A fast method for solving both the time-dependent Schrödinger equationin angular coordinates and its associated “m-mixing” problem

Matthew G. Reuter,a� Mark A. Ratner, and Tamar SeidemanDepartment of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, USA

�Received 26 June 2009; accepted 8 August 2009; published online 3 September 2009�

An efficient split-operator technique for solving the time-dependent Schrödinger equation in anangular coordinate system is presented, where a fast spherical harmonics transform accelerates theconversions between angle and angular momentum representations. Unlike previous techniques, thismethod features facile inclusion of azimuthal asymmetries �solving the “m-mixing” problem�,adaptive time stepping, and favorable scaling, while simultaneously avoiding the need for bothkinetic and potential energy matrix elements. Several examples are presented. © 2009 AmericanInstitute of Physics. �doi:10.1063/1.3213436�

I. INTRODUCTION

The direct, numerical integration of the time-dependentSchrödinger equation �TDSE�,

i�d

dt���t�� = H�t����t�� = �T + V�t�����t�� ,

�1����t0�� = ��0� ,

has become an essential component of molecular quantumdynamics. In such studies, the computational effort is spentpropagating ���t0�� to ���t0+�t��,

���t0 + �t�� = U�t0,�t����t0�� , �2�

where the propagator U is generally not known or is compu-tationally expensive due to the noncommutivity of the ki-

netic energy T and the potential energy V�t�. One thus resorts

to approximations of U �Refs. 1 and 2� which balance severalproperties including unitarity, computational efficiency, theuse of kinetic and potential energy matrix elements, and,although not critical, adaptive time stepping.

The second-order, split-operator propagator,

U�t0,�t� � e−iT�t/�2��e−iV�t0+�t/2��t/�e−iT�t/�2��, �3�

achieves unitarity, the obviation of matrix elements, andadaptive time stepping, provided �t is sufficiently small.3

Each factor is evaluated in its local representation—

momentum space for T and position space for

V�t0+�t /2�—necessitating a change in representation be-tween each factor. As such, these transformations become thecomputational bottlenecks for this propagator; they generallyscale quadratically with the number of basis states, N. Fur-thermore, since these transformations depend on the choice

of coordinate system �often related to T � and not on thespecific physical system, the optimization of a particulartransform �a coordinate-dependent fast transform� would

make this propagator efficient for a broad class of problems.General methods for algebraically deriving fast transformshave been proposed.4

For instance, when Cartesian coordinates are used, themomentum eigenstates are the Fourier modes, and the fastFourier transform �FFT� accelerates the change ofrepresentation.5–7 In this case, the split-operator techniquealso becomes computationally efficient, scaling asO�N log N�.

Unfortunately, not all systems of interest are naturallydescribed by Cartesian coordinates. For example, investiga-tions of atomic systems, molecular bending, and molecularalignment all include at least one angular variable. In thesesystems, the normalized spherical harmonics,

Y�m��,�� =� 1

2�P�

m�cos����eim�, �4�

rather than the Fourier modes, are the angular momentumeigenstates, and the computational efficiency from the FFT islost.8 Here, 0���� is the polar Euler angle, 0���2� is

the azimuthal angle, and P�m�x� is the normalized associated

Legendre function of degree � and order m.Angular variables also introduce two new computational

problems. First, the kinetic energy operator includes csc���terms that trouble numerical schemes as � tends to 0 or �.Second is the so-called “m-mixing” problem, which occurs

when V�t� couples states of differing m values �as is often the

case when V�t� lacks azimuthal symmetry�. Originating fromthe �m�-dependence of optimal Gauss–Jacobi quadraturerules, the m-mixing problem is numerical: the nonconserva-tion of m introduces suboptimal accuracy in numericalintegration and/or necessitates interpolation between�m�-dependent quadrature grids. Recent work has overcomethe m-mixing problem for potentials expressible as finitesums of m-conserving potentials;9 however, a general solu-tion has proven elusive.

With these two new considerations in mind, severalpropagators have been suggested for solving the TDSE withangular variables.8–11 Most of these techniques successfullya�Electronic mail: mgreuter@u.northwestern.edu.

THE JOURNAL OF CHEMICAL PHYSICS 131, 094108 �2009�

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handle the csc��� terms and are unitary, but compromise be-tween the use of matrix elements, computational efficiency,and adaptive time stepping. As implied, none solve them-mixing problem in full generality.

The development of a fast spherical harmonicstransform12 �FSHT� in recent years holds great promise forsolving the TDSE in an angular coordinate system. We firstassume our wavefunction is band limited, that is, we canchoose a positive integer �max, such that

���,�,t� = �=0

�max−1

m=−�

�

��m�t�Y�

m��,��

= m=−�max+1

�max−1

eim� �=�m�

�max−1��

m�t��2�

P�m�cos���� , �5�

where

��m�t� = Y�

m���t�� = �0

2�

d��0

�

d� sin������,�,t�

�Y�m��,���*.

Given a set of ��m�t� coefficients, the FSHT calculates

��� ,� , t� at specific ��i ,� j grid points in O��max2 log �max�

time. The inverse operation �grid points to coefficients�scales similarly. For comparison to the Cartesian equivalent,we note that there are N=�max

2 basis states here, so the scal-ing is O�N log N�.

Returning to the split-operator propagator, the FSHT isthe angular analog of the Cartesian FFT as it accelerates thelikewise transformation between angle and angular momen-tum representations �Eq. �5��. The method presented in thiswork employs the FSHT for such transformations, yielding apropagator �Eq. �3�� that �i� is unitary, �ii� handles the csc���terms indirectly through the spherical harmonics, �iii� scaleswell, �iv� obviates the need for both kinetic and potentialenergy matrix elements, �v� exhibits adaptive time stepping,and �vi� solves the m-mixing problem. The layout of thispaper is as follows. Section II summarizes the FSHT algo-rithm from Ref. 12. Section III presents several examples ofthis propagator, highlighting its functionality under mixed m.Finally, Sec. IV draws several conclusions.

II. THE FSHT

Reference 12 presents a FSHT algorithm that efficientlycomputes the ��

m�t� coefficients in Eq. �5� from the values of��� ,� , t� at specific ��i ,� j grid points and vice versa. Herewe present an overview of this algorithm, referring the readerto Ref. 12 and the references within it for additional details.Some of these references are duplicated here.

For convenience, we require �max to be an even, positiveinteger. The grid uses �i from the �2�max�th order Gauss–Legendre quadrature rule,

P�max�cos��i�� = 0, �6�

i=1,2 , . . . ,�max, where P��x� is the Legendre polynomial ofdegree � �m=0�. The � j are evenly distributed,

� j = 2��j − 1�/�2�max − 1� , �7�

j=1,2 , . . . , �2�max−1�.Much of the efficiency of the FSHT is obtained through

the one-dimensional fast multipole method13 �FMM� and anefficient algorithm for manipulating symmetric, tridiagonalmatrices.14 Briefly, the FMM is an approximate, althougharbitrarily accurate, method to compute matrix-vector prod-ucts of the form

v j = k

ukK�xk − yj� �8�

in linear time, as opposed to quadratic time needed for gen-eral matrix-vector products. Two kernels are needed for theFSHT: K�x−y�= �x−y�−1 and K�x−y�= �x−y�−2. The tridi-agonal matrix algorithm utilizes the FMM to apply the ei-genvector matrix �or its transpose� of a symmetric, tridiago-nal M M matrix to an arbitrary vector in O�M log M� time.Finally, the FSHT requires an extensive amount of precom-putations, which need only be performed once for each �max.Before discussing the FSHT algorithm, we outline the pre-computations and introduce our notation.15

A. FSHT precomputations

As will be seen in the discussion of the FSHT algorithm,the FSHT works with linear combinations of exclusively oddor exclusively even associated Legendre functions withoutloss in generality. As such, we will use x and y as the inde-pendent variables for the even and odd cases, respectively�0�x ,y�1�. Note that �=arccos�x� or �=arccos�y� for usein angular coordinate systems.

For each �m�=0,1 , . . . , ��max−1� we need the followingdata.

�1� The Gauss–Jacobi quadrature abscissas and weights

from P�m�+2nx�m�

�m� �x� and P�m�+2ny�m�+1

�m� �y�, where

nx�m� = ceil���max − �m��/2�

is the number of roots of P�m�+2nx�m�

�m� �x� in �0, 1� and

ny�m� = floor���max − �m��/2�

is similarly the number of roots of P�m�+2ny�m�+1

�m� �y� in �0,

1�. We will denote the even quadrature rule as�xi

�m� ,i�m� �i=1,2 , . . . ,nx

�m�� and the odd quadrature rule

as �yj�m� ,� j

�m� �j=1,2 , . . . ,ny�m��,

0 = P�m�+2nx�m�

�m� �xi�m�� , �9�

0 = P�m�+2ny�m�+1

�m� �yj�m�� , �10�

and

i�m� =

�2�m� + 4nx�m� + 1��1 − �xi

�m��2��m�−1

��P�m�+2nx�m�

�m� ���xi�m���2

, �11�

094108-2 Reuter, Ratner, and Seideman J. Chem. Phys. 131, 094108 �2009�

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� j�m� =

�2�m� + 4ny�m� + 3��1 − �yj

�m��2��m�−1

��P�m�+2ny�m�+1

�m� ���yj�m���2

. �12�

Equations �11� and �12� use f� to denote the derivativeof f . Furthermore, due to its relation to the FSHT � gridpoints, the even quadrature rule for m=0 will be de-noted �zk ,wk �k=1,2 , . . . , ��max /2��. An efficient algo-rithm for the computation of these quadrature rules ispresented in Ref. 16.

�2� Interpolation data for converting between quadrature

grids: �P�m�+2nx�m�

�m� �zk� , �P�m�+2nx�m�−2

�m� �xi�m�� ,

�P�m�+2ny�m�+1

�m� �zk� , and �P�m�+2ny�m�−1

�m� �yj�m�� .

�3� Precomputations for the tridiagonal matrix algorithm.The even tridiagonal matrix S�m� has diagonal �d�m�+2i

�m� �i=0,1 , . . . , �nx

�m�−1��, where

d�m =

2��� + 1� − 2m2 − 1

�2� − 1��2� + 3�, �13�

sub- �super-� diagonal �c�m�+2i�m� �i=0,1 , . . . , �nx

�m�−2��,

c�m =��� − m + 1��� − m + 2��� + m + 1��� + m + 2�

�2� + 1��2� + 3�2�2� + 5�,

�14�

and eigenvector matrix U�m�, such that S�m�

=U�m���m��U�m��T. The corresponding odd tridiagonalmatrix T�m� has diagonal �d�m�+2i+1

�m� �i=0,1 , . . . , �ny�m�

−1��, sub- �super-� diagonal �c�m�+2i+1�m� �i

=0,1 , . . . , �ny�m�−2��, and eigenvector matrix V�m�, such

that T�m�=V�m���m��V�m��T. We note that ��m� and ��m� arediagonal matrices with elements related to �xi

�m� and�yj

�m� , respectively.�4� Normalization matrices,

�A�m�� j,k = � j,k� i=0

nx�m�−1

�P�m�+2i�m� �xj

�m���2 �15�

and

�B�m�� j,k = � j,k� i=0

ny�m�−1

�P�m�+2i+1�m� �yj

�m���2. �16�

B. The FSHT algorithm

Figure 1 displays a schematic of the FSHT algorithm forconverting angular grid data ��� ,�� into the coefficients ofthe spherical harmonics expansion, ��

m. The inverse opera-tion can be performed by backtracing the flowchart in Fig. 1.

The FSHT algorithm starts with the angular grid data,��zi ,� j�, where −1�zi=cos��i��1 and the ��i and �� j aregiven by Eqs. �6� and �7�. The first step is to separate theeven and odd components �with respect to z� of �, denotedby and �, respectively. Accordingly,

�zi,� j� =��zi,� j� + ��− zi,� j�

2�17�

and

��zi,� j� =��zi,� j� − ��− zi,� j�

2, �18�

where 0�zi�1 for and �.The second step utilizes the FFT over � to determine

populations in each m-level. A total of �max FFTs are per-formed, one for each zi value for both and �.This produces m�zi� and �m�zi�, where m=−��max

−1� , . . . ,0 , . . . , ��max−1�.The third step interpolates the functions from the �zi

points to the �xi�m� or �yi

�m� points, depending on parity. Re-sults from the Christoffel–Darboux identities are used here,producing

m�xj�m�� = c�m�+2nx

�m�−2�m�

P�m�+2nx�m�−2

�m� �xj�m��

i=1

�max/2 2wiP�m�+2nx�m�

�m� �zi� m�zi�

zi2 − �xj

�m��2, �19�

�m�yj�m�� = c�m�+2nx

�m�−1�m�

P�m�+2nx�m�−1

�m� �yj�m��

i=1

�max/2 2wiP�m�+2nx�m�+1

�m� �zi��m�zi�

zi2 − �yj

�m��2. �20�

While this interpolation generally scales quadratically, theFMM can be used to make it scale linearly. The reverseinterpolation �needed for the inverse FSHT operation� can beperformed using

�z, ��

Χm�z�

Χm�x�

Χ�m

Eve

nPa

rity

��z, ��

�m�z�

�m�y�

��m

Odd

Parity

��m

Mixed Parity

��z, ��

Mixed Parity

1

2 2

3 3

4 4

5

FIG. 1. Schematic outline of the FSHT algorithm. The procedure is reversedfor the inverse operation. �1� Split the grid points into even and odd com-ponents with respect to z=cos���. �2� Use the FFT over the azimuthal angle� to obtain m populations. �3� Interpolate from the even, m=0 quadraturegrid, �zk , to the appropriate quadrature grid. �4� Perform an associated Leg-endre transform over the polar angle �. �5� Merge the separated even- andodd-parity data.

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m�zi� = c�m�+2nx�m�−2

�m�P�m�+2nx

�m��m� �zi�

j=1

nx�m�

j�m�P�m�+2nx

�m�−2�m� �xj

�m�� m�xj�m��

zi2 − �xj

�m��2, �21�

�m�zi� = c�m�+2nx�m�−1

�m�P�m�+2nx

�m�+1�m� �zi�

j=1

ny�m�

� j�m�P�m�+2nx

�m�−1�m� �yj

�m���m�yj�m��

zi2 − �yj

�m��2, �22�

where

j�m� = 2 j

�m��1 − �xj�m��2�−�m�, �23�

� j�m� = 2� j

�m��1 − �yj�m��2�−�m�. �24�

The fourth step transforms the remaining x or y grid to �using an associated Legendre transform. If we define thevectors �� m and �� m to have elements �i

m= m�xi�m�� and � j

m

=�m�yj�m��, the associated Legendre transform can be calcu-

lated by

�� m = U�m��A�m��−1�� m, �25�

��m = V�m��B�m��−1�� m. �26�

Here A�m� and B�m� are diagonal, and U�m� and V�m� are theeigenvector matrices of symmetric, tridiagonal matrices, al-lowing these matrix-vector products to be computed in sub-quadratic time. The coefficients �

m and ��m can be obtained

from �� m and ��m, respectively. When m�0, �m�+2im =�i+1

m , and��m�+2j+1

m =� j+1m for i=0,1 , . . . , �nx

�m�−1� and j=0,1 , . . . , �ny�m�

−1�. Since

P�−�m��z� = �− 1�mP�

�m��z� ,

�m�+2im = �−1�m�i+1

m and ��m�+2j+1m = �−1�m� j+1

m when m�0. Theinverse operations,

�� m = A�m��U�m��T�� m, �27�

�� m = B�m��V�m��T��m, �28�

are similar and can also be computed quickly. The fifth andfinal step of the FSHT merges the odd and even expansioncoefficients, which is an exercise in computer memory man-agement.

III. EXAMPLES

The FSHT was implemented in C using the CVODE

library17 in the precomputations and the FFTW package18 forthe FFT. We now present two examples demonstrating howthis split-operator propagator addresses the six propertiesoutlined in Sec. I. We begin by noting that this FSHT-accelerated propagator, by construction, does not require ki-netic and potential energy matrix elements. Furthermore, theoriginal presentation of the FSHT algorithm12 providesample substantiation of its favorable scaling.

Our first example demonstates the propagator’sunitarity19 and its adequate management of the csc � terms.Sets of ��

m coefficients for numerous �max�1600 were ran-domly generated, normalized, and subjected to the backwardand forward FSHT operations 100 times. After the series oftransformations, the average absolute error in each coeffi-cient was �10−7 with a corresponding average relative errorof �10−4. The small errors realized through numerous appli-cations of the FSHT show that the propagator is unitary andis not troubled by the csc � terms.

We proceed to show that the split-operator propagatorwith the FSHT is capable of adaptive time stepping and gen-erally solves the m-mixing problem. For our second ex-ample, we consider the alignment of Cl2 in response to alinearly polarized electric field. When the electric field ispolarized along the z-axis, the field-induced potential is20

Vz�t� = − �2�t��� cos2��z�/4, �29�

where ��t� is the electric field amplitude, �� is the polariz-ability anisotropy of the linear top molecule, and ��z ,�z� arethe polar and azimuthal angles in this coordinate system. Aphysical motivation �such as a second electromagnetic fieldor a surface� may instead orient the laser field along thex-axis, producing a potential

Vx�t� = − �2�t��� sin2��x�cos2��x�/4, �30�

where ��x ,�x� are the canonical angles in this second coor-dinate system.

The potential Vz�t� is azimuthally symmetric and does

not mix m states, whereas this symmetry is broken in Vx�t�,subjecting the simulation to the m-mixing problem. Since thephysics is independent of the choice of coordinate system,use of the propagator from equivalent initial states shouldproduce identical results. We choose the initial state to be theground state,

�x��x,�x,0� = Y00��x,�x� ,

�z��z,�z,0� = Y00��z,�z� ,

and the electric field amplitude to be a Gaussian peaked at anintensity of 51013 W cm−2 with a standard deviation of133 ps.

Figure 2�a� shows the degree of alignment, cos2 �z�, as afunction of time. As expected, the degree of alignment in-creases from the isotropic value of 1 /3 as the field turns onand evolves back to the isotropic value as the field turns off.The degree of alignment is comparably sin2 �x cos2 �x� inthe x-coordinate system, and the absolute error between thetwo degrees of alignment is displayed in Fig. 2�b�. The nu-merically insignificant errors suggest that this propagatorsolves the m-mixing problem. To further substantiate thisclaim, Fig. 2�c� plots the populations in various m levels inthe two coordinate systems. As expected, all population re-mains in mz=0 for the z-coordinate system. In the m-mixedx-coordinate system, however, all population starts and endsin mx=0, but transiently appears in other mx-levels.

094108-4 Reuter, Ratner, and Seideman J. Chem. Phys. 131, 094108 �2009�

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Finally, Fig. 2�d� shows the time step used during thepropagation. The time step was chosen by satisfying

max�� T�t

��,� V�t + �t/2��t

��� =

�

3, �31�

where T is the largest eigenvalue �in magnitude� of T whosecorresponding eigenstate has non-negligible population, and

similarly for V�t+�t /2� and Vx,z�t+�t /2�.

IV. CONCLUSIONS

Solutions to the TDSE in angular coordinates are impor-tant to a large variety of areas, ranging from atomic physicsand molecular spectroscopy to chemical reactions and spin-tronics. Since the exact propagator is not easily obtainable,approximations that balance �i� unitarity, �ii� the use of ki-netic and potential energy matrix elements, �iii� computa-tional efficiency, �iv� adaptive time stepping, �v� the handlingof csc � terms, and �vi� the inclusion of azimuthal asymme-tries must be considered. This work uses a split-operator for-malism and a FSHT to present the first propagator to simul-taneously overcome all six problems. As such, thispropagator exhibits a general solution to the m-mixing prob-lem, which has hampered the studies of systems with azi-muthally asymmetric potentials.

We think that this propagator will be especially useful inseveral applications. First, the obviation of kinetic and po-tential matrix elements makes this propagator ideal for angu-lar systems with numerical potentials. Such cases have re-cently been reported in studies of molecular alignment withsurface-adsorbed molecules.21 Second, imaginary time stepscan be used with this propagator to easily diagonalize angu-lar Hamiltonians with azimuthal asymmetries and/or numeri-cal potentials. Third, we expect the incorporation of radialvariables into this technique to be straightforward since theycan be accelerated by the FFT.7 Finally, we note that similarfast transforms for various canonical special functions havealso been reported,22 and can be combined with the split-operator formalism to yield efficient propagators for largeclasses of chemical problems.

ACKNOWLEDGMENTS

We are grateful to Dr. Jeff R. Hammond and Dr.Thorsten Hansen for interesting conversations and to theNSF Chemistry Division �Grant No. CHE-0616927� for theirgenerous support. M.G.R. thanks the DoE ComputationalScience Graduate Fellowship Program �Grant No. DE-FG02-97ER25308� for a graduate fellowship.

1 R. Kosloff, J. Phys. Chem. 92, 2087 �1988�.2 D. Lauvergnat, S. Blasco, X. Chapuisat, and A. Nauts, J. Chem. Phys.

126, 204103 �2007�.3 There are three factors limiting the size of �t: �i� The propagator needs toremain accurate to second order. Higher-order split-operator schemesusually sacrifice unitarity, time reversibility, or both �Refs. 23 and 24�.�ii� V�t� needs to be approximately constant over the time step. �iii�max��T�t / �2��� , �V�t+�t /2��t /�� �2�, where T is the largest eigen-

value �in magnitude� of T whose corresponding eigenstate has non-

negligible population, and similarly for V�t+�t /2�.4 S. Egner and M. Püschel, IEEE Trans. Signal Process. 49, 1992 �2001�.5 M. D. Feit, J. A. Fleck, Jr., and A. Steiger, J. Comput. Phys. 47, 412�1982�.

6 M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 78, 301 �1983�.7 M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 80, 2578 �1984�.8 M. R. Hermann and J. A. Fleck, Jr., Phys. Rev. A 38, 6000 �1988�.9 T. K. Kjeldsen, L. A. A. Nikolopoulos, and L. B. Madsen, Phys. Rev. A

75, 063427 �2007�.10 C. E. Dateo and H. Metiu, J. Chem. Phys. 95, 7392 �1991�.11 G. C. Corey and D. Lemoine, J. Chem. Phys. 97, 4115 �1992�.12 M. Tygert, J. Comput. Phys. 227, 4260 �2008�.13 See, e.g., P. G. Martinsson and V. Rokhlin, SIAM J. Sci. Comput. �USA�

29, 1160 �2007�.

�a�

�b�

�c�

�d�

�t�

fs�

m�

Popu

latio

n�x�

zE

rror�

�cos

2 Θz�

0.5

1.0

1.5

2.0

10�10

10�6

10�2

0.97

0.98

0.99

1.00

10�15

10�14

10�13

10�12

10�11

0.34

0.36

0.38

0.40

0.42

0.44

0.46

mz � 0 mx � 0

mx � 2

mx � 4

4002000�200�400t �ps�

FIG. 2. Calculated solutions of the TDSE for the alignment of Cl2. �a� Thedegree of alignment as a function of time for a Gaussian electric field am-plitude with maximum intensity of 51013 W cm−2 and standard deviationof 133 ps. �b� The absolute error �in magnitude� in the degree of alignmentbetween calculations in the two coordinate systems. The negligible errorshows that this propagator is not affected by azimuthal asymmetries, solvingthe m-mixing problem. �c� Populations in various m levels for the two co-ordinate systems. All population remains in mz=0 for the azimuthally sym-metric potential, as expected. Without the azimuthal symmetry, populationtransfers between mx levels are observed. �d� Demonstration of the adaptivetime-stepping capability of the propagator. When only low-energy states arepopulated, larger time steps can be taken.

094108-5 Fast angular wavepacket propagation J. Chem. Phys. 131, 094108 �2009�

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14 M. Gu and S. C. Eisenstat, SIAM J. Matrix Anal. Appl. 16, 172 �1995�.15 The notation used in this work is similar to that of Ref. 12; however,

some changes have been made in an attempt to increase clarity.16 A. Glaser, X. Liu, and V. Rokhlin, SIAM J. Sci. Comput. �USA� 29,

1420 �2007�.17 S. D. Cohen and A. C. Hindmarsh, Comput. Phys. 10, 138 �1996�.18 M. Frigo and S. G. Johnson, Proc. IEEE 93, 216 �2005�.19 Because the FMM is only exact in the limit of an infinite sum �arbitrary

accuracy�, the FSHT is also approximate but arbitrarily accurate. Our

claim of unitarity should be interpreted as numerical unitarity within thespecified tolerance.

20 T. Seideman and E. Hamilton, Adv. At., Mol., Opt. Phys. 52, 289 �2006�.21 M. G. Reuter, M. Sukharev, and T. Seideman, Phys. Rev. Lett. 101,

208303 �2008�.22 M. Tygert, “Recurrence relations and fast algorithms,” Department of

Computer Science, Yale University Technical Report No. 1343, 2005.23 A. Bandrauk and H. Shen, Chem. Phys. Lett. 176, 428 �1991�.24 M. Suzuki, J. Math. Phys. 32, 400 �1991�.

094108-6 Reuter, Ratner, and Seideman J. Chem. Phys. 131, 094108 �2009�

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