Electron-impact ionization of diatomic molecules using a configuration-average distorted-wave method

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1 PHYSICAL REVIEW A 76, Eectron-impact ionization of diatomic moecues using a configuration-average distorted-wave method M. S. Pindzoa and F. Robicheaux Department of Physics, Auburn University, Auburn, Aabama 36849, USA J. Cogan Theoretica Division, Los Aamos Nationa Laboratory, Los Aamos, New Mexico 87545, USA C. P. Baance Department of Physics, Roins Coege, Winter Park, Forida 32789, USA Received 7 March 27; pubished 19 Juy 27 Eectron-impact ionization cross sections for diatomic moecues are cacuated in a configuration-average distorted-wave method. Core bound orbitas for the moecuar ion are cacuated using a singe-configuration sef-consistent-fied method based on a inear combination of Sater-type orbitas. The core bound orbitas are then transformed onto a two-dimensiona r, numerica attice from which a Hartree potentia with oca exchange is constructed. The singe-partice Schrödinger equation is then soved for the vaence bound orbita and continuum distorted-wave orbitas with S-matrix boundary conditions. Tota cross section resuts for H 2 and N 2 are compared with those from semiempirica cacuations and experimenta measurements. DOI: 1.113/PhysRevA PACS numbers: 34.5.Gb I. INTRODUCTION Eectron-impact ionization of diatomic moecues is an important coision process in many different areas of physics and chemistry. For exampe, eectron ionization of H 2 and other first-row-eement diatomic moecues is important in understanding divertor pasma dynamics in controed fusion experiments 1. In addition, the eectron ionization of diatomic moecues is an exampe of the quanta three-body Couomb breakup probem invoving a nonspherica nucear fied. As such, it serves as a testing ground for the deveopment of new theoretica methods to expain the increasingy detaied observations of the ow-energy three-body breakup process 2. Perturbative distorted-wave cacuations of energy and ange differentia cross sections for the eectron ionization of diatomic moecues rey on accurate singe-partice moecuar continuum wave functions 3. Nonperturbative R-matrix-with-pseudostates and time-dependent cosecouping cacuations of tota cross sections have been recenty competed for the eectron ionization of H and H 2 5,6. Extensions of the nonperturbative time-dependent cose-couping method to cacuate energy and ange differentia cross sections for the eectron ionization of diatomic moecues wi aso rey on accurate moecuar continuum wave functions as projection states. As widey used for atomic targets, singe-partice continuum wave functions may be obtained by expansion in spherica harmonics and numerica soution of the resuting one-dimensiona 1D radia Schrödinger equation. Athough certainy viabe for simpe targets ike H 2, such singe-center spherica expansions for heavier moecues may become difficut to converge. Beginning with the pioneering work of Tuy and Berry 7, singe-partice continuum wave functions may aso be obtained by expansion in rotationa functions and numerica soution of the resuting 2D radia and anguar Schrödinger equation 8,9. Recenty, we extended a configuration-average distortedwave CADW method for atoms 1 to hande the cacuation of eectron-impact excitation and ionization cross sections for diatomic moecues 11. The moecuar CADW method reies on singe-partice continuum wave functions obtained by soution of a 2D radia and anguar Schrödinger equation. The CADW ionization cross sections for H were found to be in exceent agreement with previous distorted-wave cacuations using singe-partice continuum wave functions cacuated in separabe proate spheroida coordinates 12. However, recent CADW ionization cross sections for H 2 6 were found to be somewhat arger than expected, based on comparisons of CADW ionization cross sections with nonperturbative theory and experiment for He 13. In this paper, we repeat our CADW cacuations for the singe ionization of H 2 using a modified version of the agorithm used to obtain singe-partice continuum wave functions by soution of a 2D radia and anguar Schrödinger equation. The key modification is the incusion of the fu S matrix in the numerica soution of the 2D radia and anguar Schrödinger equation. This appears to competey eiminate a mixing for neutra moecue scattering, no matter what the choice of box size. In addition, we make further CADW cacuations for the singe ionization of N g N g. In Sec. II we review the CADW cross section expressions and then present the new theory for the singe-partice continuum wave functions. In Sec. III we present ionization cross sections for H 2 and N 2 and compare them with binary encounter Bethe cacuations and experiment. In Sec. IV we give a brief summary. Uness otherwise stated, a quantities are given in atomic units /27/761/ The American Physica Society

2 PINDZOLA et a. II. THEORY A. Distorted-wave ionization cross section For atoms and diatomic moecues, the direct ionization cross section may be cacuated using a configurationaverage distorted-wave method 1,11. The most genera transition between configurations of a diatomic moecue is of the form n w i i i n w 1 e e e f f f, w is the occupation number, and n, i i i, e e e, and f f f are the quantum numbers of the bound vaence eectron and the incident, ejected, and fina continuum eectrons =m is the absoute vaue of the magnetic quantum number. The configuration-average ionization cross section is given by ion = E/2 64 w d e k 3 i k e k f S i i e e 1 M d + M e M x, f 2 the tota energy E= n + i = e + f, =k 2 /2, S =22, is the statistica weight of the n orbita, and the continuum normaization is chosen as one times a sine function. The direct scattering term is given by PHYSICAL REVIEW A 76, k q! k q! M d = m i m e m k k + q! k + q! k R kq* f f f, e e e ; i i i,n R kq f f f, e e e ; i i i,n, 3 q=m e m k and q=m e m k. The exchange scattering term is given by k q! k q! M e = m i m e m k k + q! k + q! k R kq* e e e, f f f ; i i i,n R kq e e e, f f f ; i i i,n, 4 q=m i m e k and q=m i m e k. The cross scattering term is given by k q! k q! M x = m i m e m k k + q! k + q! k R kq* f f f, e e e ; i i i,n R kq e e e, f f f ; i i i,n, 5 q=m e m k and q=m i m e k. In a three scattering terms f =m i +m m e f and the r, poar coordinate integra is given by k R kq r f f f, e e e ; i i i,n dr dr 2 d d 2 P = k q cos 1 P k 1 r qcos 2 k+1 1 * * u f f r f 1, 1 u e e r e 2, 2 u i i i r 1, 1 u n r 2, 2, 6 P q k cos = 1 q 4k+q!/2k+1k q!y kq, = are associated Legendre functions, u n r, are bound reduced orbitas, and u r, are continuum reduced orbitas. The bound and continuum reduced orbitas are defined by r,, = ur, r sin e im 2, r,, is the spatia part of the singe-partice wave function and the bound normaization is dr dur, 2 =1. B. Moecuar ion potentia 7 The bound orbitas for the moecuar ion are cacuated using a singe-configuration sef-consistent-fied method based on a inear combination of Sater-type orbitas 14. The moecuar bound orbitas are then transformed 15 onto a two-dimensiona r, numerica attice to yied bound reduced orbitas u n r,. Using the moecuar bound orbitas we construct a Hartree potentia with oca exchange given by V HX r, = V direct r, + V exchange r,, is an adjustabe parameter used to achieve the experimenta ionization potentia for the vaence bound orbita of the moecue. The direct potentia is given by V direct r, = V k rp k cos, 9 k V k r = w du n r, n n dr 2 r k k+1 r P k cos. 1 The oca exchange potentia is given by

3 ELECTRON-IMPACT IONIZATION OF DIATOMIC V exchange r, = r, u r, = n r, 2 w n n 2r 2 sin. 1/3, The sums in Eqs. 1 and 12 are over a bound orbitas for the moecuar ion, weighted by their occupation numbers w n. C. Moecuar wave functions The bound and continuum orbitas needed to cacuate the configuration-average ionization cross section of Eq. 2 are found by soution of the singe-partice Schrödinger equation given by V nucear r, + V HX r, r,, =. The nucear potentia is given by V nucear r, = Z r R2 rr cos Z r R2 + rr cos 13, 14 Z is the charge on each nuceus of a homonucear diatomic moecue and R is the internucear separation distance. The variationa principe appied to Eq. 13 for a reduced orbita on a two-dimensiona r, numerica attice yieds Kur i, j + V centrifuga r i, j + V nucear r i, j + V HX r i, j ur i, j =. 15 For ow-order finite differences with uniform attice spacings r and, the effect of the kinetic energy operator is given by Kur i, j = 2 1 c iur i+1, j + c i 1 ur i 1, j c iur i, j r 2 1 2r i 2 d jur i, j+1 + d j 1 ur i, j 1 d jur i, j 2, 16 2 c i =r i+1/2 2 /r i r i+1, c i=r i+1/2 2 +r i 1/2 /r 2 i, d j =sin j+1/2 / sin j sin j+1, and d j=sin j+1/2 +sin j 1/2 /sin j. The centrifuga potentia is given by 2 V centrifuga r, = 2r 2 sin The vaence bound orbita needed to cacuate the configuration-average ionization cross section of Eq. 2 is found by matrix diagonaization of the one-eectron Hamitonian found in Eq. 15. For N r =3 and N =2 the Hamitonian matrix has the bock tridiagona form D11 C1 D 11 T 12 C 1 C 1 T 21 D 21 C 2 H =T11 C 1 D 21 T 22 C 2 C 2 T 31 D 31 C 2 D 31 T 32, 18 T ij =c i/2r 2 +d j/2r i 2 2 +Vr i, j, D ij = d j /2r i 2 2, and C i = c i /2r 2. The parameter is adjusted in the Hartree potentia with oca exchange unti the matrix diagonaization yieds an eigenenergy that agrees with the experimenta ionization potentia for the vaence bound orbita. The many continuum orbitas needed to cacuate the configuration-average ionization cross section of Eq. 2 are found by recasting the one-eectron Schrödinger equation of Eq. 15 and the arge-r boundary conditions for the continuum orbita as a system of inear equations Au=b. Foowing standard WKB theory, we choose S-matrix boundary conditions with ong-range unit normaization such that PHYSICAL REVIEW A 76, u k r, = F k r, + S G k r,, F k r, = k 4z r ei r P, G k r, = k 4z r e i r P, P = 2 sin Y,= is a normaized associated Legendre function, z r=d r/dr, r=kr+q/kn 2kr /2+, k= 2 is the inear momentum, is the anguar momentum, q is the asymptotic charge, and is the Couomb phase shift. For N r =3 and N =2 the continuum inear equations have the bock tridiagona form

4 PINDZOLA et a. PHYSICAL REVIEW A 76, T11 D11 C1 D 11 T 12 C 1 C 1 T 21 D 21 C 2 G 1 31 C 2 G 2 31 C 1 D 21 T 22 C 2 G 1 32 C 2 G 2 32 C 2 X 1 31 X 2 31 C 2 X 1 32 X 2 32 u 12 u 21 u 22 u11 S 1 S 2= C 2 F 31 C 2 F 32 Y 31 Y 32, 22 u ij =ur i, j, F ij =F k r i, j, and G ij =G k r i, j. The eements of the X and Y matrices are given by X 31 =T 31 G 31 +C 3 G 41 +D 31 G 32, X 32 =T 32 G 32 +C 3 G 42 +D 31 G 31, Y 31 =T 31 F 31 +C 3 F 41 +D 31 F 32, and Y 32 =T 32 F 32 +C 3 F 42 +D 31 F 31. We derive Eq. 22 from Eq. 15, using Eq. 19 for the radia attice points i=n r and N r +1 in the form u ij = F ij + S 1 G 1 ij + S 2 G 2 ij. 23 We sove for u by standard LU decomposition of the matrix A. We note that u contains a discrete representation of the compex continuum orbita u n r,, and eements of the S matrix. A key eement in guarding against mixing in the continuum orbitas is to sove for the normaized associated Legendre functions P by matrix diagonaization of the anguar part of the one-eectron Hamitonian found in Eq. 15. For N r =N =2 the anguar Hamitonian matrix has the tridiagona form = d d 1 sin H anguar 24 d 1 2 III. RESULTS d sin 2 2. The configuration-average distorted-wave method was used to cacuate the eectron-impact ionization cross section for H 2 at an internucear separation of R=1.4. We empoy a 332 point attice in r, spherica poar coordinates with a uniform mesh spacing of r=.1 and = Using a singe-configuration sef-consistent-fied 1s moecuar orbita for H 2 + at R=1.4 14,15, the exchange potentia parameter is adjusted in the diagonaization of Eq. 15 to yied a 1s vaence orbita for H 2 with an ionization potentia of 15.4 ev. Partia cross sections are cacuated for i = 9 and i = 4 and extrapoated to higher i and i using appropriate fitting functions. Tota cross sections are cacuated at incident energies from i =2 to 1 ev and then interpoated over intermediate energies using a B-spine fitting function. Eectron-impact ionization cross section resuts for H 2 at R=1.4 are presented in Fig. 1. The CADW resuts, shown as the soid ine, are about 3% higher than resuts of an eectron beam and gas ce experiment 16 near the peak of the cross section. We note that simiar CADW resuts for the He atom are aso about 3% higher than experiment near the peak of the cross section. For comparison, we aso present the ionization cross section for H 2 as cacuated using a oneparameter binary encounter Bethe BEB method, shown as the dashed ine. The BEB ionization cross section is given by 17 ion =4w I H I s nu u u nu u +1 u +2, 25 u=e/i s, E is the incident energy, I H =13.6 ev is the ionization potentia of atomic hydrogen, and I s =15.4 ev is the ionization potentia of moecuar hydrogen. The semiempirica binary encounter method has been extensivey generaized to accuratey predict tota ionization cross sections for many atoms and moecues and their ions 18. We aso note that recent ab initio nonperturbative R-matrix-with-pseudostates 5 and time-dependent cosecouping 6 cacuations of tota cross sections for the eectron-impact ionization of H 2 are in exceent agreement with experiment. Cross Section (Mb) Eectron Energy (ev) FIG. 1. Eectron-impact ionization of H 2. Soid curve, configuration-average distorted-wave method; dashed curve, binary encounter Bethe method; soid circes, experiment 16 1Mb =1 18 cm

5 ELECTRON-IMPACT IONIZATION OF DIATOMIC Cross Section (Mb) Eectron Energy (ev) FIG. 2. Eectron-impact ionization of N 2. Soid curve, configuration-average distorted-wave method; dashed curve, binary encounter Bethe method; soid circes, experiment 19 1Mb =1 18 cm 2. The configuration-average distorted-wave method was used to cacuate the eectron-impact ionization cross section for N 2 1 g + N g + at an internucear separation of R=2.1. We empoy a 15,32 point attice in r, spherica poar coordinates with a uniform mesh spacing of r=.2 and = Using singe-configuration sef-consistent-fied 1s, 2s, 2p, 2p, 3s, and 3p moecuar orbitas for N 2 + at R=2.1 14,15, the exchange potentia parameter is adjusted in the diagonaization of Eq. 15 to yied a 3s vaence orbita for N 2 with an ionization potentia of 15.7 ev. Partia cross sections are cacuated for i = 1 and i = 6, and extrapoated to higher i and i using appropriate fitting functions. Tota cross sections are cacuated at incident energies from i =2 to 1 ev and then interpoated over intermediate energies using a threeparameter fitting function. Eectron-impact ionization cross section resuts for N 2 at R=2.1 are presented in Fig. 2. The CADW resuts, shown as the soid ine, are about 5% higher than an eectron beam, gas ce, and aser-induced fuorescence experiment 19 near the peak of the cross section. We note that simiar CADW resuts for the Si atom are aso about 5% higher than experiment near the peak of the cross section. For comparison we aso present BEB resuts with I s =15.7 ev, shown as the dashed ine. PHYSICAL REVIEW A 76, IV. SUMMARY In concusion, we have appied a configuration-average distorted-wave method to cacuate eectron-impact ionization cross sections for H 2 and N 2. A singe-configuration sefconsistent-fied method was used to generate the moecuar orbitas for H 2 + and N 2 + needed to construct a Hartree potentia with oca exchange. Vaence bound orbitas for H 2 and N 2 were obtained by matrix diagonaization of the singepartice Schrödinger equation. Distorted-wave continuum orbitas were obtained using a new agorithm to sove the 2D radia and anguar Schrödinger equation. The key modification is the incusion of the fu S matrix in the standard LU decomposition numerica method. Tota distorted-wave cross section resuts for H 2 and N 2 are found to be between 3% and 5% higher than recent experimenta measurements, in keeping with the usua range of accuracy found for distortedwave cacuations of the eectron ionization of ground-state neutra atoms. In the future, we pan to appy the moecuar CADW method to cacuate eectron-impact excitation and ionization processes in a wide variety of diatomic moecues and their ions. Athough the number of radia grid points needed to adequatey describe the heavier diatomic moecues, ike N 2, may be significanty reduced by empoying a variabe mesh spacing, the overa accuracy of a first-order perturbative method remains imited. On the other hand, the further deveopment and appication of nonperturbative approaches, ike the R-matrix-with-pseudostates and time-dependent cose-couping methods, wi improve the accuracy of truy ab initio predictions of eectron ionization cross sections for moecues. In addition, the new CADW method for soving the 2D radia and anguar Schrödinger equation for continuum distorted-waves is being used with great success to produce projection states to extract energy and ange differentia cross sections in nonperturbative time-dependent cose-couping cacuations for the doube photoionization of H 2 2 and the eectron singe ionization of H 2 and other diatomic moecues. ACKNOWLEDGMENTS This work was supported in part by grants from the U.S. Department of Energy. Computationa work was carried out at the Nationa Energy Research Scientific Computing Center in Oakand, Caifornia and at the Nationa Center for Computationa Sciences in Oak Ridge, Tennessee. 1 Nucear Fusion Research, edited by R. E. H. Cark and D. H. Reiter, Chemica Physics Series Vo. 78 Springer, Berin, A. J. Murray, J. Phys. B 38, D. S. Mine-Brownie, M. Foster, J. Gao, B. Lohmann, and D. H. Madison, Phys. Rev. Lett. 96, M. S. Pindzoa, F. Robicheaux, and J. Cogan, J. Phys. B 38, L J. D. Gorfinkie and J. Tennyson, J. Phys. B 38, M. S. Pindzoa, F. Robicheaux, S. D. Loch, and J. P. Cogan, Phys. Rev. A 73, J. C. Tuy and R. S. Berry, J. Chem. Phys. 51, J. A. Richards and F. P. Larkins, J. Phys. B 17, ; 19, C. A. Weatherford, M. Dong, and B. C. Saha, Int. J. Quantum Chem. 65,

6 PINDZOLA et a. 1 M. S. Pindzoa, D. C. Griffin, and C. Bottcher, in Atomic Processes in Eectron-Ion and Ion-Ion Coisions, edited by F. Brouiard, NATO Advanced Studies Institute, Series B: Physics Penum Press, New York, 1986, Vo. 145, p M. S. Pindzoa, F. Robicheaux, J. A. Ludow, J. Cogan, and D. C. Griffin, Phys. Rev. A 72, F. Robicheaux, J. Phys. B 29, M. S. Pindzoa and F. J. Robicheaux, Phys. Rev. A 61, Theoretica Chemistry Group, Computer code ALCHEMY IBM PHYSICAL REVIEW A 76, Research Laboratory, San Jose, CA, M. A. Morrison, Comput. Phys. Commun. 21, H. C. Straub, P. Renaut, B. G. Lindsay, K. A. Smith, and R. F. Stebbings, Phys. Rev. A 54, Y. K. Kim and M. E. Rudd, Phys. Rev. A 5, W. M. Huo, Phys. Rev. A 64, N. Abramzon, R. B. Siege, and K. Becker, J. Phys. B 32, L J. Cogan, M. S. Pindzoa, and F. Robicheaux, Phys. Rev. Lett. 98,