Forward-Backward Quantum Dynamics for Time Correlation Functions

Transcription

1 806 J. Phys. Chem. A 2004, 108, Forward-Backward Quantum Dynamics or Time Correlation Functions Nancy Makri Departments o Chemistry and Physics, UniVersity o Illinois, 601 South Goodwin AVenue, Urbana, Illinois ReceiVed: May 23, 2003; In Final Form: October 8, 2003 A novel quantum mechanical ormulation o time correlation unctions is derived, based on Bohmian trajectories integrated along a orward-backward time contour. The derived expression involves a smooth integrand amenable to Monte Carlo integration with a readily available position space density. The Bohmian potential governing the motion o the quantum trajectories is weak and well-behaved, acilitating the computation o Bohmian trajectories. I. Introduction Time correlation unctions provide important inormation on the dynamics and spectroscopy o chemical systems. For example, the autocorrelation unction o the dipole-moment operator leads directly to vibrational or electronic spectra, lux correlation unctions provide reaction rate inormation, and velocity correlation unctions yield diusion constants in the bulk. The inclusion o quantum mechanical eects in the computation o time correlation unctions presents a challenging problem that continues to receive much attention. At present, the dynamics o small molecules are treatable by a variety o basis-set or grid-based methods, and low-dimensional subsystems coupled to dissipative harmonic baths can be simulated via numerically exact path integral methods. In recent years, eorts to devise simulation tools capable o including quantum mechanical eects to the dynamics o polyatomic systems dominated by arbitrary anharmonic interactions have moved to the oreront o theoretical chemistry research. Among the most successul approaches currently available are the centroid molecular dynamics scheme, techniques based on surace hopping, and semiclassical initial value methods. The semiclassical approximation to quantum mechanics 1,2 oers a rigorous and intuitively appealing way o including quantum eects via classical trajectory input. Semiclassical theory is based on classical trajectories and their stability properties, which can be obtained by inexpensive calculations even or very large systems. 3 These characteristics make semiclassical methods very appealing, as they can account or important quantum mechanical eatures o a system, while oering an intuitive understanding o dynamical processes. 4-7 In recent years, the semiclassical method has experienced a rebirth o interest with the development o semiclassical initial value representations (IVR), 4,8-20 which circumvent the troublesome root-search problem o the original ormulation. Numerical calculations exploiting these advances have demonstrated the ability o the semiclassical approximation to provide an accurate description o essentially all types o quantum eects in chemical dynamics, 13,20-22 and a number o successul applications to polyatomic molecules have been presented. 19,20,23-31 To date, the highly oscillatory character o the semiclassical integrand remains the main drawback o semiclassical methods, preventing integration by Monte Carlo methods in condensed phase systems. A signiicant development in the past ew years has been the ormulation o orward-backward semiclassical dynamics (FBSD) methods, which circumvent the problem by combining (and thus partially canceling) the phases o orward and reverse time trajectories entering the semiclassical representation o a correlation unction, albeit with the loss o some quantum intereerence. In recent years, an intriguing ormulation o quantum dynamics based on quantum trajectories has been pursued by several groups The so-called Bohmian or hydrodynamic ormulation o quantum mechanics assumes a orm very similar to the semiclassical approximation, and recent work 68 has discussed the apparent similarities and undamental dierences o the two ormulations. Even though the wave unction (or propagator) is written as an amplitude multiplied by a phase in both theories, the semiclassical method relies on cross terms between distinct trajectories in order to account or quantum intererence, while the Bohmian prescription employs a single trajectory or each point in space whose motion is governed by a quantum potential. The neglect o such cross terms lies at the heart o FBSD 13,32-37,40,43,45 and quasiclassical methods, 44,69-72 making them easible by avoiding oscillatory phases at the expense o neglecting quantum intererence. In a recent paper by my group, 68 it was shown that the Bohmian expression or a time-dependent expectation value can take a similar quasiclassical (initial value) orm, ree o undesirable phases. Interestingly, no phase actors enter the Bohmian expectation value, and yet all quantum eects are ully accounted or! Provided that the quantum potential can be evaluated reliably along a quantum trajectory, this quasiclassical expression allows ull quantum mechanical calculations via Monte Carlo sampling o a readily available initial density. However, accurate determination o the strong, rugged Bohmian orce continues to present serious diiculties in bound anharmonic systems. Further, numerical determination o the quantum potential based on inite-dierence or polynomial-itting procedures requires the knowledge o trajectory inormation in the vicinity o the given phase space point. This interdependence o the Bohmian trajectories prevents the propagation o a single trajectory by itsel. Recent work in my group has shown that this problem can be overcome in principle through the determination o the Bohmian orce rom the stability properties o a given quantum trajectory. The Bohmian trajectory stability (BTS) method 73 determines the quantum orce along a trajectory on the ly, /jp CCC: $ American Chemical Society Published on Web 01/07/2004

2 Time Correlation Functions J. Phys. Chem. A, Vol. 108, No. 5, by solving a set o dierential equations or the coordinates o the quantum trajectory, the quantum orce, and the ininite-order stability properties. In the ininite hierarchy o BTS equations, the nonlocality o quantum mechanics maniests itsel as an ininite set o quantum potential derivatives at any given point in space that provide inormation about the global properties o this unction; thus, the BTS ormulation is ully equivalent to the exact quantum mechanical problem. Under certain conditions, the BTS hierarchy can be truncated, leading to a local scheme or determining approximate Bohmian trajectories that can capture some quantum mechanical eatures o a process. 73 The possibility o integrating individual quantum trajectories allows the use o Monte Carlo methods to evaluate Bohmian initial value representations. The present paper introduces a orward-backward quantum dynamical (FBQD) representation o time correlation unctions based on Bohmian trajectories. The derived expression assumes an initial value orm that contains a slowly Varying phase. In this sense, the obtained FBQD result enjoys the main advantage o a smooth integrand characteristic o FBSD methods and yet is a ully quantum mechanical result that correctly accounts or quantum intererence and tunneling. Another important eature o the FBQD correlation unction is that the underlying quantum orce is weak and well behaved, greatly acilitating the computation o Bohmian trajectories. Section II presents the theoretical ormulation o FBQD. Several analytic and numerical examples are given in section III, and section IV presents some concluding remarks. II. Theory For simplicity, the theory that ollows is presented or a onedimensional system. Extension o the ormalism to many dimensions is straightorward. For a general (possibly time-dependent) Hamiltonian, the Bohmian orm o the solution to the time-dependent Schrödinger equation is written in the orm Here, R(x;t) is a real-valued amplitude and the phase S(x;t) satisies the quantum Hamilton-Jacobi equation The latter diers rom the ordinary equation o classical mechanics through the presence o a quantum potential that is proportional to the curvature o the amplitude The initial condition or eq 2.3 is the phase S 0 (x 0 ) o the initial wave unction at the coordinate x 0, which reaches the position x at the time t upon integration according to the equations with an initial momentum ip Ψ(x;t) ) Ĥ(t)Ψ(x;t) (2.1) t Ψ(x;t) ) R(x;t)e is(x;t)/p (2.2) - S(x;t) ) 1 t 2m ( S(x;t) x ) 2 + V(x;t) + Q(x;t) (2.3) Q(x;t) ) -p2 2m R(x;t)-1 2 R(x;t) x 2 (2.4) x (t ) ) m -1 p(t ) p (t ) )-V (x(t )) - Q (x(t )) (2.5) p 0 ) S 0 (x 0 ) (2.6) The amplitude obeys the ollowing continuity equation with R(x;t) ) R 0 (x 0 ) x 1/2 R 0 (x) ) Ψ 0 (x) Consider the time correlation unction or a pure state where F 0 ) Ψ 0 Ψ 0. For simplicity, it is assumed throughout this paper that the operators  and Bˆ are unctions o position. Such correlation unctions are o particular importance to the vibrational spectroscopy o polyatomic molecules, where one is interested in the autocorrelation unction o the dipole moment operator µˆ ) µ(xˆ). The procedure that ollows can easily be extended to correlation unctions o more general operators and to mixed states (e.g., inite-temperature correlation unctions). Equation 2.8 can be written as In principle, each o the time-evolution steps in eq 2.9 can be evaluated using the Bohmian prescription with an appropriate initial condition. Speciically, the initial state Ψ 0 is propagated to time t, and a dierent initial state given by ÂΨ 0 is propagated backward in time. However, the quantum potentials governing these two dierent dynamics are not the same because the initial states are dierent. As a result, orward and backward trajectories must be launched with dierent initial conditions in order to reach the same coordinate x. Changing the integration variable to the initial value x 0 o the orward trajectory converts the correlation unction to the orm In this, quantum trajectories with initial position x 0 (and a momentum derived rom the derivative o the phase o Ψ 0 ) must be integrated orward in time, reaching the coordinate x at time t under the Bohmian potential mentioned above. However, one must solve a root-search problem to ind the appropriate initial condition o the backward trajectory that reaches the same inal position x. Solution o the boundary value problem is already impractical in the context o purely classical propagation and even more so in the present case where the dynamics is governed by the Bohmian quantum potential. To alleviate the root-search problem, the correlation unction is converted to a orward-backward representation. For this purpose, the initial wave unction is propagated sequentially by the irst three operators in the correlation unction. Thus, we deine the state Using this deinition, the correlation unction becomes which is readily converted to an initial value orm (2.7) C AB (t) ) Tr(Fˆ 0Âe iĥt/p Be -iĥt/p ) (2.8) C(t) ) dx Ψ 0  e iĥt/p x B(x) x -iĥt/p Ψ 0 (2.9) C AB (t) ) dx 0 x Ψ 0 A eiĥt/p x B(x) x e -iĥt/p Ψ 0 (2.10) Ψ 0rt,Bˆ,tr0 e iĥt/p Bˆ e -iĥt/p Ψ 0 (2.11) C AB (t) ) dx Ψ 0  x x Ψ 0rt, Bˆ, tr0 (2.12)

3 808 J. Phys. Chem. A, Vol. 108, No. 5, 2004 Makri C AB (t) ) dx 0 x Ψ 0 Â x Ψ 0rt,Bˆ,tr0 (2.13) Because all the dynamics is now included in the propagation o a single state, a single initial condition is involved, and thus evaluation o eq 2.13 does not require a root search. Equation 2.11 involves three propagation steps, which are perormed according to the Bohmian prescription. First, a trajectory launched rom the position x 0 with a momentum satisying eq 2.6 is propagated to the time t ollowing the Bohmian dynamics speciied by the Hamiltonian operator and the given initial state. At the end o this step, the wave unction is given by the expression Ψ tr0 (x t ) ) R (x t ;t)e is (x t :t)/p (2.14) where x t is the coordinate reached by the Bohmian trajectory, and R and S are the amplitude and action speciied by the dynamics along the orward trajectory. Subsequently, eq 2.14 is multiplied by B(x t ), and the resulting unction Ψ Bˆ,tr0 (x t ) ) B(x t )Ψ tr0 (x t ) (2.15) serves as the initial condition or a new Bohmian propagation in the negative time direction. The initial momentum o the backward trajectory is given by S / x t ) p t. At the end o this backward time evolution, i.e., when the time equals zero, the Bohmian trajectory has reached the coordinate x, and the corresponding wave unction is Ψ 0rt,Bˆ,tr0 (x) ) R b (x;0)e is b(x:0)/p where R b and S b are the amplitude and action speciied by the dynamics along the backward Bohmian trajectory. Use o the hydrodynamic continuity equation implies that R b (x;0) ) R b (x t ;t) 1/2 t x b (2.16) (2.17) where the subscript b indicates that the derivative is evaluated at points along the backward trajectory. Recalling that the initial amplitude o the backward trajectory is R b (x t ;t) ) B(x t )R (x t ;t) (2.18) and making use o the continuity relation along the orward trajectory R (x t ;t) ) R (x 0 ;0) x 1/2 0 x t ) Ψ 0 (x 0 )e -is 0(x 0 )/p x 1/2 0 x t leads to the ollowing expression or the correlation unction C AB (t) ) dx 0 x Ψ 0 Â x B(x t ) x 0 Ψ 0 x t 1/2 x t x b (2.19) where S(x) ) S (x t ;t) + S b (x;0) - S 0 (x 0 ) is the net action accumulated along the orward-backward trajectory. Finally, the product o stability elements can be simpliied, leading to the expression 1/2 e is(x)/p (2.20) C AB (t) ) dx 0 Ψ 0 Â x B(x t ) x 0 Ψ 0 x 1/2 e is(x)/p (2.21) Equation 2.21 is the main result o this paper. The correlation unction is obtained rom a orward-backward quantum trajectory procedure, which involves a trajectory propagated along orward and reverse time directions, interrupted by the action o the operator Bˆ. Its attractive eatures can be summarized as ollows: (i) The FBQD correlation unction represents an exact quantum mechanical result and thus leads to an exact description o tunneling, intererence, and zero-point energy eects. (ii) Equation 2.21 is an initial Value representation. As such, it does not require numerical generation and storage o the timedependent wave unction on a grid (as long as the Bohmian orce can be generated by a local or semilocal procedure) 73 nor does it involve a root search problem. (iii) The correlation unction is expressed as the result o a orward-backward time propagation. In this sense, it enjoys all the attractive eatures o orward-backward semiclassical dynamics (FBSD). Speciically, the phase arising rom the orward-backward action is a smooth unction o the integration Variables and thus amenable to Monte Carlo procedures. Unlike FBSD expressions, which cannot account or quantum intererence eects, the present FBQD ormulation incorporates quantum intererence results exactly. (iv) FBQD can also be compared to the ull semiclassical representation o a correlation unction expressed in terms o two separate semiclassical propagators or the orward and backward evolution parts, usually reerred to as double IVR. Such expressions are capable o accounting or nonclassical intererence eects through o-diagonal contributions rom distinct orward and backward classical trajectories, but the classical nature o the dynamics does not allow or the inclusion o deep tunneling. As is well-known, the phase arising rom the combined action integrals is a highly oscillatory unction that prevents the use o Monte Carlo methods or the evaluation o semiclassical correlation unctions. By contrast, the FBQD expression contains the net action along orward-backward quantum trajectories, which (as argued above) is a smooth unction. Thus, FBQD is not only superior to the ull semiclassical (double IVR) correlation unction in its ability to describe strictly quantum mechanical eects but also easier or Monte Carlo methods. (v) FBQD is a coordinate-space ormulation and thus does not require the use o coherent states 9 or Wigner transorms 69 to obtain a sampling unction or the initial conditions o the trajectories. Thus the diicult numerical issue o evaluating these phase space representations is avoided. An added bonus is the reduction in the dimension o the integrals compared to quasiclassical or ull semiclassical expressions. (vi) The FBQD ormulation closely resembles FBSD expressions and thus can be combined with a FBSD treatment in a single calculation. As a result, FBQD may be employed to upgrade a semiclassical orward-backward treatment o a large system by including important quantum eects (such as intererence and tunneling) originating rom a small number o light particles. (vii) Finally, the orm o eq 2.21 greatly acilitates computation o the quantum potential. The initial density in an equilibrium correlation unction commutes with the system s time evolution operator. In the speciic case o a pure state, the orward part o the time evolution involves propagation o an eigenstate o the Hamiltonian. It can be shown that Bohmian trajectories remain stationary under the action o the quantum orce arising rom eigenstates o the Hamiltonian, and thus quantum trajectories remain ixed at their initial positions during

4 Time Correlation Functions J. Phys. Chem. A, Vol. 108, No. 5, the orward part o the dynamics. Subsequently, the eigenstate is multiplied by the operator Bˆ. I the latter is the dipole operator responsible or vibrational transitions, this process produces a state that is to a irst approximation similar to an excited state o the same Hamiltonian. Thus, evolution o that state is mild, leading to a sot, slowly Varying quantum orce that is ree o sharp variations, and the dynamics does not suer rom the instability problems that plague the Bohmian method when applied to bound anharmonic potentials. The desirable eatures outlined above make the FBQD ormulation o time correlation unctions a very attractive one. It should be emphasized, however, that numerical evaluation o the quantum trajectories entering the FBQD expression requires the concurrent generation o the underlying Bohmian orce. Circumventing the various numerical diiculties surrounding the latter is not within the scopes o the present paper, but it is hoped that the dramatic smoothing and weakening o the Bohmian potential achieved through the propagation o FBQD trajectories corresponding to near eigenstates o the Hamiltonian will enable this task in a variety o systems o practical interest. The next section illustrates these promising eatures o the FBQD ormulation with analytic and numerical examples. III. Examples The attractive eatures o the FBQD representation o a time correlation unction are illustrated with two examples. The irst example involves the position autocorrelation unction  )Bˆ ) xˆ or a system described by a harmonic potential o requency ω and eigenstates Φ n at zero temperature. During the orward part o the dynamics, the ground-state wave unction evolves to time t by acquiring a phase Ψ tr0 (x t ) ) Φ 0 (x 0 )e -iωt/2 (3.1) During this stage the quantum potential is easily ound to be Q (x) )- 1 2 mω2 x pω (3.2) 2 Because the total (classical plus quantum) orce equals zero, the Bohmian particles remain stationary (x t ) x 0, p t ) p 0 ) 0) and the orward action is S )- 1 / 2 pωt. At the end o the orward propagation, the position operator acts on the wave unction, changing it to the irst excited state o the Hamiltonian Ψ Bˆ,tr0 (x) ) p 2mω Φ 1 (x) (3.3) The backward propagation is governed by the quantum orce generated by this wave unction, so again trajectories remain ixed, such that x ) x 0, while the backward action is S b ) 3 2 pωt Substitution o these relations in eq 2.21 gives C xx (t) Φ 0 xˆ e iĥt/p xˆe -iĥt/p Φ 0 ) p dx 2mω 0 Φ 1 (x)x 0 Φ 0 (x 0 )e iωt ) p 2mω eiωt (3.4) which is recognized as the exact result. In this idealized model, Figure 1. The position autocorrelation unction or the quartic oscillator described by eq 3.5. Filled circles and hollow squares show real and imaginary parts, respectively. (a) FBQD results using a basis set method to evaluate the quantum potential. (b) FBSD results. The solid and dashed lines display the real and imaginary parts o the exact results obtained by a basis set calculation. the total orce acting on the Bohmian particles remained equal to zero at all times, and the phase was independent o coordinates. The second example involves a one-dimensional oscillator with quartic anharmonicity Ĥ ) pˆ2 2m mω2 x x 4 (3.5) with m ) ω ) 1. The three lowest energy levels in this system have values equal to 0.559, 1.769, and 3.139pω. Comparison with the eigenvalues o the corresponding harmonic potential (equal to 0.5, 1.5, and 2.5pω) reveals the very signiicant role o the anharmonic term in this system. The quantum potential required to integrate the Bohmian trajectories was calculated accurately via a basis-set method. The FBQT correlation unction shown in Figure 1 is seen to be in excellent agreement with that obtained via a basis-set method. Despite the regularity o the oscillations observed in this igure, it should be emphasized again that the potential in eq 3.5 is a strongly anharmonic one, and the requency o the 0 1 and 1 2 transitions exceed ω by 21 and 37%, respectively. The eects

5 810 J. Phys. Chem. A, Vol. 108, No. 5, 2004 Makri Figure 2. Total (classical plus quantum) potentials or the quartic oscillator described by eq 3.5. (a) and (b) Forward evolution o a Gaussian wavepacket displaced by 0.7 units rom the potential minimum. (c) and (d) FBQD evolution or the zero-temperature position autocorrelation unction. (The legend measures elapsed time rom the beginning o the backward propagation.) Notice the smaller range o FBQD potential axis. o nonlinearity are strongly elt in the evolution o most relevant trajectories. To urther substantiate this claim, Figure 1 also displays the results o the derivative FBSD ormulation, 36 where the correlation unction takes the orm C FBSD AB (t) ) dx 0 dp 0 A (x 0, p 0 )B(x t, p t ) (3.6) where A is a complex-valued unction related to the initial wave unction and the operator Â. Equation 3.6 is similar in appearance to the FBQD expression, as it is also an initial value orm lacking an oscillatory phase. The observed dephasing o the FBSD result is a consequence o strong quantum intererence or this system. As discussed in recent work, the inluence o the strong quartic term in this system leads to a classical dynamics that is dominated by caustic (ocal) points. This eature leads to sharp variations o the quantum potential corresponding to wave packets undergoing orward time evolution. The resulting extremely strong orces render propagation unstable. 68 The quantum potential (calculated via basis set methods) corresponding to propagation o a Gaussian wave packet in this anharmonic system displaced by 0.7 units rom the potential minimum is shown in parts a and b o Figure 2. Shown in parts c and d o Figure 2 is the quantum potential arising in the FBQD calculation o the position autocorrelation unction or the same system. (The position operator is again chosen as a rough approximation o the dipole-moment unction or a diatomic molecule.) Even though this operator does not promote the wave unction o an anharmonic system exclusively to the irst excited eigenstate, the wave unction generated this way bears close resemblance to the eigenunction, such that the resulting quantum potential is smooth and well behaved. As a result, the Bohmian trajectories employed in the calculation o Figure 1 were integrated with a time step comparable to that or integration o purely classical trajectories, whereas orward wave packet propagation or the same system became possible only by decreasing the time step by a actor o 200. It is emphasized, however, that the quantum potential employed in the calculation shown in Figure 1 was obtained by using a nonlocal basis set expansion. The dramatic smoothing o the quantum potential

6 Time Correlation Functions J. Phys. Chem. A, Vol. 108, No. 5, Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant No. NSF CHE Figure 3. The integrand o the FBQD expression or the quartic oscillator model at ωt ) 0.5. Solid and dashed lines show the real and imaginary parts o the integrand, respectively. achieved in the FBQD ormulation is encouraging and suggests that its determination by local schemes may become possible in the near uture. Finally, Figure 3 shows the real and imaginary parts o the integrand o eq 2.21 at a speciic time. The real and imaginary parts are seen to be smooth unctions ree o oscillatory components. This eature is a result o the slow variation o the net orward-backward action in the FBQD expression. The absence o a rapid oscillatory phase rom the integrand implies that Monte Carlo methods, which provide the only viable approach or systems o several degrees o reedom, are suitable or the evaluation o FBQD correlation unctions. IV. Concluding Remarks In summary, the FBQD ormulation introduced in this paper oers an attractive, ully quantum mechanical representation o time correlation unctions. Because the FBQD integrand is a smooth unction, the present ormulation enjoys the advantages o orward-backward semiclassical approximations without the neglect o quantum mechanical eatures such as phase intererence and tunneling. Unlike FBSD, which requires the calculation o an appropriate phase space transorm o the density operator, FBQD is implemented directly in position space with a density directly obtainable rom the raw wave unction (or density matrix) describing the initial condition. Last, a very important eature o FBQD correlation unctions is the smoothness and small magnitude o the quantum potential governing the dynamics. Bohmian calculations on bound anharmonic systems have encountered severe diiculties in the past because o the strong quantum orce and rugged shape o the quantum potential. The absence o these undesirable eatures rom the present ormulation suggests that the numerical issues surrounding the evaluation o the quantum orce may become suiciently mild to make Bohmian calculations with a sel-consistent determination o the quantum orce practical. Finally, the attractive eatures o FBQD also apply to correlation unctions o more general operators and to systems at thermal equilibrium. 74 Given the advantageous eatures o the FBQD ormulation, the problem o accurate determination o the quantum orce is the single most serious obstacle that currently prevents application o this methodology to general multidimensional systems. 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