Wave Packet Dynamics onMultiply-Valued Potential Surfaces:Report on Work in Progress

BENJAMIN HALL, ERIK DEUMENS, YNGVE ÖHRN, JOHN R. SABINQuantum Theory Project, University of Florida, Gainesville FL 32611

Received 5 May 2011; accepted 20 May 2011Published online 13 January 2011 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.23190

ABSTRACT: A perennial problem in quantum scattering calculations is reducing the3N degrees of freedom (three for each atom or electron present) to a more computationallymanageable number, while still retaining the information present in each degree offreedom. We present a method of extracting a nonadiabatic, multiply-valuedpotential-energy surface from electron nuclear dynamics (Deumens et al., Rev Mod Phys,1994, 66, 917) trajectories; we then perform nuclear wave packet dynamics on that surfaceand calculate differential cross sections for two-center, one-active-electron systems. © 2011Wiley Periodicals, Inc. Int J Quantum Chem 112: 247–252, 2012

Key words: wave packet; sattering; electron nuclear dynamics

1. Introduction

Q uantum scattering calculations involvingheavy projectiles have only rarely been

completed using the fully coupled Schrödingerequation, in either its time-dependent (TDSE) ortime-independent (TISE) form. Instead, almost allapproaches approximate some part of the problem.Common approximations include those based onthe Born–Oppenheimer (BO) separation of slowlyvarying (nuclear) and quickly varying (electronic)degrees of freedom (see Refs. [1–5] for details)

Correspondence to: B. Hall; e-mail: hall@qtp.ufl.edu

and, more recently, the approach of Deumens andcoworkers to calculate the fully coupled dynamicsof electrons with “classical” nuclei [6]. The BO-based methods will be referred collectively as pre-computed potential energy surface (PPES) methods,whereas the latter goes by the name of electronnuclear dynamics (END) and is the starting pointfor the present work.

At the core, all of the approximation schemes andmodels reduce to the need to reduce the dimension-ality of the problem. Fully coupled TDSE requires3N + 1 degrees (three for each nucleus or electronplus one time coordinate). Transforming to center-of-momentum coordinates allows one to neglectthe three degrees of freedom relating to the cen-ter of mass; incorporation of symmetry can further

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reduce the dimensionality of the nuclear degrees offreedom. For the simplest collision systems, thoseconsisting of two nuclei and one electron (e.g., H+ +H), this reduction still leaves six degrees of free-dom: two nuclear, three electronic, and one for time.More complex systems have much higher dimen-sionality. Reduction of dimensionality is requiredas the computational complexity of the Schrödingerequation scales exponentially with the dimension-ality. This reduction of (electronic) dimensionalityis accomplished in PPES methods by expansionin adiabatic or diabatic states, leading to coupledequations where the only gridded coordinate is thereaction coordinate. For END, the electronic dimen-sionality is handled via the expansion in coherentstates but at the cost of requiring classical nuclearmotion.

In this report, we describe a method being devel-oped to partially correct a major deficiency inEND calculations, namely the restriction to classicalnuclei. This method and its computer implementa-tion (working title ENDwave) involve using a bundleof END trajectories to sample both the electronicdynamics and the effective potential energy sur-face experienced by the nuclear wave function ateach point along the (classical) END trajectories.This effective potential is then made suitable fornumerical calculations and is used to propagatea wave packet corresponding to the “true” pro-jectile nuclear wave function. In turn, this yieldsthe nuclear scattering amplitude, which combineswith the angular electronic transition amplitudes(from the END trajectories) to form the doublydifferential electron-transfer cross-section. The cur-rent implementation is limited to two-center, one-(active)-electron problems—the theory should bestraightforwardly extensible to larger systems. Asthis is a work in progress, we report the benchmark-ing results on a limited set of analytic (but realistic)potential surfaces.

2. Methodology

Addition of wave packet dynamics to END-derived potential surfaces and electronic dynamicsrequires three major steps: First, a “suitable” bun-dle of END trajectories must be calculated. Themaximum impact parameter and initial separation(relative to the target atom) must be large enoughto cover the entire interaction region, and the spac-ing of impact parameters within this region must befine enough to sample the entire classically accessible

region.Abrief discussion of END theory is presentedin subsection 2.1 as this has been presented in manyother articles. See Refs [6–9] for more details.

Second, from these sampled points along the ENDtrajectories R(t, b), we extract the effective poten-tial experienced by the nuclei and form it into apotential surface suitable for wave packet evolution.As described in subsection 2.2, these END effectivepotential surfaces are multiply-valued (as a conse-quence of removing the explicit time dependence inthe trajectories).

Finally, wave packet evolution is performed onthe surface(s) constructed in the second step andthe nuclear scattering amplitude is calculated, asdescribed in subsection 2.3.

All equations are presented in Hartree atomicunits (e = � = me = c = 1).

2.1. ELECTRON NUCLEAR DYNAMICS

END is a systematic, theoretically elegant methodfor constructing fully coupled equations of motionfrom the TDSE for both electrons and classical nuclei[6]. The fundamental idea behind END is the expres-sion of the total wave function as

�(t) = |z(t, R), R(t), P(t)〉|R(t), P(t)〉, (1)

where the parameters z(t, R) are the Thouless param-eters [10] corresponding to the electronic state attime t and R(t) and P(t) are the position and momen-tum vectors for the nuclei. Substitution of this wavefunction into the quantum-mechanical Lagrangian[Eq. (2)] and application of the time-dependent vari-ational principle yields the equations of motion forthe trajectory and electronic state (parameterized byzph).

L =⟨�

∣∣∣∣ı� ∂

∂t− H

∣∣∣∣ �⟩〈�|�〉−1 (2)

The electronic states are not expanded in the adi-abatic (nor in diabatic) molecular orbitals; instead,they are written as coherent linear combinations ofatomic orbitals:

|ψh〉 = |φh〉 +∑

p

zph|φp〉, (3)

where the states |φ〉 are the atomic orbitals and zph

is the (complex) coefficient for dynamic state h in(unnoccupied) orbital p. This expansion removes theneed to precalculate the adiabatic molecular orbitalsand select a basis of channels. The electronic state

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is able to distort and reshape itself as demanded bythe instantaneous coupling between electrons andnuclei. In the current implementation of END, thevariational principle is restricted to the space of sin-gle Slater determinants; this restriction will be liftedin future implementations. The restriction to classi-cal nuclei, however, cannot be lifted in situ withoutfurther severe approximations that would unaccept-ably reduce the accuracy and applicability of thetheory.

We use the implementation of END (ENDyne) tocalculate a bundle of independent trajectories withdifferent initial conditions R(t = 0) = R0 ∈ �0 ={b, Z0}. The direction of the initial velocity is definedto be the +z. The spacing and extent of b depend onthe system but should in all cases cover the inter-action region with sufficient density to sample theentire available space.

2.2. MULTIPLY-VALUED POTENTIALENERGY SURFACES

The electronic energy of the system is not used inthe END evolution; it is computed as an output alongeach trajectory. This is not the adiabatic potential(i.e., is not an eigenvalue of the static hamiltonian)—it is the total energy (excluding the nuclear kineticenergy) of the molecular system at each value of thesystem parameters z(t), R(t), and P(t).

E(R(t), z(R, P)) = Enn(R(t)) + Ene(R(t), z(t, R))

+ Eee(z(t, R)) + Tel(z(t, R)), (4)

where Enn, Ene, Eee, and Tel are the nuclear–nuclearrepulsion, electron–nuclear attraction, electron–electron repulsion, and electronic kinetic energies,respectively. Derivation of expressions for theseenergies in terms of the wave function parametersis beyond the scope of this work—see Ref. [6] for afull derivation.

Note that all terms in the energy depend (eitherexplicitly or through the Thouless parameters) onthe classical positions along the trajectory R(t) and,in theory, the classical momenta P. For quantumwave packet evolution, we need an expression forthe energy as a function of the nuclear coordinatesX. Transformation from R(t) to X requires care aseach point in space-fixed space X may correspond tomany R(t′, b), either at different times along the sametrajectory (only for head-on collisions) or where tra-jectories intersect (i.e., R(t1, b1) = R(t2, b2)). As theelectronic state depends on the history of the trajec-tory, the energies corresponding to these different

FIGURE 1. Illustration of initial conditions for oneimpact parameter, showing the extent of the bundle usedto construct the potential surface. Selected particles areshown as solid circles. Note that the last projectile(labeled N ) enters the interaction region (dashed circle)after the first particle (labeled 1) leaves the region.

trajectories at the same value of R will, in general,differ.

This multiplicity in energy leads to the idea thatthe potential surfaces extracted from END trajec-tories are “folded” or multiple valued. It must bestressed here that these multiple values are not theresult of multiple simultaneous eigensolutions toa hamiltonian. No coupling elements can be writ-ten between “surfaces.” In space-fixed coordinates,they form a set of surfaces that are connected bycontinuity and smoothness and join along a singleseam.

The time dependance of the trajectories meansthat at any time t, only a single curve is being sam-pled. Wave packet calculations require a potentialsurface at all X at all times. We remove the timedependance by constructing (at least in principle) abundle of trajectories extending both perpendicularto P0 (differing in impact parameter) as well as in ini-tial distance Z0. The extent in Z is large enough thatthe final trajectory (at fixed b) enters the interactionregion after the first trajectory (at that same b) leavesthe interaction region (see Fig. 1). As these trajecto-ries are assumed to be asymptotic, this extent in Z canbe mapped to a time delay: ti = (Zinteraction − Z0i)v−1.Assume the first trajectory (which started on the edgeof the interaction region) spent time T before exit-ing the interaction region. As all the trajectories areindependant, all trajectories with the same impactparameter follow the same path, only with a delayin time. Thus, the potential experienced by the firsttrajectory at t = t1 will be experienced (at that sameposition) by the ith trajectory at t = ti + t1. As a

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result, at time t = T, all accessible points will be beingsampled. This implies that we can simply drop thetime dependance and take the potential values expe-rienced by the single line of impact parameters at alltimes as the values at the corresponding coordinatesX. Numerically, trajectories lying “close” in energyat a given position are binned together and assignedthe average value to form discrete surfaces.

As an illustration of the differences in approachbetween the present method and the standard BOapproach, we detail the steps for a model calcula-tion using both methods. For both the methods, weassume that the potentials can be parameterized asYukawa potentials [11]:

V(R) = ge−µR

R+ Vasympt (5)

where g and mu are positive parameters definingthe height and range of the potential and Vasympt isthe value of the potential at large R and should beequal to the purely internal (electronic plus rotovi-brational) energy of each collision partner. In thecase of the BO approximation, we generate a set ofcoupled surfaces, each corresponding to an eigen-value of the electronic Hamiltonian, coupled at everypoint by the off-diagonal Hamiltonian matrix ele-ments. These coupling elements are largest whenthe surfaces are close in energy (the “avoided cross-ings”) and are generally Lorentzian in shape. For theEND-potential approach, electronic transitions giverise to multiple independent surfaces, coupled onlyby continuity along a single seam. This is shownin Figures 2(a) and 2(b). Note that the upper sur-face in Figure 2(b) stops when it intersects with thelower surface at R = R0 � 0.651. This point isthe “fold” between surfaces; for R < R0 the uppersurface is not accessible. All transitions between sur-faces happen at R = R0—the surfaces are completelyuncoupled for all other geometries. This contrastswith the BO model where transfer can occur at anypoint, although it is more probable near R = R0.

An alternate perspective on the folded potentialsurfaces can be gained by examining the poten-tial from the point-of-view of the projectile (thisis the standard “reaction coordinate” description).Figure 2(c) shows the same potentials used in Figures2(a) and 2(b) parameterized by the reaction coordi-nate, in effect unfolding and stretching the potentialsurface. In this view, transfer to the upper surfacecan be described as a tunneling process controlledby the collision energy relative to the asymptoticsurface separation (labeled V ′) and by the relative

slopes of the potentials. In the region of Figure 2(c)labeled “1” (ie E < V ′), no transmission is possi-ble. The wave packet density has nowhere to go. Inregion 2 (E > V ′), both transmission and reflectionoccur, with transmission saturating to a value thatdepends on the (local) shapes of the two potentialsas the energy increase. As we see, the description asfolded surfaces is equivalent to a transformation to asystem-dependent reaction coordinate, without thecomplexities inherent in such a transformation andwhile still preserving the time-dependent characterof the reaction. This procedure generalizes smoothlyto higher dimensionalities as will be discussed andexplored in ongoing work.

2.3. WAVE PACKET DYNAMICS

We use the second-order split-operator [12]method for time evolution, with a Fourier Trans-form representation of the kinetic energy operator.In future implementations, more advanced iterators(such as the Lanczos [13] method) can be included forincreased accuracy. Absorbing imaginary potentials[14] are placed at the edge of the main compu-tational region to absorb spurious reflections andperiodicities. Wave function density in the asymp-totic region is transfered to a separate asymptoticgrid and allowed to evolve freely until the total inte-grated population of the primary grid is less than athreshold (usually 1%).

As the individual sheets of the folded surface arenot coupled except at the juncture, the wave packetcan only move between surfaces through the seams.Population transfer between surfaces is enforcedby ensuring continuity and first-order smoothnessacross the join. Currently, this is done using a two-point centered difference approximation and seemsto suffice—transfer between surfaces is smooth anddoes not create numerical instabilities.

3. Application to One-DimensionalAnalytic Surfaces

At the current time, the algorithm and methodsdescribed above have been implemented and arebeing tested against both model systems and simplereal systems. Some of the benchmarking and model-potential test results in one spatial dimensions arepresented here.

The first test presented is a simple single flatsurface. The energy distribution of the “scattered”wave packet should be identical to that of the initial

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FIGURE 2. (a) Two-surface model Born–Oppenheimer potentials, with coupling (short dashes) overlaid. (b) Singlyfolded model potential for ENDwave propagation. (c) Reaction coordinate parameterization of folded potential; in region1 no transmission can occur, in region 2 both transmission and reflection can occur. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

wave packet. Down-shifting in energy would sig-nal an improperly constructed absorbing potentialand/or a loss of conservation of energy. Figure 3shows the momentum distribution of an initiallygaussian packet before and after evolution, show-ing that both the evolution and absorbing potentialare constructed properly.

A second test, sligtly more complex, shows theeffect of having multiple surfaces. Figure 4 showsthe probability of the incident particle being foundat the given energy with the indicated internal con-figuration (surface). This particular example wascalculated using two Yukawa [11] type potentials,offset by 0.1 hartree. As the figure shows, the energydistributions are very similar, except that the prob-abilities of being found on the higher surface areoffset by the same 0.1 hartree as the potentials. Thiseffect is exactly what one would expect: as energy

FIGURE 3. Momentum distribution of initial (solid) andevolved (dashed) wave packet on flat surface. [Colorfigure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

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FIGURE 4. Energy distribution of probability for elastic(solid) and rearrangement (dashed) reactions on twoYukawa potentials asymptotically separated by 0.1hartree. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

must be conserved, the greater asymptotic poten-tial on the upper surface implies that the (average)kinetic energy should be lower. The cause of thesmall (�10−10) low energy peak on the rearrange-ment curve is unknown at this time and may bean artifact of the absorbing potential or aliasing inthe transfer to the asymptotic grid. Note that, exceptfor the different asymptotic separation, these are thesame potentials shown in Figure 2(b).

4. Direction of Future Efforts

In the near future, we plan to apply this ENDwavemethod to real systems, starting with simple systemssuch as H+ + H and getting more complex fromthere. More, however, can be done to extend boththe accuracy and the range of applicability of themethod. An increase of accuracy (and reduction oftotal computational effort) could be had by replacingthe split-operator integrator with a more sophis-ticated integrator such as the Lanczos method orone based on Chebyshev polynomials. One pressingmatter of accuracy is replacing the simplistic nearest-neighbor interpolation of the END surfaces with amore robust and accurate interpolation method.

More interesting is the extension to three or morecenters as well as multiple active electrons. Theprocess of extracting the potential surfaces for multi-electron systems is more numerically involved (andwill involve multiple branchings of the surface) butposes no significant theoretical problem. All of the

electronic degrees of freedom are condensed intothe multiply-branched potential surface, and ENDhandles multiple electrons perfectly well.Additionalnuclei, on the other hand, requires either a trans-formation to some type of internal coordinates orthe extension of ENDwave to 3+ dimensions. Wealso expect the number and complexity of the foldedsurface to increase dramatically.

5. Conclusions

There is an unoccupied niche in the scattering lit-erature for non-BO quantum-nuclear calculations.The method presented above (ENDwave) holds sig-nificant promise in filling that gap. We hope toprovide a new perspective on long-standing prob-lems in low-energy atomic, ionic, and molecularscattering theory.

ACKNOWLEDGMENTS

The authors acknowledge the University ofFlorida, the Quantum Theory Project, and the HighPerformance Computing Center at the Universityof Florida for making available resources (computerand otherwise) that were essential in completing thepresent work.

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