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Iteratively-coupled propagating exterior complex scaling method for electron–hydrogen collisions
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2004 J. Phys. B: At. Mol. Opt. Phys. 37 L69 (http://iopscience.iop.org/0953-4075/37/4/L01) View the table of contents for this issue, or go to the journal homepage for more
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INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 37 (2004) L69–L76
PII: S0953-4075(04)70352-1
LETTER TO THE EDITOR
Iteratively-coupled propagating exterior complex scaling method for electron–hydrogen collisions Philip L Bartlett, Andris T Stelbovics and Igor Bray Centre for Atomic, Molecular and Surface Physics, Murdoch University, Perth 6150, Australia E-mail:
[email protected]
Received 10 October 2003 Published 30 January 2004 Online at stacks.iop.org/JPhysB/37/L69 (DOI: 10.1088/0953-4075/37/4/L01) Abstract A newly-derived iterative coupling procedure for the propagating exterior complex scaling (PECS) method is used to efficiently calculate the electronimpact wavefunctions for atomic hydrogen. An overview of this method is given along with methods for extracting scattering cross sections. Differential scattering cross sections at 30 eV are presented for the electron-impact excitation to the n = 1, 2, 3 and 4 final states, for both PECS and convergent close coupling (CCC), which are in excellent agreement with each other and with experiment. PECS results are presented at 27.2 eV and 30 eV for symmetric and asymmetric energy-sharing triple differential cross sections, which are in excellent agreement with CCC and exterior complex scaling calculations, and with experimental data. At these intermediate energies, the efficiency of the PECS method with iterative coupling has allowed highly accurate partial-wave solutions of the full Schr¨odinger equation, for L 50 and a large number of coupled angular momentum states, to be obtained with minimal computing resources.
The method for evaluating the scattered wavefunction for the electron-impact of atomic hydrogen using exterior complex scaling (ECS) was pioneered by Rescigno et al (1999), and using integral methods to obtain ionization cross sections (Baertschy et al 2001a), an excellent agreement with experimental results was obtained. The absence of any systematic approximations in the ECS method, and its universal nature (by directly calculating the underlying Schr¨odinger equation of the collision processes), gives tremendous potential for its application to collisions more complex than electron–hydrogen. Existing implementations of the ECS method for electron–hydrogen, however, are very computationally intensive. So, in order to apply the method to larger systems, it is necessary to find strategies to reduce its computational complexity. 0953-4075/04/040069+08$30.00 © 2004 IOP Publishing Ltd Printed in the UK
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Letter to the Editor
The purpose of this letter is to outline a newly developed implementation of ECS that is significantly more efficient than previous implementations. To differentiate our method from other ECS implementations, we will refer to it as the propagating exterior complex scaling (PECS) method with iterative coupling. We will then demonstrate the utility of this method by applying it to a wide range of e–H collision kinematics: elastic and inelastic discrete final-state scattering collisions, and symmetric and asymmetric energy-sharing ionizing collisions. ECS results for discrete final-state scattering have not been previously reported. The underlying equations of the PECS method are based on the ECS method (Rescigno et al 1997, McCurdy et al 2001, Baertschy et al 2001b), where the scattered (+) is solved directly from the rearranged Schr¨odinger equation outgoing wavefunction sc (+) (E − Hˆ )sc (r 1 , r 2 ) = (Hˆ − E)k0 (r 1 , r 2 ),
(1)
and where the symmetrized initial-state wavefunction is given by k0 (r 1 , r 2 ) =
√1 [1s (r 1 ) eik0 ·r 2 2
+ (−1)S (1 ↔ 2)].
(2)
1s is the hydrogen ground-state wavefunction and k0 is the momentum of the incident electron, directed along the z-axis. Though the following equations are derived for a groundstate target, they can be easily adapted for excited-state targets. We use the partial wave expansion 1 LMS S(+) sc (r 1 , r 2 ) = ψ (r1 , r2 )Y LM ˆ 1 , rˆ 2 ), (3) l1 l2 (r r1 r2 l l LM l1 l2 1 2
where YlLM is the bipolar spherical harmonic function, |l1 − l2 | L l1 + l2 , and since parity 1 l2 is conserved L + l1 + l2 must be even. Also note that M = 0 for ground-state collisions. Each partial wave is solved independently (in atomic units) using 1 LS E − Hˆ l1 (r1 ) − Hˆ l2 (r2 ) ψlLS (r1 , r2 ) − l1 l2 l1 l2 L ψlLS (4) (r1 , r2 ) = χl l (r1 , r2 ), 1 l2 1 2 1 l2 r 12 l1 l2
where 1 ∂2 1 l(l + 1) − + , Hˆ l (r) = − 2 ∂r 2 r 2r 2 and where (r1 , r2 ) = χlLS 1 l2
1 1 l1 l2 0LL − δl1 0 δl2 L r12 r2 × φ1s (r1 )jˆL (k0 r2 ) + (−1)S (1 ↔ 2) ,
(5)
iL 2π(2L + 1) k0
(6)
and jˆL is the Riccati–Bessel function. Equation (4) is solved numerically on a finite grid by using exterior complex scaling, where the radial coordinates are rotated into the complex plane at R0 using the transformation r, r < R0 z(r) = (7) R0 + (r − R0 ) eiθ , r R0 . Under this transformation, outgoing waves diminish exponentially, so the boundary condition (r1 , r2 ) = 0. However, as incoming waves diverge under at r1 , r2 = Rmax > R0 becomes ψlLS 1 l2 this transformation, χlLS (r , r ) must be truncated at r1 , r2 > R0 . This is the only systematic, 1 2 1 l2 though controlled, approximation of this method.
Letter to the Editor
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Previous ECS implementations for electron–hydrogen ionization (Baertschy et al 2001a, 2001b) solve equation (4) using a 7-point finite difference method. This requires the solution of a very large (N ≈ 5 × 106 ) and sparse system of complex linear equations for each partial wave, which is extremely computationally intensive (Baertschy and Li 2001). In contrast, our PECS method uses a propagation technique, whereby a small (1 N 1000), dense, propagation matrix (D) is solved for each grid column. Our twodimensional grid is symmetric with respect to its r1 and r2 axes, and we define a column, arbitrarily, as being all the r2 grid points for a constant r1 . This method has been detailed previously (Bartlett and Stelbovics 2004) for the uncoupled electron–hydrogen ionization model problems, and proved to be computationally efficient, numerically stable and accurate (Bartlett et al 2003a, 2003b). For each column (i 1), the method requires equation (4) to be formed into a matrix equation of the form − →(i) − →(i−1) − →(i) − →(i+1) + B (i) · ψ + C (i) · ψ = χ , (8) A(i) · ψ − → where the sparse matrices, A, B and C, and the χ vector are determined from the coefficients of a two-dimensional 3-point Numerov finite difference formula, modified to allow for arbitrary changes in grid spacing, including the change from real to complex coordinates at R0 . This propagation technique is based on that used by Poet (1980) for e–H scattering and more recently by Jones and Stelbovics (1999) and Jones and Stelbovics (2002) for their benchmark calculations for the Temkin–Poet model for e–H ionization. We have previously given details for the l = 0 modified Numerov formula (Bartlett and Stelbovics 2004), which uses the eight nearest-neighbour grid points in its calculations. The l > 0 formula will be detailed in a later publication. A propagation relation for the ith column can then be given as → − →(i+1) − − →(i) ψ = D (i) · ψ + E (i) (9) and is solved by substituting (9) into (8), giving B˜ (i) = (B (i) + A(i) · D (i−1) )−1 ˜ (i) · C (i) D (i) = −B − → − → →(i) ˜ (i) · ( − E (i) = B χ − A(i) · E (i−1) ).
(10) (11)
(12) − →(i) The vector E , and its recurrence relation (12), is an enhancement to the propagation method to allow for the inhomogeneous nature of equation (8). The most significant feature of the propagation method is that the vast majority of the computational effort is devoted to performing the matrix inversion in equation (10), which − → is independent of χ . Utilizing this feature, we have developed a highly efficient iterative coupling scheme, and though developed independently, it is similar to that used by Allison (1970) for one-dimensional problems. To simplify the labelling of our iterative scheme, the subscripts a, b are used to represent the bth iteration of the ath coupled angular momentum state (l1 (a), l2 (a), L, S), where 1 a c and c is the number of states required to achieve convergence of the LS partial wave. Also, we assume that for each iteration, the wavefunctions are calculated in ascending order of a. The iterative equivalent of equation (4) therefore becomes 1 ˆ ˆ E − H l1 (a) (r1 ) − H l2 (a) (r2 ) − l1 (a)l2 (a) l1 (a)l2 (a)L ψ˜ a,b+1 (r1 , r2 ) = χ˜ a,b (r1 , r2 ), r12 (13)
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Letter to the Editor
where b 0, and χ˜ a,b (r1 , r2 ) = χlLS (r1 , r2 ) + 1 l2 +
a >a
l1 (a)l2 (a)
a 150◦ ). The agreement of the component l-orbitals for each n-state (not shown on plots) is also very good. For n = 1, both theoretical results are in
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scattering angle (degrees) Figure 1. Spin-averaged differential scattering cross sections for the n = 1, 2, 3 and 4 discrete final states of atomic hydrogen with 30 eV incident electrons. Present results are represented by: PECS (——), CCC (- - - -) (which are mostly indistinguishable from the PECS results), and are compared with the experimental results of Williams (1975) (), Shyn and Cho (1989) ( ), Grafe et al (2001) () and Sweeney et al (2001) ().
◦
good agreement with the older experimental results of Williams (1975), and do not exhibit the increasing cross section of the back-scattered electrons, θ > 120◦ , as suggested by Shyn and Cho (1989). Our n = 2 results are in excellent agreement with experiment, and our n = 3 and n = 4 results are in reasonable agreement. For all energies presented, our PECS results were calculated using R0 = 100 au and include partial waves L 50. Our raw results are plotted, without smoothing or extrapolation. A total of 26 h of computer time was required to calculate the wavefunctions and cross sections, for all partial waves 0 L 50, both S = 0 and S = 1, using a single compute node (4 Alpha 1 GHz processors with 8 gigabytes memory). The compute time ranged from 2 min for the L = 0, S = 1 partial wave, using six coupled angular momentum states, to 45 min for the L = 0, S = 0 partial wave, using 32 coupled angular momentum states (where l1 , l2 and l2 , l1 pairs are counted separately). The 30 eV CCC results were calculated using the ‘box-state’ method (Bray et al 2003) at 80 au, and include partial waves L 30. A larger number of partial waves were required for convergence of the PECS results as no extrapolation methods were used, whereas the CCC results include an analytic extrapolation approximation for the elastic cross-section, and a Born approximation for the L > 30 partial waves of the n > 1 discrete cross sections. Based on our PECS results, the very high partial waves L > 15 are not required for the ionization cross sections and are not yet convergent at R0 = 100 au. However, for discrete scattering cross sections, though they are of small magnitude, they are convergent at R0 = 100 au and are required to give convergent differential cross sections for small scattering angles, and to
Letter to the Editor
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scattering angle θB Figure 2. Spin-averaged coplanar TDCS for equal-energy-sharing ionization of atomic hydrogen by electron impact at 30 eV. Present results are represented by: PECS (——); CCC (- - - -). ECS results (Baertschy et al 2001a) are represented by (· · · · · ·), and are mostly indistinguishable from the PECS and CCC results. The experimental results of R¨oder et al (1996) ( ) are internormalized and relative.
◦
remove oscillations at large backward scattering angles. We have chosen to explicitly calculate these high partial waves using the PECS method so as to retain the ab initio character of our calculations, and to emphasize the low computational requirements of the iterative coupling PECS method. Figure 2 shows excellent agreement between PECS, CCC and ECS results for the 30 eV ionization triple-differential cross section (TDCS) results at symmetric energy sharing. There are however differences of up to 5% in the peak TDCS of the theoretical methods, for some kinematics, but no systematic variation. All methods compare favourably with the experimental results. The PECS TDCS required L 15 to achieve convergence. Figure 3 plots PECS and CCC results for the 27.2 eV ionization TDCS at both symmetric and asymmetric energy sharing. As far as we are aware, no published asymmetric energysharing TDCS results for the ECS method are available for comparison. Both PECS and CCC compare favourably with the available experimental data; however, there are some minor differences in the magnitude of the calculations for some kinematics. The PECS results tend to be flatter and more suppressed than the CCC results for geometries where θA ≈ θB . This is most evident in the θA = −θB → 0 region of the equal energy-sharing kinematics, which should have a highly suppressed TDCS. Whereas the PECS method is fully ab initio, for asymmetric energy-sharing the CCC method requires an estimate of the underlying singledifferential cross section (Bray et al 2003). Our PECS results show that this estimating procedure is reasonably accurate for the kinematics investigated. The present study has demonstrated that the PECS method yields excellent agreement with ECS, CCC and experiment for e–H fully differential excitation and ionization cross sections. We emphasize that the PECS method is able to solve the full non-relativistic Schr¨odinger equation to very high accuracy, given sufficient computing resources, and we estimate an error of better than 0.5% in the total ionization cross sections of the present calculations. We also
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scattering angle θB Figure 3. Spin-averaged coplanar TDCS for the ionization of atomic hydrogen by electron impact at 27.2 eV, and secondary electron energies of 2, 4 and 6.8 eV. Results are represented by: PECS (——); CCC (Bray 2003) (- - - -). The experimental results, as presented by Berakdar et al (1999), ( ) are internormalized for a given secondary energy.
◦
emphasize that the iterative coupling procedure speeds up the calculations for each partial wave by one to two orders of magnitude, and permits larger numbers of angular momentum coupled states and partial waves to be included compared to the solution of the equations without iterative coupling. The computational efficiency of the propagation and iteration techniques presented, and the minimal approximations of the ECS method, should allow PECS to be applied to scattering systems more complex than e–H ionization, with present supercomputing technology. Acknowledgments We gratefully acknowledge the support of the Australian Research Council, the Australian Partnership for Advanced Computing and the West Australian Interactive Virtual Environments Centre.
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References Allison A C 1970 J. Comput. Phys. 6 378 Baertschy M and Li X 2001 Proc. ACM/IEEE Conf. on Supercomputing (Denver, CO: ACM) Baertschy M, Rescigno T N and McCurdy C W 2001a Phys. Rev. A 64 022709 Baertschy M, Rescigno T N, Isaacs W A, Li X and McCurdy C W 2001b Phys. Rev. A 63 022712 Bartlett P L and Stelbovics A T 2004 Phys. Rev. A 69 at press Bartlett P L, Bray I, Jones S, Stelbovics A T, Kadyrov A S, Bartschat K, Ver Steeg G L, Scott M P and Burke P G 2003a Phys. Rev. A 68 020702(R) Bartlett P L, Stelbovics A T and Bray I 2003b Phys. Rev. A 68 030701(R) Berakdar J, Briggs J S, Bray I and Fursa D V 1999 J. Phys. B: At. Mol. Opt. Phys. 32 895 Bray I 2003 J. Phys. B: At. Mol. Opt. Phys. 36 2203 Bray I, Bartschat K and Stelbovics A T 2003 Phys. Rev. A 67 060704(R) Grafe A, Sweeney C J and Shyn T W 2001 Phys. Rev. A 63 052715 Jones S and Stelbovics A T 1999 Aust. J. Phys. 52 621 Jones S and Stelbovics A T 2002 Phys. Rev. A 66 032717 McCurdy C W, Horner D A and Rescigno T N 2001 Phys. Rev. A 63 022711 Poet R 1980 J. Phys. B: At. Mol. Phys. 13 2995 Rescigno T N, Baertschy M, Byrum D and McCurdy C W 1997 Phys. Rev. A 55 4253 Rescigno T N, Baertschy M, Isaacs W A and McCurdy C W 1999 Science 286 2474 R¨oder J, Rasch J, Jung K, Whelan C T, Ehrhardt H, Allan R J and Walters H R J 1996 Phys. Rev. A 53 225 Shyn T W and Cho S Y 1989 Phys. Rev. A 40 1315 Sweeney C J, Grafe A and Shyn T W 2001 Phys. Rev. A 64 032704 Williams J F 1975 J. Phys. B: At. Mol. Phys. 8 2191
