Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: application to the photodissociation dynamics of NOCl.

Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: Application to the photodissociation dynamics of NOCl Chia-Chun Chou Citation: The Journal of Chemical Physics 140, 104307 (2014); doi: 10.1063/1.4867636 View online: http://dx.doi.org/10.1063/1.4867636 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum vortices within the complex quantum Hamilton–Jacobi formalism J. Chem. Phys. 128, 234106 (2008); 10.1063/1.2937905 Excited state electronic structures and dynamics of NOCl: A new potential function set, absorption spectrum, and photodissociation mechanism J. Chem. Phys. 121, 2105 (2004); 10.1063/1.1768158 Regularity in highly excited vibrational dynamics of NOCl (X 1 A′): Quantum mechanical calculations on a new potential energy surface J. Chem. Phys. 119, 4251 (2003); 10.1063/1.1592503 Integrating the quantum Hamilton–Jacobi equations by wavefront expansion and phase space analysis J. Chem. Phys. 113, 8888 (2000); 10.1063/1.1319987 Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems J. Chem. Phys. 111, 2423 (1999); 10.1063/1.479520

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THE JOURNAL OF CHEMICAL PHYSICS 140, 104307 (2014)

Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: Application to the photodissociation dynamics of NOCl Chia-Chun Choua) Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan

(Received 23 November 2013; accepted 24 February 2014; published online 12 March 2014) The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867636] I. INTRODUCTION

Bohmian mechanics provides a trajectory description of quantum phenomena from a hydrodynamic point of view.1–3 Substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation (TDSE), we obtain a system of two coupled partial differential equations, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE). In Bohm’s formalism, quantum trajectories are generated by integrating the equations of motion including the contribution of the quantum potential determined from a precomputed wave function. This analytical approach has been employed to analyze and interpret a diverse range of physical processes using quantum trajectories.4–13 In addition, Bohmian mechanics offers a novel computational method for solving quantum dynamical problems. As a synthetic approach, the quantum trajectory method14–16 (QTM) has been developed to solve the TDSE by evolving ensembles of correlated quantum trajectories. The moving least squares (MLS) algorithm was used to evaluate spatial derivatives of the density and the velocity that are needed in the time integration of the hydrodynamic equations. Remarkable progress has been made in the development and application of the QTM. For example, the artificial viscosity method has been developed to improve the accuracy and stability of the QTM.17–20 The semiclassical QTM has been developed and applied to nonadiabatic dynamics.21–26 The bipolar decomposition of the total wave function has been used to reca) Electronic mail: [email protected]

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oncile semiclassical and Bohmian mechanics for stationary bound and scattering states and wave packet dynamics.27–35 Recently, the imaginary time quantum trajectory formulation has been developed to determine the ground state energies and wave functions of quantum systems.36–40 Moreover, the derivative propagation method (DPM) that circumvents the use of fitting techniques to evaluate spatial derivatives was introduced to solve quantum dynamical problems by propagating approximate independent quantum trajectories.41, 42 Quantum trajectories in complex space in the framework of the quantum Hamilton-Jacobi formalism, developed by Leacock and Padgett in 1983,43, 44 have recently attracted considerable interest. This variant of the Bohmian approach is based on substitution of the wave function expressed by the complex action function into the TDSE to obtain the complex QHJE (this version is not the same as that in Bohm’s formalism). This complex quantum hydrodynamic representation provides conceptual novelty and also leads to new trajectorybased pictures of quantum mechanics that prove useful in computational applications. As an analytical approach, complex quantum trajectories determined from the analytical form of the wave function have been analyzed for several stationary and nonstationary problems.45–56 For computational applications, the complex quantum Hamilton-Jacobi equation (CQHJE) has been solved using the Möbius integrator to obtain the wave function for stationary states.57–60 As a synthetic approach of the complex QTM to quantum dynamics, the DPM has also been used to obtain approximate complex quantum trajectories and the wave function for one-dimensional and multidimensional barrier scattering problems.61–72 In addition, this approach has been

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utilized to describe the interference effects and node formation in the wave function,73, 74 to determine energy eigenvalues,75 and to improve the complex time-dependent Wentzel-Kramers-Brillouin (WKB) method.76, 77 The pathintegral description has been presented for the complex QTM and the complex WKB method.78 Recently, the complex QTM has been applied to the dissipative dynamics described by a stochastic Liouville-von Neumann equation with complex noise forces79 and to nonadiabatic molecular dynamics.80, 81 In a recent study,82 we briefly described a computational method (the CQHJE-BT method) for solving the CQHJE by evolving an ensemble of correlated Bohmian trajectories in real space. Analogous to the case in the quantum HamiltonJacobi formalism, the wave function is expressed in exponential form in terms of the complex action, and we substitute this expression into the TDSE to obtain the CQHJE. We follow the same technique in Ref. 73 to transform the CQHJE into the arbitrary Lagrangian–Eulerian (ALE) version by selecting a grid velocity equal to the flow velocity of the probability fluid. The ALE version of the CQHJE describes the rate of change in the complex action transported along Bohmian trajectories. The ALE version of the CQHJE and the guidance equation for Bohmian trajectories are simultaneously integrated, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are computed using the MLS algorithm. The purposes of the current study are to provide a detailed comparison between theoretical formulations of the CQHJE-BT method and the real and complex QTMs and to apply the CQHJE-BT method to the photodissociation of NOCl. In contrast with the real QTM, the wave packet dynamics in the CQHJE-BT method is obtained by integrating one single CQHJE for the complex action rather than solving a system of two coupled hydrodynamic equations of motion for the real amplitude and action. The spatial derivatives of the complex action are computed by solving only one MLS matrix equation, whereas two hydrodynamic fields must be fit in the QTM, respectively. In the complex QTM, approximate quantum trajectories evolving in complex space are determined by solving a hierarchy of equations of motion involving spatial derivatives of the complex action. Information transported by the trajectories launched from isochrones in complex space (surfaces of equal arrival times) is required to synthesize the time-dependent wave function on the real axis. The propagation of complex quantum trajectories involves the analytical continuation of a potential from the real axis, and this requires an exact analytical expression for the potential. However, in the CQHJE-BT method, the exact CQHJE is solved by propagating an ensemble of Bohmian trajectories that follow the evolving probability density in real space. Thus, both the analytical continuation of a quantum system from real space to complex space and the search for isochrones in complex space are completely avoided. Moreover, as a computational implementation, the CQHJE-BT is applied to the photodissociation of NOCl. Accurate computational results indicate that the photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories.

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The organization of the remainder of this study is as follows. In Sec. II, we briefly review the real and complex QTMs and present similarities and differences between these two trajectory methods. In Sec. III, we derive the equations of motion in the CQHJE-BT method and describe the MLS algorithm for the evaluation of the spatial derivatives of the complex action. Comparisons between the CQHJE-BT method and the real and complex QTMs are presented. In Sec. IV, the CQHJE-BT method is applied to the photodissociation of NOCl, and computational results are presented and discussed. Finally, in Sec. V, concluding remarks and topics for further research are presented. II. TRAJECTORY FORMULATIONS OF QUANTUM MECHANICS

In order to compare real and complex quantum trajectory methods, we briefly review two trajectory formulations of quantum mechanics, including Bohmian mechanics and the complex quantum Hamilton-Jacobi formalism. A. Bohmian mechanics and the quantum trajectory method

Without loss of generality, we concentrate on the onedimensional version of Bohmian mechanics.1–3 First, we express the wave function in polar form in terms of the real amplitude and the real action function as i S(x, t) , (1) ψ(x, t) = R(x, t) exp ¯ where R(x, t) and S(x, t) denote the amplitude and the action function, respectively. Substituting this expression into the TDSE and separating the resulting equation into real and imaginary parts, we obtain a system of two coupled partial differential equations 1 ∂S 2 ∂S = + V (x) + QB (x, t), (2) − ∂t 2m ∂x ∂ 1 ∂S ∂ρ =− ρ , (3) ∂t ∂x m ∂x where ρ(x, t) = R(x, t)2 is the probability density. The Lagrangian flow velocity of the probability fluid is identified as 1 ∂S(x, t) . (4) vB (x, t) = m ∂x The probability flux is then given by j (x, t) = ρ(x, t)vB (x, t), and this flux accords with the conventional flux definition in quantum mechanics. Thus, Eq. (3) has the form of the classical continuity equation. In addition, Eq. (2) is referred to as the QHJE. The right side of this equation contains not only the flow kinetic energy and the classical potential but also Bohm’s time-dependent quantum potential given by ¯2 1 ∂ 2 R . (5) 2m R ∂x 2 This quantum potential brings all quantum effects into the hydrodynamic formulation. Equations (2) and (3) form the basic equations of quantum hydrodynamics. QB (x, t) = −


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In order to derive the Lagrangian version of the QHJE and the continuity equation, we consider a suitable transformation from the Eulerian partial time derivative to the total time derivative. By introduction of a grid velocity dx/dt, the relationship between these two time derivatives becomes ∂ dx ∂ d = + . (6) dt ∂t dt ∂x Upon substitution of this total time derivative with the grid velocity equal to the flow velocity of the probability fluid, we obtain the Lagrangian version of the hydrodynamic equations dS 1 ∂S 2 = − (V + QB ) = LB (x, t), (7) dt 2m ∂x dρ ∂vB = −ρ , dt ∂x

(8)

where LB (x, t) is the quantum Lagrangian. As a synthetic approach, the QTM has been developed as a computational tool to solve the TDSE. Equations (7), (8) (or an equivalent equation for the amplitude), and the trajectory equation 1 ∂S dx = dt m ∂x

(9)

form the set of equations of motion for quantum trajectories. These equations can be integrated on the fly to generate the density, action, and complex-valued wave function by evolving an ensemble of real-valued quantum trajectories.83

p(z, t) dz = . dt m

(13)

By introducing complex coordinates, the wave function, the complex action, and relevant functions are analytically extended to complex space. Continuation of the QHJE into complex space is accomplished by replacing x with z; hence, Eq. (11) becomes the complex-extended QHJE, 1 ∂A = − ∂t 2m

∂A ∂z

2 + V (z) +

¯ ∂ 2A . 2mi ∂z2

The complex QTM61, 64, 65, 73, 74 is based on the quantum Hamilton-Jacobi formalism, developed by Leacock and Padgett.43, 44 In contrast with Eq. (1), the time-dependent wave function is expressed in terms of the complex action by i A(x, t) , (10) ψ(x, t) = exp ¯ where A(x, t) is the complex action. Substituting Eq. (10) into the TDSE yields the complex QHJE (CQHJE), ∂A 1 ∂A 2 ¯ ∂ 2A − = + V (x) + . (11) ∂t 2m ∂x 2mi ∂x 2 In this formalism, we obtain a single equation for the complex action, as opposed to the coupled equations for the real phase and amplitude given in Eqs. (2) and (3). As in Bohmian mechanics, the quantum momentum function (QMF) is defined by the guidance equation p(x, t) = ∂A(x, t)/∂x. To find a quantum trajectory, we may rearrange this equation as (12)

However, because the action A(x, t) is complex-valued and the time remains real-valued, the trajectory requires a complexvalued coordinate. Accordingly, the QMF p(x, t) and the complex action A(x, t) are extended to the complex space by replacing x with z. Thus, complex quantum trajectories are

(14)

The terms on the right correspond to the kinetic energy, the classical potential, and the complex quantum potential defined by Q(z, t) =

¯ ∂ 2A . 2mi ∂z2

(15)

The complex-extended QHJE in Eq. (14) is almost identical to the QHJE in Eq. (2). However, the complex quantum potential involves the second derivative of the complex action, whereas Bohm’s quantum potential in Eq. (5) displays a nonlinear dependence on the amplitude function. Through the identification of the velocity of complex quantum trajectories in Eq. (13), we can transform Eq. (14) into the Lagrangian version of the complex-extended QHJE along a trajectory using the Lagrangian time derivative in Eq. (6), p2 dA = − (V + Q) = L(z, t), dt 2m

B. Quantum Hamilton-Jacobi formalism and the complex quantum trajectory method

1 ∂A(x, t) dx = . dt m ∂x

defined by

(16)

where L(z, t) is the complex quantum Lagrangian. In addition, a comparison of Eq. (16) with Eq. (7) reveals the strong similarity between the real and complex trajectory formulations of quantum mechanics. However, the most significant difference between these two formalisms arises from the fact that an obvious probability flux continuity equation does not exist in the complex quantum Hamilton-Jacobi formalism.84–89 In the complex QTM,61–70, 73–75, 77 the complex-extended QHJE in Eq. (14) is approximately solved by propagating individual Lagrangian quantum trajectories in complex space. Equations of motion for these trajectories are derived through the use of the DPM,41 which leads to a hierarchy of coupled differential equations for the complex action and its spatial derivatives along each trajectory. The Lagrangian form of the complex-extended QHJE in Eq. (16) can be expressed in terms of the complex action and its spatial derivatives as 1 2 ¯ dA = A1 − V − A2 , dt 2m 2mi

(17)

where the order of the spatial derivative is denoted by the subscript, An = ∂ n A/∂zn . For example, in order to develop the second-order DPM equations for complex trajectories, we take the first and second spatial derivative of the Eulerian version of the complex-extended QHJE in Eq. (14) and convert these equations into the Lagrangian frame. The system of coupled Lagrangian equations is truncated at order two by


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dropping third- and fourth-order derivatives, and we obtain dA1 = −V1 , dt

(18)

dA2 1 = − A22 − V2 . dt m

(19)

Thus, Eqs. (17)–(19) constitute the system of equations for complex-valued DPM(2), and these equations describe the approximate complex action and its first and second derivatives transported along the second-order complex trajectories.

f (r ) =

III. COMPLEX QUANTUM HAMILTON-JACOBI EQUATION WITH BOHMIAN TRAJECTORIES

In Sec. II, in order to compare the real-valued and complex-valued QTMs, we focused on the one-dimensional version of these two trajectory formulations. In this section, we present a synthetic trajectory approach, which is called the CQHJE-BT method,82 to quantum wave packet dynamics through the integration of the exact CQHJE by propagating an ensemble of correlated Bohmian trajectories in real multidimensional configuration space. As in the complex QTM, substituting the exponential form of the wave function, ψ(r , t) = exp[iA(r , t)/¯], into the TDSE yields the CQHJE, 1 2 ¯ 2 ∂A = (∇A) + V (r ) + ∇ A. ∂t 2m 2mi

(20)

The terms in this equation have the same physical meaning as those in the one-dimensional version in Eq. (11). Rather than analytically extending Eq. (20) to complex space, we convert this equation into the ALE version through the transformation in Eq. (6) by choosing the grid velocity equal to the flow velocity of the probability density given by Eq. (4), 1 d r = Re[∇A], dt m

(21)

= Re[∇A] has been where Re represents the real part and ∇S used. Then, the resulting equation describes the rate of change in the complex action along Bohmian trajectories dA ∂A d r · ∇A = + dt ∂t dt =−

1 2 ¯ 2 1 · ∇A. (∇A) − V (r ) − ∇ A + Re[∇A] 2m 2mi m (22)

The one-dimensional version of this equation has been presented in Ref. 73, and the DPM has been employed to solve the CQHJE with several specific choices of trajectory velocity fields. Equations (21) and (22) form the equations of motion in the CQHJE-BT method. No approximations have been made to obtain these equations; hence, they are exact formulations of the TDSE. Expressing the complex action in terms of the real amplitude and action by A(r , t) = S(r , t) − i¯C(r , t),

nb

aj pj (r − rk ),

(24)

j =1

A. Theoretical formulation

−

where C(r , t) = ln R(r , t) is the C-amplitude, we can show that Eq. (22) is equivalent to the coupled equations of motion for the hydrodynamic fields in Eqs. (7) and (8). The first and second derivatives of the complex action at the positions of the moving fluid elements are needed to integrate the equations of motion in Eqs. (21) and (22). As in the QTM,14, 15, 83 we utilize the MLS algorithm to compute these derivatives through a local fit calculated around each of the probability fluid elements. We assume that the complex action can be approximated by a function f (r ) which is locally expanded in a basis set of local polynomials of dimension nb ,

(23)

where each individual basis function pj (r − rk ) depends on the locations of both the observation point r and the reference point rk . Because the action function A(r ) is complex valued, we allow the expansion coefficients {aj } to be complex, but the variable r and the basis functions pj (r ) remain real. In order to find the coefficients {aj }, we consider the residual sum of the absolute squares of the errors for np points within the stencil surrounding point rk defined by k =

np

|A(ri ) − f (ri )|2 .

(25)

i=1

The least squares error minimization leads to one single normal matrix equation for the complex expansion coefficients {aj }: PT Pa = PT A, where P (dimension np × nb ) is the rectangular collocation matrix with elements Pij = pj (xi − xk ), a (nb × 1) is the solution vector, and A (np × 1) is the complex action function vector.82 The solution vector of this equation is formally given by a = (PT P)−1 PT A. This implies that we only need to solve one matrix equation for the complex expansion coefficients {aj } to obtain the approximate spatial derivatives of the complex action A(r ) at point rk . B. Comparisons with the real and complex quantum trajectory methods

The real QTM83 and other relevant methods are based on the polar decomposition of the wave function, including the linearized quantum force method,90–92 the covering function method,93, 94 the artificial viscosity method,17–20 the hybrid method,95, 96 and the moving boundary truncation method.97–99 In these methods, the time-dependent wave function is obtained by solving a system of two coupled hydrodynamic equations of motion for the real amplitude and the real action function. The spatial derivatives of two hydrodynamic fields must be evaluated to integrate the hydrodynamic equations of motion. Although the real amplitude and action functions are separately computed, these two fields can be approximated with different basis sets and even with different approaches. In contrast with the real QTM, the wave packet dynamics in the CQHJE-BT method is obtained by integrating one single CQHJE for the complex action. The spatial derivatives of the complex action are computed by solving only one MLS matrix equation. If the real amplitude and action functions are approximated to the same order with


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the same approach, the CQHJE-BT method provides a simplified and compact set of equations of motion for complicated systems. In addition, because Bohmian trajectories follow the main features of the evolving probability density, we choose to integrate the CQHJE by propagating an ensemble of Bohmian trajectories in this study. Adaptive moving grids for non-Bohmian trajectories can be directly applied to the CQHJE-BT method. In Bohmian mechanics, the wave function is expressed in polar form in terms of the real amplitude and the real action function, ψ = R exp (iS/¯). By contrast, substituting the wave function expressed in terms of the complex action, ψ = exp (iA/¯), into the TDSE yields the CQHJE. Schrödinger himself recognized the necessity for a complex action function.100 It has been pointed out that the CQHJE shows the exact classical-quantum correspondence for Gaussian wave packet dynamics in a harmonic potential.101 However, for the simple case, Bohm’s quantum force is not only nonvanishing but is the same magnitude as the classical force. We consider a Gaussian wave packet evolving in the harmonic potential V (x) = mω2 x 2 /2. The analytical expression for the Gaussian wave packet corresponding to a coherent state is given by101 i i 2 ψ(x, t) = N exp −α0 (x − xt ) + pt (x − xt ) + γt , ¯ ¯ (26) where α 0 = mω/2¯ and N = (2α 0 /π )1/4 . The time-dependent parameters are given by p0 sin ωt, (27) xt = mω pt = p0 cos ωt,

(28)

p02 1 sin 2ωt − ¯ωt. (29) 4mω 2 From the analytical expression in Eq. (26), the real amplitude and action function in Eq. (1) are given by R(x, t) = N exp [−α 0 (x − xt )2 ] and S(x, t) = pt (x − xt ) + γ t . Thus, Bohm’s quantum potential in Eq. (5) becomes γt =

1 1 (30) QB (x, t) = − mω2 (x − xt )2 + ¯ω. 2 2 This equation indicates that Bohm’s quantum potential is an inverted harmonic potential. As discussed in Ref. 101, although Bohm’s quantum potential can be regarded as a “correction” to classical mechanics, this term does not tend to zero when we take the limit ¯ → 0. By contrast, the analytical expression in Eq. (26) leads to the complex action given by i i ¯ 2 ln N − α0 (x − xt ) + pt (x − xt ) + γt . A(x, t) = i ¯ ¯ (31) Accordingly, the complex quantum potential becomes Q(x, t) = ¯ω/2, and it is equal to the ground state energy of the harmonic oscillator. In addition, this quantum correction term is independent of coordinates, and it vanishes when we take

the classical limit ¯ → 0.101 Moreover, unlike the real-valued equations of motion in Bohmian mechanics, the CQHJE contains all of the information present in the wave function, leading some authors to conclude that the complex version is the more fundamental.46 In quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator whose dynamics most closely resembles the oscillating behavior of a classical harmonic oscillator. This wave packet actually behaves quite like a classical particle because its average position and momentum follow Hamilton’s equations and its width is constant in time. In addition, the quantumclassical correspondence is exact for a Gaussian in a harmonic potential.101 In complex space, the constant complex quantum potential in Eq. (15) indicates that the quantum force is equal to zero and the complex quantum trajectories for a coherent state are completely governed by the classical force.50 This fact implies that complex quantum trajectories in a coherent state are identical to the corresponding complex classical trajectories.50, 102, 103 This remarkable consistency establishes the correspondence of a quantum harmonic oscillator in a coherent state with the classical harmonic oscillator. In the previous studies associated with the complexvalued QTM,61–70, 73–75, 77 the complex-extended QHJE in Eq. (14) is reformulated into a hierarchy of equations of motion involving spatial derivatives of the complex action. Various orders of spatial derivatives are coupled together in an infinite hierarchy, and a truncation must be implemented so that a closed set of equations can be solved. Truncation of the infinite hierarchy of equations of motion leads to individual approximate quantum trajectories evolving in complex space. Although low-order truncations of this set may yield useful and accurate approximations, there also exists inherent error originating from the truncation scheme. In addition, high-order truncations may not improve the accuracy of the approximate DPM solutions. For example, there is no algorithm for determining the best order of the DPM for barrier scattering problems.64, 66 Moreover, the propagation of complex trajectories requires analytical continuation of the original quantum dynamical problem to complex space. Information transported by trajectories launched from isochrones (surfaces of equal arrival times) is required to synthesize the time-dependent wave function on the real axis. The computational task of locating isochrones for evolving trajectories in complex space is time consuming for multidimensional applications. However, in the CQHJE-BT method, the exact CQHJE is integrated by propagating an ensemble of Bohmian trajectories that follow the evolving probability density in real space. Thus, both the analytical continuation of a quantum system from real space to complex space and the search for isochrones in complex space are completely avoided. IV. COMPUTATIONAL RESULTS AND DISCUSSION

We apply the CQHJE-BT method to the photodissociation of NOCl where the system is reduced to two coordinates by freezing the angular variable at the equilibrium value.104–106 The photodissociation dynamics of NOCl


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has also been studied through the integration of the quantum hydrodynamic equations in the Eulerian and Lagrangian representations.107, 108 Finite element techniques have been used to evaluate spatial derivatives of the hydrodynamic fields that are required for the wave packet evolution in the twodimensional photodissociation problem.108 This approach involves a mapping process between the physical grid and a computational grid and local interpolation on the finite element in the computational space for approximations of the numerical evaluation of the derivatives. Moreover, for edge points, approximate function values have to be extrapolated at additional grid points outside the grid boundaries. By contrast, the MLS method is utilized to approximate the spatial derivatives of the complex action in the CQHJE-BT method, and the extrapolation process, which is subject to a high level of uncertainty, is avoided. The photodissociation dynamics of NOCl is described by the Jacobi coordinates R, r, and θ , where R is the distance (dissociative coordinate) from Cl to the center of mass of NO, r is the bond length (vibrational coordinate) of NO, and θ indicates the angle between r and R. In this coordinate system, the Hamiltonian acting on the wave function for the NOCl molecule is written as 1 ∂2 1 ∂2 ¯2 (Rψ) (rψ) + Hψ = − 2 md R ∂R 2 mv r ∂r 2 ∂ ∂ψ 1 sin θ + V (R, r, θ )ψ. (32) + Iθ sin θ ∂θ ∂θ The reduced masses md and mv and the moment of inertia Iθ for the NOCl molecule are given by 1 1 1 = + , md mN + mO mCl

(33)

1 1 1 = + , mv mN mO

(34)

1 1 1 = + , 2 Iθ md R mv r 2

(35)

where the dimensionless coordinates x and y range from zero to infinity. Thus, all calculations will be carried out in the dimensionless representation. In the time-dependent study of photodissociation, the initial ground-state vibrational wave function on the ground S0 electronic potential surface is vertically excited to the S1 electronic potential surface within the Franck-Condon approximation. The excitation from the S0 to S1 surface is assumed to occur instantaneously via a laser field. Thus, the initial wave packet on the S1 surface has the same shape as that on the S0 surface. Then, the dynamics of photodissociation is described by the time evolution of this initial bound state on the excited electronic surface. Because the S0 surface has a deep well, the initial wave packet determined by the ground state wave function of the S0 surface under the quadratic approximation is given by106–108 ω ω 1/4 1 2 φ0 (x, y) = exp{−(1/2)[a(x − xeq )2 π2 + 2b(x − xeq )(y − yeq ) + c(y − yeq )2 ]}, (38) where xeq = 2.944, yeq = 1, ω1 = 527.1, ω2 = 498.6, a = 506.1, b = 12.53, and c = 519.7. The S1 surface for NOCl is taken as108 U (x, y) = a2 (y − y1eq )2 + c00 (1 − Q(x))[1 + c01 Q(x) + c10 (y − y1eq ) + c20 (y − y1eq )2 + c11 Q(x)(y − y1eq )

where mN , mO , and mCl are the masses of atoms N, O, and Cl, respectively. The coordinates and the Hamiltonian are the same for both the S0 and the S1 potential energy surfaces. For simplicity, the angle variable θ is frozen at the equilibrium value θ eq = 127.4◦ , and the S0 and S1 surfaces for NOCl have the same equilibrium value.106 Then, the threedimensional TDSE is reduced to a two-dimensional version given by ¯2 1 ∂2 ∂ψ 1 ∂2 (Rψ) + (rψ) =− i¯ ∂t 2 md R ∂R 2 mv r ∂r 2 + V (R, r, θeq )ψ.

where req is a length scale taken to be the value of the vibrational coordinate at the molecular equilibrium on the S0 potential energy surface. In this study, md = 29 661, mv = 13 710, and req = 2.155. Unless otherwise stated, all quantities are given in atomic units (¯ = 1). In the dimensionless representation, the TDSE in Eq. (36) becomes 1 ∂ 2φ ∂ 2φ ∂φ + U (x, y)φ, (37) =− + i ∂τ 2 ∂x 2 ∂y 2

(36)

Moreover, dimensionless expressions for the relevant physical quantities can be introduced through the transformations:107, 108 2 2 mv req mv req , φ = Rrψ, U = V , t = 0 ¯2 ¯ 1/2 md R r t x= , y= , τ= , mv req req t0

+ c21 Q(x)(y − y1eq )2 ],

(39)

where Q(x) = 1 − exp [−α(x − x1eq )]. The associated parameters are given by x1eq = 2.945,

y1eq = 0.991,

a2 = 0.202 × 106 , c00 = 2450,

α = 2.198, c10 = −3.129,

c01 = 0.217,

c20 = −19.69,

c11 = 3.260,

c21 = −3.508.

This simplified S1 potential and the parameters are identical to those used in Ref. 108, and the more general expression of the S1 surface is given in Ref. 106. We now integrate the equations of motion in Eqs. (21) and (22) by evolving ensembles of Bohmian trajectories representing the photodissociation of NOCl on the electronically excited potential energy surface. From the initial wave packet in Eq. (38), the initial complex action is given by A(x, y) = (¯/i) ln φ 0 (x, y). To improve the accuracy of the calculations, the initial locations of trajectories were sampled to be inversely proportional to the initial Gaussian for the density expressed as the product of Gaussians in one variable in the


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FIG. 1. Snapshots of an ensemble of Bohmian trajectories with the probability density at six times for the photodissociation of NOCl on the excited potential energy surface. Contours of the simplified S1 potential energy surface as a function of the coordinates R and r at the equilibrium angle θ eq = 127.4◦ are shown as black curves.

principal axes of the quadratic potential.108–111 At the initial time, the trajectories were launched from a 20 × 20 Cartesian grid located around the center of the Gaussian wave packet on the upper potential surface. As shown in Fig. 1, particles are densely packed near the center of the initial wave packet and sparse near the edges of the ensemble. The particle ensemble was then propagated with the time step t = 0.5 from t = 0 to t = 2000 using the explicit first-order Euler method to integrate the equations of motion in Eqs. (21) and (22). At each

time step, the complex action was locally fit to a quadratic surface with the local polynomial basis B = {1, ξ , η, ξ 2 /2, ξ η, η2 /2}, where ξ = x − xk and η = y − yk , to approximate the derivatives around each grid point with the 35 nearestneighbor grid points. Figure 1 displays the time evolution of the ensemble of Bohmian trajectories and the probability densities generated by the linear interpolation for the complex action using the routine griddata from MATLAB. Contours of the


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400

2000

380

1500 t 1000

360

500 Ntraj

340

0

320

2.4 r

300

2.2 2

280 0

250

500

FIG. 2. The number of trajectories in the ensemble decreases over time during the ensemble evolution.

simplified S1 potential energy surface as a function of the coordinates R and r at the equilibrium angle θ eq = 127.4◦ are also shown in this figure. The S1 surface of the NOCl molecule consists of a simple valley with a downward slope. Initially, the compact wave packet is centered at R = 4.314 and r = 2.155. As time progresses, this wave packet gradually spreads and rapidly propagates along the dissociative coordinate with a slight oscillatory motion along the vibrational coordinate of NO. Finally, the entire wave packet escapes from the inner region and evolves into the NO + Cl dissociation channel. As shown in this figure, the compact form of the wave packet is retained during the entire process, and the Bohmian trajectories follow the main features of the evolving probability density. In order to increase the numerical stability of the propagation scheme of the CQHJE-BT method, trajectories with a probability density less than one (ρ i < 1) were dropped from the particle ensemble as time progressed. Figure 2 indicates that the number of trajectories in the ensemble gradually decreases over time. At the final time, only 291 trajectories were kept in the ensemble. The removal of trajectories with

(a)

1.8

750 1000 1250 1500 1750 2000 t

1

4

4.5

5

5.5

R

6

6.5

7

7.5

FIG. 3. Time evolution of Bohmian trajectories for the photodissociation of NOCl with contours of the simplified S1 potential energy surface. Only 97 trajectories are shown in this figure for clarity.

an extremely low probability density can eliminate divergent trajectories and avoid the computational breakdown resulting from numerical instabilities. As shown in Fig. 3, the ensemble of trajectories was launched on the upper electronic potential energy surface. As time advances, these trajectories travel along the potential valley with a downward slope toward the NO + Cl dissociation channel. During the propagation process, the trajectory ensemble evolves as a unified whole on the S1 potential energy surface. Figure 4(a) shows the autocorrelation function defined by the overlap integral between the evolving wave packet at time t and the initial wave packet at t = 0, C(t) = φ(0)|φ(t). This function reflects the dynamics of the system on the S1 surface.106–108, 112 As displayed in Fig. 1, after the initial wave packet is launched, the wave packet immediately leaves its place of birth and propagates in the direction of the steepest descent. Because of this motion, the autocorrelation function rapidly decays to zero, as illustrated in Fig. 4(a). As time proceeds, the wave packet travels along the dissociative coordinate and never returns to its

(b)

0.8 0.6 0.4 0.2 C(t)

σ

0

−0.2 −0.4 −0.6 −0.8 0

250

500

750 1000 1250 1500 1750 2000 t

0

0.5

1.0 1.5 E (eV)

2.0

FIG. 4. (a) Time-dependence of the autocorrelation function C(t) for NOCl. The absolute value of the autocorrelation function |C(t)| is shown as a black curve. The real and imaginary parts of the autocorrelation function are shown by red dashed and blue dashed-dotted curves, respectively. (b) The energy-resolved absorption spectrum for NOCl.


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origin; thus, the autocorrelation function remains zero for all times. In addition, the energy-dependent absorption spectrum can be obtained by the Fourier transform of the timedependent autocorrelation function given by 2 2 σ (ω) = ω Re (40) C(t) eiωt e−t /τw dt . The quantity τw = 67 fs characterizes the finite resolution of the spectrum because of the limited propagation time.112 The choice of the window function corresponds to a convolution of the spectrum with a Gaussian. Figure 4(b) presents the spectrum for NOCl, and the broad spectrum corresponds to fast and direct dissociation. Although some differences exist between the spectrum and those in Refs. 105 and 106 obtained from the exact full three-dimensional calculations, the main peak position near 1.2 eV is in good agreement with the exact result. In order to assess the accuracy of the computational results, we consider the overlap integral between the exact and the computational wave functions φ exact |φ comp , which is a sensitive measure of the errors.106, 112 The real part of the overlap integral is strictly correlated with the norm of the difference between both wave functions106 φcomp − φexact 2 = 2[1 − Re(φexact |φcomp )].

(41)

Thus, the real part of the overlap integral serves as a measure for the difference between these two wave functions. The real part of the overlap integral is equal to one for identical wave functions, and the difference between the real part of the overlap integral and unity defines the most restrictive error criterion. For this case, the exact results were obtained from the split-operator method using a large space fixed grid.101 Figure 5 presents the real part of the overlap integral between the exact and the computational wave functions. At t = 0, the initial wave packet was discretized in terms of 400 fluid elements. This discretization leads to the deviation of the real part of the overlap integral from unity at the begin-

1.00

0.98

0.96

0.94

0.92

0.90

250

500

750

1000 1250 1500 1750 t

FIG. 5. The real part of the overlap integral between the computational and exact wave functions as a function of time. The overlap integral was computed every 50 time steps.

ning of the propagation of the ensemble of trajectories. As time proceeds, the real part of the overlap integral gradually decreases; however, its value is larger than 0.92 for all times. This indicates that approximately 92% of the wave packet are correctly described during the evolution of the trajectory ensemble. Therefore, the photodissociation dynamics of NOCl can be obtained by propagating a small ensemble of trajectories.

V. SUMMARY AND CONCLUDING REMARKS

In the CQHJE-BT method, the wave function is expressed in exponential form in terms of the complex action. Substituting this expression into the TDSE, we obtain the CQHJE. By selecting a grid velocity equal to the velocity of the probability flow, we can transform the CQHJE to the ALE version describing the rate of change in the complex action transported along Bohmian trajectories. In order to simultaneously integrate the ALE version of the CQHJE and the trajectory guidance equation, the spatial derivatives of the complex action are computed using the MLS algorithm. Thus, the time-dependent wave function is readily synthesized through the time integration of the CQHJE by propagating an ensemble of Bohmian trajectories. As a computational illustration, the CQHJE-BT method was employed to analyze the photodissociation dynamics of NOCl. The wave packet dynamics on the excited electronic potential energy surface can be obtained through the integration of the CQHJE by evolving a small ensemble of trajectories. As time progresses, the wave packet described by the ensemble of trajectories gradually spreads and rapidly propagates into the NO + Cl dissociation channel. During the ensemble propagation, trajectories with an extremely low probability density were dropped from the particle ensemble. The removal process can eliminate divergent trajectories, increase the computational stability of the propagation scheme, and avoid computational breakdown originating from the accumulation of numerical errors. Moreover, the energy-resolved absorption spectrum calculated by the Fourier transform of the autocorrelation function indicates that the broad spectrum corresponds to fast and direct dissociation. The main peak position is in good agreement with the exact full threedimensional calculations.105, 106 Finally, the real part of the overlap integral between the exact and the computational wave functions was employed to assess the accuracy of the computational results. In this case, over 92% of the wave packet were correctly described during the evolution of the trajectory ensemble. Therefore, the photodissociation dynamics of NOCl can be accurately obtained by propagating a small ensemble of trajectories. In the real-valued QTM,14, 83 wave packet dynamics is obtained by solving a system of two coupled hydrodynamic equations of motion for the real amplitude and action function. The spatial derivatives of two hydrodynamic fields required for the integration of the hydrodynamic equations of motion must be computed. In contrast, in the CQHJE-BT method, the wave packet dynamics is obtained by integrating one single CQHJE for the complex action. The spatial


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derivatives of the complex action are evaluated by solving only one MLS matrix equation. In the complex-valued QTM,61, 64, 65, 73, 74 the complexextended QHJE is approximately solved by propagating individual Lagrangian quantum trajectories in complex space. Approximate complex action and quantum trajectories are obtained by solving a hierarchy of equations of motion involving spatial derivatives of the complex action. Truncation of the infinite hierarchy of equations of motion leads to inherent error, and there is no algorithm for determining the best order of the truncation scheme.64, 66 In addition, the propagation of complex trajectories requires analytical continuation of a quantum system to complex space. Singularities in the complex extension of the potential energy surface may present difficulties for the propagation of complex quantum trajectories.65 Moreover, information transported by the trajectories launched from isochrones in complex space is required to synthesize the time-dependent wave function on the real axis. It is difficult and computationally demanding to locate isochrones even for two-dimensional systems. As opposed to the complex-valued QTM, the CQHJEBT method provides a computational method for integrating the exact CQHJE by propagating Bohmian trajectories that follow the evolving probability density in real space. Wave packet dynamics is obtained from the CQHJE without involving the propagation of complex trajectories, and the analytical continuation of a potential to complex space is not required. Thus, potential energy surfaces represented by a discrete set of points in real space, as generated by density functional theory calculations, can be directly used in the CQHJE-BT method. Furthermore, the search for isochrones in complex space is completely avoided. Therefore, this study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex QTMs for wave packet dynamics. The CQHJE-BT method presents a novel synthetic trajectory approach to quantum wave packet dynamics. As with the node problem in the QTM,83 singularities originating in the spatial derivatives of the complex action near nodal regions may lead to numerical instability and computational breakdown in the CQHJE-BT method. It is expected that computational methods for coping with the node problem in the QTM, such as the bipolar decomposition approach,27–29, 33 the covering function method,93, 94 and the mixed wave function representation,113 can be applied to the CQHJE-BT method. In addition, the CQHJE-BT method may be readily applied to computational techniques previously developed for the QTM based on the polar decomposition of the wave function, such as adaptive grid methods,83 the artificial viscosity method,17 and the mixed quantum–classical Bohmian method.114 Computational methods that are specifically suitable for the accurate evaluation of the spatial derivatives of the complex action should be developed. In addition to theoretical and methodological developments, applications of the CQHJEBT method to more realistic problems, such as the full three-dimensional study of the photodissociation of NOCl, deserve further investigation. Additional analysis and applications will be reported in future studies.

J. Chem. Phys. 140, 104307 (2014)

ACKNOWLEDGMENTS

We gratefully acknowledge National Tsing Hua University (Grant Nos. 102N1200I1 and 102N2102E1) and the National Science Council (Grant No. NSC 102-2113-M-007016-MY2) for their financial support of this research. 1 D.

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