN [bohr] 5.5

5 2.1 2.2 2.3 2.4 2.5 CN [bohr]

(b)

100

50 LQF parameters

0 1000 2000 Time Figure 5 The LQF parameters for ICN: (a) the average position, y vs x, of the wave- (x) (y) packet and initial positions of trajectories; (b) the width parameters c2 (solid line), c3 (x) = (y) (dash), and c3 c2 (dot-dash).

(x) = (y) spreading in the dissociation mode; c3 c2 reflects the correlation between the degrees of freedom in the course of dynamics. Figure 6 shows the real part of the autocorrelation function on panel (a) and the corresponding spectra on panel (b). The two plotted curves were obtained with the same set of quantum trajectories; the results were obtained using Eq. [76] and using the Gaussian fitting of (x, y, t) as outlined in “Dynamics with the Globally Approximated Quantum Potential” section. The advantage of Eq. [76], in which C(2t) is obtained from the propagation up to t, is apparent; the spectrum is in excellent agreement with exact results. C(t) obtained with the wave function fitting shows deviations from the QM result after 800 a.u. of time. This is because the propagation error grows with time (not because of the fitting). Nevertheless, the agreement of the spectra is good, showing that, overall, the LQF method is accurate for systems with fast dynamics. Dynamics with the Globally Approximated Quantum Potential (AQP) 323

0.5 (a)

0 Re(C(t))

-0.5 0 1000 5

(b) 4

I(E) 2

0 0 0.02 0.04 Energy Figure 6 Photodissociation cross-section of ICN: (a) real part of the autocorrelation function; (b) the corresponding spectra. The LQF results are shown with a solid line when using Eq. [76] and with a dash when a Gaussian ﬁtting of (t) is made in the C(t) computation. Quantum results are marked with circles.

There are several ways to improve the description of non-Gaussian wave functions evolving in general potentials: (1) More functions can be added to the basis f . For example, the Chebyshev basis of up to six polynomials has been used in Ref. 151. Representation of r in a complete basis will give exact QM dynamics. (2) The basis can be kept small but made to be system-speciﬁc. For example, a two-function basis consisting of a constant and an exponent gives an exact description of r for the eigenstate of the Morse oscillator. Such a basis, with the parameter in the exponent tailored to the potential, has been used in Ref. 156 to describe the zero-point energy of H2 in the reaction channel of the O + H2 reaction. Another approach, discussed immediately after, is based on (3) the linear approximations to r on subspaces. Recall that in the global AQP procedure, spatially separated parts of the wave function are coupled as a result of averaging over the entire ensemble of quantum trajectories. The issue 324 Semiclassical Bohmian Dynamics of unphysical coupling can be resolved by using an approach based on the linear approximations to r on subspaces. Additionally, such an approach gives a more ﬂexible description of r.

Approximation on Subspaces or Spatial Domains Theory The globally determined LQF describes QM effects on a short time scale adequate for direct dynamics reactions. After the wave packet bifurcates, the LQF quickly goes to zero because of a large variance of the wave packet on the entire space. At the same time, parts of the wave function in the reactant or product channels might still be localized. To remove this “unphysical” long-distance interaction, one can divide the coordinate space onto several subspaces, or domains, labeled l = 1 ...L. Each subspace is deﬁned by a “domain” function ≥ l(x): l(x) 0 for all x. These subspaces may correspond to physically significant regions such as reactants, products, and transition-state regions in reactive dynamics. Subspaces also can be based on other considerations such as shape of the wave functions and features of the potential V. The domains are ﬁxed in time and, in general, have nonzero overlap in space. The last domain with index L complements the rest of the domains to unity, = − L(x) 1 l(x) [78] l=1,L−1

This requirement ensures that the sum of exact solutions of the time-dependent Schrodinger¨ equation on subspaces is equivalent to the solution on full space. The kinetic energy operator on the subspace has to be Hermitian, that is, its matrix element i|Kˆ |j is 1 1 ∇ (x) ×∇ (x) × (x)dx =− (x) (x) 2m i j l 2m l i ∇ 2 l(x) × ∇ + ×∇ j(x)dx [79] l(x)

With this form of Kˆ in the Bohmian formulation of time-dependent Schrodinger¨ equation, the quantum potential Q in Eq. [12] is replaced by its modiﬁed version Ql with the additional interface term, ∇2 ∇ ∇ =− 1 A(x, t) + l(x) × A(x, t) Ql [80] 2m A(x, t) l(x) A(x, t) Dynamics with the Globally Approximated Quantum Potential (AQP) 325 for each domain. The density on the lth domain (Eq. [19]) also is modiﬁed as follows:

d(x, t) ∇ (x) =−∇v × (x, t) − l v(x, t) [81] dt l(x)

Note that (x, t) and S(x, t) still are defined on the full space and not on the domains (i.e., there are no domain-specific wave functions). The summation of = Eq. [81] across all domains weighted by l gives Eq. [19], because l(x) 1 ∇ = and l(x) 0, provided domain functions satisfy Eq. [78]. This formulation is equivalent to the Schrodinger¨ equation on the full space. No advantage is offered if the quantum potential is determined exactly. But it allows one to define domain-specific approximations to the quantum potential, improving accuracy of the AQP. The nonclassical momentum r given by Eq. [13] can be approximated on each domain by minimizing a functional = − 2 Il (r(x, t) r˜l(x, t)) l(x)(x, t)dx [82]

= where the approximating function is now domain-speciﬁc, r˜l f (x)cl(t). The contribution to the AQP from each domain is determined by Eq. [80],

1 Q˜ =− (r˜2 +∇r˜ ) + r˜ ∇ [83] l 2m l l l l l

The total quantum potential is a sum over domains ˜ = ˜ Q Ql [84] l=1,L

˜ Ql should be evaluated at the optimal values of cl that minimize Eq. [82] and ∇ = for which cIl 0. Then, the AQP defined on domains rigorously will conserve energy in a closed system, as was the case with the full space approximation (Eq. [71]). Derivation of the minimization procedure in multidimensional coordinates is given in Appendix C. Because Eq. [81] for the density summed over the domains is equivalent to Eq. [19], one can use the weight conservation property, given by Eq. [20], as before. The wave function still is defined on the full space according to Eq. [2]. The subspaces are introduced only to approximate better the nonclassical momentum and to define the AQP. Implementation with the linear basis on multiple domains is given subsequently. 326 Semiclassical Bohmian Dynamics

Application: Dynamics of the Collinear H3 System The collinear hydrogen exchange reaction, HA+HBHC →HAHB+HC, is a standard test in reaction dynamics and presents a considerable challenge for semiclassical approaches. The system is described in the Jacobi coordinates of reactants in which the kinetic energy is diagonal. The Hamiltonian, coordinates, and potential surface are the same as in Ref. 157. The initial wave packet is deﬁned as 2 (x, y, 0) = (˛ ˛ )1/2 exp(−˛ (x − x¯ )2 − ˛ (y − y¯)2 + ıP (x − x¯ )) 1 2 1 2 0 [85] where y is the vibrational coordinate of the diatomic (the distance between HB and HC), and x is the translational degree of freedom (the distance between HA and the diatomic). In the Jacobi coordinates of reactants, the angle = arctan(y/x) divides the potential energy surface into reactant region, 0 <≤ /6, and product region, /6 << /3. The surface is symmetric with respect to 0 = /6. Values of the parameters in atomic units (scaled by the reduced mass of the diatomic, mH/2 = 1) are x¯ = 4.5, y¯ = 1.3, ˛1 = 4.0, ˛2 = 9.73, and P0 = [−15, −1]. The scaled unit of time is equal to 918 a.u. The initial positions for the quantum trajectories {x(i),y(i)} are chosen on a rectangular grid with spacings dx = 0.017 and dy = 0.020. Trajectories with weights smaller −8 (i) (i) than 10 are discarded. The initial momenta {px ,py } are px(i) = P0 and (i) = (i) (i) = (i) − py 0; the initial classical actions are S(x0 ,y0 ) p0(x x¯ ). The wave- packet transmission probability P(t), y(i) P(t) = w H arctan t − [86] i (i) 6 i xt is analyzed. The summation goes over the trajectories in the product region; H denotes the Heavyside function in angle. For optimization using domains, the reactant and product channels are deﬁned by the switching functions,

1 arctan(z( − /6)) 1 arctan(z( − /6)) Á = − ,Á = + [87] reac 2 2 arctan(z /6) prod 2 2 arctan(z /6)

The slope parameter is z = 10. The subspaces in the diatomic vibrational coordinate, which “uncouple” the two wall regions from the low-energy region, are introduced in each channel. In the reactant channel, there are two Gaussian domains in y

= − − 2 k exp( ˇ(y qk) ) [88] Dynamics with the Globally Approximated Quantum Potential (AQP) 327

centered at q1 = 0.9 and q2 = 1.4 bohr with the width parameter ˇ = 6.0. There is also a complimentary domain covering the bottom of the well. The reactant domain functions are multiplied by Áreac. The product domains are sym- metrical to the reactant domains with respect to = /6 and are multiplied by Áprod. In addition, three two-dimensional Gaussian domains are specified in the transition state with appropriate modification of the reactant and product domains satisfying Eq. [78]. This brings the total number of domains to L = 9. The transition-state domain functions are centered in the transition state around the minimum along = /6atq7 = 2.1,q8 = 2.6, and q9 = 3.1 bohr. The width parameter along the transition state is ˇ = 5.0 in both coordinates. Ensembles of 2500 quantum trajectories have been propagated for each wave packet. The wave-packet reaction probability as a function of the total energy is presented in Figure 7a, which also shows the classical (Q = 0) and the full space LQF results. Optimization on the full space gives qualitative agreement with the quantum result. Optimization on the domains leads to the improved probabilities, which are now within 0.04 of the QM values. This system exhibits significant anharmonicity in the diatomic potential. Thus, the main shortcoming of the LQF method is conversion of the zero-point energy into classical energy of the trajectories on a short time-scale. This has been corrected by treating high-energy regions of the potential surface within separate domains. This kind

1 1 (a)QM (b)

QM L=7 Q=0 QM LQF L=7 L=9 I(E) |C(t)| Probability

0 0.4 0.8 1.2 E, eV

0 0 0 0.4 0.8 1.2 0 0.5 1 1.5 2 E, eV Time

Figure 7 Dynamics of the collinear H3 system. (a) The wave-packet transmission probability as a function of the total energy of the wave-packet: results for the quantum (thick solid line), classical (thin solid line), and LQF on full space (thin dash) and on domains (thick dash) propagations are shown. (b) The absolute value of the auto-correlation function for the transition state wave-packet as a function of time. The quantum and the domain results are shown with the solid and dashed lines respectively. The insert shows the corresponding spectra (the semiclassical spectrum is marked with circles). 328 Semiclassical Bohmian Dynamics of treatment “uncouples” the high-energy regions from the low-energy regions where most trajectories evolve. The collinear H3 system is known for resonances in the transition state. The transition-state dynamics provides a wave function phase-sensitive test of the LQF method on domains. An initial Gaussian wave packet, given by Eq. [85], is defined in the transition state and displaced from its minimum. The initial parameters are {y0 = 2.5,x0 = 1.3,˛1 = 6.3,˛2 = 6.3,P0 = 0.0}. Figure 7c shows the wave packet autocorrelation function for t = [0.0, 2.0] computed by Eq. [76] and the corresponding energy spectrum. Five two- dimensional Gaussian domains in the transition region with centers placed = ={ } along /6 direction, ql 2.1, 2.6, 3.1, 3.6, 4.1 bohr, were used in this calculation. The width parameter in the direction of the transition state was ˇ = 8.0. The width parameter for the domain functions in the perpendicular direction was smaller, ˇ = 2.0, to make these functions more delocalized because the channel regions were described within a single domain each, defined by Áreac and Áprod. In this calculation, 4281 trajectories with weights greater than 10−6 have been propagated. The peaks in the spectrum are within 7% of the converged values. The position of the spectrum peaks is converged within its resolution. The number of trajectories is greater compared with the wave- packet probability calculations because the autocorrelation function C(t)isa complex quantity. The semiclassical method reproduced one recurrence in C(t) around t = 1.0. Low-amplitude recurrences at later times are not reproduced. The insert on the figure shows that the exact and the semiclassical spectra of C(t) are in good agreement with each other. A general observation is that the LQF description follows the same trend as other semiclassical methods; the accuracy of the semiclassical description depends on the desired level of detail (i.e., average values tend to be more accurate than the quantities derived from the wave function projections onto eigenstates).

Nonadiabatic Dynamics Arguably, nonadiabatic dynamics is the most important quantum effect on the motion of nuclei in chemical reaction dynamics. Its importance is ob- served in reactive scattering, photochemistry, and enzymatic reactions.158–160 Quantum tunneling becomes negligible for heavy nuclei, and quantum interference often is quenched in large molecular systems at short times because of the wave function decoherence. However, nonadiabatic effects still can inﬂuence such systems. The most widely used method for including nonadiabatic effects into the dynamics of molecules is the trajectory surface hopping method,94,161 which is based on classical evolution of trajectories that can switch between different electronic states. The method has numerous applications including photodissociation of ozone,162 formic acid,163 and azobenzene164 to name a few. In the area of reactive scattering, one question of theoretical and ex- perimental interest is the effect of the spin-orbit coupling between electronic Dynamics with the Globally Approximated Quantum Potential (AQP) 329

3 1 states on reactivity. Such a study of O( P, D) + H2 → OH + H performed with quantum trajectories will be described in the section “Application: The 3 1 Four Electron State Dynamics for O( P, D) + H2.” The system is of interest in combustion and atmospheric chemistry, and it has been investigated thoroughly both experimentally and theoretically.165–169 Accurate electronic potential energy surfaces correlated to the triplet and singlet states of atomic oxygen have been developed by Rogers et al.170 and Dobbyn and Knowles.171 The spin- orbit couplings have been determined by Hoffmann and Schatz.172 Maiti and Schatz have used trajectory surface hopping of quasi-classical trajectories to study spin-orbit interaction-induced intersystem crossing effects in the O + H2 reaction within a four electronic state model.173 A complete QM calculation of this size is still a challenge for the standard QM methods; the ﬁrst QM study using the wave-packet technique was reported in 2005.169 In the quasi-classical trajectory surface hopping study,173 it was estimated that intersystem crossing effects enhance total reaction cross sections by up to 20%. Essentially, no such effect was found in the full quantum result.169 Therefore, nonadiabatic dynamics of O + H2 presents a stringent test for any approximate method. The AQP investigation of the spin-orbit coupling effect 156 in O + H2 reaction is reviewed subsequently, preceded by the summary of the theoretical approach.

The Nonadiabatic Generalization of Bohmian Formulation The original Bohmian formulation given by Eqs. [2]–[4] can be extended straightforwardly to dynamics on multiple electronic states. For clarity, consider a wave packet evolving in the presence of two one-dimensional potential surfaces, V1 and V2, coupled by a potential V12. A typical curve-crossing system in the diabatic representation is sketched in Figure 8. This system is described by a two-component wave function, ={ 1, 2}, governed by the time-dependent Schrodinger¨ equation, 2 ∂ − ∇2 + V − ı (x, t) + V (x, t) = 0 [89] 2m 1 ∂t 1 12 2 2 ∂ − ∇2 + V − ı (x, t) + V (x, t) = 0 [90] 2m 2 ∂t 2 12 1

Here and throughout the subscript of functions refers to the two surfaces, V1 and V2. A multisurface quantum trajectory description is based on the substitution of the wave function in terms of its real amplitude and phase into Eq. [89], = ı = i(x, t) Ai(x, t) exp Si(x, t) ,i 1, 2,... [91] 330 Semiclassical Bohmian Dynamics

Figure 8 The curve crossing model: diabatic potentials, V1 and V2, and coupling V12 are shown with the thin solid, dashed and thick solid lines, respectively. The wave- packet 1 initially located in the reactant region of V1 is propagated toward the product region. Analysis of its overlap with a stationary wave-packet 0 gives the energy-resolved reaction probabilities. followed by the separation of real and imaginary parts. The subscript i labels electronic states. This Bohmian formulation was presented and implemented using the exact quantum trajectories by Wyatt et al.43 The polar representation of i is also a starting point in the derivation of a widely used surface-hopping method.123 The surface-hopping method is characterized by purely classical dynamics of trajectories probabilistically “hopping” between the surfaces and the quantum-classical mixing approaches of Refs. 20, 24, and 25. Other methods of nonadiabatic dynamics based on classical or semiclassical mechanics are discussed in Ref. 174. In the quantum trajectory framework, the concept of trajectory weights, given by Eq. [20], should be generalized to a nonadiabatic formulation. Using = 2 =∇ the wave function densities, i(x, t) Ai (x, t), and identifying pi Si(x, t)in the moving frames of reference deﬁned by appropriate momenta pi according to Eq. [9], the real parts of Eqs. [89–90] give the time evolution of the action functions Si(xt,i) 2 dSi(xt,i) pt,i = − (V + Q + Q ) x =x [92] dt 2m i i ij i t,i where i = 1, 2 and j = 1, 2, and i =/ j. Qi is the single-surface quantum potential given by Eq. [5]. Qij is an additional potential resulting Dynamics with the Globally Approximated Quantum Potential (AQP) 331 from coupling,

Qij = V12ji cos (S) [93] j(x, t) 1 ji = ,S= (S1(x, t) − S2(x, t)) [94] i(x, t)

The imaginary parts of Eqs. [89–90] give the following expression for the trajectory weights:

dw i =−2V sin (S) w [95] dt 12 ji i

In this formulation, there are two sets of trajectories—one set on each surface evolving in time. Coupling affects both the dynamics of trajectories and the time evolution of their weights. The AQP method has been adapted in a practical way44,45 for the nonadiabatic formulation of Eqs. [92]–[95], and adequate if the trajectory dynamics was smooth, such as for asymptotically degenerate V1 43 and V2 coupled by a localized V12. Nevertheless, the formulation of Eqs. [92]–[95], though formally exact, has drawbacks. Qij, given by Eq. [93], involves a possibly singular ratio of the densities. For instance, a typical initial condition for a multisurface problem is a wave packet occupying a single electronic state, such as 2(x, 0) = 0. In Ref. 43, this problem was circumvented by propagating two sets of initial wave packets with nonzero population on both surfaces, whose linear combination gives the desired initial condition. Alternatively, the singularity can be canceled 44 at t =0 by the choice of initial conditions for the trajectories. If 2(x, 0) = 0, then one can set up trajectories on the second surface with the same initial positions and momenta as on the lower surface. A phase shift between 1(x, 0) and 2(x, 0) can be introduced, S(x, 0)=± /2, so that the singularity in Q21 cancels at t =0. The subsequent time evolution is stable. The sign of the phase shift depends on the derivative of V2. In the context of the semiclassical dynamics, there is also a conceptual problem with the Bohmian formulation—nonadiabatic behavior is an intrinsi- cally quantum effect that does not vanish in the semiclassical limit. In practice, this is manifested through the behavior of the coupling terms in Eqs. [92] and [95] as →0. The force on trajectories, derived from Q12, has a contribution proportional to

−1 ∇ cos(S) = (p2(x) − p1(x)) sin(S) [96] which does not go to zero in this limit. In fact, it becomes large and oscillatory because of the −1 prefactor. Apart from computational challenges, this 332 Semiclassical Bohmian Dynamics asymptotic behavior shows that the Bohmian formulation is incompatible with semiclassical dynamics, where propagation is expected to become classical as → 0. A practical and conceptually appealing alternative is to use a mixed coordinate space/polar wave function representation,44,175

i(x, t) = i(x, t)i(x, t) [97]

The “semiclassical” part of the dynamics, which is smooth and nonsingular, can be represented by the quantum trajectories of the polar part i(x, t). The “hard” quantum effects, such as interference or nonadiabatic transitions, are included through a prefactor i(x, t). The prefactor can be treated either as a trajectory- speciﬁc quantity i(x(i,t)) computed along each trajectory of the ith surface or as a spatial function i(x, t) represented in a small basis. For nonadiabatic processes i can be interpreted as a complex population amplitude of the ith electronic state. As an illustration of the mixed representation implementation, let us sub- stitute Eq. [97] into the time-dependent Schrodinger¨ Eq. [89]. Partitioning of i into the polar part i and a prefactor i is arbitrary. Let us assume that i evolves on a single surface Vi and use this fact in Eq. [89]. The time-evolution of i is given by

2 di(x(i,t)) 2 j(x(i,t)) ı =− ∇ i(x(i,t)) + 2r(i,t)i(xt) + V12 j(x(i,t)) dt 2m i(x(i,t)) [98]

The kinetic energy terms in Eq. [98] have explicit dependence on 2 therefore, they are small in the classical limit and can be negelected or cheaply estimated. The assumption that dynamics of i is governed by Vi provides the closest analogy with the single surface propagation. This assumption is reasonable for systems with localized coupling V12, as in the example below; however, it is not necessarily the best choice in other situations. Figure 8 refers to the curve crossing model of Tully;161 full details of the nonadiabatic calculations using the AQP and mixed representation can be found in Ref. 44. Two sets of trajectories are propagated on V1 and V2, and the amplitude prefactors i are found from Eq. [98]. The kinetic energy terms are estimated from a small basis expansion of i. The initial population of V2 is zero. Figure 9 illustrates the population transfer between the surfaces. Figure 10 shows the energy-resolved reaction probabilities computed from the wave-packet correlation functions 0| i(t), 176 which are in good agreement with QM probabilities. Theory relevant to the O + H2 application is presented below. Dynamics with the Globally Approximated Quantum Potential (AQP) 333

|χ | 50 a.u. 1 0.5 100 a.u. 150 a.u. 0.9 200 a.u.

|χ | 2

0 0.8 -4 -2 0 2 4 Coordinate [a 0]

Figure 9 Snapshots of the population function amplitudes, |1| and |2|, at times 44 t ={50, 100, 150, 200} a.u. for the high-energy wave-packet 1. Initially 1 = 1 and 2 = 0.

Theory: Nonadiabatic Trajectory Formulation in the Mixed Wavefunction Representation The theoretical formulation in this section is given for the three-dimensional Jacobi coordinates (x, y, Â) describing a nonrotating triatomic system, which will be used in the next section (see Appendix B for a generalized formulation).

1 QM LQF QM P LQF 11

0.5 Reaction probability

P12

0 0 200 400 600 800 Energy [kJ/mol]

Figure 10 The energy resolved reaction probabilities for the diabatic transition, P11, and for the nonadiabatic process, P12. 334 Semiclassical Bohmian Dynamics

Consider dynamics on multiple coupled potential surfaces in the diabatic representation ( = 1) ∂ ı = Tˆ I + V [99] ∂t

Here, I is the identity matrix whose size is given by the number of potential energy surfaces (i.e., the number of electronic states). The kinetic energy operator is 1 ∂2 1 ∂2 1 ∂2 ∂ Tˆ =− − − + cot Â [100] 2M ∂x2 2m ∂y2 2 ∂Â2 ∂Â

The moment of inertia is included using

1 1 1 = + [101] Mx2 my2

The matrix V is a symmetric matrix that contains the diabatic potential energy surfaces and couplings. The mixed representation approach, in its most general form, represents the total wave function as a matrix-vector product = × . The polar parts describe the overall dynamics, possibly semiclassically, in coordinate space. The coordinate space prefactors describe the amplitude changes caused by coupling between the surfaces. The ith component of is i = ij(x, y, Â, t)j(x, y, Â, t) [102] j

Indexes i and j label the electronic states. The polar parts j(x, y, Â, t) can evolve on the diabatic or nonadiabatic potential energy surfaces or on effective potential surfaces combining features of both diabatic and nonadiabatic representations. The potentials governing dynamics of j should be chosen to minimize the semiclassical propagation error. At the same time, the spatial derivatives of ij should be kept small so that they can be neglected or cheaply estimated. Examples of the wave function representation Eq. [102] can be found in Ref. 45. 3 1 For the O + H2 system, the lowest P2,1,0 and D potential energy surfaces become degenerate in the product region.172 The product region is also the region of nonzero spin-orbit coupling. Therefore, the simplest treatment of the same in Eq. [102] for all electronic states is sufﬁciently accurate, as has been veriﬁed in one-dimensional model studies,45

i(x, y, Â, t) = i(x, y, Â, t)(x, y, Â, t) [103] Dynamics with the Globally Approximated Quantum Potential (AQP) 335

The dynamics of is governed by a so-far unspeciﬁed potential, Vd,

∂ ı = (Tˆ + V ) [104] ∂t d

From Eq. [99] the time-evolution of the “population” prefactor is governed by

∂ ı =−ı(vT ∇) + (Tˆ + Tˆ )I + (V − V I) [105] ∂t c d

The operator Tˆ c couples r with the ﬁrst derivatives of ,

r ∂ ry ∂ r ∂ Tˆ =− x − − Â [106] c M ∂x m ∂y ∂Â

For an efficient trajectory implementation, the evolution of and is determined approximately. Equation [104] is solved using the AQP approach of the section “Dynamics with the Globally Approximated Quantum Potential”. Equation [105] is simplified as follows: (1) The first term on the RHS is combined with the LHS to give the time-derivative of along a trajectory; (2) The ˆ ˆ effective potential Vd is chosen to minimize the effects of Tc, and T acting on , and these derivative terms are neglected. Typical initial conditions for a wave th { = = } function—a single (k ) populated electronic state—are k,i ıik .In this case, will be smoother if the elements of the potential part in Eq. [105] are small. Minimization of these elements suggests the following form of the effective potential: | V = ij i j ij Vd | [107] i i i

The simpliﬁed equation for becomes

d ı = (V − V I) [108] dt d

A single ensemble of trajectories evolves under the combined inﬂuence of the quantum potential as well as the “average” classical potential, which includes all matrix elements of V weighted by their respective populations i|j. The formulation of Eqs. [104] and [105] is equivalent to the time-dependent Schrodinger¨ Eq. [99] if solved exactly. 336 Semiclassical Bohmian Dynamics

3 1 Application: The Four Electronic State Dynamics for O( P, D) + H2 The formalism of the previous section is applied to the four state model describing the spin-orbit interaction induced intersystem crossing of the O + H2 → OH + H reaction, developed by Hoffmann and Schatz.172 The total number of the oxygen atom electronic states, including the spin-orbit interaction, is 15— 1 1 3 ﬁve D2, one S0, and nine P states. This number is reduced to four by ignoring high-energy states and parity decoupling. The three triplet states and one singlet states of the reactants convert into four doublet states of the products—two 2 2 1/2 and two 3/2 states. The diabatic Hamiltonian of Hoffmann and Schatz 172 is used in calculations. Two of the three triplet surfaces are the 3A surface (both identical in the diabatic representation) of Ref. 170. The other triplet is the 3A surface from the same paper. The singlet surface is the 1A of Ref. 171. The surfaces are sketched in Figure 11. The wave-packet reaction probabilities on the diabatic surfaces are computed as sums across trajectories in the product region, d = | (k) |2 Pi i(x ) wk [109] k, prod where the index k labels trajectories and i labels the surfaces. The ﬁrst and third 3 3 surfaces correlate with P2 and P0 states of oxygen. They remain coupled 2 2 in the product region, correlating with the 3/2 and 1/2 states of OH. 3 1 The other two surfaces, correlating with P1 and D2 states of oxygen, also

60 O+H OH+H 1 2 D Vc x 5000 40 3 2Π P0 1/2 0.2 3 0.4 P1 0.2 2 3 Π

Energy [kcal/mol] 20 3/2 P2

2Π 3 P 0 -10 -5 0 5 10

Reaction coordinate [a0] Figure 11 Diabatic electronic states. 1A correlates with 1D state (thin solid line). 3A 3 3 3 3 correlates with P2 and P1 states (thick solid line). A correlates with P0 state (dashed line). The asymptotic coupling is shown with the dot-dashed line. Asymptotic splittings of the adiabatic potential energy surfaces are indicated in the inserts in kcal/mol. Dynamics with the Globally Approximated Quantum Potential (AQP) 337

2 2 are coupled in the product region, correlating with 3/2 and 1/2 states of OH. Taking the sign of the asymptotic couplings into account, the adiabatic 2 wave functions of the lower doubly degenerate state, 3/2, are

− + a = 1√ 3 , a = 2√ 4 [110] 1 2 2 2

2 and of the higher doubly degenerate state, 1/2, are

+ − a = 1√ 3 , a = 2√ 4 [111] 3 2 4 2

The propagation is terminated once the adiabatic probabilities, a = | a |2 Pi i (xk) wk [112] k, prod reach constant values. The initial wave packet is a direct product of a Gaussian in the translational coordinate and the ground state of the ith potential energy surface in the internal degrees of freedom 2˛ 1/4 (x, y, Â, 0) = exp −˛(x − x )2 + ıp (x − x ) (y) [113] i 0 0 0

Wave packets initially in the triplet state, i = 1, 2, 3, were considered. The remaining wave-function components are zeros— j =0,j=/ i. The parameters of the translational wave packet are ˛=4, x0 =7 and p0 =16. The vibrational 177 eigenstate is taken as the ground state of a Morse potential, Vm, approximating the following H2 interaction:

2 Vm = D(1 − ) ,(y) = exp[−z(y − ym)] [114]

D = 0.169,z= 1.06,ym = 1.41 [115]

The initial vibrational wave function is √ − / (y) = 2k 1 2 exp(−),= 2Dm/z [116] where k is the normalization constant, k = 8.6286 × 10−2. The AQP and classical calculations were performed using 2000–4000 trajectories with the Sobol pseudo-random sampling of the initial positions in three dimensions using normal deviates in x and y and uniform deviates in 338 Semiclassical Bohmian Dynamics

178 179 cos Â. A single surface calculation has shown that rÂ is small compared with the radial components; therefore, it was set to zero in the trajectory calculations. The classical results are obtained by setting the quantum potential to zero, Q=0. Approximation to the radial components rx,ry were made using the basis Á ={1,x− x0,y− ym,(y)} This basis is exact for the Morse poten- 179 tial eigenstates, which improves the asymptotic description of H2 fragment. Evaluation of the potential energy surfaces and couplings was the most expensive part of the trajectory calculation. The time-dependent quantum calculations were performed using the split- operator method180,181 on a 256 × 256 grid for the distances and using 60 discrete variable representation points182 for the angle. The action of the potential part of the Hamiltonian,

exp(−ıVdt) = V˜ , V˜ = MMT [117] is accomplished by diagonalizing the potential matrix. The matrix M consists of the eigenvectors of V. The elements of the diagonal matrix are functions of the eigenvalues i of the potential matrix V, ii = exp(−ıidt). The matrices V˜ were stored for each grid point, whose total number was around 4 × 106. The time-dependent reaction probabilities and populations of a wave 3 packet initialized on the second surface ( P1) with translational momentum p0 = 16 are shown in Figure 12. The diabatic probabilities of the surfaces two and four obtained using QM and approximate propagation are shown in Fig- ure 12(a). The probabilities begin to oscillate between these two diabatic surfaces as the reactive part of the wave packet evolves in the product channel. The reaction probability on the other two diabatic surfaces is negligible. The adiabatic probabilities, shown in Figure 12(b), approach constant values after approximately t =2300. Panels (c) and (d) show the population of the diabatic surfaces as functions of time. Note that population on the surfaces one and three remains below a few percent at all times. The AQP results with the exponential basis function are in good agreement with the QM results. For this high-energy wave packet, the classical dynamics of also agree well with the QM results at long times. Overall, it was found (for a range of the collision energies) that the coupling of the triplet surfaces to the singlet in O + H2 has essentially no effect on the total reaction probabilities. The coupling affects only the relative populations of the doublet states.

TOWARD REACTIVE DYNAMICS IN CONDENSED PHASE

Semiclassical methodology presented in this review ultimately is directed toward the description of QM effects in large reactive molecular systems with Toward Reactive Dynamics in Condensed Phase 339

1 0.3 (a) (c) 3 3 AQP P P 1 0.2 1 AQP QM 0.5 diabatic P 0.1 Q=0 Population 1 D 1 Q=0 2 0 D 2 0

2 (b) Π (d) 3 1/2 P0 2 Π 0.02 0.1 3/2 adiabatic P Population 3 P2 0 0 1000 2000 3000 0 1000 2000 3000 Time [a.u.] Time [a.u.] 3 Figure 12 Dynamics of the wave-packet initialized on P1 with p0 = 16 (solid lines 3 show QM results): (a) Reaction probabilities on the diabatic surfaces correlating with P1 1 (AQP/classical results are shown with circles/dash) and with D2 (AQP/classical results are shown as squares/dot-dash); (b) Reaction probabilities on the adiabatic surfaces 2 2 correlating with 3/2 (AQP/classical results are shown with circles/dash) and with 1/2 3 1 (AQP/classical results are shown as squares/dot-dash); (c) Populations on P1 and D2 3 (legend is the same as in a)); (d) Populations on P2 (AQP/classical results are shown 3 with circles/dash) and with P0 (AQP/classical results are shown as squares/dot-dash). hundreds of atoms. Classical molecular dynamics provides a reasonable general picture of chemical reaction dynamics in most systems of practical interest. However, the isotope effect measurements and the comparison of typical reaction energies and zero point energies show that quantum mechanical effects play an important role in many systems, particularly in reactions of proton transfer (some examples can be found in Refs. 2, 83, 183). The adequate theoretical description of quantum effects is proved to be a very challenging task. The collective research during the last three decades points to three primary reasons: (1) many reactions occur at the time scale much longer than that of a typical quantum dynamics simulation; (2) the forces acting on the reactive species are not well represented by simple harmonic approximations; and (3) quantum effects, such as the zero point energy, require an ensemble description rather than an individual trajectory description. This section describes a computational approach that incorporates QM effects into the molecular dynamics framework and is based on Bohmian trajectories. A typical view of a reaction occurring in a condensed phase, in other 340 Semiclassical Bohmian Dynamics

Reactants Products

Reaction coordinate

Environment Figure 13 Schematic representation of a reaction occurring in a molecular environment, such as in a liquid, nanostructure, or a biological system. words a reactive coordinate is coupled to a molecular environment or a “bath,” is sketched on Figure 13. The main QM effects that can be significant in such a reactive system include the following: (1) the motion along the reactive coordinate can be influenced by QM tunneling; and (2) the zero point energy—or, more generally, localization energy—in the reactive and bath degrees of freedom and energy flow among them. The energy changes in the reactive mode obvi- ously will influence the probability of the reaction. The next section describes the approximate treatment of the zero point energy in a high-dimensional anharmonic bath on a long timescale.184 The “Bound Dynamics with Tunneling” section presents a mixed wave function approach to describe tunneling in a bound reactive coordinate.185 This approach is compatible with the trajectory framework and extendable toward full QM description. In the future, these two approaches will be combined to model dynamics in condensed phase systems.

Stabilization of Dynamics by Balancing Approximation Errors Formally, the zero point energy is the energy of the lowest eigenstate. It is a sum of the kinetic and potential energy contributions because of localization, or “ﬁnite size,” of the eigenstate. In the Bohmian formulation, the kinetic energy contribution to the zero point energy is given by the expectation value of the quantum potential,

2 2 2 Q=− A|∇2A= ∇A|∇A= r2 [118] 2m 2m 2m Toward Reactive Dynamics in Condensed Phase 341

This can be called the “quantum” energy in contrast to the “classical” energy,

p2 Hˆ − Q= +V [119] 2m

The concept of “quantum” energy can be applied to any localized function, not just to the ground state. Efﬁcient—scalable to high-dimensionality—and stable description of the quantum energy, Q, is the goal of this section. The LQF description is the cheapest and is exact for Gaussian wave packets. It gives the exact quantum energy for all eigenstates and coherent states of the harmonic oscillator. In anharmonic systems, the LQF describes Q only on a short timescale (depending on the anharmonicity). For an eigenfunction in Bohmian formalism, the quantum force exactly cancels the classical one, resulting in stationary trajectories. In the LQF, exact cancellation generally does not happen; a net force acting on the trajectories, representing wave function tails, might be large causing these “fringe” trajectories to start moving. Sooner or later, the movement will affect the moments of the trajectory distribution, resulting in a decoherence of the LQF trajectories and in a loss of the zero point energy (or quantum energy) description. Because the total energy is conserved in the LQF, as shown in the previous section, the quantum energy will be transferred into classical energy of the trajectory motion. Below, the LQF ideas are extended to prevent this “loss” of quantum energy for small anharmonic- ities, or more precisely for small nonlinearity of the classical and nonclassical momenta. Thus, a cheap and stable zero point energy description over many oscillation periods is provided.

Approximation of Gradients = The derivation is given in atomic units ( 1) for a system described in Ndim Cartesian coordinates x = (x,y...) in vector notations. The classical and nonclassical momenta are vectors

− p =∇S, r = A 1∇A [120]

In LQF, linear approximation to each component of A−1∇A is made. The trajectory momenta are known but never used in the ﬁtting procedure. The nonlinearity of p, however, quickly translates into nonlinearity of r. In this section, r and p are treated on equal footing. The deviations of both quantities from linearity are used to limit accumulation of propagation error with time. In the multidimensional case, the time evolution of r and p along a trajectory, given by Eq. [17] in one dimension, is determined as

dp (∇∇)r m∇V + m = (r ∇)r + [121] dt 2 342 Semiclassical Bohmian Dynamics

dr (∇∇)p −m = (r ∇)p + [122] dt 2

For practical reasons, the spatial derivatives of r and p in Eq. [122] need to be determined from the global approximations to these quantities. The trajectory- specific quantities r and p are found by solving Eq. [122]. In general, the re- lations Eq. [120] will not be fulfilled in dynamics with approximations to the RHS of Eq. [122]. Similar to the AQP method, p and r are expanded in a basis set of functions for the purpose of derivative evaluations. The expansion coefficients are determined from the minimization of the error functional using the total energy conservation as a constraint. The total energy can be defined without spatial derivatives as

p p r r E = +V+ [123] 2m 2m

The energy conservation constraint will couple the fitting procedures of A−1∇A and p. The linear basis f =(x,y...,1) is considered here. Formulation for a general basis is given in Ref. 184. The functions = r = r r˜x cx f, r˜y cy f ... [124] approximate components of the vector A−1∇A. The functions p p p˜ x =cx f, p˜ y =cy f ... [125] approximate components of the vector p. To express the energy conservation condition, dE/dt = 0, the fitting coefficients are arranged into matrices Cr and Cp,

r = r r C [cx , cy ...] [126]

p p p C = [cx , cy ...] [127]

Differentiating Eq. [123] with respect to time and using Eq. [122] with the derivatives obtained from Eqs. [124] and [125], the energy conservation condition becomes dE r 0 (Crp − Cpr) = = 0 [128] dt m Toward Reactive Dynamics in Condensed Phase 343

Quantity r 0 denotes a vector of nonclassical momentum extended to the size + of the basis Ndim 1

0 r = (rx,ry ...,0) [129]

The matrices Cr and Cp are symmetric. The least-squares ﬁt of A−1∇A and p in terms of a linear basis with the constraint of Eq. [128] is minimization of the functional

− I =A 1∇A − Crf 2+p − Cpf 2+2r 0 × (Crp − Cpr) [130] with respect to the fitting coefficients and with respect to the Lagrange multiplier . The optimal coefficients solve a system of linear equations, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ MODp C r B r ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ OMDr ⎠ ⎝ C p ⎠ = ⎝ B p ⎠ [131] D p D r 0 0

In Eq. [131] the following matrices and vectors are introduced: (1) M is the × block-diagonal matrix of the dimensionality NdimNb NdimNb with the basis =⊗ function overlap matrix S f f as Ndim blocks on the diagonal and zeros otherwise; (2) O is a zero matrix of the same size as M; and (3) the elements of the vectors C r, C p, B r, B p, D r, and D p are the elements of the matrix Cr, Cp, Br, Bp, Dr, and Dp, respectively, listed in a column after column order. Cr and Cp are given by Eqs. [126] and [127]. The remaining four matrices are deﬁned as

r =−1 ∇⊗ T p = ⊗ B 2 ( f ) , B f p [132] Dr =−r 0 ⊗ r, Dp =p 0 ⊗ r [133] where p 0 denotes a vector of classical momentum extended to the size of the basis

0 p =(px,py ...,0) [134]

The fitting of A−1∇A is the same as in the LQF procedure, except that now it is coupled to the least-square fit of p. Formally, the total size of the matrix in + Eq. [131] is 2NdimNb 1. Its structure allows one to invert the matrix on the left-hand side by performing a single matrix inversion of the block S of the size 178 2 Nb. Thus, the cost of the quantum force computation scales as NtrajNdim. This is essential for efficient high-dimensional implementation. 344 Semiclassical Bohmian Dynamics

From the conceptual point of view, the appealing features of the outlined approximation scheme are the energy conservation of Eq. [123] and equal- footing treatment of r and p. These features lead to a fuller use of trajectory information in the approximation. However, because of the ﬁnite basis representation, Eq. [122] are truncated effectively; for the linear basis, the laplacian terms are zeros. Another deﬁciency is that r computed along the trajectories is, in general, different from A−1∇A computable, in principle, from the trajectory positions and Eq. [20]. Both are addressed in the next section.

Correcting for the Effect of Linearization on Dynamics Approximation based on the linear basis f is cheap and gives exact dynamics in the important limit of Gaussian wave functions evolving in locally harmonic potentials. However,it results in a “cold” truncation of the time-evolution equations [122] because the second derivatives of the basis functions are zeros. Such truncation of differential equations leads to dynamics that are unstable with respect to small deviations of r and p from nonlinearity. It can be compensated by introducing additional terms into Eq. [122]. These additional terms depend on the difference of exact and approximated values of p and r and balance errors because of the linear basis in the ﬁrst order of the nonlinearity parameters. The explicit form is determined from the analytical models and has no adjustable parameters. Consider the lowest order nonlinearities of the classical and nonclassical momenta (in one dimension)

∇A − 2 p = p + p x + x2,r= ,A= e ˛x |1 + ı × (x − x )| [135] 0 1 A 0

Analysis of the short-time dynamics with spatial derivatives obtained from linearization of quantities given by Eq. [135] has shown that the following approximate equations of motion: dp ∇2r m +∇V = r∇r + = r∇r˜ + 2∇r˜ × (r − r˜) + O(ı4) [136] dt 2 dr ∇2p −m = r∇p + = r∇p˜ + 2∇r˜ × (p − p˜ ) + O(ı3) [137] dt 2 cancel the leading errors in the nonlinearity parameters ı and . In the multidimensional case, derivatives ∇r˜ and ∇p˜ of the approximate functions generalize into matrices Cr and Cp, given by Eqs. [124] and [125]. The approximate time-evolution equations become

dr −m = Cpr + 2Cr p − pﬁt [138] dt Toward Reactive Dynamics in Condensed Phase 345

p m +∇V = Crr + 2Cr(r − rﬁt) [139] dt

In the previous section functions (r˜x, r˜y ...) approximate components of A−1∇A, and their determination is coupled to the approximation of p in terms fit of (p˜ x, p˜ y ...) by the energy conservation condition. On the other hand, r ap- fit fit proximates r. In general, there is a difference between the fittings (rx ,ry ...) fit fit and (r˜x, r˜y ...). Approximations r and p should be such that the stabilization terms do not contribute to the total energy of the trajectory ensemble and do not change the normalization of the wave function. The latter condition can be written as dr/dt=0. Separate least squares fits of r and p in terms of the linear basis by minimizing r − rfit2 and p − pfit2, respectively, satisfy both these requirements. (General basis is discussed in Ref. 184.) Physically, the stabilization terms provide “friction” opposing the growth of the discrepancies between the trajectory-dependent r and p and their linear fits with time.

Numerical Examples of the Long-Time Zero-Point Energy Description The ﬁrst illustration of the stabilization method is the description of zero point energy, or more generally, of the quantum energy Q for a nonrotating hydrogen molecule, as in Ref. 42. The classical potential V is the Morse oscillator with 17 bound states. The system is one dimensional and is described in atomic units scaled by the reduced mass of H2 to have m=1. The initial wave packet is a Gaussian wave function mimicking the ground state of H2, 2˛ 1/4 (x, 0) = exp −˛(x − x )2 [140] m

= −2 = where ˛ 9.33 a0 and xm 1.4 is the minimum of the Morse potential. Figure 14(a) shows positions of the trajectories obtained with the LQF method described in the “Approximate of Gradients” section and the stabilized AQP dynamics of this section. The plot illustrates the effect of runaway trajectories leading to the trajectory “decoherence” and transfer of quantum potential energy into classical energy, which is described at the beginning of “Stabilization of Dynamics by Balancing Equation Errors”. Note the “fringe” LQF trajectory immediately leaving the trajectory ensemble toward the dissociation region. This behavior quickly drives variance of the wave packet and, thus, the quantum potential and the quantum force to zero, as shown on the lower panel. Trajec- tories obtained with the stabilization procedure maintain their coherence and describe the quantum energy Q very well. The effects of stabilization terms are shown in Figure 14(a). The oscillatory behavior of the two central trajectories is a consequence of propagating the Gaussian function of Eq. [140] rather than the eigenstate. These oscillations correlate with the oscillations in Q. Behavior of the outlying trajectories of the stabilized AQP dynamics shows 346 Semiclassical Bohmian Dynamics

2.5 LQF (a) stabilized 2

Position 1.5

1 021

QM 2 LQF (b) stabilized

0 021 Time

Figure 14 Quantum energy for the H2 molecule: (a) Trajectory positions as functions of time obtained using the LQF and the stabilized dynamics; (b) Average quantum potential as a function of time obtained using the LQF (dash), the stabilized dynamics (solid line), and the exact QM propagation (circles).

higher frequency oscillations superimposed on the oscillations of the central trajectories. These additional oscillations are a result of the corrections of the “friction force” introduced into the equations of motion. The trajectory propagation was performed with the third order Milne predictor-corrector algorithm,178 and the accuracy of Q was checked for 200 oscillation periods of H2. Application of approximate methods to high-dimensional systems must be validated by tests that can be compared with exact QM results, which generally implies separable Hamiltonians or harmonic potentials. For multidimensional testing of the stabilized AQP method, the average quantum energy has been computed for a model potential. The separable model potential consists of the Eckart barrier, centered at the zero of the reaction coordinate, and the Morse oscillators in the vibrational degrees of freedom. The vibrational degrees of freedom are the same as in the one-dimensional application. The parameters of the barrier mimic the H + H2 system and are given in Ref. 42. The initial multidimensional wave packet is deﬁned as a direct product of a Gaussian in the reaction coordinate −1 1/4 2 (x, 0) = (2˛ ) exp −(x − x0) + ıp0(x − x0) [141] Toward Reactive Dynamics in Condensed Phase 347

with parameter values { =6,x0 =4,p0 =6}, and Gaussian functions in the vibrational degrees of freedom, given by Eq. [140]. After the wave packet in the reaction coordinate bifurcates, Q of the system is equal to that of the vibrational modes. In general, the AQP ﬁtting depends on the choice of the basis. Therefore, rotation of the system of coordinates relative to the normal mode coordinates introduces effective coupling between the degrees of freedom. Dynamics in the rotated system of coordinates provides a more stringent test of the method. In the rotated system of coordinates, the wave packet and the classical potential are nonseparable. In addition, the numerical procedure of the quantum force computation and trajectory propagation in the rotated system used no information about the separability of the original Hamiltonian. To compare the quantum energy description for different numbers of vibrational degrees of freedom, the rotation matrix, written here for clarity as = Ndim 4, is speciﬁed by the parameter Ä, ⎛ ⎞ ˛ −Ä −Ä −Ä ⎜ ⎟ ⎜ Ä 1 + ˇˇ ˇ⎟ = ⎜ ⎟ [142] ⎝ Äˇ1 + ˇˇ⎠ Äˇ ˇ1 + ˇ

= − − 2 = − − with ˛ 1 (Ndim 1)Ä and ˇ (˛ 1)/(Ndim 1). This transformation does not change the diagonal kinetic energy operator, provided that masses for all dimensions are equal. The stabilized dynamics implemented with the linear basis is invariant under such transformation. Numerical performance of the stabilized dynamics has been tested for up to 40 dimensions with random Gaussian sampling of initial positions.178 Calculation of the quantum potential and force is dominated by the computa- 2 tion of the moments of the trajectory distribution, which scales as NtrajNdim. Calculation of the global linearization parameters is performed at each time 4 step for the ensemble of trajectories with the cost of Ndim. The average quantum potential divided by the number of the vibrational degrees of freedom, − Q /(Ndim 1) is shown in Figure 15. The difference in the quantum energy at short times is from the quantum energy in the reactive coordinate, which is not included in the QM calculation. At later times, the trajectory results reproduce the changes in localization of the vibrational wave functions well, and the semiclassical accuracy is essentially independent of the dimensionality. Propagation of 2 × 104 trajectories gave a relative difference in the quantum energy of around 0.5%, with a standard deviation of about 1%, for all numbers of vibrational degrees of freedom. The largest calculation for = Ndim 40 took two hours on a single processor of a desktop workstation. 348 Semiclassical Bohmian Dynamics

5 -1) dim

4.5 /(N

4 0 1 2 Time Figure 15 Average quantum potential per vibrational degree of freedom for a Gaus- sian wave-packet scattering on the Eckart barrier in the presence of Ndim − 1 Morse oscillators. Semiclassical results are shown for Ndim ={20, 40} with circles and dash, respectively. The QM result for long times is shown with a solid line.

Bound Dynamics with Tunneling In general, semiclassical methods are not intended for quantitative description of “hard” quantum effects, such as tunneling and interference. In this regime, the exact quantum trajectory dynamics is numerically unstable because the exact quantum potential of Eq. [5] is singular because of the nodes of (x, t). The AQP trajectories are stable, but quantum forces quickly become inaccurate, trajectories decohere, and dynamics becomes essentially classical. For an initially localized wave function evolving in the double well potential—a prototype model of the proton transfer in condensed phase—the key dynamics feature is tunneling between the the “reactant” and the “product” wells. 185 To capture this behavior in the trajectory framework compatible with the trajectory representation of high-dimensional bath, the mixed form of the wave function can be used similar to nonadiabatic dynamics,

(x, t) = 1(x, t)1(x, t) + 2(x, t)2(x, t) [143]

Wave functions 1 and 2 evolve independently of each other in the asymptotic potentials of reactants and products, V1 and V2, respectively, according to the Schrodinger¨ equation

2 2 ∂j ∂ j ı =− + V ,j= 1, 2 [144] ∂t 2m ∂x2 j j

For chemical systems, asymptotic dynamics can be sufﬁciently simple to be accomplished semiclassically with two ensembles of the AQP trajectories. Toward Reactive Dynamics in Condensed Phase 349

Subsequently, we will use the superscripts to label quantities associated with trajectories in these two ensembles. The tunneling or population transfer between the two wells under the inﬂuence of the full potential V is accomplished through the complex amplitudes 1 and 2. These two functions will be represented in a basis f of Nb polynomials, for example, in the following Taylor basis f = (1,x,x2,...): n−1 n−1 = = + 1 cn(t)x ,2 cn Nb (t)x [145] = = n 1,Nb n 1,Nb

Multiplication of the Schrodinger¨ equation for the function , given by Eq. [143], by the components of the total basis of ,

= F (f11,...,fNb 1,f12,...,fNb 2) [146] and integration over x gives a linear system of differential equations for the basis coefﬁcients c,

dc ıS = Hc [147] dt

Equation [144] has been used to obtain the last result. Matrix S

S =F ⊗ F [148] is the time-dependent overlap matrix of the size 2Nb. The right-hand side of Eq. [147] is a “partial” Hamiltonian matrix H,

H =F ⊗ (K + ) [149] containing the derivatives of i as well as the potential energy terms. The matrix is constructed as the outer product of the basis function vector F, given by Eq. [146], and the vector K + whose elements are given by 2 2 ∂ f ∂f ∂j K =− i + 2 i [150] i 2m ∂x2 j ∂x ∂x

i = (V − Vj)Fi [151]

= ≤ ≤ = + ≤ ≤ where j 1 for 1 i Nb and j 2 for Nb 1 i 2Nb. 350 Semiclassical Bohmian Dynamics

In the AQP implementation, the integrals in the diagonal blocks of the matrices can be expressed readily as sums over trajectory weights,

| |2 = (k) j(x, t) o(x)dx o(xt,j )wk,j [152] k k indexing the trajectories and j the asymptotic channel. There are no kinetic = energy terms in the minimal basis set Nb 1 related to the two-state representation of the system.38 The AQP dynamics is exact for Gaussian wave functions governed by the harmonic asymptotic potentials V1,2. Functions 1,2 and their derivatives are known, analytically, in this case, and the whole approach becomes exact for a sufﬁciently large basis for 1,2. To use the minimal basis for in the semiclassical regime, we can deﬁne anharmonic V1,2 and use the stabilized dynamics of r and p of “Stabilization of Dynamics by Balancing Equation Errors” implemented with the linear basis. Then the derivatives of j at the trajectory positions are ∂j = (j) + (j) rk ıpk j [153] ∂x x=x(j) k

In the off-diagonal matrix elements 1|o(xˆ )|2, the linearization parameters (1) of r and p are used for evaluation of 2 at the positions xk of the ﬁrst trajectory set and vice versa. The linearization parameters of r and p already are available from the propagation. The mixed-type integrals are evaluated in a symmetrized fashion, | | = (j) (j) (j) 2 1 o(xˆ ) 2 o xk wk z xk [154] j=1,2 k

The ratio of the wave functions, z(x) = 2(x)/1(x), is found from the lineariza- tions of r(j) and p(j) as well. The overlap matrix S is Hermitian and, generally, time dependent. The Hamiltonian matrix H is complex and, generally, nonsym- metric. Conservation of the total wave function normalization is not guaranteed with the semiclassical propagation of j. The numerical example is given for a strongly anharmonic double well. This is the potential of the proton transfer coordinate from the model of Topaler and Makri,186 which became the benchmark for approximate and semiclassical methods. With the particle mass rescaled to m = 1, the potential is

V = 14x4 − 20x2 + 50/7 [155] Toward Reactive Dynamics in Condensed Phase 351

The initial wave function is a Gaussian wave packet localized in the left well, 2 1/4 2 (x, 0) = exp − x − q(1) + ıp(1) x − q(1) [156] 1 0 0 0

(2) =− (1) The initial “image” Gaussian 2 has the same form as 1 with q0 q0 (2) =− (1) = and p0 p0 . The corresponding population functions are 1(x, 0) 1 and 2(x, 0) = 0. The wave packet parameters are chosen so that (x, 0) mimics the ground state localized in the left well: = 4.47,x0 =−0.77, and p0 = 0. The parameters of the asymptotic potential V1 are chosen to minimize its 2 deviation from the full potential, (V − V1) , weighted by the initial wave 2 function density | (x, 0)| . The product asymptotic potential V2 is a reﬂection of V1, V2(x) = V1(−x). Two asymptotic potentials – (A) the quadratic potential,

k(x + q )2 VA = 0 [157] 1 2 and (B) the Morse oscillator, − + 2 B = z(x q0) − V1 D e 1 [158] illustrate full QM and semiclassical implementations. The parameters are k = 34.500,q0 = 0.741, and D = 28.204,z= 1.114,q0 = 0.836. Figure 16(a) shows the full potential and its asymptotic approximations, A B V1 and V1 , as well as the initial wave function density in arbitrary scale. The initial energy of the wave packet constitutes 59% of the barrier height. The probability for the particle to be in the right well

∞ P(t) = | (x, t)|2dx [159] 0 obtained with the asymptotic dynamics in the quadratic potential, is shown in Figure 16b. To account for the difference of the full potential V and its harmonic asymptotes in the region of the steep wall, a relatively large basis for 1 and 2 of four-ﬁve functions is necessary to approach the exact QM results. A The advantage of the quadratic functional form for V1 is that the trajectory dynamics of the Gaussians 1,2 and, consequently, the evaluation of matrix elements are exact. Therefore, this is the full QM limit of the mixed wave function representation approach. The semiclassical description of the same system consists of the stabilized B AQP dynamics in the asymptotic potential—the Morse oscillator of V1 , which 352 Semiclassical Bohmian Dynamics

Coordinate

10 Energy (a) 0 -1 0 1 1 (b) Probability

0 0 10 20 1 (c) Probability

0 0 10 20 Time Figure 16 Dynamics in the double well potential. The oscillation period in the asymptotic well is approximately 1.4 atomic units. Panel (a) Full potential V (solid line) and A B its quadratic, V1 (dot-dash), and Morse, V1 (circles), asymptotes. Initial wave function density, | (x, 0)|2, in arbitrary units is also shown with a dash; Panel (b) Probability of A finding the particle in the product well as a function of time with dynamics defined by V1 . The quantum and mixed representation results for Nb = 1 and Nb = 5 are shown with the solid line, dash and circles, respectively; Panel (c) Probability of finding the particle B in the product well with trajectory dynamics defined by V1 . The quantum and mixed representation results with semiclassical stabilized dynamics are shown for Nb = 1 and Nb = 2 using thick solid line, dash and thin solid line, respectively.

A is closer to the full potential in the well region than V1 , and a small basis for B 1,2. Dynamics under the inﬂuence of V1,2 accounts for the anharmonicity of a single well through 1,2, as illustrated in Figure 16(c). Simple coordinate- = independent prefactors (Nb 1) capture oscillations in the probabilities rather = accurately. Nb 2 gives agreement with the quantum result of about the same = quality as Nb 1. To summarize, a combination of small basis set for with the stabilized AQP dynamics should be appropriate for coupled system/bath semiclassical systems. The normalization conservation, in principle, can be included into the ﬁtting of r and p in the AQP dynamics, providing a connection between the basis set and the trajectory components of the total wave function. This issue as well as multidimensional applications will be investigated in the future. Conclusions 353

CONCLUSIONS

The presented review focused on the semiclassical approximations to quantum dynamics of chemical systems developed in recent years and is based on the Madelung–de Broglie–Bohm formulation of the time-dependent quantum mechanics. The appeal of the Bohmian formulation stems from the classical-like picture of quantum-mechanical evolutions, which are interpreted using the trajectory language. The wave function is replaced by an ensemble of point particles that follow deterministic trajectories and obey a Newtonian equation of motion. The quantum effects are represented by a nonlocal quantum potential that enters the Newton equation and couples the evolution of different trajectories in the Bohmian ensemble. This classical-like trajectory point of view allows one to treat various degrees of freedom, for instance light electrons and heavy nuclei, on the same footing. This naturally leads to families of semiclassical and mixed quantum-classical approximations. The trajectory representation offers several advantages over the traditional grids and basis sets. Bohmian trajectories follow the quantum-mechanical distributions, avoiding the unnecessary computational effort that often is taken to treat regions of very low quantum density. This feature of the Bohmian formulation allows one to eliminate the exponential scaling of the computational effort with system dimensionality, which is a common feature of the conventional approaches. Trajectories are easy to propagate using the tools developed for classical molecular dynamics. In contrast to the semiclassical techniques that also use the trajectory language, Bohmian mechanics is, in principle, exact and can be converged arbitrarily close to the accurate quantum-mechanical answer. At the same time, the exact Bohmian trajectory dynamics has proven expensive and unstable for general systems. This is a result of difficulties in evaluating the quantum force on an unstructured trajectory grid and because the behavior of the quantum trajectories is sensitive to the accuracy of the force. The Bohmian formulation of quantum dynamics is most useful as a computational tool when it is implemented approximately with semiclassical systems. There, it affords a description of the leading quantum effects in high dimensions and elegantly couples quantum and classical degrees of freedom in a unified framework. Bohmian mechanics offers a solution to the trajectory branching problem in the quantum-classical simulation by creating a new type of the quantum backreaction on the classical subsystem. The Bohmian backreaction uniquely is defined, computationally simple, and directly relates to the full classical limit. Branching of the quantum-classical trajectories is achieved in the Bohmian approach by coupling the classical subsystem to a single quantum particle in the Bohmian ensemble. In the quantum-classical Ehrenfest approximation, which is the most common approach, a single average classical trajectory is generated. In contrast, an ensemble of quantum-classical Bohmian trajectories is created 354 Semiclassical Bohmian Dynamics for a single initial quantum-mechanical wave function. Traditionally, trajectory ensembles are produced using a variety of ad hoc surface hopping procedures. The Bohmian quantum-classical method uniquely is defined and gives results that are similar to surface hopping. A related and conceptually appealing hybrid quantum-classical theory can be derived using the phase space representation of quantum-mechanical density matrices. The phase space picture provides an excellent starting point for a hierarchy of approximations generated by closures; the reference state is provided by the phase space description of the quantum harmonic oscillator with classical trajectories. The formulation in terms of the partial hydrodynamics moments reproduces the dynamics of the harmonic system of coupled light and heavy particle, for which the exact closure can be obtained. The independent Bohmian trajectory methods involve propagation of high-order derivatives of the wave function phase and amplitude. The propagation is carried out in real space with the derivative propagation method and in complex coordinate space using Bohmian mechanics with complex action. Similarly, the Bohmian trajectory stability approach evolves the wave function phase and stability matrix. Independent trajectories are particularly appealing because of the trivial parallelization of the computational effort. The true computational cost develops with the need for at least the second derivatives of the potential and because of the polynomial scaling of the equations with the system dimensionality. Truncation of the independent trajectory hierarchy at a low order directly relates to to the semiclassical methods, such as the WKB approximation. In contrast to the semiclassical schemes, the higher orders of the Bohmian mechanics converge to the exact quantum mechanics. The independent Bohmian trajectory methods can be used to compute a stationary wave function using a single or very few trajectories to obtain energy eigenvalues in the spirit of Diffusion Monte Carlo and to capture quantum interference from multiple paths in the complex plane. The approximate quantum potential technique is designed to describe the dominant quantum effects with essentially linear scaling. Here, the quantum potential is defined from the moments of an ensemble of trajectories for the entire space or for a few subspaces. An approximate quantum potential defined this way allows one to compute the quantum force analytically. Simple global approximations to quantum dynamics describe zero-point energy in anharmonic systems very well. Combined with the space-prefactor functions, such global approximations capture “hard” quantum effects, such as deep tunneling and nonadiabatic dynamics. This strategy enables progress from gas-phase reactions to studies of quantum chemical dynamics in condensed phase. All-in-all, the Bohmian formulation of time-dependent quantum mechanics generates a very intuitive picture of quantum dynamics, provides a straightforward connection to classical mechanics, and creates exciting opportunities for the development of semiclassical approximations with a great potential for applications to complex chemical systems. Appendix A: Conservation of Density within a Volume Element 355

ACKNOWLEDGMENTS

The authors thank Tammie Nelson for proofreading the manuscript. OVP is grateful to Bob Wyatt for including the Bohmian formulation of quantum mechanics into his quantum chemistry class and acknowledges ﬁnancial support provided by the USA National Science Foundation, Department of Energy, and Petroleum Research Fund of the American Chemical Society. SG and VR acknowledge the Donors of the American Chemical Society Petroleum Research Fund and the Chemistry Division of the National Science Foundation for ﬁnancial support.

APPENDIX A: CONSERVATION OF DENSITY WITHIN A VOLUME ELEMENT

In a closed system, the probability of finding a particle within a volume element, (t), associated with a quantum trajectory remains constant in time, (x, t)ı(t) = w. This is demonstrated below for a multidimensional system of coordinates x = (x1,x2,...). To determine the time dependence of ı(t), one makes an infinitesimal displacement of position and velocity of a trajectory, v = dx/dt , defined by Hamilton’s equations of motion, to obtain

d m ıv =−∇2(V + U)ıx [A1] dt d ıx = ıv [A2] dt

Differentiating w with respect to time t and using Eqs. [A2] and [4] and the noncrossing property of quantum trajectories, ı = ıx1 × ıx2 × ... =/ 0, one obtains d d(x, t) dı(t) d(x, t) ((x, t)ı(t)) = ı(t) + (x, t) = ı(t) + (x, t) dt dt dt dt [A3] dıx 1 dıx 1 d(x, t) × 1 + 2 + ... ı(t) = + (x, t)∇× v ı(t) = 0 dt ıx1 dt ıx2 dt [A4] where ıv ∇× v = n [A5] ıx n n

This means that after discretizing the initial wave function (x, 0) through a set of trajectories with initial positions {x(i)}, velocities {v(i) =∇S(x(i), 0)/m}, (i) 2 (i) densities {(x , 0) = A (x , 0)}, and corresponding volume elements {ıi(0)} 356 Semiclassical Bohmian Dynamics for each trajectory, the probability in its volume element will be conserved: (i) (i) (x ,t)ıi(t) = (x , 0)ıi(0) = wi. In principle, Eqs. [A1] and [A2] give an independent way of ﬁnding the gradient of velocity and the volume element for a trajectory from the stability matrix evolution given by Eqs. [56] and [57]. In practice, their implementation might be too expensive; it requires the second derivative of the quantum potential (i.e., the fourth derivative of the density), as well as the second derivatives of the classical potential.

APPENDIX B: QUANTUM TRAJECTORIES IN ARBITRARY COORDINATES

For a general curvilinear system of coordinates {x}, the kinetic energy operator Tˆ is

1 † Tˆ = ∇ G∇ [B1] 2

Here ∇ is the gradient operator of a general form

∂ ∇i = fi(x) [B2] ∂xi and ∇† acts on the left. G is an inverse matrix of masses and moments of inertia; in general, G can have off-diagonal elements. In chemical applications, a system of coordinates often is chosen to eliminate derivative cross-terms in the Hamiltonian. This allows the asymptotic motion of fragments to be uncoupled. This means that the matrix G is diagonal. Typical coordinates for these applications are the Jacobi or Radau coordinates in spectroscopy or reactive scattering calculations.187 The following derivation is given for the diagonal form of G.It can be extended to the nondiagonal case in a straightforward manner. We use atomic units, = 1, throughout, and the -dependence is noted in which it is important for interpretation. For square-integrable wave functions, the Jaco- bian J = J(x) of the transformation from Cartesian coordinates to a given set of coordinates is taken into account. This allows Eq. [B1] to be rewritten 1 Tˆ =− ∇T G∇+dT G∇ [B3] 2 The components of the vector d are

∂fi fi ∂J di = + [B4] ∂xi J ∂xi Appendix B: Quantum Trajectories in Arbitrary Coordinates 357

The hydrodynamic or Bohmian form of the time-dependent Schrodinger¨ equation is based on the representation of a wave function in terms of real phase and amplitude or density (x, t) = A(x, t) exp (ıS(x, t)) = (x, t) exp (ıS(x, t)) [B5]

Substitution of Eq. [B5] into the Schrodinger¨ equation and separation into real and imaginary parts gives the following equations:

∂S 1 + (∇S)T G(∇S) + V + U = 0 [B6] ∂t 2 1 U =− ∇T G(∇A) + dT G(∇A) [B7] 2A and

∂ + (∇S)T G(∇) +∇T G∇S + dT G(∇S) = 0 [B8] ∂t

U is the quantum potential, which is, formally, the only 2 term becoming small in the classical limit →0. All other terms do not depend explicitly on .We can deﬁne a full time derivative

d ∂ ∂ ∂ = + (∇S)T G(∇) = + v [B9] dt ∂t ∂t i ∂x i i in the frame of reference moving with the velocity

= ∂S 2 vi Giifi [B10] ∂xi

Then, Eq. [B8] becomes d =− ∇T G(∇S) + dT G∇ [B11] dt

By differentiating Eq. [B6] with respect to xi and using Eq. [B9] one obtains 2 d ∂S + ∂fk ∂S + ∂ + = Gkkfk (V U) 0 [B12] dt ∂xi ∂xi ∂x ∂xi k k 358 Semiclassical Bohmian Dynamics

If derivatives of the phase are identiﬁed with the momentum,

∂S pi = [B13] ∂xi and the phase S is identiﬁed with the classical action function, then for any form of the gradient operator, Eqs. [B6], [B10], and [B12] give time-dependence of x, p, and S,

dx i = G f 2p [B14] dt ii i i dpi =− ∂ + − ∂fk 2 (V U) Gkkfk pk [B15] dt ∂xi ∂xi k

dS 1 = G f 2p2 − (V + U) [B16] dt 2 kk k k k

Equations [B14]–[B16] are consistent with classical equations of motion of a trajectory governed by the Hamiltonian

1 H = G f 2p2 + V + U [B17] 2 kk k k k

For a nondiagonal form of the matrix G, the equations of motion correspond to the Hamiltonian of Eq. [B17] with the single summation replaced by the double sum, kl Gklfkflpkpl. Importantly, the density of a wave function “carried” by a trajectory within the associated volume element J, = iıxi, or the trajectory weight

w = J [B18] is conserved in closed systems as has been the case in Cartesian coordinates,150

dw d dJ d = J + + J = 0 [B19] dt dt dt dt

This can be veriﬁed by using Eqs. [B9]–[B11] to deﬁne time derivatives in Eq. [B19],

dJ = (∇S)T G(∇J) [B20] dt Appendix C: Optimal Parameters of the Linearized Momentum 359

d ıv ∂f = k = ∇T G(∇S) + k G f p [B21] dt ıx ∂x kk k k k k k k

The weight conservation property means that one does not need to solve Eq. [B11] involving gradients of p and to determine the time dependence of the density. Moreover, the expectation value of an operator that is local in the coordinate representation, such as the wave packet probability, can be ˆ = found by simple summation across the trajectory weights, O n O(xn)wn. Equations [B14]–[B16] and [B19] give a local description of quantum dynamics with the exception of the nonlocal quantum potential U given by Eq. [B7]. This is the quantity that vanishes in the classical limit of small and large mass for nodeless wave function densities. We approximate U to make the quantum trajectory framework practical in large systems while retaining the dominant quantum effects. It is convenient to deﬁne U in terms of the nonclassical component of the gradient operator, ∇A(x, t) r = [B22] A(x, t)

The quantum potential in an arbitrary system of coordinates becomes 1 U =− rT G r +∇T G r + dT G r [B23] 2

APPENDIX C: OPTIMAL PARAMETERS OF THE LINEARIZED MOMENTUM ON SPATIAL DOMAINS IN MANY DIMENSIONS

In N dimensions for the linear ﬁtting functions, minimization of Eq. [82] with respect to expansions coefﬁcients can be written as a matrix equation for each spatial domain. The domain label l is omitted below and the index n labels dimensions. A general linear function r˜(n)(x) is represented as a scalar product of two vectors of length (N + 1)—a vector of basis functions,

T f = (x1,x2,...,xN, 1) [C1] and a vector of parameters (n) T c = (c1n,c2n,...,cNn,c0n) [C2] Then, r˜(n)(x) = f × c(n). Organizing vectors c(n) into a rectangular matrix C of the size N × (N + 1), C = c(1), c(2),...,c(N) [C3] 360 Semiclassical Bohmian Dynamics the condition on the minimal deviation on the domain, ∇ = = (n) − (n) 2 c(n) I 0,I r r˜ [C4] can be rewritten as a linear matrix equation 2SC + B = 0 [C5]

The matrix equation then can be solved for C. The dimension of the overlap matrix S =f ⊗ f is (N + 1) × (N + 1). Its elements are the ﬁrst and second moments of the trajectory distribution weighted by the domain function, (i) (j) sij = f f (x)(x, t)d [C6]

Here, d denotes the volume element, d = dx1dx2 .... The size N × (N + 1) of the matrix B containing the interface terms, B =∇⊗ (f), is the same as the size of C. The matrix elements of B are d (j) bij = f (x) × (x, t)d [C7] dxi

In numerical implementation, integrals in Eqs. [C6] and [C7] are replaced by a summation across trajectories, with (x, t)d represented by the trajectory weights according to Eq. [20].

REFERENCES

1. J. C. Light and T. Carrington, Jr., Adv. Chem. Phys., 114, 263 (2000). Discrete Variable Representations and their Utilization. 2. B. R. Ussing, C. Hang, and D. A. Singleton, J. Am. Chem. Soc., 128, 7594 (2006). Dynamic Effects on the Periselectivity, Rate, Isotope Effects, and Mechanism of Cycloadditions of Ketenes with Cyclopentadiene. 3. C. Dekker and M. A. Ratner, Physics World, 14, 29 (2001). Electronic Properties of DNA. 4. J. D. Lear, Z. R. Wasserman, and W. F. DeGrado, Science, 240, 1177 (1988). Synthetic Am- phiphilic Peptide Models for Protein Ion Channels. 5. Y. Cha, C. J. Murray, and J. P. Klinman, Science, 243, 1325 (1989). Hydrogen Tunneling in Enzyme Reactions. 6. M. J. Knapp and J. P. Klinman, Eur. J. Biochem., 269, 3113 (2002). Environmentally Coupled Hydrogen Tunneling—Linking Catalysis to Dynamics. 7. O. V. Prezhdo and P. J. Rossky, J. Chem. Phys., 107, 5863 (1997). Evaluation of Quantum Transition Rates from Quantum-Classical Non-Adiabatic Molecular Dynamics Simulations. 8. C. Brooksby, O. Prezhdo, and P. Reid, J. Chem. Phys., 119, 9111 (2003). Molecular Dynamics Study of the Weakly Solvent Dependent Relaxation Dynamics following Chlorine Dioxide Photoexcitation. References 361

9. B. Habenicht, C. Craig, and O. Prezhdo, Phys. Rev. Lett., 96, 187401 (2006). Time-Domain Ab Initio Simulation of Electron and Hole Relaxation Dynamics in a Single-Wall Semicon- ducting Carbon Nanotube. 10. O. V. Prezhdo, Chem. Phys. Lett., 460, 1 (2008). Multiple Excitons and the Electron-Phonon Bottleneck in Semiconductor Quantum Dots: An Ab Initio Perspective. 11. O. V. Prezhdo, W. R. Duncan, and V. V. Prezhdo, Acc. Chem. Res., 41, 339 (2008). Dynamics of the Photoexcited Electron at the Chromophore-Semiconductor Interface. 12. S. Hammes-Schiffer and J. C. Tully, J. Chem. Phys., 101, 4657 (1994). Proton Transfer in Solution: Molecular Dynamics with Quantum Transitions. 13. H. D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett., 165, 73 (1990). The Multi-Configurational Time-Dependent Hartree Approach. 14. H. D. Meyer and G. A. Worth, Theor. Chem. Acc., 109, 251 (2003). Quantum Molecular Dynamics: Propagating Wavepackets and Density Operators Using the Multiconfiguration Time-Dependent Hartree Method. 15. H. B. Wang and M. Thoss, J. Chem. Phys., 119, 1289 (2003). Multilayer Formulation of the Multiconfiguration Time-Dependent Hartree Theory. 16. D. V. Shalashilin and M. S. Child, J. Chem. Phys., 121, 3563 (2004). Real Time Quantum Propagation on a Monte Carlo Trajectory Guided Grids of Coupled Coherent States: 26D Simulation of Pyrazine Absorption Spectrum. 17. Y. H. Wu and V. S. Batista, J. Chem. Phys., 124, 224305 (2006). Matching-Pursuit Split- Operator Fourier-Transform Simulations of Excited-State Intramolecular Proton Transfer in 2-(2’-Hydroxyphenyl)-Oxazole. 18. M. Ben-Nun, J. Quenneville, and T. J. Martinez, J. Chem. Phys., 104, 5161 (2001). Ab Initio Multiple Spawning: Photochemistry from First Principles Quantum Molecular Dynamics. 19. S. Y. Kim and S. Hammes-Schiffer, J. Chem. Phys., 124, 244102 (2006). Hybrid Quantum/Classical Molecular Dynamics for a Proton Transfer Reaction Coupled to a Dis- sipative Bath. 20. O. Prezhdo and V. Kisil, Phys. Rev. A, 56, 162 (1997). Mixing Quantum and Classical Mechanics. 21. T. D. Hone, S. Izvekov, and G. A. Voth, J. Chem. Phys., 122, 054105 (2005). Fast Centroid Molecular Dynamics: A Force-Matching Approach for the Predetermination of the Effective Centroid Forces. 22. J. Gao and D. G. Truhlar, Annu. Rev. Phys. Chem., 53, 467 (2002). Quantum-Mechanical Methods for Enzyme Kinetics. 23. G. Naray-Szab´ o´ and A. Warshel, Eds. Computational Approaches to Biochemical Reactivity, Vol. 19 of Understanding Chemical Reactivity. Kluwer Academic Publishers, Reading, MA, 1997. 24. E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys., 113, 9369 (2000). Mixing Quantum and Classical Dynamics Using Bohmian Trajectories. 25. O. V. Prezhdo and C. Brooksby, Phys. Rev. Lett., 86, 3215 (2001). Quantum Backreaction through the Bohmian Particle. 26. M. Karplus, R. D. Sharma, and R. N. Porter, J. Chem. Phys., 40, 2033 (1964). Dynamics of Reactive Collisions—H + H2 Exchange Reaction. 27. D. A. McCormack and K. F. Lim, J. Chem. Phys., 106, 572 (1997). The Zero-Point Energy Problem in Classical Trajectory Simulations at Dissociation Threshold. 28. Y. Guo and D. L. Thompson, J. Chem. Phys., 104, 576 (1996). Analysis of the Zero-Point Energy Problem in Classical Trajectory Simulations. 29. M. F. Herman and E. Kluk, Chem. Phys., 91, 27 (1984). A Semiclassical Justification for the Use of Non-Spreading Wavepackets in Dynamics Calculations. 30. K. G. Kay, J. Chem. Phys., 100, 4377 (1994). Integral Expressions for the Semiclassical Time- Dependent Propagator. 362 Semiclassical Bohmian Dynamics

31. W. H. Miller, J. Phys. Chem. A, 105, 2942 (2001). The Semiclassical Initial Value Repre- sentation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations. 32. E. Madelung, Z. Phys., 40 (1926). Quantum Theory in Hydrodynamic Form. 33. L. de Broglie, Nature, 118, 441 (1926). Interference and Corpuscular Light. 34. D. Bohm, Phys. Rev., 85, 166 (1952). A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables, I and II. 35. S. Garashchuk and V. A. Rassolov, Chem. Phys. Lett., 364, 562 (2002). Semiclassical Dy- namics Based on Quantum Trajectories. 36. D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, New York, 2006. 37. P. R. Holland, The Quantum Theory of Motion. Cambridge University Press, Cambridge, UK, 1993. 38. L. D. Landau and E. M. Lifshitz, Quantum Mechanics. Butterworth-Heinemann, Oxford, UK, 1999. 39. W. H. Miller, J. Chem. Phys., 48, 464 (1968). Uniform Semiclassical Approximations for Elastic Scattering and Eigenvalue Problems. 40. J. H. van Vleck, Proc. Nat. Acad. Sci. U.S.A., 14, 178 (1928). The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics. 41. M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1990. 42. V. A. Rassolov and S. Garashchuk, J. Chem. Phys., 120, 6815 (2004). Bohmian Dynamics on Subspaces Using Linearized Quantum Force. 43. R. E. Wyatt, C. L. Lopreore, and G. Parlant, J. Chem. Phys., 114, 5113 (2001). Electronic Transitions with Quantum Trajectories. 44. V. A. Rassolov and S. Garashchuk, Phys. Rev. A, 71, 032511 (2005). Semiclassical Nonadi- abatic Dynamics with Quantum Trajectories. 45. S. Garashchuk, V. A. Rassolov, and G. C. Schatz, J. Chem. Phys., 123, 174108 (2005). Semi- classical Nonadiabatic Dynamics Using a Mixed Wavefunction Representation. 46. I. Burghardt and L. S. Cederbaum, J. Chem. Phys., 115, 10303 (2001). Hydrodynamic Equa- tions for Mixed Quantum States. I. General Formulation. 47. I. Burghardt and L. S. Cederbaum, J. Chem. Phys., 115, 10312 (2001). Hydrodynamic Equa- tions for Mixed Quantum States. II. Coupled Electronic States. 48. I. Burghardt and K. B. Moller, J. Chem. Phys., 117, 7409 (2002). Quantum Dynamics for Dissipative Systems: A Hydrodynamic Perspective. 49. J. B. Maddox and E. R. Bittner, J. Phys. Chem. B, 106, 7981 (2002). Quantum Dissipation in the Hydrodynamic Moment Hierarchy: A Semiclassical Truncation Strategy. 50. E. R. Bittner, J. B. Maddox, and I. Burghardt, Int. J. Quantum Chem., 89, 313 (2002). Relaxation of Quantum Hydrodynamic Modes. 51. A. Donoso and C. C. Martens, Phys. Rev. Lett., 87, 223202 (2001). Quantum Tunneling Using Entangled Classical Trajectories. 52. A. Donoso and C. C. Martens, J. Chem. Phys., 116, 10598 (2002). Solution of Phase Space Diffusion Equations Using Interacting Trajectory Ensembles. 53. C. J. Trahan and R. E. Wyatt, J. Chem. Phys., 119, 7017 (2003). Evolution of Classical and Quantum Phase-Space Distributions: A New Trajectory Approach for Phase Space Hydrodynamics. 54. J. Liu and N. Makri, J. Phys. Chem. A, 108, 5408 (2004). Monte Carlo Bohmian Dynamics from Trajectory Stability Properties. 55. Y. Goldfarb, I. Degani, and D. J. Tannor, J. Chem. Phys., 125, 231103 (2006). Bohmian Mechanics with Complex Action: A New Trajectory-Based Formulation of Quantum Mechanics. References 363

56. B. Poirier, J. Chem. Phys., 121, 4501 (2004). Reconciling Semiclassical and Bohmian Me- chanics. I. Stationary States. 57. K. Park, B. Poirier, and G. Parlant, J. Chem. Phys., 129, 194112 (2008). Quantum Trajectory Calculations for Bipolar Wavepacket Dynamics in One Dimension. 58. A. S. Sanz, F. Borondo, and S. Miret-Artes, J. Phys., 14, 6109 (2002). Particle Diffraction Studied Using Quantum Trajectories. 59. A. S. Sanz, F. Borondo, and S. Miret-Artes,´ J. Chem. Phys., 120, 8794 (2004). Quantum Trajectories in Atom–Surface Scattering with Single Adsorbates: The Role of Quantum Vortices. 60. A. S. Sanz and S. Miret-Artes,´ J. Chem. Phys., 122, 014702 (2005). Quantum Trajectories in Elastic Atom-Surface Scattering: Threshold and Selective Adsorption Resonances. 61. A. S. Sanz and S. Miret-Artes,´ J. Chem. Phys., 126, 234106 (2007). A Causal Look into the Quantum Talbot Effect. 62. R. E. Wyatt and E. R. Bittner, J. Chem. Phys., 113, 8898 (2000). Quantum Wave Packet Dynamics with Trajectories: Implementation with Adaptive Lagrangian Grids. 63. L. R. Pettey and R. E. Wyatt, J. Phys. Chem. A, 112, 13335 (2008). Application of the Moving Boundary Truncation Method to Reactive Scattering: H + H2,O+ H2,O+ HD. 64. S. S. Iyengar, H. B. Schlegel, and G. A. Voth, J. Phys. Chem. A, 107, 7269 (2003). Atom- Centered Density Matrix Propagation (ADMP): Generalizations Using Bohmian Mechanics. 65. J. Jakowski, I. Sumner, and S. S. Iyengar, J. Chem. Theory Comput., 2, 1203 (2006). Com- putational Improvements to Quantum Wave Packet ab Initio Molecular Dynamics Using a Potential-Adapted, Time-Dependent Deterministic Sampling Technique. 66. I. Sumner and S. S. Iyengar, J. Phys. Chem. A, 111, 10313 (2007). Quantum Wavepacket Ab Initio Molecular Dynamics: An Approach for Computing Dynamically Averaged Vibrational Spectra Including Critical Nuclear Quantum Effects. 67. R. E. Wyatt and K. Na, Phys. Rev. E, 65, 016702 (2002). Quantum Trajectory Analysis of Multimode Subsystem-Bath Dynamics. 68. D. Babyuk and R. E. Wyatt, J. Chem. Phys., 124, 214109 (2006). Multidimensional Reactive Scattering with Quantum Trajectories: Dynamics with 50-200 Vibrational Modes. 69. E. R. Bittner, J. Chem. Phys., 112, 9703 (2000). Quantum Tunneling Dynamics Using Hy- drodynamic Trajectories. 70. B. K. Kendrick, J. Chem. Phys., 119, 5805 (2003). A New Method for Solving the Quantum Hydrodynamic Equations of Motion. 71. R. E. Wyatt, CNLS Workshop: Quantum and Semiclassical Molecular Dynamics of Nanostruc- tures, Los Alamos, NM 2004. How to Cope with Problems in Quantum Hydrodynamics. 72. C. J. Trahan, K. Hughes, and R. E. Wyatt, J. Chem. Phys., 118, 9911 (2003). A New Method for Wave Packet Dynamics: Derivative Propagation Along Quantum Trajectories. 73. Y. Goldfarb, J. Schiff, and D. J. Tannor, J. Chem. Phys., 128, 164114 (2008). Complex Trajectory Method in Time-Dependent WKB. 74. S. Garashchuk and V. A. Rassolov, Chem. Phys. Lett., 376, 358 (2003). Quantum Dynamics with Bohmian Trajectories: Energy Conserving Approximation to the Quantum Potential. 75. S. Garashchuk and V. A. Rassolov, J. Chem. Phys., 120, 1181 (2004). Energy Conserv- ing Approximations to the Quantum Potential: Dynamics with Linearized Quantum Force. 76. M. Jalaie and K. Lipkowitz, in Reviews in Computational Chemistry, Vol. 14, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 2000, pp. 441–486. Published Force Field Parameters for Molecular Mechanics, Molecular Dynamics, and Monte Carlo Simulations. 77. J. Hill, C. Freeman, and L. Subramanian, in Reviews in Computational Chemistry, Vol. 16, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 2000, pp. 141–216. Use of Force Fields in Materials Modeling. 78. D. Sholl, Acc. Chem. Res., 39, 403 (2006). Understanding Macroscopic Diffusion of Adsorbed Molecules in Crystalline Nanoporous Materials via Atomistic Simulations. 364 Semiclassical Bohmian Dynamics

79. O. N. Kalugin, V. V. Chaban, V. V. Loskutov, and O. V. Prezhdo, Nano Lett., 8, 2126 (2008). Uniform Diffusion of Acetonitrile Inside Carbon Nanotubes Favors Supercapacitor Perfor- mance. 80. M. Robb, M. Garavelli, M. Olivucci, and F.Bernardi, in Reviews in Computational Chemistry, Vol. 15, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 2000, pp. 87–146. A Computational Strategy for Organic Photochemistry. 81. T. Martinez, Acc. Chem. Res., 39, 119 (2006). Insights for Light-Driven Molecular Devices from Ab Initio Multiple Spawning Excited-State Dynamics of Organic and Biological Chro- mophores. 82. W. R. Duncan and O. V. Prezhdo, Ann. Rev. Phys. Chem., 58, 143 (2007). Theoretical Studies of Photoinduced Electron Transfer in Dye-Sensitized TiO2. 83. S. Hammes-Schiffer, Acc. Chem. Res., 39, 93 (2006). Hydrogen Tunneling and Protein Motion in Enzyme Reactions. 84. G. Voth, Acc. Chem. Res., 39, 143 (2006). Computer Simulation of Proton Solvation and Transport in Aqueous and Biomolecular Systems. 85. M. Ouyang and D. Awschalom, Science, 301, 1074 (2003). Coherent Spin Transfer Between Molecularly Bridged Quantum Dots. 86. J. Yu, D. Hu, and P. Barbara, Science, 289, 1327 (2000). Unmasking Electronic Energy Trans- fer of Conjugated Polymers by Suppression of O2 Quenching. 87. D. S. Kilin, K. Tsemekhman, O. V. Prezhdo, E. I. Zenkevich, and C. von Borczyskowski, J. Photochem. Photobio. A—Chem., 190, 342 (2007). Ab Initio Study of Exciton Transfer Dynamics from a Core-Shell Semiconductor Quantum Dot to a Porphyrin-Sensitizer. 88. H. Tamura, J. G. S. Ramon, E. R. Bittner, and I. Burghardt, Phys. Rev. Lett., 100, 107402 (2008). Phonon-Driven Ultrafast Exciton Dissociation at Donor-Acceptor Polymer Hetero- junctions. 89. D. S. Kilin, O. V. Prezhdo, and M. Schreiber, J. Phys. Chem. A, 111, 10212 (2007). Photoin- duced Vibrational Coherence Transfer in Molecular Dimers. 90. W. R. Duncan, C. F. Craig, and O. V. Prezhdo, J. Am. Chem. Soc., 129, 8528 (2007). Time- Domain Ab Initio Study of Charge Relaxation and Recombination in Dye-Sensitized TiO2. 91. B. F. Habenicht and O. V. Prezhdo, Phys. Rev. Lett., 100, 197402 (2008). Nonradiative Quenching of Fluorescence in a Semiconducting Carbon Nanotube: A Time-Domain Ab Initio Study. 92. O. Prezhdo and P. Rossky, J. Phys. Chem., 100, 17094 (1996). Solvent Mode Participation in the Nonradiative Relaxation of the Hydrated Electron. 93. A. E. Bragg, M. C. Cavanagh, and B. J. Schwartz, Science, 321, 1817 (2008). Linear Response Breakdown in Solvation Dynamics Induced by Atomic Electron-Transfer Reactions. 94. J. Tully, J. Chem. Phys., 93, 1061 (1990). Molecular-Dynamics with Electronic-Transitions. 95. D. Coker and L. Xiao, J. Chem. Phys., 102, 496 (1995). Methods for Molecular-Dynamics with Nonadiabatic Transitions. 96. O. Prezhdo and P. Rossky, J. Chem. Phys., 107, 825 (1997). Mean-Field Molecular Dynamics with Surface Hopping. 97. O. Prezhdo, J. Chem. Phys., 111, 8366 (1999). Mean Field Approximation for the Stochastic Schrodinger Equation. 98. A. Horsﬁeld, D. Bowler, A. Fisher, T. Todorov, and C. Sanchez, J. Phys.—Cond. Mat., 16, 8251 (2004). Beyond Ehrenfest: Correlated Non-Adiabatic Molecular Dynamics. 99. X. Li, J. Tully, H. Schlegel, and M. Frisch, J. Chem. Phys., 123, 084106 (2005). Ab Initio Ehrenfest Dynamics. 100. C. Craig, W.Duncan, and O. Prezhdo, Phys. Rev. Lett., 95, 163001 (2005). Trajectory Surface Hopping in the Time-Dependent Kohn-Sham Approach for Electron-Nuclear Dynamics. 101. A. Jasper, S. Nangia, C. Zhu, and D. Truhlar, Acc. Chem. Res., 39, 101 (2006). Non- Born-Oppenheimer Molecular Dynamics. References 365

102. O. V. Prezhdo, Theor. Chem. Acc., 116, 206 (2006). Quantized Hamilton Dynamics. 103. G. A. Worth, H. D. Meyer, H. Koeppel, L. S. Cederbaum, and I. Burghardt, Int. Rev. Phys. Chem., 27, 569 (2008). Using the MCTDH Wavepacket Propagation Method to Describe Multimode Non-Adiabatic Dynamics. 104. W. H. Miller, J. Phys. Chem. A, 113, 1405 (2009). Electronically Nonadiabatic Dynamics via Semiclassical Initial Value Methods. 105. I. V. Aleksandrov, Z. Naturforsch., 36A, 902 (1981). The Statistical Dynamics of a System Consisting of a Classical and a Quantum Subsystem. 106. W. Boucher and J. Traschen, Phys. Rev. D, 37, 3522 (1988). Semiclassical Physics and Quan- tum Fluctuations. 107. A. Anderson, Phys. Rev. Lett., 74, 621 (1995). Quantum Backreaction on Classical Variables. 108. R. Kapral and G. Ciccotti, J. Chem. Phys., 110, 8919 (1999). Mixed Quantum-Classical Dynamics. 109. V. Kisil, Europhys. Lett., 72, 873 (2005). A Quantum-Classical Bracket From P-Mechanics. 110. O. Prezhdo, J. Chem. Phys., 124, 201104 (2006). A Quantum-Classical Bracket that Satisﬁes the Jacobi Identity. 111. P. Ehrenfest, Z. Physik, 45, 455 (1927). Bemerkung uber¨ die Angenaherte¨ Gultigkeit¨ der Klassischen Mechanik Innerhalb der Quantenmechanik. 112. S. T. Epstein, Force Concept in Chemistry, Van Nostrand Reinhold, New York 1981, pp. 1–38. 113. E. Gindensperger, C. Meier, and J. Beswick, Adv. Quant. Chem., 47, 331 (2004). Hybrid Quantum/Classical Dynamics Using Bohmian Trajectories. 114. J. Stromquist and S. Gao, J. Chem. Phys., 106, 5751 (1996). Quantum-Mechanical Wave Packet Calculation of Photoinduced Surface Reaction: O2/Pt(111). 115. D. S. Sholl and J. C. Tully, J. Chem. Phys., 109, 7702 (1998). A Generalized Surface Hopping Method. 116. M. Reed, C. Zhou, C. Muller, T. Burgin, and J. Tour, Science, 278, 252 (1997). Conductance of a Molecular Junction. 117. C. Brooksby and O. V. Prezhdo, Chem. Phys. Lett., 346, 463 (2001). Quantized Mean-Field Approximation. 118. E. Gindensperger, C. Meier, J. Beswick, and M. Heitz, J. Chem. Phys., 116, 10051 (2002). Quantum-Classical Description of Rotational Diffractive Scattering Using Bohmian Trajec- tories: Comparison with Full Quantum Wave Packet Results. 119. C. Meier and J. Beswick, J. Chem. Phys., 121, 4550 (2004). Femtosecond Pump-Probe Spec- troscopy of I2 in a Dense Rare Gas Environment: A Mixed Quantum/Classical Study of Vibrational Decoherence. 120. C. Meier, Phys. Rev. Lett., 93, 173003 (2004). Mixed Quantum-Classical Treatment of Vibrational Decoherence. 121. J. Li, C. Woywod, V. Vallet, and C. Meier, J. Chem. Phys., 124, 184105 (2006). Investigation of the Dynamics of Two Coupled Oscillators with Mixed Quantum-Classical Methods. 122. H. Azzouz and D. Borgis, J. Chem. Phys., 98, 7361 (1993). A Quantum Molecular-Dynamics Study of Proton-Transfer Reactions Along Asymetrical H-Bonds in Solution. 123. J. Burant and J. Tully, J. Chem. Phys., 112, 6097 (2000). Nonadiabatic Dynamics via the Classical Limit Schrodinger¨ Equation. 124. I. Burghardt and G. Parlant, J. Chem. Phys., 120, 3055 (2004). On the Dynamics of Coupled Bohmian and Phase-Space Variables: A New Hybrid Quantum-Classical Approach. 125. I. Burghardt, J. Chem. Phys., 122, 094103 (2005). Dynamics of Coupled Bohmian and Phase- Space Variables: A Moment Approach to Mixed Quantum-Classical Dynamics. 126. E. Wigner, Phys. Rev., 40, 749 (1932). On the Quantum Correction for Thermodynamic Equilibrium. 366 Semiclassical Bohmian Dynamics

127. R. G. Littlejohn, Physics Reports, 138, 193 (1986). The Semiclassical Evolution of Wave Packets. 128. C. C. Martens and J. Y.Fang, J. Chem. Phys., 106, 4918 (1997). Semiclassical-Limit Molecular Dynamics on Multiple Electronic Surfaces. 129. T. Dittrich, C. Viviescas, and L. Sandoval, Phys. Rev. Lett., 96, 070403+ (2006). Semiclassical Propagator of the Wigner Function. 130. E. J. Heller, J. Chem. Phys., 67, 3339 (1977). Phase Space Interpretation of Semiclassical Theory. 131. K. H. Hughes, S. M. Parry, and I. Burghardt, J. Chem. Phys., 130, 054115+ (2009). Closure of Quantum Hydrodynamic Moment Equations. 132. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Physics Reports, 106, 121 (1984). Distribution Functions in Physics: Fundamentals. 133. W. P. Schleich, Quantum Optics in Phase Space, Wiley, New York, 2001. 134. D. Kohen, F. H. Stillinger, and J. C. Tully, J. Chem. Phys., 109, 4713 (1998). Model Studies of Nonadiabatic Dynamics. 135. O. V. Prezhdo and Y. V. Pereverzev, J. Chem. Phys., 113, 6557 (2000). Quantized Hamilton Dynamics. 136. O. V. Prezhdo, J. Chem. Phys., 117, 2995 (2002). Classical Mapping for Second-Order Quan- tized Hamiltonian Dynamics. 137. R. E. Wyatt, Quantum Dynamics with Trajectories: An Introduction to Quantum Hydrody- namics. Springer-Verlag, New York, 2005. 138. F. Grossmann and E. Heller, Chem. Phys. Lett., 241, 45 (1995). A Semiclassical Correlation- Function Approach to Barrier Tunneling. 139. J. Liu and N. Makri, Molec. Phys., 103, 1083 (2005). Bohm’s Formulation in Imaginary Time: Estimation of Energy Eigenvalues. 140. K. Kay, J. Chem. Phys., 100, 4432 (1994). Numerical Study of Semiclassical Initial-Value Methods for Dynamics. 141. K. G. Kay, J. Chem. Phys., 101, 2250 (1994). Semiclassical Propagation for Multidimensional Systems by an Initial-Value Method. 142. D. M. Ceperley and L. Mitas, Adv. Chem. Phys. (1996). Monte Carlo Methods in Quantum Chemistry. 143. Y. Goldfarb and D. J. Tannor, J. Chem. Phys., 127, 161101 (2007). Interference in Bohmian Mechanics with Complex Action. 144. D. Huber and E. J. Heller, J. Chem. Phys., 87, 5302 (1987). Generalized Gaussian Wave Packet Dynamics. 145. C. Trahan and B. Poirier, J. Chem. Phys., 124, 034115 (2006). Reconciling Semiclassical and Bohmian Mechanics. II. Scattering States for Discontinuous Potentials. 146. C. Trahan and B. Poirier, J. Chem. Phys., 124, 034116 (2006). Reconciling Semiclassical and Bohmian Mechanics. III. Scattering States for Continuous Potentials. 147. B. Poirier and G. Parlant, J. Phys. Chem. A, 111, 10400 (2007). Reconciling Semiclassical and Bohmian Mechanics. IV. Multisurface Dynamics. 148. B. Poirier, J. Chem. Phys., 128, 164115 (2008). Reconciling Semiclassical and Bohmian Me- chanics. V. Wavepacket Dynamics. 149. B. Poirier, J. Chem. Phys., 129, 084103 (2008). Reconciling Semiclassical and Bohmian Me- chanics. VI. Multidimensional Dynamics. 150. S. Garashchuk and V. A. Rassolov, J. Chem. Phys., 118, 2482 (2003). Semiclassical Dynamics with Quantum Trajectories: Formulation and Comparison with the Semiclassical Initial Value Representation Propagator. 151. S. Garashchuk and V. A. Rassolov, Chem. Phys. Lett., 446, 395 (2007). Semiclassical Nona- diabatic Dynamics of NaFH with Quantum Trajectories. References 367

152. S. Garashchuk, J. Chem. Phys., 126, 154104 (2007). Computation of Correlation Functions and Wave Function Projections in the Context of Quantum Trajectory Dynamics. 153. J. A. Beswick and J. Jortner, Chem. Phys., 24, 1 (1977). Absorption Lineshapes for the Pho- todissociation of Polyatomic Molecules. 154. R. C. Brown and E. J. Heller, J. Chem. Phys., 75, 186 (1981). Classical Trajectory Approach to Photodissociation: The Wigner Method. 155. R. D. Coalson and M. Karplus, J. Chem. Phys., 93, 3919 (1990). Multidimensional Variational Gaussian Wave Packet Dynamics with Application to Photodissociation Spectroscopy. 156. S. Garashchuk, V. A. Rassolov, and G. C. Schatz, J. Chem. Phys., 124, 244307 (2006). Semi- 3 1 classical Nonadiabatic Dynamics based on Quantum Trajectories for the O( P, D) + H2 System. 157. S. Garashchuk and D. J. Tannor, Chem. Phys. Lett., 262, 477 (1996). Wave Packet Correlation Function Approach to H2() + H → H + H2( ): Semiclassical Implementation. 158. L. J. Butler, Annu. Rev. Phys. Chem., 49, 125 (1998). Chemical Reaction Dynamics Beyond the Born-Oppenheimer Approximation.

159. L. Sun, K. Song, and W. L. Hase, Science, 296, 875 (2002). A SN 2 Reaction that Avoids its Deep Potential Energy Minimum. 160. S. C. Ammal, H. Yamataka, M. Aida, and M. Dupuis, Science, 299, 1555 (2003). Dynamics- Driven Pathway in an Intramolecular Rearrangement. 161. J. C. Tully, Faraday Discuss., 110, 407 (1998). Mixed Quantum-Classical Dynamics. 162. Z. Qu, H. Zhu, S. Y. Grebenshchikov, and R. Schinke, J. Chem. Phys., 122, 191102 (2005). The Triplet Channel in the Photodissociation of Ozone in the Hartley Band: Classical Tra- jectory Surface Hopping Analysis. 163. E. Martinez-Nunez, S. Vazquez, G. Granucci, M. Persico, and C. M. Estevez, Chem. Phys. Lett., 412, 35 (2005). Photodissociation of Formic Acid: A Trajectory Surface Hopping Study. 164. A. Toniolo, C. Ciminelli, M. Persico, and T. J. Martınez,´ J. Chem. Phys., 123, 234308 (2005). Simulation of the Photodynamics of Azobenzene on its First Excited State: Comparison of Full Multiple Spawning and Surface Hopping Treatments. 165. D. C. Robie, S. Arepalli, N. Presser, T. Kitsopoulos, and R. J. Gordon, J. Chem. Phys., 92, 7382 (1990). The Intramolecular Kinetic Isotope Effect for the Reaction O(3P) + HD. 166. D. J. Garton, T. K. Minton, B. Maiti, D. Troya, and G. C. Schatz, J. Chem. Phys., 3 118, 1585 (2003). A Crossed Molecular Beams Study of the O( P) + H2 Reaction: Comparison of Excitation Function with Accurate Quantum Reactive Scattering Calcu- lations. 167. M. Braunstein, S. Adler-Golden, B. Maiti, and G. C. Schatz, J. Chem. Phys., 121, 4316 (2004). 3 Quantum and Classical Studies of the O( P) + H2( = 0 − 3,j = 0) → OH + H Reaction Using Benchmark Potential Surfaces. 168. N. Balakrishnan, J. Chem. Phys., 119, 195 (2003). Quantum-Mechanical Investigation of the O + H2 → OH + H Reaction. 169. T.-S. Chu, X. Zhang, and K.-L. Han, J. Chem. Phys., 122, 213201 (2005). A Quantum Wave- 3 1 Packet Study of Intersystem Crossing Effects in the O( P2,1,0, D2) + H2 Reaction. 170. S. Rogers, D. Wang, A. Kuppermann, and S. Walch, J. Phys. Chem. A, 104, 2308 (2000). Chemically Accurate ab Initio Potential Energy Surfaces for the Lowest 3A and 3A Elec- 3 tronically Adiabatic States of O( P) + H2. 171. F. J. Aoiz, L. Banares,˜ J. F. Castillo, M. Brouard, W. Denzer, C. Vallance, P. Honvault, et al., Phys. Rev. Lett., 86, 1729 (2001). Insertion and Abstraction Pathways in the Reaction 1 O( D2) + H2 → OH + H. 172. M. R. Hoffmann and G. C. Schatz, J. Chem. Phys., 113, 9456 (2000). Theoretical Studies of Intersystem Crossing Effects in the O + H2 Reaction. 368 Semiclassical Bohmian Dynamics

173. F. J. Aoiz, L. Banares,˜ J. F. Castillo, M. Brouard, W. Denzer, C. Vallance, P. Honvault, J.-M. Launay, A. J. Dobbyn, and P. J. Knowles, Phys. Rev. Lett., 86, 1729 (2001). Insertion 1 and Abstraction Pathways in the Reaction O( D2) + H2 → OH + H. 174. A. Donoso, D. Kohen, and C. C. Martens, J. Chem. Phys., 112, 7345 (2000). Simulation of Nonadiabatic Wave Packet Interferometry Using Classical Trajectories. 175. S. Garashchuk and V. A. Rassolov, J. Chem. Phys., 121, 8711 (2004). Modiﬁed Quantum Trajectory Dynamics Using a Mixed Wavefunction Representation. 176. D. J. Tannor and D. E. Weeks, J. Chem. Phys., 98, 3884 (1993). Wave Packet Correlation- Function Formulation of Scattering Theory—The Quantum Analog of Classical S-Matrix Theory. 177. P. M. Morse, Phys. Rev., 34, 57 (1929). Diatomic Molecules According to the Wave Mechan- ics. II. Vibrational Levels. 178. W.Press, B. Flannery, S. Teukolsky, and W.Vetterling, Numerical Recipes: The Art of Scientiﬁc Computing, Second edition, Cambridge University Press, Cambridge, UK, 1992. 179. V. A. Rassolov, S. Garashchuk, and G. C. Schatz, J. Phys. Chem. A, 110, 5530 (2006). Quantum Trajectory Dynamics in Arbitrary Coordinates. 180. M. Feit, J. Fleck Jr., and A. Steiger, J. Comput. Phys., 47, 412 (1982). Solution of the Schrodinger¨ Equation by a Spectral Method. 181. C. Leforestier, R. H. Bisselling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, et al., J. Comput. Phys., 94, 59 (1991). A Comparison of Different Propagation Schemes for the Time-Dependent Schrodinger¨ Equation. 182. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys., 82, 1400 (1985). Generalized Discrete Variable Approximation in Quantum Mechanics. 183. M. J. Knapp, K. Rickert, and J. P. Klinman, J. Am. Chem. Soc., 124, 3865 (2002). Temperature-Dependent Isotope Effects in Soybean Lipoxygenase-1: Correlating Hydrogen Tunneling with Protein Dynamics. 184. S. Garashchuk and V. A. Rassolov, J. Chem. Phys., 129, 024109 (2008). Stable Long-Time Semiclassical Description of Zero-Point Energy in High-Dimensional Molecular Systems. 185. S. Garashchuk, J. Phys. Chem. A, 113, 4451–4456 (2009). Description of Bound Reactive Dynamics within the Approximate Quantum Trajectory Framework. 186. N. Makri, J. Phys. Chem. A, 102, 4414 (1998). Quantum Dissipative Dynamics: A Numeri- cally Exact Methodology. 187. M. Mladenovic, J. Chem. Phys., 112, 1070 (2000). Rovibrational Hamiltonian for General Polyatomic Molecules in Spherical Polar Parameterization. I. Orthogonal Representation.