semiclassical bohmian dynamics · 288 semiclassical bohmian dynamics...

CHAPTER 6

Semiclassical Bohmian Dynamics

Sophya Garashchuk,a Vitaly Rassolov,a and OlegPrezhdob

a Department of Chemistry and Biochemistry, University ofSouth Carolina, Columbia, South CarolinabDepartment 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 ener-gies of chemical reactions, reaction rates and branching ratios, and other chemi-cal phenomena without a quantum-mechanical (QM) description. In particular,the dynamics of molecular systems often involve QM effects such as zero-pointenergy, tunneling, and nonadiabatic transitions. QM effects are essential foraccurate description and understanding of reactions in complex chemical envi-ronments. For example, the zero-point energy stored in the vibrational modesof chemical reactants, products, and transition state species modifies reactionenergy barriers. Reaction rates and branching ratios can be affected greatlyby such changes. QM tunneling can be critical in proton transfer reactions.However, the conventional methods of solving the time-dependent Schrodingerequation1 scale exponentially with the system size. In addition, the relevant dy-namics occur on a long time-scale.2 Therefore, it is extremely difficult to modelQM tunneling in condensed phase chemical systems.3–6 Nonadiabatic dynam-ics involving transitions between different electronic or vibrational energy levels

Reviews in Computational Chemistry, Volume 27edited by Kenny B. Lipkowitz

Copyright © 2011 John Wiley & Sons, Inc.

287

288 Semiclassical Bohmian Dynamics

is ubiquitous in photochemistry.7–11 In this process, a chemical system is excitedby light, undergoes a nonequilibrium evolution, and ultimately relaxes to theground state. QM tunneling can be viewed as a particular kind of nonadiabatictransition.12

Several multidimensional quantum approaches have been proposed in-cluding those using basis set contractions,13–15 coherent state representations,16,17 and mixed quantum-classical strategies.18–24 A trajectory representationof large molecular systems carries a special appeal because of several favor-able factors: (1) the initial conditions of a trajectory simulation can be sampledwith Monte Carlo techniques. This allows the exponential scaling of the exactwave function or density matrix with system size to be circumvented. (2) Var-ious degrees of freedom involving light and heavy particles can be treated onequal footing, and quantum-classical separation issues can be avoided.25 (3)Wave functions are highly oscillatory close to the classical �→ 0 limit. As aresult, a trajectory description of heavy particles, such as nuclei, is often moreappropriate than a grid or a basis set representation. (4) Classical equationsof motion are simple to solve. Numerous molecular dynamics techniques26

for propagating classical trajectories are applied routinely to chemical systemscomposing hundreds of thousands of atoms. Incorporating the dominant QMeffects caused by wave function localization27,28 constitutes a challenge for thetrajectory methods. Such effects can be considered naturally by representingwave functions in terms of ensembles of trajectories. In comparison, semiclas-sical methods commonly use independent trajectories.29–31

A great interest in the Bohmian interpretation of quantum dynamics hasbeen witnessed during the last decade. In particular, its potential to generatecomputational tools for solving the time-dependent Schrodinger equation hasattracted considerable attention. The Bohmian formulation of quantum dynam-ics promises a better than exponential scaling of the computational effort withsystem 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 widerrecognition after David Bohm34 used it to develop an alternative interpretationof quantum mechanics. This review describes the semiclassical methodologiesinspired by the Bohmian formulation of quantum mechanics. These methodsare designed to represent the complex dynamics of chemical systems.

The review is constructed as follows: the next section introduces theMadelung-de Broglie–Bohm formalism. This is done by drawing an analogywith classical mechanics and explicitly highlighting the non-classical featuresof the Bohmian mechanics. The nonclassical contributions to the momentum,energy, and force are introduced. The fundamental properties of Bohmian quan-tum 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 dy-namics based on the Bohmian formalism is derived and illustrated with anexample involving a light and a heavy particle. At this point, the Bohmian rep-resentation is used as a tool to couple the quantum and classical subsystems. Thequantum 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 derivean alternative quantum-classical coupling scheme. This resolves the so-calledquantum backreaction problem, also known as the trajectory branching prob-lem. Next, the partial hydrodynamic moment approach to coupling classicaland quantum systems is outlined. The hydrodynamic moments provide a con-nection between the Bohmian and phase-space descriptions of quantum me-chanics. The “Independent Trajectory Methods” Section describes approachesbased on independent Bohmian trajectories. It includes a discussion of thederivative propagation method, the Bohmian trajectory stability approach, andBohmian trajectories with complex action. Truncation of these hierarchies atthe second order reveals a connection to other semiclassical methods. The focusthen shifts toward Bohmian dynamics with globally approximated quantumpotentials. Separate subsections are devoted to the global energy-conservingapproximation for the nonclassical momentum, approximations on subspacesand nonadiabatic dynamics. Each approach is introduced at the formal theo-retical level and then is illustrated by an example. The Section “Towards Reac-tive Dynamics in Condensed Phase” deals with computational issues includingnumerical stability, error cancellation, dynamics linearization, and long-timebehavior. 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 mechan-ics. The three appendices prove the quantum density conservation, introducequantum trajectories in arbitrary coordinates, and explain optimization of sim-ulation parameters in many dimensions.

THE FORMALISM AND ITS FEATURES

The Trajectory Formulation

For simplicity, let us start with a derivation of the Bohmian equations inone spatial dimension x for a particle of massm. For notation clarity, let us use∇ to denote differentiation with respect to x. Arguments of functions will beomitted where unambiguous. Differentiation (or Derivatives) with respect to avariable other than x will be indicated as a subscript; for example, ∇c denotesdifferentiation with respect to c. The multidimensional generalization to an


arbitrary coordinate system is given in Appendix B. The conventional form ofthe time-dependent Schrodinger equation is

(− �

2

2m∇2 + V

) (x, t) = ı�

∂ (x, t)∂t

[1]

After Madelung, the complex time-dependent wave function is represented inpolar form as

(x, t) = A(x, t) exp( ı�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 simplemanipulations results in a system of two equations

∂S(x, t)∂t

= − 12m

(∇S(x, t))2 − V −Q [3]

∂�(x, t)∂t

= −∇(

1m

∇S(x, t)�(x, t))

[4]

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

Q = − �2

2m∇2A(x, t)A(x, t)

[5]

The quantum potential Q enters the equation on par with the external “classi-cal” potential V = V(x, t), which is generally a function of x and t as well asQ = 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

j(x, t) = �(x, t)1m

∇S(x, t) = �m

� ( ∗(x, t)∇ (x, t)

)[7]

Equation [4] becomes the usual continuity equation. Analogy with fluid me-chanics suggests the name “hydrodynamic” formulation of the Schrodingerequation. Note that Eqs. [3] and [4] are formally equivalent to the originalSchrodinger Eq. [1] except that the polar form Eq. [2] is problematic at thenodes of the wave function. At the nodes, the phase S is undefined, A(x, t) = 0,


and Q is generally singular. The singularity in Q cancels for excited eigenstatesas will be explained.

Equations [3] and [4] describe the flow of the probability fluid throughthe stationary points x. Transition to the trajectory framework is made byidentifying

p(x, t) = ∇S(x, t) [8]

and switching to the Lagrangian frame of reference

d

dt= ∂

∂t+ p

m∇ [9]

Subsequently, the subscript t will be used to define trajectory-dependent quan-tities, 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 asS(xt) and �(xt). Variables without subscripts will refer to functions of coordi-nate x and time t. For example, S(x, t) is the phase of a wave function at timet, whereas S(xt) is the action function computed along the trajectory describedby the position xt and momentum pt.

Differentiation of Eq. [3] with respect to x gives Newton’s equations ofmotion for a trajectory characterized by momentum pt and position xt

dxt

dt= pt

m[10]

dpt

dt= −∇(V +Q)

∣∣x=xt [11]

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

dS(xt)dt

= p2t

2m− (V +Q)

∣∣x=xt [12]

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 initialmomenta of the quantum trajectories are defined according to Eq. [8].

The quantities xt and pt fully define the quantum trajectory. Quantumeffects are incorporated into its behavior through the nonlocal quantum force.The force depends on the wave function amplitude and its derivatives up to the


third order. Often it is useful to consider an additional function attributable tothe quantum trajectory, namely, the nonclassical momentum component r(x, t),

r(x, t) = ∇A(x, t)A(x, t)

[13]

Formally, it is complementary to the classical component p(x, t) because bothresult from the action of the QM momentum operator on the wave functiongiven in the polar form [2]

p =(

−ı�∇A(x, t)A(x, t)

+ ∇S(x, t)) = (−ı�r(x, t) + p(x, t)) [14]

The quantum potential expressed in terms of r is

Q = − �2

2m

(r2(x, t) + ∇r(x, t)

)[15]

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

〈Q〉 = �2〈r2〉2m

[16]

In particular, 〈Q〉 is one half of the zero-point energy for the ground state ofthe harmonic oscillator.

The time-dependence of r can be derived from Eq. [4]. Combined withEqs. [11] and [5], it gives the following evolution equations, which emphasizethe common structure of the differential operator on the right-hand-side (RHS):

m∇V ∣∣x=xt +m

dpt

dt= �2

(rt + ∇

2

)∇rt [17]

−mdrtdt

=(rt + ∇

2

)∇pt [18]

Features of the Bohmian Formulation

Conservation of Probability and NormalizationTransformation of Eq. [4] into the Lagrangian frame of reference gives theevolution of the wave function density

d

dt�(xt) =

(∂

∂t+ pt

m∇

)�(xt) = −∇pt × �(xt) [19]


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

w(xt) = �(xt)dxt,dw(xt)dt

= 0 [20]

This is consistent with the standard “classical” continuity equation. The cor-responding multidimensional derivation is given in Appendix A. Therefore, allQM effects in the evolution of quantum trajectories result from the quantumforce, 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-pointenergy in bound systems.

Conservation of the trajectory “weight” implies that the Bohmiantrajectories define the most efficient grid representation for the wave functionwith time-dependent grid points. The wave function density will remainnegligible at these time-dependent grid points provided that it was negligi-ble at time t = 0. Equation [20] also helps to interpret the wave functionusing the time-dependence of the trajectory positions; because (x, t) issingle-valued, the quantum trajectories cannot cross. Wide separation ofinitially equidistant trajectories indicates regions of low wave function density.Conversely, closely spaced trajectories correspond to high wave functiondensity.

A Gaussian wave packet evolving in a harmonic potential provides a sim-ple illustration of quantum trajectory dynamics. The time-dependence of (x, t)is analytic,36 and the trajectories can be constructed easily. The center of thewave packet moves purely classically, whereas the time dependence of the over-all wave packet is influenced by the quantum force, as illustrated in Figure 1.A comparison of the trajectories propagated with and without the quantumpotential Q demonstrates the noncrossing rule—classical trajectories cross atfocal points, whereas the quantum trajectories do not cross. The noncrossingrule is a manifestation of the single-valued wave function and of the Heisenberguncertainty principle. An initially wide Gaussian wave packet with the initialwidth equal to 25% of the coherent value, Figure 1b, produces a small quantumpotentialQ. In this case, the dynamics of the quantum and classical trajectoriesare very similar except at the focal points. In contrast, an initially narrow wavepacket 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 classicaldynamics differ at all times.

The quantum force acts to make the wave packet “flat.” This is the infinitetime limit for a Gaussian wave packet evolving in a constant classical poten-tial. For more complicated classical potentials, with ∇V /= 0, there will be an


0 2 4 6

-10

0

10

Posi

tion

0 2 4 6Time

-10

0

10

Posi

tion

(a)

(b)

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

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

The Quantum Trajectory Ensemble, Expectation Values, and EnergyTo solve the time-dependent Schrodinger equation using trajectories, or sim-ply to visualize the wave function dynamics, an ensemble of trajectories isinitialized at t = 0. For each initial position x = x0, the trajectory weight,w = A2(x,0)dx0, and the classical momentum, p0 = ∇S(x,0), are determinedfrom the initial wave function. The trajectories are propagated in time accord-ing 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 expec-tation values of x-dependent operators can be computed readily. This is done


by integrating over the time-dependent trajectory positions. In a discretizedtrajectory representation, x(i)

t , the integration is replaced with the summation

〈o(x)〉t =∫o(x)�(x, t)dx =

∑i

o(x(i)t )wi [21]

where the index i enumerates the trajectories. It can be seen easily that the nor-malization of the wave function is conserved; in the Eq. [21], this correspondsto the unit operator o = 1. Operators dependent on p can be evaluated in thesame fashion. For example, the kinetic energy is

〈T〉t = 12m

∑i

(p(i)t )2wi [22]

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

Stationary States and Behavior at the NodesA stationary state is a special solution of the time-dependent Schrodinger equa-tion. It can be written as a product of the spatial and temporal factors. Substi-tuting the polar form of the wave function,

(x, t) = exp(− ı�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 Schrodingerequation

dS(t)dt

= E [24]

− �2

2m∇2A(x)A(x)

+ V = Q+ V = E [25]

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 aparticular condition on its amplitude. The amplitude of an eigenfunction givesthe quantum potential Q which differs from the negative of classical potential−V only by a constant. Therefore, the quantum force corresponding to aneigenfunction exactly cancels the classical force. Given zero initial momenta,(i.e., ∇S = 0) the trajectory positions do not change with time.


This picture of stationary trajectories also applies to excited eigenstatesimplying that at the nodes where A(x) = 0 the singularity in the quantum po-tential, given by Eq. [5], always cancels. For example, for the eigenstates of theharmonic oscillator, V = mω2x2/2, the ground-state wave function,

�0(x) = exp(−mω

2�x2

)[26]

substituted into Eq. [25] gives E0 = �ω/2. The first excited state wavefunction �1(x) = x�0(x) gives E1 = 3�ω/2 and so on. Note that the wavefunction normalization factors have been omitted for clarity. In contrast,for nonstationary wave functions with nodes, the singularities in Q do notcancel. In general, this makes the direct numerical solution of Eqs. [4] and[12] impractical. Nevertheless, the behavior of the quantum trajectoriesis very intuitive; the trajectories “flow” coherently, avoid the nodes, andnever cross. The trajectories corresponding to the time-evolution of a linearcombination of �0 and �1 are shown on Figure 2 around the density nodeat t = 2.5. Avoiding the nodal region, Figure 2a, is accomplished by rapidchanges in the trajectory momenta, Figure 2b. As a consequence, numer-ical implementation of such unstable dynamics is very expensive. Manyexamples of the quantum trajectory dynamics can be found in the book byHolland.37

1 2 3 4

Time

-1

0

1

Posi

tion

1 2 3 4Time

-5

0

5

Mom

entu

m

(a) (b)

Figure 2 Bohmian dynamics in the presence of the density nodes: (a) position of thetrajectories 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 bothpanels.


The Classical Limit of the Schrodinger Equation andthe Semiclassical Regime of Bohmian Trajectories

The Bohmian form of the time-dependent Schrodinger equation gives astraightforward route to classical mechanics. In the heavy particle limit, m →∞, or equivalently, when a typical action becomes large compared with Planck’sconstant, �→ 0, the quantum potentialQ vanishes. This is consistent with therepresentation 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 aclassical trajectory, and the changes in the wave-packet width can be neglected.

An alternative connection between Bohmian, classical, and semiclassicaldynamics is through the Wentzel–Kramers–Brillouin (WKB)38 treatments thatare based on the � expansions of the exponentiated wave function, as in Eq. [60],which is described later. The traditional semiclassical condition is based on theWKB approximation to solutions of the time-independent Schrodinger equa-tion. This condition states that the action function must be much larger thanPlank’s constant. The classical momentum p entering the action function isdefined as

pWKB =√

2m(E− V),∫pdx � [27]

Identification of the classical and nonclassical momenta, Eqs. [8] and [13] re-spectively, suggests a similar criterion that is applicable to time-dependent wavefunctions:

� |r| |p| [28]

or in terms of the energy given in Eq. [15],

|Q| p2

2m[29]

The momentum condition of Eq. [28] is more convenient than the energycondition Eq. [29] the former is expressed in terms of simple quantities, whichare linear in the semiclassical picture of a moving particle (i.e., for Gaussianwave-packets).

In the context of trajectory dynamics, the momentum semiclassical condi-tion Eq. [28] is more general than the WKB expression Eq. [27]. This is becauseit is not based on a particular approximate solution to the Schrodinger equation.Moreover, the condition of Eq. [28] is more convenient because it is expressedin terms of r and p, which are natural attributes of quantum trajectories. Forsemiclassical systems with small Q, the Bohmian momentum and the WKBmomentum are close to each other. Therefore, the momentum semiclassicalcondition and the WKB expression are related closely.


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

�〈|r|〉〈|p|〉 [30]

The singularities in r have negligible contributions to the dynamics because ofvanishing wave function density.

A general semiclassical method must satisfy the semiclassical condition atall times during time evolution; therefore, it cannot be defined for any specificform of the initial wave function. In practice, this means that approximationsmust be made only for quantities that are negligible in the m → ∞ or �→ 0limits. In particular, removing singularities in the quantum potential by con-straining density, phase, or momentum generally would violate the semiclassi-cal condition. The approximate quantum potential method of “Global Energy-Conserving Approximation of the Nonclassical Momentum” constrains thefunctional 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 trajectorymethods, are defined through the � expansion of the solution to the Schrodingerequation. Independent Bohmian trajectory methods, such as the derivativepropagation method, the Bohmian trajectory stability method, and Bohmianmechanics 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 approximatequantum potential method, can be considered semiclassical if they can be im-proved systematically in the limit of large mass for an arbitrary physically rea-sonable initial wave function and kinetic energy density. The kinetic energy den-sity is defined as −�2∇2 /(2m ). “Improved systematically” implies that thereis a general, unambiguous numerical prescription for convergence toward theexact solution. The approximate quantum potential approach can be improvedsystematically if the linearization of the nonclassical momentum is accom-plished over subspaces,42 or if r(x, t) is represented in terms of a complete basis.


Using Quantum Trajectories in Dynamics of ChemicalSystems

Conceptually, the quantum trajectory formalism has been extended tononadiabatic dynamics,43–45 the phase-space representation, and the densitymatrix approaches.46–53 There exist developments on imaginary time propa-gation,54 complex space Bohmian trajectories,55 and dynamics that are basedon the bipolar rather than polar decomposition of wave functions.56,57

On the practical side, quantum trajectories have been used in the-oretical and computational chemistry for three distinct purposes: (1) tointerpret the wave function computed by the conventional wave functionpropagation techniques; (2) to monitor the wave function density flow fordynamical grid adjustments; and (3) to solve the time-dependent Schrodingerequation, and to obtain (x, t) or quantities of interest, directly from thetrajectories.

Quantum trajectories have been used to interpret and to draw quantum-classical analogies in the area of surface scattering. Phenomena includingFresnel and Fraunhofer regimes, rainbow scattering, the quantum Talbot effect,and others have been explored.58–61 Some ideas from the Bohmian dynamicslead to the moving grid techniques62 in which positions of grid points aretime-dependent but are different from the Bohmian trajectories. The numericalgoals of moving grid techniques are to gain stability of the dynamics of gridpoints and to improve accuracy in derivative evaluations. Instead of movingthe grid points, grids can be optimized by adding or eliminating the grid pointsby reconstructing Bohmian trajectories at grid edges.63 A similar criterionis used in the ab initio wave-packet dynamics of Iyengar to optimize thewave-packet representation and to minimize ab initio evaluations of classicalforces.64–66

Using quantum trajectories as a practical way of solving the multidimen-sional time-dependent Schrodinger equation is an exciting prospect. Severalhigh-dimensional applications of the exact quantum trajectory method, includ-ing up to 200 degrees of freedom, have been reported.67,68 There, the quantumforce is evaluated on the fly with the moving least-squares fitting of the wavefunction amplitude. However, a general implementation of the exact numericalBohmian 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” implementa-tions based on the Taylor expansion of the equations of motion truncated ata low order. Both real-valued54,72 and complex-valued trajectories55,73 havebeen used. The approximate quantum potential (AQP) approach35,74,75 in-volves propagation of the trajectory ensemble together with a global evalua-tion of the quantum force from the moments of the trajectory distribution. This“mean-field” type of approximation gives quantum force for all trajectories si-multaneously.


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

BOHMIAN QUANTUM-CLASSICAL DYNAMICS

Dynamics of most chemical reactions are typically very complex for afully quantum-mechanical analysis. Fortunately often, it is possible to distin-guish between particles, such as electrons and protons, that require a quantumdescription and particles, such as heavy nuclei, which can be described accu-rately using classical mechanics. When quantum particles remain in the samequantum state throughout the reaction, the Born–Oppenheimer (adiabatic) ap-proximation is invoked. The quantum state merely provides an external poten-tial 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. Manyother types of chemical reactions involve several quantum states. Examplesinclude photochemical reactions,80,81 transfers of electrons,11,82 protons,83,84

spins,85 energy86–88 and quantum phase,89 electron-vibrational relaxation pro-cesses,9,10,90,91 and solvation dynamics.92,93 These phenomena extend beyondthe Born–Oppenheimer approximation and are modeled by the nonadiabaticgeneralizations of molecular dynamics.94–104

Coupling between quantum and classical degrees of freedom constitutesthe key question in mixed quantum-classical approaches. It raises central issuesthat do not admit unique solutions. Numerous coupling schemes have beenproposed ranging from formal mathematical solutions20,105–110 to specific al-gorithms that have been applied to many problems in chemistry, physics, andbiology.9–11,80–84,87–92 Historically, the first and the most straightforward ofthe quantum-classical approaches is based on the Ehrenfest theorem. The theo-rem states that the equations of motions for the average values of the quantumposition and momentum operators coincide with the classical equations of mo-tion.99,111 This leads to the mean-field approximation in which the classicalvariables are coupled to the expectation values of the quantum observables. Ifthe quantum system remains in a single state, then the Ehrenfest approach re-duces to adiabatic molecular dynamics. In general, the quantum system formsa superposition of several states, and the classical dynamics evolve in the mean-field quantum potential. The average Ehrenfest trajectory is inappropriate whenseveral reaction channels exist and involve substantially different potential en-ergy surfaces.94,96 In a corresponding quantum description, the wave packetssplit and follow different reaction channels. The branching difficulty in thecoupling of quantum and classical mechanics is known as the quantum back-reaction 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 al-gorithm. Other, more computationally demanding, quantum-classical approx-imations dealing with the trajectory branching include the multiconfigurationmean-field theory,20,98,103 partial Wigner transform dynamics,20,108 and semi-classical treatments.104 The Bohmian interpretation of quantum mechanics34

provides an alternative means of achieving the branching of the classical tra-jectories.24,25 By correlating each classical trajectory with an individual parti-cle, an ensemble of trajectories can be generated. Trajectories associated withdifferent quantum states are represented by different Bohmian particles. Tra-jectories evolve independently and branch as in the fully quantum-mechanicaldescription. Subsequently, we describe the Ehrenfest and Bohmian quantum-classical approaches. Their properties are illustrated by a model representingscattering of a light particle off a surface containing slow phonon modes. Thephotoinduced electron transfer from a molecule to a semiconductor surface indye-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

H(x;X) = − �2

2m∇2x + V(x;X) [31]

The classical subsystem generates an external field contributing to the potentialV(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 theHamiltonian [31] with the purely classical kinetic and potential W(X) energies

Eq−cl = Eq + Ecl =∫d3x �∗(x)H(x;X)�(x) + MX2

2+W(X) [32]

The wave function �(x) evolves according to the time-dependent Schrodingerequation

i�∂�(x)∂t

=(

− �2

2m∇ 2x + V(x;X(t))

)�(x) [33]

in which the potential V depends on time through the dynamics of classicalvariables X(t). The evolution of the classical coordinates obeys the Newtonequation

MX = −∇XW(X) + Fq [34]


which contains the quantum force Fq in addition to the ordinary classical force−∇XW(X). The definition of the quantum force constitutes the quantum back-reaction problem.20,105–110 The quantum force of the Ehrenfest approach isgiven by the quantum mechanical expectation value of the gradient of the quan-tum Hamiltonian

Fq = −∫d3x �∗(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 approxima-tion are illustrated in Figure 3. For a given wave function of the quantumsubsystem, 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, remainslocalized throughout the time of an experiment. Here, the Ehrenfest force ofEq. [35] generates an optimal, classical description of the wave packet. On theother hand, if the wave packet branches, as illustrated in Figure 3a, then theEhrenfest approach fails to capture the branching. It asymptotically cannot de-scribe the distinct reaction channels of the classical subsystem associated withdifferent quantum states.

Quantum-Classical Coupling via Bohmian Particles

The branching of the classical subsystem is reproduced by the Bohmianquantum-classical approach. This is done by generating an ensemble of classi-cal trajectories correlated with different members of the Bohmian ensemble ofquantum particles. Consider the Bohmian formulation of the Ehrenfest force ofEq. [35]. The polar form of the wave function �(x) = R(x) exp(ıS(x)/�) leadsto the following expression for the quantum mechanical expectation value ofthe Hamiltonian Eq. [31]:∫

d3x �∗(x)

[− �

2

2m∇2x + V(x;X)

]�(x) =

∫d3x R2(x)

×[

(∇xS(x))2

2m+Q(x) + V(x;X)

][36]

where Q(x) is the quantum potential

Q(x) = − �2

2m∇ 2x R(x)R(x)

[37]


Figure 3 Schematic representation of the evolution of a heavy particle moving in the po-tential created by light particles. For example, the X coordinate can represent the bondlength of a diatomic molecule. The black solid lines depict the potential energy sur-faces for the molecular ground and excited electronic states. (a) A quantum-mechanicalwave-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 greysolid line directly above the dashed line. While in the excited state, the wave-packetmoves to infinite X, and the diatomic dissociates. The nonadiabatic coupling betweenthe electronic states causes transfer of a fraction of the wave-packet back to the groundstate 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 sev-eral components, corresponding to different outcomes of the excitation dynamics. (b)In the Ehrenfest approximation, the classical particle cannot split and evolves on a sin-gle trajectory. The mean-field potential energy surface (grey line) is an average of theground and excited state potentials. The particle evolution corresponds to neither ofthe quantum-mechanical outcomes; the diatomic can be artificially trapped between thebound 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 ensembleof classical-like particles. Each Bohmian particle is coupled independently to the heavyparticle. This coupling generates an ensemble of classical trajectories evolving on differ-ent potential energy surfaces. This treatment mimics the quantum-mechanical branchingof the wave-packet describing the heavy particle.

The quantum probability distribution is R2(x) = �∗(x)�(x). As a result, inBohmian mechanics, the quantum energy Eq. [36] is interpreted as the energy ofan ensemble of particles with the probability distribution R2(x). The energy ofeach particle is equal to [(∇xS(x))2/2m+Q(x) + V(x;X)]. The Ehrenfest force,


generated by the quantum subsystem and acting on the classical subsystem,takes the following form in the Bohmian representation:

Fq = −∫d3x R2(x)∇XV(x;X) [38]

It can be viewed as the average of the forces −∇XV(x;X) resulting from theensemble of Bohmian particles with the probability distribution R2(x). Theensemble averaged Bohmian force Eq. [38] is identical to the Ehrenfest force ofEq. [35], as indicated by the time-dependent Hellmann-Feynman theorem.112

The theorem states that the time derivative of the expectation value of the quan-tum energy involves only the derivative of the quantum Hamiltonian, providedthat the wave function evolves according to the time-dependent SchrodingerEq. [33].

The Bohmian quantum-classical approach solves the branching problemby moving the

∫d3x R2(x) ensemble averaging outside the quantum-classical

dynamics. The initial conditions for the Bohmian ensemble of quantum particlesare sampled from R2(x). The averaging is performed only at the final time. Inparticular, the quantum force acting on the classical subsystem at any giveninstance does not involve the integration overR2(x) as in Eq. [38]. The quantumforce is calculated for a single representative of the Bohmian ensemble. Thistreatment of the quantum-classical coupling generates a distribution of classicaltrajectories correlated with different Bohmian particles, Figure 3c. As a result,the classical trajectories evolve differently depending on whether the correlatedBohmian particles correspond to the excited state wave function, Figure 3c, orto the ground state.

The Bohmian quantum-classical simulation runs according to the follow-ing algorithm. First, initial conditions for the wave function and classical tra-jectories are chosen in the usual manner. Positions x of Bohmian particles aresampled from the initial distribution R2(x). For each initial coordinateX of theclassical particle, an ensemble of initial coordinates x for the Bohmian particlesis sampled fromR2(x). Each Bohmian particle is correlated with a separate copyof the classical subsystem. Second, for each member of the quantum-classicalensemble, the wave function is propagated by the time-dependent SchrodingerEq. [33]. Simultaneously, the classical trajectory is evolved by the NewtonEq. [34] with the quantum force

Fq = −∇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 Newtonequation, including the quantum potential Eq. [37]

mx = −∇x[Q(x) + V(x;X)] [40]


or, equivalently, directly by

x = ∇xS/m [41]

The first option can be used with the semiclassical schemes for propagatingthe Bohmian ensemble described in the sections “The Independent TrajectoryMethods” and “Dynamics with the Globally Approximated Quantom Poten-tial” of this review. The second option is advantageous if the time-dependentwave function is available by a direct quantum-mechanical propagation. Third,the results are averaged over the ensemble of the Bohmian quantum-classicaltrajectories. The Bohmian quantum-classical method is defined by Eqs. [33],[34], [39], and [40]. The fully classical limit for both subsystems is achievedeasily by setting �→ 0 in Eq. [37] and eliminating the quantum potential fromEq. [40]. An alternative derivation of this approach is given in Refs. 24 and113. The derivation starts with a fully quantum description of both subsys-tems. The quantum potential then is dropped from the equation of motion forthe classical subsystem.

Numerical Illustration of the BohmianQuantum-Classical Dynamics

The Bohmian quantum-classical approach is illustrated here with a modelintended originally as a simplified representation of gaseous oxygen interactingwith a platinum surface.114,115 Alternatively, it can be viewed as a model forthe photoinduced electron transfer in a molecular chromophore adsorbed ona surface of a solid-state bulk material. Systems of this type form the basis fordye-sensitized semiconductor solar cells, also known as Gratzel cells,11,82 andvarious molecular electronics devices.116 The model consists of a light particlex with mass m colliding with a heavier particle X with mass M. The heavyparticle is bound to an immobile surface, Figure 4. In the molecule-bulk electrontransfer process, the light particle can be viewed as the electron coming fromthe 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]

V1(x) = a[e−2b(x−c) − 2e−b(x−c)] [42b]

V2(X) = 12M�2X2 [42c]

V(x,X) = Ae−B(x−X) [42d]

where T1 and T2 are the kinetic energy operators. The harmonic potentialV2 describes the interaction of the heavy particle with the surface. The Morse


Figure 4 A model illustrating the advantages of the Bohmian quantum-classical dy-namics over the Ehrenfest approach. The top panel depicts the system. The light particleapproaches and scatters off a surface which contains a phonon mode involving the heavyparticle. The Hamiltonian and its parameters are given in Eq. [42a] and Table I, respec-tively. The bottom panels show the time-dependent probability for the light particle tomove a certain distance away from the surface following the scattering event. The datashown 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 asufficiently long time. The light particle wave-packet splits, and a part of it remains tem-porarily trapped with the heavy particle. In the exact solution, the trapped part of thewave-packet follows the scattered part and eventually decays. The Ehrenfest approacherrs both in the scattering onset and the asymptotic scattering probability, as highlightedby 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, acceleratingthe scattering. At longer times, the lack of branching creates an artificial constraint onthe energy exchange between the light and heavy particles. The excess energy transferredfrom the heavy to the light particle during the early evolution leaves the heavy particlewith insufficient energy to continue promoting the scattering of the light particle.


Table 1 Parameters Used in Simulationof the Scattering Problem, Eq. [42a] andFigure 4

m 1 amu a 700 kJ/molM 10 amu b 5.0 A−1

� 4 × 1014 s−1 c 0.7A 104 kJ/mol x0 6.0 AB 4.25 A−1 � 0.5 A

potential V1 describes the interaction of the light particle with the surface. Thetwo particles interact by the exponentially repulsive potential V . Parametersparticular 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 lightparticle is described by a Gaussian wave-packet

�(x, t = 0) = exp[ik0x

�

]exp

[− (x− x0)2

�2

][43]

located 6 A away from the surface. The initial momentum of the light particlek0 = −√

2mE0 corresponds to an incident energyE0 shown in Figure 4. The ini-tial conditions of the heavy particle represent the ground state of the harmonicoscillator. Quantum mechanically, the heavy particle is described by the ground-state wave function of the harmonic potential V2, Eq. [42a]. In the Ehrenfestand Bohmian quantum-classical approaches, the ground state of the harmonicoscillator is represented by an ensemble of classical particles. The particle en-semble is generated microcanonically at the energy of the quantum-mechanicalground state. The Bohmian ensemble representing the light particle is sampledfrom the density R2(x) = �∗(x)�(x) generated for the Gaussian Eq. [43]. Thecoordinate of the Bohmian particle is evolved in time according to Eq. [41] andis used to compute the force on the heavy particle using Eq. [39].

←(Continued) Similar to the Ehrenfest scheme, the quantum-classical Bohmian approacherrs at the early times by mistreating zero-point energy. However, it correctly reproducesthe asymptotic value of the scattering probability, since the interactions of the immedi-ately scattered and transiently trapped parts of the light particle with the heavy phononare not averaged as in the Ehrenfest approach but are treated independently.


The simulation is characterized by the time-dependent scattering proba-bility for the light particle to leave the region within x0 = 6 A of the surface

Ps(t) =∫ ∞

x0

|�(x, t)|2dx [44]

The results of the Ehrenfest and Bohmian quantum-classical approaches aresimilar to the fully quantum-mechanical simulation data at high-incident en-ergies (Figure 4). The disagreement increases at lower energies. The Ehrenfestmethod deviates from exact quantum mechanics both at short and long times.Asymptotically, the failure of the Ehrenfest method to reproduce the completescattering of the wave packet stems from its mean-field nature. In the fullyquantum description, the wave packet describing the light particle splits. Partof the wave packet remains temporarily trapped with the heavy particle nearthe surface. In the exact solution, the trapped part of the wave packet followsthe scattered part and eventually decays. The trapped and scattered portionsof the wave packet become correlated with different parts of the wave-packetdescribing the heavy particle. The heavy particle wave packet also branchesinto several components. A single mean-field trajectory of the heavy particleproduced by the Ehrenfest approach cannot describe this correlation. The en-ergy 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 scat-tering probability in the Ehrenfest method increases too rapidly, relative tothe fully quantum scattering probability. This occurs because the energy of theheavy particle, including zero-point energy, is fully available for exchange withthe light particle. Quantum treatment of the phonon results in preservation ofzero-point energy. However, when the phonon is described classicly, zero-pointenergy is transferred to the light particle. The excess energy transfer from theheavy to the light particle during the early evolution causes the light particle toaccelerate too fast. In addition, the scattering probability in Figure 4 increasestoo rapidly. Later, the heavy particle is left with insufficient energy to continuepromoting the scattering of the light particle at longer times. This leads to theerroneous scattering asymptotes in Figure 4.

The quantum-classical Bohmian approach uses an altered treatment of theinteractions of the scattered and transiently trapped parts of the light particlewith the heavy particle. The asymptotic value of the scattering probability isreproduced correctly by this approach. Interactions are treated separately in thequantum-classical Bohmian approach, whereas the interactions are averaged inthe Ehrenfest approach. The interaction explicitly depends on the position of arandomly selected Bohmian particle representing the light particle subsystem.An ensemble of heavy particle trajectories is generated for a single evolution ofthe light particle wave function. The energy associated with a single quantum-classical Bohmian trajectory is not required to be conserved. This feature of theBohmian approach is analogous to the quantum-mechanical property that the


energy associated with different branches of a wave-packet is not conserved. Asa result, the energy can continue to flow from the heavy to the light particle inthe trajectories describing the transiently trapped part of the light particle, andscattering proceeds to full completion. Because the heavy particle is describedclassicly 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 proceedstoo fast. Thus, in the example considered here, the Bohmian quantum-classicalapproximation improves over the Ehrenfest approach in the properties associ-ated with the branching of the trajectories of the heavy particle. The conserva-tion of zero-point energy cannot be resolved properly with classical dynamicsand requires semiclassical corrections, such as in the quantized Ehrenfest app-roach.89,102,117

Properties of the Bohmian Quantum-ClassicalDynamics

The Bohmian interpretation of quantum mechanics provides a classical-like view on the evolution of quantum particles and creates an alternative routefor generating quantum-classical approximations to the dynamics of complexsystems. In particular, Bohmian mechanics offers a solution to the trajectorybranching problem in the quantum-classical simulation. The problem is re-solved by creating a new type of the quantum back-reaction acting on theclassical subsystem. The Bohmian back-reaction is defined uniquely and is com-putationally simple. It provides a straightforward connection to the full clas-sical limit. The branching of quantum-classical trajectories is achieved in theBohmian approach by coupling of the classical subsystem to a single quantumparticle in the Bohmian ensemble. An ensemble of quantum-classical trajecto-ries is generated for a single, initial quantum-mechanical wave function. Thisis in contrast to the quantum-classical Ehrenfest approximation in which a sin-gle average classical trajectory is generated. The Bohmian quantum-classicalapproach succeeds where the Ehrenfest method fails. In particular, it succeedsfor cases in which distinct classical trajectories are required for different statesof the quantum subsystem. Distinct Bohmian quantum-classical trajectoriesemerge for different quantum states. Traditionally, the branching problem issolved by modification of the Ehrenfest approach with a surface-hopping pro-cedure.94–97,100,101 Although the Bohmian quantum-classical method producesresults that are similar to surface hopping, the latter procedure is ad hoc andvaries between implementations. In contrast, the Bohmian approach is uniquelydefined.

Bohmian quantum-classical dynamics can be implemented easily by a mi-nor modification of a standard Ehrenfest code. In the Ehrenfest method, thequantum force operator is averaged across the whole quantum-mechanicaldensity distribution. In contrast, the Bohmian force is computed for a singlepoint in the quantum-mechanical density function, avoiding the averaging. The


quantum subsystem can be evolved using the Bohmian equations of motion forthe ensemble of Bohmian particles (Eqs. [37] and [40]). Alternatively, the evolu-tion of the Bohmian particles can be obtained directly from the time-dependentwave function (Eq. [33]). The latter route eliminates calculation of the quan-tum potential [37] and its derivative, which may be ill-behaved in the regionsof low quantum density. The total quantum-classical energy is not conservedalong a single quantum-classical Bohmian trajectory. This is expected becausethe energy of a branch of a quantum-mechanical wave packet also has no con-servation requirement. Potentially, the energy nonconservation can present aproblem in a quantum-classical simulation, although applications consideredthus far find no such problem.24,25,113,118–121

In addition to resolving the branching problem in a unique way, theBohmian approach to the coupling of quantum and classical mechanics providesa new opportunity that is not attainable by the traditional approaches such asEhrenfest and surface hopping. Consider a situation, such as in Refs. 122 and12, in which a molecular complex AH-B is coupled to a solvent. The protonH has to be treated quantum mechanically, whereas both A and B atoms andthe solvent can be treated classicly. In the traditional approaches, the quantum-classical coupling of the proton to the solvent is formulated in exactly the sameway as the proton coupling to the atoms A and B. If the proton tunnels betweenA and B but does not tunnel into the solvent, then the quantum behavior of theproton 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 quantummechanically and to the solvent purely classically. The evolution of the protonwill 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-solventcoupling terms. The time-dependent Schrodinger equation, which determinesthe quantum potential, will be decoupled from the solvent and will dependonly on A and B. To couple the quantum-mechanical proton to the classicalsolvent, the traditional methods must include solvent-dependent terms in theSchrodinger equation for the proton. The Bohmian quantum-classical approachcan provide significant savings by ignoring the multiple solvent terms in theSchrodinger equation. At the same time, solvent terms must be included in theNewton Eq. [40] for the proton.

In addition to the approach described here,24,25,113 several relatedschemes for quantum-classical dynamics based on the Bohmian interpretationof 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 combinethe Bohmian description for the quantum subsystem with a Liouville-space de-scription for the classical subsystem. The Bohmian description is recast in thelanguage of hydrodynamics, partial hydrodynamic moments are introduced,and a hierarchy of equations is derived using closures as in quantized Hamiltondynamics in Refs. 102, 117 and 89.

Hybrid Bohmian Quantum-Classical Phase–Space Dynamics 311

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

HYBRID BOHMIAN QUANTUM-CLASSICALPHASE–SPACE DYNAMICS

The previous section presented a mixed quantum-classical scheme thatcollapses to a single trajectory in the classical limit for the heavy particle. Thisis the simplest and often adequate quantum-classical description. A more accu-rate representation can be obtained in a phase–space picture. The doubling ofthe number of independent variables improves the accuracy of the classical pic-ture. For example, a purely classical propagation of a distribution function thatcorresponds to a Gaussian wave function gives an exact description of quantumharmonic motion.37,126 Therefore, the mixed quantum-classical description inthe phase–space representation is particularly relevant for semiclassical systems.The coupling of approximate quantum and phase–space classical descriptionsis explored in many publications.51,127–129 One challenge of this approach isthe phase–space description of the quantum part. The expansion of quantumequations in powers of Plank’s constant suffers from a convergence problem.130 The coupling of the Bohmian and phase–space descriptions is a promisingalternative. 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

∂�W

∂t= −(H,�W )qu = i

�(H exp �qp/(2i)�W − �W exp �qp/(2i)H)

[45]

In Eq. [45] �W = �W (q, p,Q, P) is the Wigner distribution126 correspondingto the wave function �(x,Q, P). The Wigner transform of the wave function istaken with respect to the quantum coordinate x. The dependence on the classicalvariables (Q,P) is parametric. The Hamiltonian H = Hq(q, p) + Vint(q,Q) +HQ(Q,P) involves the Hamiltonians of the quantum and classical subsystems,Hq and HQ, respectively, together with the interaction potential Vint(q,Q).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(HQP�W − �WQPH). The imbalance between the classicalPoisson bracket and the quantum Poisson bracket operator in Eq. [45] leads toproblems with operator ordering. These problems result in a violation of theJacobi identity. Recent work by one of us demonstrates that it is possible to


restore proper behavior by performing all calculations quantum mechanicallyand taking the �→ 0 limit at the very end of the calculations.110

In contrast to the full phase space distribution, the Bohmian picture pro-vides a unique momentum p = pq associated with every position q. The con-nection between the Bohmian and phase space descriptions is established viapartial hydrodynamic moments

〈pn〉 = 〈pn�W 〉(q,Q, P) =∫dp pn�W (q, p,Q, P) [46]

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

q = p

m[47]

p = − ∂

∂q(Vq(q) + Vint(q,Q)) + Fhyd(q,Q, P) [48]

Q = P

M[49]

P = − ∂

∂q(VQ(Q) + Vint(q,Q)) [50]

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

Fhyd(q,Q, P) = −1m〈1〉

∂(q,Q, P)∂q

[51a]

(q,Q, P) = 〈p2〉 − 〈p〉2

〈1〉 [51b]

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 spatialvariance with respect to the quantum position q gives rise to the quantum force.

Unlike traditional Bohmian mechanics formulated in the coordinatespace, 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 notinteracting with the classical subsystem, the hydrodynamic force reduces46 tothe negative gradient of the quantum potential (Eq. [3]). For harmonic systems,the higher order partial hydrodynamic moments can be expressed through thelower 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 theclassical hydrodynamics force given in the coordinate space representation. Anoscillator 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 quantizedhamilton dynamics.102,135,136

THE INDEPENDENT TRAJECTORY METHODS

This section describes approaches based on independent Bohmian trajec-tories and derived from the hierarchy of time-evolution equations for the wavefunction phase and amplitude as well as their spatial derivatives. For practi-cal reasons, the hierarchy is truncated by setting higher order derivatives tozero. Truncation at the second order reveals a connection to other semiclassicalmethods.

The Derivative Propagation Method

The derivative propagation method is based on the hierarchy of equationsfor the evolution of the wave function phase, amplitude, and their derivatives.The wave function is represented as an exponent with the real and imaginaryfunctions as its argument

(x, t) = exp(C(x, t) + ı

�S(x, t)

)[52]

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

∂C

∂t= − 1

2m[∇2S + 2∇C × ∇S] [53]

∂S

∂t= − 1

2m∇S × ∇S + �

2

2m[∇2C + ∇C × ∇C] [54]

and subsequent differentiation with respect to x gives an infinite hierarchy ofequations for the derivatives. The equations are coupled because the equationsfor the lower order derivatives contain higher order derivatives. The termswith C in the RHS of Eq. [54] constitute the quantum potential. By truncatingthe Taylor expansion of S and C around position x at a certain order, oneobtains a nonlinear system of evolution equations for the trajectory-specificexpansion coefficients. For example, truncation at the second order results in asystem of six equations involving the coefficients, classical potential V , and its


first and second derivatives. Truncation through the second order is exact forGaussian wave packets evolving in a quadratic potential. The equations can betransformed into the Lagrangian frame of reference in which the trajectoriesand their associated systems of equations are propagated independently.

The advantage of the derivative propagation method is that it is insensitiveto the trajectory crossings, unlike the exact implementations. In addition, a fewor even one trajectory may be sufficient for some applications. However, theindependence of the trajectories comes at the cost of requiring high derivativesof V . Moreover, the scaling on the number of equations with dimensionalitymakes implementation beyond the quadratic order prohibitively expensivefor high-dimensional systems. We refer you to the book of Wyatt137 for moredetails and applications of the derivative propagation method. Of conceptualinterest is the calculation of the energy-resolved transmission probabilityover the Eckart barrier. The calculation is carried out with the second andthird order of the derivative propagation method. The second-order methodshows a “typical semiclassical” accuracy that is surprisingly similar to theresults of Grossmann and Heller138 obtained with the Van Vleck–Gutzwillerpropagator. This result suggests a formal connection to the semiclassicaltheories. The third-order method improves agreement with the exact QMresult.

The Bohmian Trajectory Stability Approach.Calculation of Energy Eigenvalues by Imaginary TimePropagation

A different independent quantum trajectory scheme has been developedby Liu and Makri.139 Similar to the derivative propagation method, it formsan infinite hierarchy of equations. Unlike the derivative propagation method, itexplicitly uses conservation of the wave function probability given by Eq. [20].According to wave function probability conservation, the wave function den-sity �(x, t) is related to the initial density by the stability matrix element of atrajectory, namely the Jacobian computed along the trajectory

�(xt) = �(x0)∂x0

∂xt, J(xt, x0) = ∂xt

∂x0[55]

The elements of the trajectory stability or monodromy matrices

M(t) =

⎛⎜⎜⎝∂pt

∂p0

∂pt

∂x0∂xt

∂p0

∂xt

∂x0

⎞⎟⎟⎠ [56]

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 bythe quantum potential in addition to the classical potential. Evolution of thestability matrix

dM(t)dt

=

⎛⎜⎜⎝

0 −∂2H

∂x2

∂2H

∂p2 0

⎞⎟⎟⎠ M(t) [57]

gives J; however, it is coupled to the higher derivatives of J. Differentiation ofEq. [57] with respect to x results in an infinite hierarchy of equations which inpractice, are truncated at a low order. In general, the features of the Bohmiantrajectory stability approach are similar to those of the derivative propagationmethod. An important difference is that, formally the stability method doesnot 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 theBohmian formulation into imaginary139 time constitutes a very promising ap-plication of the stability matrix approach.54 It is known that transformationfrom real to imaginary time, t → −ı�, converts the time-dependent Schrodingerequation into the diffusion equation

−� ∂∂� (x, �) = H (x, �) [58]

This equation offers a convenient way to obtain low-energy eigenvalues. Anywave function propagated under Eq. [58] converges to the eigenstate of the samesymmetry, as in the diffusion Monte Carlo method.142 The same substitutioncan 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 trajectorydynamics. Nonetheless, Liu and Makri show that one can ensure smooth be-havior of the trajectories by repartitioning A and S at each time step, because (x, �) is real and division intoA and S is arbitrary. The authors proceed to com-pute the ground-state energies for HCN and H2O in two and three dimensionsrespectively. Accurate ground-state energies were obtained with the fourth- andsixth-order of the Bohmian trajectory stability propagation method.


Bohmian Mechanics with Complex Action

Bohmian mechanics with complex action55,73,143 is an extension of thetime-dependent Schrodinger equation into the complex plane. This constitutesa drastic departure from the methods discussed, yet the implementation of thecomplex action technique is related directly to the derivative propagation andthe Bohmian trajectory stability methods discussed. The starting point is theexponentiation of the wave function

(x, t) = exp( ı�S(x, t)

)[60]

substituted into the time-dependent Schrodinger equation. The result is an equa-tion of the Hamilton-Jacobi type, now containing a complex action functionS(x, t),

∂S

dt= −mv

2

2− V + ı�

2∇v [61]

where mv(x, t) = ∇S(x, t). Transformation into the Lagrangian frame of refer-ence is accomplished formally as before

dx

dt= v(x, t) = ∇S(x, t)

m[62]

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

mdv

dt= −∇V + ı�

2∇2v [63]

Similar to the derivative propagation and Bohmian trajectory stability methods,Eq. [63] involves an unknown spatial derivative of the phase. Thus, Eq. [63] isdifferentiated with respect to x and so forth to arrive at an infinite hierarchyof equations. For practical reasons, the hierarchy is truncated at a low order.The first order is equivalent to the complex Gaussian wave-packet methodof Huber and Heller.144 The second order is equivalent to the complex WKBmethod.73 The complex quantum force acting on a Gaussian wave packet iszero, which is simpler than the linear force of the real-space formulation. Thesimplification comes at a price. First, analytic continuation of all quantities,including V and its gradients, Hessians and so on, into the complex plane isneeded. 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 ofall 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 allowa “semiclassical” description of interference through multiple paths.143 Thebipolar decomposition of the wave function, presented in a series of papersby Poirier,56,145–149 also describes interference but within the real Bohmiantrajectory framework.

Bohmian mechanics with complex action provides a new angle on thesemiclassical description of quantum effects and representation of nonlocality.It creates insightful connections between several semiclassical methods. Theneed for analytic continuations and the rapid scaling of the efforts with sys-tem dimensionality constitute major challenges to numerical applications forchemical systems.

DYNAMICS WITH THE GLOBALLYAPPROXIMATED QUANTUM POTENTIAL (AQP)

The independent trajectory methods described in the previous section pro-vide a semiclassical description of quantum effects already at the second order ofthe hierarchy truncation. This feature can be interpreted as a “local” descriptionof quantum nonlocality based on the first and second derivatives of the wavefunction at point x, through the phase, amplitude, classical, and quantum po-tentials. After all, if the wave function is smooth, one can define it accuratelyin a large region of space by knowing its value and high-order derivatives at asingle point. In this section, we will describe a different approach for estimatingnonlocal quantum effects in semiclassical dynamics in an “average” sense fromthe trajectory ensemble. We will use atomic units with � = 1 below and willdrop � 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, ideallylinear, such that the computation of the quantum force is a small addition tothe classical trajectory propagation. The polynomial scaling allows the methodto be applied to hundreds of degrees of freedom. “Robust” implies numericalstability. In particular, the numerical propagation should be insensitive totrajectory crossings and should switch to classical propagation if the quantumforce becomes inaccurate. “Semiclassical” means that the method shoulddescribe the leading QM effects. However, it should not aim for exact QMdynamics. The latter requirement is a necessary trade-off for a better thanexponential scaling with system dimensionality. In addition, it is desirable for amethod to have an accuracy criterion and a systematic procedure for achievingthe exact QM description. The first development of this kind of method wasbased on expansion of the wave function density �(x, t) in terms of Gaussianfunctions with optimized parameters.35,150 The number of Gaussian functions


can vary but must remain small. The procedure was expensive as a result ofnonlinear parameter optimization and was unstable toward the addition ofnew Gaussians in the course of dynamics. This was the proof of concept for theAQP idea and the first demonstration of quantum trajectory dynamics in thesemiclassical implementation. A more numerically efficient and conceptuallyintuitive approximation based on the nonclassical momentum is presentedsubsequently.

Global Energy-Conserving Approximation of theNonclassical Momentum

TheoryAn analytical representation of the AQP, Q is important for efficiency andaccuracy of the quantum force computation. One physically appealing way ofconstructing Q is to consider the nonclassical component, r(x, t), of the QMmomentum operator defined by Eq. [13]. The representation of this single objectwithin a basis set,

r(x, t) ≈ r(x, t) = �c(t) × �f (x) [64]

then can be used to obtain both Q,

Q = − 12m

(r2 + ∇ r) [65]

and the analytical quantum force Fq = −∇Q. A detailed derivation for multidi-mensional systems is given in Ref. 74. The vector �c contains the time-dependentbasis expansion coefficients. The coefficients �c are found from the minimizationof the average deviation between r(x, t) and r(x, t)

I = 〈(r− r)2〉t = I0 +∫ (

∇ r(x, t) + r2(x, t))�(x, t)dx [66]

Differentiation by parts was used to obtain Eq. [66]. I0 denotes the term thatdoes not depend on �c. By changing the integration variable to xt, using Eq. [20],and replacing the integration with summation over discrete trajectories, oneobtains

I = I0 +∫ (

∇ r(xt) + r2(xt))�(xt)dxt = I0 +

∑i

(∇ r(x(i)

t ) + r2(x(i)t )

)wi

[67]

The fact that neither �(x, t) nor its derivatives appear in Eq. [67] is of centralimportance for efficient implementation. The algorithm scales linearly with


respect to the number of trajectories, as evidenced by the single summationover the trajectories.

Minimization of I with respect to the expansion coefficients that are, theelements of �c, gives a system of linear equations

∇cI = 2S�c + �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

sjk = 〈fj|fk〉 =∑i

wifjfk

∣∣∣x=x(i)

t, bj = 〈dfj

dx〉 =

∑i

widfj

dx

∣∣∣x=x(i)

t[69]

The solution in vector form is

�c = −12

S−1�b [70]

One also can obtain the evolution equations for the expansion �c in terms ofvarious moments of the trajectory distribution (an interested reader is referredto 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

dt= 〈 p

m

dp

dt+ ∇(V + Q) × dx

dt+ ∂Q

∂t〉 = 〈∇cQ〉d�c

dt= 0 [71]

The energy is conserved because Q is proportional to I − I0. Because of the latterproperty, the optimization procedure becomes a variational determination ofQ.

The smallest physically meaningful basis �f consists of just two functions�f = (1, x). For this case, the approximation can be written in a particularlytransparent form

r = − x− 〈x〉2(〈x2〉 − 〈x〉2)

[72]

The wave function is normalized to 1, and the nonclassical momentum is ap-proximated with a linear function centered at the average position of the wavepacket and inversely proportional to its variance, = 〈x2〉 − 〈x〉2. This func-tional form of r results in a quadratic Q and, consequently, a linear quantumforce. Hence, the approximation is termed the linearized quantum force (LQF).Note that Fq is inversely proportional to 2; therefore, Fq quickly vanishes fordelocalized wave packets.


From the physical point of view, the linear nonclassical momentum exactlycorresponds to a Gaussian wave packet evolving in time in a locally quadraticpotential. For general potentials, it can describe the dominant quantum effectsin semiclassical systems, such as wave-packet bifurcation, wave-packet spreadin energy, and moderate tunneling. Zero-point energy effects are reproducedfor short times, including a few vibrational periods and depending on the an-harmonicity of the system. This is adequate for direct gas phase reactions asdemonstrated 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 byexplicitly using the trajectory weights Eq. [20] instead of propagating the wavefunction density �(xt) along the trajectories according to Eq. [19]. Therefore, toextract useful information, one has to reconstruct (x, t) through interpolationor fitting. In this way, information other than expectation values, for instance,correlation functions involving time-dependent overlaps of wave functions orinternal eigenstate projections given by integrals of the type 〈�(0)| (t)〉, can beobtained. Note that the reconstruction procedure is not part of the propagationprocess; therefore, it does not affect the accuracy of the dynamics. If �(x,0) islocalized, then one can use the cheapest strategy and approximate (x, t) as acomplex Gaussian. Parameters for the Gaussian are found from the momentsof the trajectory distribution using |�(x,0)| as a weighting function.150 If moreaccurate information about (x, t) is required, then one can find (x, t) usinga convolution with a narrow Gaussian with the width parameter, ˇ. For eachtrajectory position xt, the wave function phase S(xt) and its gradient pt areknown. The wave function density can be determined at any position x as

�(x) =√ˇ

�

∑i

exp(−ˇ(x(i)

t − x)2)wi [73]

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

Application: Photodissociation Cross Section of ICNLet us consider photodissociation of ICN treated in the Beswick–Jortner model153 following Refs. 154 and 155. A wave packet representing an ICN moleculeis excited by a laser from the ground to the excited electronic state. In the excitedstate, the ICN molecule dissociates into I and CN. The Hamiltonian and theJacobi coordinates (x, y) are described in Ref. 155. An initial wave function

(x, y,0) =√

2�

(˛11˛22 − ˛212)1/4 exp

(−˛11(y− y)2

−˛22(x− x)2 + 2˛12(y− y)(x− x))

[74]


is defined as the lowest eigenstate of the ground electronic surface with zeromomentum. The ground-state potential consists of two harmonic potentials inCN and CI stretches. Thus, (x, y,0) is a correlated Gaussian wave packet lo-cated on the repulsive wall of the excited surface. The photodissociation crosssection is computed from the Fourier transform of the wave packet autocorre-lation 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 potentialsurface yields a rather simple dissociation dynamics; C(t) decays on the timescale of about one and a half oscillations of the CN stretch. The LQF spectrumis in excellent agreement with the quantum result.74 For a more challengingtest, the value of the repulsion parameter three times larger than its physicalvalue, yielding a predissociation process (system II in Ref. 155), also has beenconsidered. Propagation of an initially real wave function makes calculation ofthe spectrum especially simple,

C(2t) = 〈 (−t)| (t)〉 = 〈 ∗(t)| (t)〉 =∑i

exp(

2ıS(x(i)t )

)wi [76]

The need to reconstruct (t) has been eliminated, and the propagation time hasbeen 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 approximatedin a linear basis �f = (1, x, y) with the component-specific expansion coefficientvectors r(j) = �c(j) × �f for j = x, y. The minimization procedure is directly anal-ogous to the one-dimensional case,

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

The only difference is that in Eq. [70], the vectors �c and �b are replaced withmatrices C = (�c(x), �c(y)), and B = (∇x�f ,∇y�f ). The full derivation can be foundin Ref. 75.

The LQF parameters are shown in Figure 5. Panel (a) shows the averageposition of the wave packet and the initial positions of some of the trajectoriessampling | (0)|2. An ensemble of 167 trajectories has been propagated forthree 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. Thedistance increases in the course of dissociation as the CN stretch undergoes threevibrations. Panel (b) shows the optimal expansion parameters: c(x)

2 represents

the changing width of the wave packet in the CN mode; c(y)3 describes the


2.1 2.2 2.3 2.4 2.5

CN [bohr]

5

5.5

6I-

CN

[bo

hr]

0 1000 2000

Time

0

50

100

LQ

F pa

ram

eter

s

(a)

(b)

Figure 5 The LQF parameters for ICN: (a) the average position, 〈y〉 vs 〈x〉, of the wave-packet and initial positions of trajectories; (b) the width parameters c(x)

2 (solid line), c(y)3

(dash), and c(x)3 = c

(y)2 (dot-dash).

spreading in the dissociation mode; c(x)3 = c

(y)2 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 ob-tained with the same set of quantum trajectories; the results were obtained usingEq. [76] and using the Gaussian fitting of (x, y, t) as outlined in “Dynamicswith the Globally Approximated Quantum Potential” section. The advantageof Eq. [76], in whichC(2t) is obtained from the propagation up to t, is apparent;the spectrum is in excellent agreement with exact results. C(t) obtained withthe 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 ofthe fitting). Nevertheless, the agreement of the spectra is good, showing that,overall, the LQF method is accurate for systems with fast dynamics.


0 1000

-0.5

0

0.5

Re(

C(t

))

0 0.02 0.04Energy

0

1

2

3

4

5

I(E

)

(a)

(b)

Figure 6 Photodissociation cross-section of ICN: (a) real part of the autocorrelationfunction; (b) the corresponding spectra. The LQF results are shown with a solid linewhen 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 wavefunctions evolving in general potentials: (1) More functions can be added to thebasis �f . For example, the Chebyshev basis of up to six polynomials has beenused in Ref. 151. Representation of r in a complete basis will give exact QMdynamics. (2) The basis can be kept small but made to be system-specific. Forexample, a two-function basis consisting of a constant and an exponent givesan 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 inRef. 156 to describe the zero-point energy of H2 in the reaction channel ofthe O + H2 reaction. Another approach, discussed immediately after, is basedon (3) the linear approximations to r on subspaces. Recall that in the globalAQP procedure, spatially separated parts of the wave function are coupled as aresult of averaging over the entire ensemble of quantum trajectories. The issue


of unphysical coupling can be resolved by using an approach based on the linearapproximations to r on subspaces. Additionally, such an approach gives a moreflexible description of r.

Approximation on Subspaces or Spatial Domains

TheoryThe globally determined LQF describes QM effects on a short time scale ade-quate for direct dynamics reactions. After the wave packet bifurcates, the LQFquickly goes to zero because of a large variance of the wave packet on theentire space. At the same time, parts of the wave function in the reactant or prod-uct channels might still be localized. To remove this “unphysical” long-distanceinteraction, one can divide the coordinate space onto several subspaces, or do-mains, 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 signif-icant regions such as reactants, products, and transition-state regions in reactivedynamics. Subspaces also can be based on other considerations such as shapeof the wave functions and features of the potential V . The domains are fixedin time and, in general, have nonzero overlap in space. The last domain withindex L complements the rest of the domains to unity,

�L(x) = 1 −∑

l=1,L−1

�l(x) [78]

This requirement ensures that the sum of exact solutions of the time-dependentSchrodinger equation on subspaces is equivalent to the solution on fullspace.

The kinetic energy operator on the subspace has to be Hermitian, that is,its matrix element 〈�i|K|�j〉 is

12m

∫∇� i (x) × ∇�j(x) ×�l(x)dx = − 1

2m

∫�l(x)� i (x)

×[∇2 + ∇�l(x)

�l(x)× ∇

]�j(x)dx [79]

With this form of K in the Bohmian formulation of time-dependent Schrodingerequation, the quantum potentialQ in Eq. [12] is replaced by its modified versionQl with the additional interface term,

Ql = − 12m

(∇2A(x, t)A(x, t)

+ ∇�l(x)�l(x)

× ∇A(x, t)A(x, t)

)[80]


for each domain. The density on the lth domain (Eq. [19]) also is modified asfollows:

d�(x, t)dt

= −∇v× �(x, t) − ∇�l(x)�l(x)

v�(x, t) [81]

Note that �(x, t) and S(x, t) still are defined on the full space and not on thedomains (i.e., there are no domain-specific wave functions). The summation ofEq. [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 quantumpotential, improving accuracy of the AQP. The nonclassical momentum r givenby Eq. [13] can be approximated on each domain by minimizing a functional

Il =∫

(r(x, t) − rl(x, t))2�l(x)�(x, t)dx [82]

where the approximating function is now domain-specific, rl = �f (x)�cl(t). Thecontribution to the AQP from each domain is determined by Eq. [80],

Ql = − 12m

((r2l + ∇ rl)�l + rl∇�l

)[83]

The total quantum potential is a sum over domains

Q =∑l=1,L

Ql [84]

Ql should be evaluated at the optimal values of �cl that minimize Eq. [82] andfor which ∇cIl = 0. Then, the AQP defined on domains rigorously will conserveenergy in a closed system, as was the case with the full space approximation(Eq. [71]). Derivation of the minimization procedure in multidimensional co-ordinates is given in Appendix C.

Because Eq. [81] for the density summed over the domains is equivalentto 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 toEq. [2]. The subspaces are introduced only to approximate better the nonclas-sical momentum and to define the AQP. Implementation with the linear basison multiple domains is given subsequently.


Application: Dynamics of the Collinear H3 SystemThe collinear hydrogen exchange reaction, HA+HBHC→HAHB+HC, is a stan-dard test in reaction dynamics and presents a considerable challenge for semi-classical approaches. The system is described in the Jacobi coordinates of re-actants in which the kinetic energy is diagonal. The Hamiltonian, coordinates,and potential surface are the same as in Ref. 157. The initial wave packet isdefined as

(x, y,0) =√

2�

(˛1˛2)1/2 exp(−˛1(x− x)2 − ˛2(y− y)2 + ıP0(x− x))

[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 betweenHA 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 respectto �0 = �/6. Values of the parameters in atomic units (scaled by the reducedmass 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 initialpositions for the quantum trajectories {x(i), y(i)} are chosen on a rectangular gridwith spacings dx = 0.017 and dy = 0.020. Trajectories with weights smallerthan 10−8 are discarded. The initial momenta {p(i)

x , p(i)y } are px(i) = P0 and

p(i)y = 0; the initial classical actions are S(x(i)

0 , y(i)0 ) = p0(x(i) − x). The wave-

packet transmission probability P(t),

P(t) =∑i

wiH

[arctan

y(i)t

x(i)t

− �

6

][86]

is analyzed. The summation goes over the trajectories in the product region; Hdenotes the Heavyside function in angle.

For optimization using domains, the reactant and product channels aredefined by the switching functions,

�reac = 12

− arctan(z(� − �/6))2 arctan(z�/6)

, �prod = 12

+ arctan(z(� − �/6))2 arctan(z�/6)

[87]

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

�k = exp(−ˇ(y− qk)2) [88]


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 re-actant domain functions are multiplied by �reac. The product domains are sym-metrical to the reactant domains with respect to � = �/6 and are multiplied by�prod. In addition, three two-dimensional Gaussian domains are specified in thetransition state with appropriate modification of the reactant and product do-mains satisfying Eq. [78]. This brings the total number of domains toL = 9. Thetransition-state domain functions are centered in the transition state around theminimum along � = �/6 at q7 = 2.1, q8 = 2.6, and q9 = 3.1 bohr. The widthparameter along the transition state is ˇ = 5.0 in both coordinates. Ensemblesof 2500 quantum trajectories have been propagated for each wave packet.

The wave-packet reaction probability as a function of the total energyis presented in Figure 7a, which also shows the classical (Q = 0) and the fullspace LQF results. Optimization on the full space gives qualitative agreementwith the quantum result. Optimization on the domains leads to the improvedprobabilities, which are now within 0.04 of the QM values. This system exhibitssignificant anharmonicity in the diatomic potential. Thus, the main shortcomingof the LQF method is conversion of the zero-point energy into classical energyof the trajectories on a short time-scale. This has been corrected by treatinghigh-energy regions of the potential surface within separate domains. This kind

0 0.4 0.8 1.2E, eV

0

1

Pro

babi

lity

QM

Q=0

LQF

L=9

0 0.5 1 1.5 2Time

0

1

|C(t

)|

QM

L=7

0 0.4 0.8 1.2E, eV

I(E

)

QM

L=7

(a) (b)

Figure 7 Dynamics of the collinear H3 system. (a) The wave-packet transmission prob-ability as a function of the total energy of the wave-packet: results for the quantum (thicksolid 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-correlationfunction for the transition state wave-packet as a function of time. The quantum andthe domain results are shown with the solid and dashed lines respectively. The insertshows the corresponding spectra (the semiclassical spectrum is marked with circles).


of treatment “uncouples” the high-energy regions from the low-energy regionswhere 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 testof the LQF method on domains. An initial Gaussian wave packet, given byEq. [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 placedalong � = �/6 direction, ql = {2.1, 2.6, 3.1, 3.6, 4.1} bohr, were used in thiscalculation. The width parameter in the direction of the transition state wasˇ = 8.0. The width parameter for the domain functions in the perpendiculardirection was smaller, ˇ = 2.0, to make these functions more delocalized be-cause the channel regions were described within a single domain each, definedby �reac and �prod. In this calculation, 4281 trajectories with weights greaterthan 10−6 have been propagated. The peaks in the spectrum are within 7% ofthe converged values. The position of the spectrum peaks is converged withinits resolution. The number of trajectories is greater compared with the wave-packet probability calculations because the autocorrelation function C(t) is acomplex 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 ofC(t) are in good agreement with each other. A general observation is that theLQF description follows the same trend as other semiclassical methods; theaccuracy of the semiclassical description depends on the desired level of detail(i.e., average values tend to be more accurate than the quantities derived fromthe wave function projections onto eigenstates).

Nonadiabatic Dynamics

Arguably, nonadiabatic dynamics is the most important quantum effecton 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 interfer-ence often is quenched in large molecular systems at short times because of thewave function decoherence. However, nonadiabatic effects still can influencesuch systems. The most widely used method for including nonadiabatic effectsinto the dynamics of molecules is the trajectory surface hopping method,94,161

which is based on classical evolution of trajectories that can switch between dif-ferent electronic states. The method has numerous applications including pho-todissociation 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


states on reactivity. Such a study of O(3P,1 D) + H2 → OH + H performedwith quantum trajectories will be described in the section “Application: TheFour Electron State Dynamics for O(3P,1 D) + H2.” The system is of interest incombustion and atmospheric chemistry, and it has been investigated thoroughlyboth experimentally and theoretically.165–169 Accurate electronic potential en-ergy surfaces correlated to the triplet and singlet states of atomic oxygen havebeen developed by Rogers et al.170 and Dobbyn and Knowles.171 The spin-orbit couplings have been determined by Hoffmann and Schatz.172 Maiti andSchatz have used trajectory surface hopping of quasi-classical trajectories tostudy spin-orbit interaction-induced intersystem crossing effects in the O + H2reaction within a four electronic state model.173 A complete QM calculationof this size is still a challenge for the standard QM methods; the first QM studyusing the wave-packet technique was reported in 2005.169

In the quasi-classical trajectory surface hopping study,173 it was estimatedthat intersystem crossing effects enhance total reaction cross sections by upto 20%. Essentially, no such effect was found in the full quantum result.169

Therefore, nonadiabatic dynamics of O + H2 presents a stringent test for anyapproximate method. The AQP investigation of the spin-orbit coupling effectin O + H2 reaction156 is reviewed subsequently, preceded by the summary ofthe theoretical approach.

The Nonadiabatic Generalization of Bohmian FormulationThe original Bohmian formulation given by Eqs. [2]–[4] can be extendedstraightforwardly to dynamics on multiple electronic states. For clarity, con-sider a wave packet evolving in the presence of two one-dimensional poten-tial surfaces, V1 and V2, coupled by a potential V12. A typical curve-crossingsystem in the diabatic representation is sketched in Figure 8. This system isdescribed by a two-component wave function, = { 1, 2}, governed by thetime-dependent Schrodinger equation,

(− �

2

2m∇2 + V1− ı�

∂

∂t

) 1(x, t) + V12 2(x, t) = 0 [89]

(− �

2

2m∇2 + V2− ı�

∂

∂t

) 2(x, t) + V12 1(x, t) = 0 [90]

Here and throughout the subscript of functions refers to the two surfaces, V1and V2.

A multisurface quantum trajectory description is based on the substitutionof the wave function in terms of its real amplitude and phase into Eq. [89],

i(x, t) = Ai(x, t) exp( ı�Si(x, t)

), i = 1,2, . . . [91]


Figure 8 The curve crossing model: diabatic potentials, V1 and V2, and coupling V12are 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 productregion. Analysis of its overlap with a stationary wave-packet 0 gives the energy-resolvedreaction probabilities.

followed by the separation of real and imaginary parts. The subscript i labelselectronic states. This Bohmian formulation was presented and implementedusing the exact quantum trajectories by Wyatt et al.43 The polar representationof i is also a starting point in the derivation of a widely used surface-hoppingmethod.123 The surface-hopping method is characterized by purely classicaldynamics of trajectories probabilistically “hopping” between the surfaces andthe quantum-classical mixing approaches of Refs. 20, 24, and 25. Other meth-ods of nonadiabatic dynamics based on classical or semiclassical mechanicsare 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. Usingthe wave function densities, �i(x, t) = A2

i (x, t), and identifying pi = ∇Si(x, t) inthe moving frames of reference defined by appropriate momenta pi accordingto Eq. [9], the real parts of Eqs. [89–90] give the time evolution of the actionfunctions Si(xt,i)

dSi(xt,i)dt

= p2t,i

2m− (Vi +Qi +Qij)

∣∣xi=xt,i [92]

where i = 1,2 and j = 1,2, and i /= j. Qi is the single-surface quan-tum potential given by Eq. [5]. Qij is an additional potential resulting


from coupling,

Qij = V12�ji cos (�S) [93]

�ji =√�j(x, t)�i(x, t)

, �S = 1�

(S1(x, t) − S2(x, t)) [94]

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

dwi

dt= −2V12�ji sin (�S)wi [95]

In this formulation, there are two sets of trajectories—one set on each surfaceevolving in time. Coupling affects both the dynamics of trajectories and the timeevolution of their weights. The AQP method has been adapted in a practicalway44,45 for the nonadiabatic formulation of Eqs. [92]–[95], and adequate ifthe trajectory dynamics was smooth, such as for asymptotically degenerate V1and V2 coupled by a localized V12.43

Nevertheless, the formulation of Eqs. [92]–[95], though formally exact,has drawbacks. Qij, given by Eq. [93], involves a possibly singular ratio of thedensities. For instance, a typical initial condition for a multisurface problemis a wave packet occupying a single electronic state, such as �2(x,0) = 0. InRef. 43, this problem was circumvented by propagating two sets of initial wavepackets with nonzero population on both surfaces, whose linear combinationgives the desired initial condition. Alternatively, the singularity can be canceledat t=0 by the choice of initial conditions for the trajectories.44 If �2(x,0) = 0,then one can set up trajectories on the second surface with the same initialpositions 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 inQ21cancels at t=0. The subsequent time evolution is stable. The sign of the phaseshift depends on the derivative of V2.

In the context of the semiclassical dynamics, there is also a conceptualproblem 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 contributionproportional to

∇ cos(�S) = �−1(p2(x) − p1(x)) sin(�S) [96]

which does not go to zero in this limit. In fact, it becomes large and oscilla-tory because of the �−1 prefactor. Apart from computational challenges, this


asymptotic behavior shows that the Bohmian formulation is incompatible withsemiclassical dynamics, where propagation is expected to become classical as�→ 0.

A practical and conceptually appealing alternative is to use a mixed co-ordinate 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, canbe represented by the quantum trajectories of the polar part �i(x, t). The “hard”quantum effects, such as interference or nonadiabatic transitions, are includedthrough 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 surfaceor as a spatial function �i(x, t) represented in a small basis. For nonadiabaticprocesses �i can be interpreted as a complex population amplitude of the ithelectronic 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 �ievolves on a single surface Vi and use this fact in Eq. [89]. The time-evolutionof �i is given by

ı�d�i(x(i,t))

dt= − �

2

2m

(∇2�i(x(i,t)) + 2r(i,t)�i(xt)

)+ V12

�j(x(i,t))�i(x(i,t))

�j(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 closestanalogy with the single surface propagation. This assumption is reasonable forsystems with localized coupling V12, as in the example below; however, it isnot necessarily the best choice in other situations. Figure 8 refers to the curvecrossing model of Tully;161 full details of the nonadiabatic calculations using theAQP and mixed representation can be found in Ref. 44. Two sets of trajectoriesare propagated on V1 and V2, and the amplitude prefactors �i are found fromEq. [98]. The kinetic energy terms are estimated from a small basis expansionof �i. The initial population of V2 is zero. Figure 9 illustrates the populationtransfer between the surfaces. Figure 10 shows the energy-resolved reactionprobabilities computed from the wave-packet correlation functions 〈 0| i(t)〉,176 which are in good agreement with QM probabilities. Theory relevant to theO + H2 application is presented below.


Coordinate [a ]0

0

0.5 50 a.u.100 a.u.150 a.u.200 a.u.

-4 -2 4200.8

0.9

1

|χ2|

|χ1|

Figure 9 Snapshots of the population function amplitudes, |�1| and |�2|, at timest = {50,100,150,200} a.u. for the high-energy wave-packet 1.44 Initially �1 = 1 and�2 = 0.

Theory: Nonadiabatic Trajectory Formulation in the Mixed WavefunctionRepresentationThe theoretical formulation in this section is given for the three-dimensionalJacobi coordinates (x, y, �) describing a nonrotating triatomic system, whichwill be used in the next section (see Appendix B for a generalized formulation).

0 200 400 600 800

Energy [kJ/mol]

0

0.5

1

Rea

ctio

n pr

obab

ility

QM LQFQMLQF

P12

P11

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


Consider dynamics on multiple coupled potential surfaces in the diabatic rep-resentation (� = 1)

ı∂

∂t� =

(TI + V

)� [99]

Here, I is the identity matrix whose size is given by the number of potentialenergy surfaces (i.e., the number of electronic states).

The kinetic energy operator is

T = − 12M

∂2

∂x2 − 12m

∂2

∂y2 − 12�

(∂2

∂�2 + cot �∂

∂�

)[100]

The moment of inertia is included using

1�

= 1Mx2 + 1

my2 [101]

The matrix V is a symmetric matrix that contains the diabatic potentialenergy surfaces and couplings. The mixed representation approach, in its mostgeneral 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 amplitudechanges caused by coupling between the surfaces. The ith component of � is

i =∑j

�ij(x, y, �, t)�j(x, y, �, t) [102]

Indexes i and j label the electronic states. The polar parts �j(x, y, �, t) can evolveon the diabatic or nonadiabatic potential energy surfaces or on effective poten-tial surfaces combining features of both diabatic and nonadiabatic representa-tions. The potentials governing dynamics of �j should be chosen to minimizethe 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 inRef. 45.

For the O + H2 system, the lowest 3P2,1,0 and 1D potential energy surfacesbecome degenerate in the product region.172 The product region is also theregion of nonzero spin-orbit coupling. Therefore, the simplest treatment of thesame � in Eq. [102] for all electronic states is sufficiently accurate, as has beenverified in one-dimensional model studies,45

i(x, y, �, t) = �i(x, y, �, t)�(x, y, �, t) [103]


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

ı∂

∂t� = (T + Vd)� [104]

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

ı∂

∂t�� = −ı(�vT∇)��+ (Tc + T)I��+ (V − VdI)�� [105]

The operator Tc couples �r with the first derivatives of ��,

Tc = − rxM

∂

∂x− ry

m

∂

∂y− r�

�

∂

∂�[106]

For an efficient trajectory implementation, the evolution of � and � is deter-mined approximately. Equation [104] is solved using the AQP approach ofthe section “Dynamics with the Globally Approximated Quantum Potential”.Equation [105] is simplified as follows: (1) The first term on the RHS is com-bined with the LHS to give the time-derivative of �� along a trajectory; (2) Theeffective 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 wavefunction—a single (kth) populated electronic state—are {� = k, �i = ıik}. Inthis 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 theeffective potential:

Vd =∑i j〈�i|�j〉Vij∑i〈�i|�i〉

[107]

The simplified equation for �� becomes

ıd

dt�� = (V − VdI)�� [108]

A single ensemble of trajectories evolves under the combined influence ofthe quantum potential as well as the “average” classical potential, which in-cludes all matrix elements of V weighted by their respective populations 〈�i|�j〉.The formulation of Eqs. [104] and [105] is equivalent to the time-dependentSchrodinger Eq. [99] if solved exactly.


Application: The Four Electronic State Dynamics for O(3P,1 D) + H2The formalism of the previous section is applied to the four state model describ-ing the spin-orbit interaction induced intersystem crossing of the O + H2 →OH + H reaction, developed by Hoffmann and Schatz.172 The total number ofthe oxygen atom electronic states, including the spin-orbit interaction, is 15—five 1D2, one 1S0, and nine 3P states. This number is reduced to four by ignoringhigh-energy states and parity decoupling. The three triplet states and one sin-glet states of the reactants convert into four doublet states of the products—two2�1/2 and two 2�3/2 states. The diabatic Hamiltonian of Hoffmann and Schatz172 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 isthe 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 com-puted as sums across trajectories in the product region,

Pdi =∑

k, prod

|�i(�x(k))|2wk [109]

where the index k labels trajectories and i labels the surfaces. The first and thirdsurfaces correlate with 3P2 and 3P0 states of oxygen. They remain coupledin the product region, correlating with the 2�3/2 and 2�1/2 states of OH.The other two surfaces, correlating with 3P1 and 1D2 states of oxygen, also

-10 -5 0 5 10

Reaction coordinate [a0]

0

20

40

60

Ene

rgy

[kca

l/mol

]

3P

2Π

0.4

3P

0

3P

1

3P

2

2Π1/2

2Π3/2

1D

0.2

0.2

O+H2

OH+H

Vc 5000x

Figure 11 Diabatic electronic states. 1A′ correlates with 1D state (thin solid line). 3A′′

correlates with 3P2 and 3P1 states (thick solid line). 3A′ correlates with 3P0 state (dashedline). The asymptotic coupling is shown with the dot-dashed line. Asymptotic splittingsof the adiabatic potential energy surfaces are indicated in the inserts in kcal/mol.


are coupled in the product region, correlating with 2�3/2 and 2�1/2 statesof OH.

Taking the sign of the asymptotic couplings into account, the adiabaticwave functions of the lower doubly degenerate state, 2�3/2, are

a1 = 1 − 3√2

, a2 = 2 + 4√2

[110]

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

a3 = 1 + 3√2

, a4 = 2 − 4√2

[111]

The propagation is terminated once the adiabatic probabilities,

Pai =∑

k, prod

|�ai (�xk)|2wk [112]

reach constant values. The initial wave packet is a direct product of a Gaussianin the translational coordinate and the ground state of the ith potential energysurface in the internal degrees of freedom

i(x, y, �,0) =(

2˛�

)1/4

exp(−˛(x− x0)2 + ıp0(x− x0)

)�(y) [113]

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

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

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

The initial vibrational wave function is

�(y) = (2��k

)�−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 inthree dimensions using normal deviates in x and y and uniform deviates in


cos �.178 A single surface calculation179 has shown that r� is small comparedwith the radial components; therefore, it was set to zero in the trajectory calcu-lations. The classical results are obtained by setting the quantum potential tozero, Q=0. Approximation to the radial components rx, ry were made usingthe basis � = {1, x− x0, y− ym, �(y)} This basis is exact for the Morse poten-tial eigenstates,179 which improves the asymptotic description of H2 fragment.Evaluation of the potential energy surfaces and couplings was the most expen-sive 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 dis-crete variable representation points182 for the angle. The action of the potentialpart of the Hamiltonian,

exp(−ıVdt) � = V � , V = M�MT [117]

is accomplished by diagonalizing the potential matrix. The matrix M consistsof the eigenvectors of V. The elements of the diagonal matrix � are functionsof the eigenvalues �i of the potential matrix V,ii = exp(−ı�idt). The matricesV were stored for each grid point, whose total number was around 4 × 106.

The time-dependent reaction probabilities and populations of a wavepacket initialized on the second surface (3P1) with translational momentump0 = 16 are shown in Figure 12. The diabatic probabilities of the surfaces twoand four obtained using QM and approximate propagation are shown in Fig-ure 12(a). The probabilities begin to oscillate between these two diabatic sur-faces as the reactive part of the wave packet evolves in the product channel.The reaction probability on the other two diabatic surfaces is negligible. Theadiabatic probabilities, shown in Figure 12(b), approach constant values afterapproximately t=2300. Panels (c) and (d) show the population of the diabaticsurfaces as functions of time. Note that population on the surfaces one and threeremains below a few percent at all times. The AQP results with the exponentialbasis function are in good agreement with the QM results. For this high-energywave packet, the classical dynamics of � also agree well with the QM resultsat long times. Overall, it was found (for a range of the collision energies) thatthe coupling of the triplet surfaces to the singlet in O + H2 has essentially noeffect on the total reaction probabilities. The coupling affects only the relativepopulations of the doublet states.

TOWARD REACTIVE DYNAMICS IN CONDENSEDPHASE

Semiclassical methodology presented in this review ultimately is directedtoward the description of QM effects in large reactive molecular systems with

Toward Reactive Dynamics in Condensed Phase 339

0

0.1

0.2

0.3

Pdiab

atic

AQP

AQP

QM

Q=0

Q=0

0

0.5

1

Popu

latio

n

1000 2000 3000

Time [a.u.]

0

0.1

Padia

batic

0 1000 2000 3000

Time [a.u.]

0

0.02Po

pula

tion

3P

1

1D

2

2Π3/2

2Π1/2(d)

(c)

(b)

(a)3P

1

1D

2

3P

0

3P

2

Figure 12 Dynamics of the wave-packet initialized on 3P1 with p0 = 16 (solid linesshow QM results): (a) Reaction probabilities on the diabatic surfaces correlating with 3P1(AQP/classical results are shown with circles/dash) and with 1D2 (AQP/classical resultsare shown as squares/dot-dash); (b) Reaction probabilities on the adiabatic surfacescorrelating with 2�3/2 (AQP/classical results are shown with circles/dash) and with 2�1/2

(AQP/classical results are shown as squares/dot-dash); (c) Populations on 3P1 and 1D2(legend is the same as in a)); (d) Populations on 3P2 (AQP/classical results are shownwith circles/dash) and with 3P0 (AQP/classical results are shown as squares/dot-dash).

hundreds of atoms. Classical molecular dynamics provides a reasonable gen-eral picture of chemical reaction dynamics in most systems of practical interest.However, the isotope effect measurements and the comparison of typical re-action energies and zero point energies show that quantum mechanical effectsplay an important role in many systems, particularly in reactions of protontransfer (some examples can be found in Refs. 2, 83, 183). The adequate theo-retical description of quantum effects is proved to be a very challenging task.The collective research during the last three decades points to three primaryreasons: (1) many reactions occur at the time scale much longer than that ofa typical quantum dynamics simulation; (2) the forces acting on the reactivespecies are not well represented by simple harmonic approximations; and (3)quantum effects, such as the zero point energy, require an ensemble descriptionrather than an individual trajectory description.

This section describes a computational approach that incorporates QMeffects into the molecular dynamics framework and is based on Bohmian tra-jectories. A typical view of a reaction occurring in a condensed phase, in other


ProductsReactants

Environment

Reaction coordinate

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 sucha reactive system include the following: (1) the motion along the reactive coor-dinate can be influenced by QM tunneling; and (2) the zero point energy—or,more generally, localization energy—in the reactive and bath degrees of freedomand energy flow among them. The energy changes in the reactive mode obvi-ously will influence the probability of the reaction. The next section describesthe approximate treatment of the zero point energy in a high-dimensional an-harmonic bath on a long timescale.184 The “Bound Dynamics with Tunneling”section presents a mixed wave function approach to describe tunneling in abound reactive coordinate.185 This approach is compatible with the trajectoryframework and extendable toward full QM description. In the future, these twoapproaches will be combined to model dynamics in condensed phase systems.

Stabilization of Dynamics by BalancingApproximation Errors

Formally, the zero point energy is the energy of the lowest eigenstate. It is asum of the kinetic and potential energy contributions because of localization, or“finite size,” of the eigenstate. In the Bohmian formulation, the kinetic energycontribution to the zero point energy is given by the expectation value of thequantum potential,

〈Q〉 = − �2

2m〈A|∇2A〉 = �

2

2m〈∇A|∇A〉 = �

2

2m〈r2〉 [118]


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

〈H −Q〉 = 〈p2〉2m

+ 〈V〉 [119]

The concept of “quantum” energy can be applied to any localized function, notjust to the ground state. Efficient—scalable to high-dimensionality—and stabledescription of the quantum energy, 〈Q〉, is the goal of this section.

The LQF description is the cheapest and is exact for Gaussian wave pack-ets. It gives the exact quantum energy for all eigenstates and coherent states ofthe harmonic oscillator. In anharmonic systems, the LQF describes 〈Q〉 onlyon a short timescale (depending on the anharmonicity). For an eigenfunctionin Bohmian formalism, the quantum force exactly cancels the classical one, re-sulting in stationary trajectories. In the LQF, exact cancellation generally doesnot happen; a net force acting on the trajectories, representing wave functiontails, might be large causing these “fringe” trajectories to start moving. Sooneror 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 zeropoint energy (or quantum energy) description. Because the total energy is con-served in the LQF, as shown in the previous section, the quantum energy will betransferred into classical energy of the trajectory motion. Below, the LQF ideasare extended to prevent this “loss” of quantum energy for small anharmonic-ities, or more precisely for small nonlinearity of the classical and nonclassicalmomenta. Thus, a cheap and stable zero point energy description over manyoscillation periods is provided.

Approximation of GradientsThe derivation is given in atomic units (� = 1) for a system described in NdimCartesian coordinates �x = (x, y . . .) in vector notations. The classical and non-classical momenta are vectors

�p = ∇S, �r = A−1∇A [120]

In LQF, linear approximation to each component of A−1∇A is made. The tra-jectory momenta are known but never used in the fitting procedure. The non-linearity 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 fromlinearity are used to limit accumulation of propagation error with time.

In the multidimensional case, the time evolution of �r and �p along a tra-jectory, given by Eq. [17] in one dimension, is determined as

m∇V +md�pdt

= (�r∇)�r+ (∇ ∇)�r2

[121]


−md�rdt

= (�r∇)�p+ (∇ ∇)�p2

[122]

For practical reasons, the spatial derivatives of �r and �p in Eq. [122] need to bedetermined 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 theRHS of Eq. [122]. Similar to the AQP method, �p and �r are expanded in a basisset of functions for the purpose of derivative evaluations. The expansion coef-ficients are determined from the minimization of the error functional using thetotal energy conservation as a constraint.

The total energy can be defined without spatial derivatives as

E = 〈�p �p〉2m

+ 〈V〉 + 〈�r �r〉2m

[123]

The energy conservation constraint will couple the fitting procedures ofA−1∇Aand �p. The linear basis �f = (x, y . . . ,1) is considered here. Formulation for ageneral basis is given in Ref. 184. The functions

{rx=�c rx �f , ry=�c ry �f . . .} [124]

approximate components of the vector A−1∇A. The functions

{px=�c px �f , py=�c py �f . . .} [125]

approximate components of the vector �p. To express the energy conserva-tion condition, dE/dt = 0, the fitting coefficients are arranged into matrices Cr

and Cp,

Cr = [�c rx , �c ry . . .] [126]

Cp = [�c px , �c py . . .] [127]

Differentiating Eq. [123] with respect to time and using Eq. [122] withthe derivatives obtained from Eqs. [124] and [125], the energy conservationcondition becomes

dE

dt=

⟨�r 0 (Cr�p− Cp�r)⟩m

= 0 [128]


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

�r 0 = (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− Cr�f ‖2〉 + 〈‖�p− Cp�f ‖2〉 + 2�〈�r 0 × (Cr�p− Cp�r)〉 [130]

with respect to the fitting coefficients and with respect to the Lagrange multiplier�. The optimal coefficients solve a system of linear equations,

⎛⎜⎝

M O �Dp

O M �Dr

�Dp �Dr 0

⎞⎟⎠

⎛⎜⎝

�Cr�Cp�

⎞⎟⎠ =

⎛⎜⎝

�Br�Bp0

⎞⎟⎠ [131]

In Eq. [131] the following matrices and vectors are introduced: (1) M is theblock-diagonal matrix of the dimensionality NdimNb×NdimNb with the basisfunction overlap matrix S=〈�f⊗�f 〉 as Ndim blocks on the diagonal and zerosotherwise; (2) O is a zero matrix of the same size as M; and (3) the elementsof the vectors �Cr, �Cp, �Br, �Bp, �Dr, and �Dp 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 aredefined as

Br = −12 〈(∇ ⊗ �f )T 〉, Bp = 〈�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 thebasis

�p 0 = (px, py . . . ,0) [134]

The fitting of A−1∇A is the same as in the LQF procedure, except that now itis coupled to the least-square fit of �p. Formally, the total size of the matrix inEq. [131] is 2NdimNb+1. Its structure allows one to invert the matrix on theleft-hand side by performing a single matrix inversion of the block S of the sizeNb.178 Thus, the cost of the quantum force computation scales as NtrajN

2dim.

This is essential for efficient high-dimensional implementation.


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

Correcting for the Effect of Linearization on DynamicsApproximation based on the linear basis �f is cheap and gives exact dynamicsin the important limit of Gaussian wave functions evolving in locally harmonicpotentials. However, it results in a “cold” truncation of the time-evolution equa-tions [122] because the second derivatives of the basis functions are zeros. Suchtruncation of differential equations leads to dynamics that are unstable with re-spect to small deviations of �r and �p from nonlinearity. It can be compensated byintroducing additional terms into Eq. [122]. These additional terms depend onthe difference of exact and approximated values of �p and �r and balance errorsbecause of the linear basis in the first order of the nonlinearity parameters. Theexplicit form is determined from the analytical models and has no adjustableparameters.

Consider the lowest order nonlinearities of the classical and nonclassicalmomenta (in one dimension)

p = p0 + p1x+ �x2, r = ∇AA, A = e−˛x

2 |1 + ı× (x− x0)| [135]

Analysis of the short-time dynamics with spatial derivatives obtained from lin-earization of quantities given by Eq. [135] has shown that the following ap-proximate equations of motion:

m

(dp

dt+ ∇V

)= r∇r+ ∇2r

2= r∇ r+ 2∇ r× (r− r) +O(ı4) [136]

−mdrdt

= r∇p+ ∇2p

2= r∇p+ 2∇ r× (p− p) +O(�ı3) [137]

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

−md�rdt

= Cp�r+ 2Cr(�p− �pfit) [138]


m

( �pdt

+ ∇V)

= Cr�r+ 2Cr(�r− �rfit) [139]

In the previous section functions (rx, ry . . .) approximate components ofA−1∇A, and their determination is coupled to the approximation of �p in termsof (px, py . . .) by the energy conservation condition. On the other hand, �rfit ap-proximates �r. In general, there is a difference between the fittings (rfit

x , rfity . . .)

and (rx, ry . . .). Approximations �rfit and �pfit should be such that the stabiliza-tion terms do not contribute to the total energy of the trajectory ensemble anddo not change the normalization of the wave function. The latter conditioncan be written as 〈d�r/dt〉 = 0. Separate least squares fits of �r and �p in terms ofthe linear basis by minimizing 〈‖�r− �rfit‖2〉 and 〈‖�p− �pfit‖2〉, respectively, satisfyboth these requirements. (General basis is discussed in Ref. 184.) Physically, thestabilization terms provide “friction” opposing the growth of the discrepanciesbetween the trajectory-dependent �r and �p and their linear fits with time.

Numerical Examples of the Long-Time Zero-Point Energy DescriptionThe first illustration of the stabilization method is the description of zero pointenergy, or more generally, of the quantum energy 〈Q〉 for a nonrotating hydro-gen molecule, as in Ref. 42. The classical potential V is the Morse oscillatorwith 17 bound states. The system is one dimensional and is described in atomicunits scaled by the reduced mass of H2 to have m=1. The initial wave packetis a Gaussian wave function mimicking the ground state of H2,

(x,0) =(

2˛�

)1/4

exp(−˛(x− xm)2

)[140]

where ˛ = 9.33 a−20 and xm = 1.4 is the minimum of the Morse potential.

Figure 14(a) shows positions of the trajectories obtained with the LQF methoddescribed in the “Approximate of Gradients” section and the stabilized AQPdynamics of this section. The plot illustrates the effect of runaway trajectoriesleading to the trajectory “decoherence” and transfer of quantum potential en-ergy into classical energy, which is described at the beginning of “Stabilization ofDynamics by Balancing Equation Errors”. Note the “fringe” LQF trajectory im-mediately leaving the trajectory ensemble toward the dissociation region. Thisbehavior quickly drives variance of the wave packet and, thus, the quantumpotential and the quantum force to zero, as shown on the lower panel. Trajec-tories obtained with the stabilization procedure maintain their coherence anddescribe the quantum energy 〈Q〉 very well. The effects of stabilization termsare shown in Figure 14(a). The oscillatory behavior of the two central trajecto-ries is a consequence of propagating the Gaussian function of Eq. [140] ratherthan the eigenstate. These oscillations correlate with the oscillations in 〈Q〉.Behavior of the outlying trajectories of the stabilized AQP dynamics shows


10 2Time

0

2

4

<Q

>

QM

LQF

stabilized

10 21

1.5

2

2.5

Posi

tion

LQFstabilized (a)

(b)

Figure 14 Quantum energy for the H2 molecule: (a) Trajectory positions as functions oftime obtained using the LQF and the stabilized dynamics; (b) Average quantum potentialas 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 centraltrajectories. These additional oscillations are a result of the corrections of the“friction force” introduced into the equations of motion. The trajectory prop-agation was performed with the third order Milne predictor-corrector algo-rithm,178 and the accuracy of 〈Q〉 was checked for 200 oscillation periodsof H2.

Application of approximate methods to high-dimensional systems must bevalidated by tests that can be compared with exact QM results, which generallyimplies separable Hamiltonians or harmonic potentials. For multidimensionaltesting of the stabilized AQP method, the average quantum energy has beencomputed for a model potential. The separable model potential consists of theEckart barrier, centered at the zero of the reaction coordinate, and the Morseoscillators in the vibrational degrees of freedom. The vibrational degrees offreedom are the same as in the one-dimensional application. The parametersof the barrier mimic the H + H2 system and are given in Ref. 42. The initialmultidimensional wave packet is defined as a direct product of a Gaussian inthe reaction coordinate

(x,0) = (2˛�−1)1/4 exp(−�(x− x0)2 + ıp0(x− x0)

)[141]


with parameter values {�=6, x0 =4, p0 =6}, and Gaussian functions in thevibrational degrees of freedom, given by Eq. [140]. After the wave packet inthe reaction coordinate bifurcates, 〈Q〉 of the system is equal to that of thevibrational 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 coordinatesintroduces effective coupling between the degrees of freedom. Dynamics in therotated system of coordinates provides a more stringent test of the method. Inthe rotated system of coordinates, the wave packet and the classical potentialare nonseparable. In addition, the numerical procedure of the quantum forcecomputation and trajectory propagation in the rotated system used no infor-mation about the separability of the original Hamiltonian.

To compare the quantum energy description for different numbers of vi-brational degrees of freedom, the rotation matrix, written here for clarity asNdim = 4, is specified by the parameter �,

� =

⎛⎜⎜⎜⎝˛ −� −� −�� 1 + ˇ ˇ ˇ

� ˇ 1 + ˇ ˇ

� ˇ ˇ 1 + ˇ

⎞⎟⎟⎟⎠ [142]

with ˛=√

1−(Ndim−1)�2 and ˇ= (˛−1)/(Ndim−1). This transformation doesnot change the diagonal kinetic energy operator, provided that masses for alldimensions are equal. The stabilized dynamics implemented with the linearbasis is invariant under such transformation.

Numerical performance of the stabilized dynamics has been tested forup to 40 dimensions with random Gaussian sampling of initial positions.178

Calculation of the quantum potential and force is dominated by the computa-tion of the moments of the trajectory distribution, which scales as NtrajN

2dim.

Calculation of the global linearization parameters is performed at each timestep for the ensemble of trajectories with the cost of N4

dim. The average quan-tum potential divided by the number of the vibrational degrees of freedom,〈Q〉/(Ndim−1) is shown in Figure 15. The difference in the quantum en-ergy at short times is from the quantum energy in the reactive coordinate,which is not included in the QM calculation. At later times, the trajectoryresults reproduce the changes in localization of the vibrational wave func-tions well, and the semiclassical accuracy is essentially independent of thedimensionality.

Propagation of 2 × 104 trajectories gave a relative difference in the quan-tum energy of around 0.5%, with a standard deviation of about 1%, forall numbers of vibrational degrees of freedom. The largest calculation forNdim = 40 took two hours on a single processor of a desktop workstation.


0 1 2Time

4

4.5

5<

Q>

/(N

dim

-1)

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 Morseoscillators. 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 de-scription of “hard” quantum effects, such as tunneling and interference. Inthis regime, the exact quantum trajectory dynamics is numerically unstable be-cause the exact quantum potential of Eq. [5] is singular because of the nodesof (x, t). The AQP trajectories are stable, but quantum forces quickly becomeinaccurate, trajectories decohere, and dynamics becomes essentially classical.For an initially localized wave function evolving in the double well potential—aprototype model of the proton transfer in condensed phase—the key dynam-ics feature is tunneling between the the “reactant” and the “product” wells.185 To capture this behavior in the trajectory framework compatible with thetrajectory representation of high-dimensional bath, the mixed form of the wavefunction 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 theasymptotic potentials of reactants and products, V1 and V2, respectively, ac-cording to the Schrodinger equation

ı�∂�j

∂t= − �

2

2m∂2�j

∂x2 + Vj�j, j = 1,2 [144]

For chemical systems, asymptotic dynamics can be sufficiently simple to beaccomplished semiclassically with two ensembles of the AQP trajectories.


Subsequently, we will use the superscripts to label quantities associated withtrajectories in these two ensembles.

The tunneling or population transfer between the two wells under theinfluence of the full potential V is accomplished through the complex ampli-tudes �1 and �2. These two functions will be represented in a basis �f of Nbpolynomials, for example, in the following Taylor basis �f = (1, x, x2, . . .):

�1 =∑

n=1,Nb

cn(t)xn−1, �2 =∑

n=1,Nb

cn+Nb (t)xn−1 [145]

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

�F = (f1�1, . . . , fNb�1, f1�2, . . . , fNb�2) [146]

and integration over x gives a linear system of differential equations for thebasis coefficients �c,

ı�Sd�cdt

= H�c [147]

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 ofEq. [147] is a “partial” Hamiltonian matrix H,

H = 〈�F ⊗ ( �K + ��)〉 [149]

containing the derivatives of �i as well as the potential energy terms. The matrixis constructed as the outer product of the basis function vector �F, given byEq. [146], and the vector �K + �� whose elements are given by

Ki = − �2

2m

(∂2fi

∂x2 �j + 2∂fi

∂x

∂�j

∂x

)[150]

�i = (V − Vj)Fi [151]

where j = 1 for 1 ≤ i ≤ Nb and j = 2 for Nb + 1 ≤ i ≤ 2Nb.


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

∫|�j(x, t)|2o(x)dx =

∑k

o(x(k)t,j )wk,j [152]

k indexing the trajectories and j the asymptotic channel. There are no kineticenergy terms in the minimal basis set Nb = 1 related to the two-state represen-tation of the system.38 The AQP dynamics is exact for Gaussian wave functionsgoverned by the harmonic asymptotic potentials V1,2. Functions �1,2 and theirderivatives are known, analytically, in this case, and the whole approach be-comes 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 stabi-lized dynamics of r and p of “Stabilization of Dynamics by Balancing EquationErrors” implemented with the linear basis. Then the derivatives of �j at thetrajectory positions are

∂�j

∂x

∣∣∣∣x=x(j)

k

=(r(j)k + ıp(j)

k

)�j [153]

In the off-diagonal matrix elements 〈�1|o(x)|�2〉, the linearization parametersof r and p are used for evaluation of �2 at the positions x(1)

k of the first trajectoryset and vice versa. The linearization parameters of r and p already are availablefrom the propagation. The mixed-type integrals are evaluated in a symmetrizedfashion,

2〈�1|o(x)|�2〉 =∑j=1,2

∑k

o(x

(j)k

)w

(j)k z

(x

(j)k

)[154]

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 guaranteedwith 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 Topalerand Makri,186 which became the benchmark for approximate and semiclassicalmethods. With the particle mass rescaled to m = 1, the potential is

V = 14x4 − 20x2 + 50/7 [155]


The initial wave function is a Gaussian wave packet localized in the leftwell,

�1(x,0) =(

2��

)1/4

exp(

−�(x− q

(1)0

)2 + ıp(1)0

(x− q

(1)0

))[156]

The initial “image” Gaussian �2 has the same form as �1 with q(2)0 = −q(1)

0

and p(2)0 = −p(1)

0 . The corresponding population functions are �1(x,0) = 1 and�2(x,0) = 0. The wave packet parameters are chosen so that (x,0) mimicsthe 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 itsdeviation from the full potential, 〈(V − V1)2〉, weighted by the initial wavefunction density | (x,0)|2. The product asymptotic potential V2 is a reflectionofV1,V2(x) = V1(−x). Two asymptotic potentials – (A) the quadratic potential,

VA1 = k(x+ q0)2

2[157]

and (B) the Morse oscillator,

VB1 = D(e−z(x+q0) − 1

)2[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,VA1 and VB1 , as well as the initial wave function density in arbitrary scale. Theinitial energy of the wave packet constitutes 59% of the barrier height. Theprobability for the particle to be in the right well

P(t) =∫ ∞

0| (x, t)|2dx [159]

obtained with the asymptotic dynamics in the quadratic potential, is shownin Figure 16b. To account for the difference of the full potential V and itsharmonic 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.The advantage of the quadratic functional form for VA1 is that the trajectorydynamics of the Gaussians �1,2 and, consequently, the evaluation of matrixelements are exact. Therefore, this is the full QM limit of the mixed wavefunction representation approach.

The semiclassical description of the same system consists of the stabilizedAQP dynamics in the asymptotic potential—the Morse oscillator of VB1 , which


-1 0 1

Coordinate

0

10

20

Ene

rgy

0 10 200

1

Prob

abili

ty

0 10 20

Time

0

1

Prob

abili

ty(a)

(b)

(c)

Figure 16 Dynamics in the double well potential. The oscillation period in the asymp-totic well is approximately 1.4 atomic units. Panel (a) Full potential V (solid line) andits quadratic, VA

1 (dot-dash), and Morse, VB1 (circles), asymptotes. Initial wave function

density, | (x,0)|2, in arbitrary units is also shown with a dash; Panel (b) Probability offinding the particle in the product well as a function of time with dynamics defined byVA

1 .The quantum and mixed representation results for Nb = 1 and Nb = 5 are shown withthe solid line, dash and circles, respectively; Panel (c) Probability of finding the particlein the product well with trajectory dynamics defined by VB

1 . The quantum and mixedrepresentation results with semiclassical stabilized dynamics are shown for Nb = 1 andNb = 2 using thick solid line, dash and thin solid line, respectively.

is closer to the full potential in the well region than VA1 , and a small basis for�1,2. Dynamics under the influence of VB1,2 accounts for the anharmonicity ofa single well through �1,2, as illustrated in Figure 16(c). Simple coordinate-independent prefactors (Nb = 1) capture oscillations in the probabilities ratheraccurately. Nb = 2 gives agreement with the quantum result of about the samequality as Nb = 1.

To summarize, a combination of small basis set for � with the stabilizedAQP dynamics should be appropriate for coupled system/bath semiclassicalsystems. The normalization conservation, in principle, can be included into thefitting of r and p in the AQP dynamics, providing a connection between thebasis set and the trajectory components of the total wave function. This issueas well as multidimensional applications will be investigated in the future.

Conclusions 353

CONCLUSIONS

The presented review focused on the semiclassical approximations toquantum dynamics of chemical systems developed in recent years and isbased on the Madelung–de Broglie–Bohm formulation of the time-dependentquantum mechanics. The appeal of the Bohmian formulation stems from theclassical-like picture of quantum-mechanical evolutions, which are interpretedusing the trajectory language. The wave function is replaced by an ensembleof point particles that follow deterministic trajectories and obey a Newtonianequation of motion. The quantum effects are represented by a nonlocal quan-tum potential that enters the Newton equation and couples the evolution ofdifferent trajectories in the Bohmian ensemble. This classical-like trajectorypoint of view allows one to treat various degrees of freedom, for instance lightelectrons and heavy nuclei, on the same footing. This naturally leads to familiesof semiclassical and mixed quantum-classical approximations.

The trajectory representation offers several advantages over the tradi-tional grids and basis sets. Bohmian trajectories follow the quantum-mechanicaldistributions, avoiding the unnecessary computational effort that often is takento treat regions of very low quantum density. This feature of the Bohmian for-mulation allows one to eliminate the exponential scaling of the computationaleffort with system dimensionality, which is a common feature of the conven-tional approaches. Trajectories are easy to propagate using the tools developedfor classical molecular dynamics. In contrast to the semiclassical techniques thatalso use the trajectory language, Bohmian mechanics is, in principle, exact andcan be converged arbitrarily close to the accurate quantum-mechanical answer.

At the same time, the exact Bohmian trajectory dynamics has provenexpensive and unstable for general systems. This is a result of difficulties inevaluating the quantum force on an unstructured trajectory grid and becausethe behavior of the quantum trajectories is sensitive to the accuracy of the force.The Bohmian formulation of quantum dynamics is most useful as a computa-tional tool when it is implemented approximately with semiclassical systems.There, it affords a description of the leading quantum effects in high dimensionsand elegantly couples quantum and classical degrees of freedom in a unifiedframework.

Bohmian mechanics offers a solution to the trajectory branching prob-lem in the quantum-classical simulation by creating a new type of the quantumbackreaction on the classical subsystem. The Bohmian backreaction uniquelyis defined, computationally simple, and directly relates to the full classical limit.Branching of the quantum-classical trajectories is achieved in the Bohmian ap-proach by coupling the classical subsystem to a single quantum particle in theBohmian ensemble. In the quantum-classical Ehrenfest approximation, whichis the most common approach, a single average classical trajectory is generated.In contrast, an ensemble of quantum-classical Bohmian trajectories is created


for a single initial quantum-mechanical wave function. Traditionally, trajectoryensembles are produced using a variety of ad hoc surface hopping procedures.The Bohmian quantum-classical method uniquely is defined and gives resultsthat are similar to surface hopping.

A related and conceptually appealing hybrid quantum-classical theory canbe derived using the phase space representation of quantum-mechanical den-sity matrices. The phase space picture provides an excellent starting point for ahierarchy of approximations generated by closures; the reference state is pro-vided by the phase space description of the quantum harmonic oscillator withclassical trajectories. The formulation in terms of the partial hydrodynamicsmoments reproduces the dynamics of the harmonic system of coupled light andheavy particle, for which the exact closure can be obtained.

The independent Bohmian trajectory methods involve propagation ofhigh-order derivatives of the wave function phase and amplitude. The prop-agation is carried out in real space with the derivative propagation method andin complex coordinate space using Bohmian mechanics with complex action.Similarly, the Bohmian trajectory stability approach evolves the wave functionphase and stability matrix. Independent trajectories are particularly appealingbecause of the trivial parallelization of the computational effort. The true com-putational cost develops with the need for at least the second derivatives ofthe potential and because of the polynomial scaling of the equations with thesystem dimensionality. Truncation of the independent trajectory hierarchy ata low order directly relates to to the semiclassical methods, such as the WKBapproximation. In contrast to the semiclassical schemes, the higher orders ofthe Bohmian mechanics converge to the exact quantum mechanics. The inde-pendent Bohmian trajectory methods can be used to compute a stationary wavefunction using a single or very few trajectories to obtain energy eigenvalues inthe spirit of Diffusion Monte Carlo and to capture quantum interference frommultiple paths in the complex plane.

The approximate quantum potential technique is designed to describe thedominant quantum effects with essentially linear scaling. Here, the quantumpotential is defined from the moments of an ensemble of trajectories for theentire space or for a few subspaces. An approximate quantum potential de-fined this way allows one to compute the quantum force analytically. Simpleglobal approximations to quantum dynamics describe zero-point energy in an-harmonic systems very well. Combined with the space-prefactor functions, suchglobal approximations capture “hard” quantum effects, such as deep tunnel-ing and nonadiabatic dynamics. This strategy enables progress from gas-phasereactions to studies of quantum chemical dynamics in condensed phase.

All-in-all, the Bohmian formulation of time-dependent quantum mechan-ics generates a very intuitive picture of quantum dynamics, provides a straight-forward connection to classical mechanics, and creates exciting opportunitiesfor the development of semiclassical approximations with a great potential forapplications 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 toBob Wyatt for including the Bohmian formulation of quantum mechanics into his quantum chem-istry 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 andVR acknowledge the Donors of the American Chemical Society Petroleum Research Fund and theChemistry Division of the National Science Foundation for financial support.

APPENDIX A: CONSERVATION OF DENSITYWITHIN A VOLUME ELEMENT

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

md

dtı�v = −∇2(V +U)ı�x [A1]

d

dtı�x = ı�v [A2]

Differentiating w with respect to time t and using Eqs. [A2] and [4] and thenoncrossing property of quantum trajectories, ı� = ıx1 × ıx2 × . . . /= 0, oneobtains

d

dt(�(�x, t)ı�(t)) = ı�(t)

d�(�x, t)dt

+ �(�x, t)dı�(t)dt

= ı�(t)d�(�x, t)dt

+ �(�x, t)[A3]

×(dıx1

dt

1ıx1

+ dıx2

dt

1ıx2

+ . . .

)ı�(t) =

(d�(�x, t)dt

+ �(�x, t) �∇ × �v)ı�(t) = 0

[A4]

where

�∇ × �v =∑n

ıvn

ıxn[A5]

This means that after discretizing the initial wave function (�x,0) through aset of trajectories with initial positions {�x(i)}, velocities {�v(i) = ∇S(�x(i),0)/m},densities {�(�x(i),0) = A2(�x(i),0)}, and corresponding volume elements {ı�i(0)}


for each trajectory, the probability in its volume element will be conserved:�(�x(i), t)ı�i(t) = �(�x(i),0)ı�i(0) = wi. In principle, Eqs. [A1] and [A2] give anindependent way of finding the gradient of velocity and the volume element fora trajectory from the stability matrix evolution given by Eqs. [56] and [57]. Inpractice, their implementation might be too expensive; it requires the secondderivative 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 INARBITRARY COORDINATES

For a general curvilinear system of coordinates {�x}, the kinetic energyoperator T is

T = 12

∇†G∇ [B1]

Here ∇ is the gradient operator of a general form

∇i = fi(�x)∂

∂xi[B2]

and ∇† acts on the left. G is an inverse matrix of masses and moments of in-ertia; in general, G can have off-diagonal elements. In chemical applications, asystem of coordinates often is chosen to eliminate derivative cross-terms in theHamiltonian. This allows the asymptotic motion of fragments to be uncoupled.This means that the matrix G is diagonal. Typical coordinates for these applica-tions are the Jacobi or Radau coordinates in spectroscopy or reactive scatteringcalculations.187 The following derivation is given for the diagonal form of G. Itcan be extended to the nondiagonal case in a straightforward manner. We useatomic units, � = 1, throughout, and the �-dependence is noted in which it isimportant for interpretation. For square-integrable wave functions, the Jaco-bian J = J(�x) of the transformation from Cartesian coordinates to a given setof coordinates is taken into account. This allows Eq. [B1] to be rewritten

T = −12

(∇TG∇ + �dTG∇

)[B3]

The components of the vector �d are

di = ∂fi

∂xi+ fi

J

∂J

∂xi[B4]

Appendix B: Quantum Trajectories in Arbitrary Coordinates 357

The hydrodynamic or Bohmian form of the time-dependent Schrodinger equa-tion is based on the representation of a wave function in terms of real phaseand 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 realand imaginary parts gives the following equations:

∂S

∂t+ 1

2(∇S)TG(∇S) + V +U = 0 [B6]

U = − 12A

(∇TG(∇A) + �dTG(∇A)

)[B7]

and

∂�

∂t+ (∇S)TG(∇�) + ∇TG∇S� + �dTG(∇S)� = 0 [B8]

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

d

dt= ∂

∂t+ (∇S)TG(∇) = ∂

∂t+

∑i

vi∂

∂xi[B9]

in the frame of reference moving with the velocity

vi = ∂S

∂xiGiif

2i [B10]

Then, Eq. [B8] becomes

d�

dt= −

(∇TG(∇S) + �dTG∇

)� [B11]

By differentiating Eq. [B6] with respect to xi and using Eq. [B9] one obtains

d

dt

∂S

∂xi+

∑k

Gkkfk∂fk∂xi

(∂S

∂xk

)2

+ ∂

∂xi(V +U) = 0 [B12]


If derivatives of the phase are identified with the momentum,

pi = ∂S

∂xi[B13]

and the phase S is identified with the classical action function, then for any formof the gradient operator, Eqs. [B6], [B10], and [B12] give time-dependence of�x, �p, and S,

dxi

dt= Giif

2i pi [B14]

dpi

dt= − ∂

∂xi(V +U) −

∑k

Gkkfk∂fk∂xi

p2k [B15]

dS

dt= 1

2

∑k

Gkkf2k p

2k − (V +U) [B16]

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

H = 12

∑k

Gkkf2k p

2k + V +U [B17]

For a nondiagonal form of the matrix G, the equations of motion correspondto the Hamiltonian of Eq. [B17] with the single summation replaced by thedouble sum,

∑kl Gklfkflpkpl.

Importantly, the density of a wave function “carried” by a trajectorywithin 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

dt= d�

dtJ� + �

dJ

dt� + �J

d�

dt= 0 [B19]

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

dJ

dt= (∇S)TG(∇J) [B20]

Appendix C: Optimal Parameters of the Linearized Momentum 359

d�

dt=

∑k

ıvkıxk

� =(

∇TG(∇S) +∑k

∂fk∂xk

Gkkfkpk

)� [B21]

The weight conservation property means that one does not need to solveEq. [B11] involving gradients of �p and � to determine the time dependenceof the density. Moreover, the expectation value of an operator that is localin the coordinate representation, such as the wave packet probability, can befound 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 byEq. [B7]. This is the quantity that vanishes in the classical limit of small � andlarge mass for nodeless wave function densities. We approximate U to makethe quantum trajectory framework practical in large systems while retainingthe dominant quantum effects. It is convenient to define U in terms of thenonclassical component of the gradient operator,

�r = ∇A(x, t)A(x, t)

[B22]

The quantum potential in an arbitrary system of coordinates becomes

U = −12

(�rTG �r+ ∇TG �r+ �dTG �r

)[B23]

APPENDIX C: OPTIMAL PARAMETERS OF THELINEARIZED MOMENTUM ON SPATIAL DOMAINSIN 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 foreach spatial domain. The domain label l is omitted below and the index n labelsdimensions. A general linear function r(n)(�x) is represented as a scalar productof two vectors of length (N + 1)—a vector of basis functions,

�f = (x1, x2, . . . , xN,1)T [C1]

and a vector of parameters

�c(n) = (c1n, c2n, . . . , cNn, c0n)T [C2]

Then, r(n)(�x) = �f × �c(n). Organizing vectors �c(n) into a rectangular matrix C ofthe size N × (N + 1),

C = (�c(1), �c(2), . . . , �c(N)) [C3]


the condition on the minimal deviation on the domain,

∇�c(n)I = 0, I = ⟨(r(n) − r(n))2

�⟩

[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 overlapmatrix S = 〈�f ⊗ �f �〉 is (N + 1) × (N + 1). Its elements are the first and secondmoments of the trajectory distribution weighted by the domain function,

sij =∫f (i)f (j)�(�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 asthe size of C. The matrix elements of B are

bij =∫

d

dxi

(f (j)�(�x)

)× �(�x, t)d� [C7]

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

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